creg davis

Large-Scale Testing and Simulation of Earthquake-Induced Ultra Low Cycle Fatigue in
Bracing Members Subjected to Cyclic Inelastic Buckling
By
BENJAMIN VINCENT FELL
B.S. (Rensselaer Polytechnic Institute) 2003
M.S. (Stanford University) 2004
DISSERTATION
Submitted in partial satisfaction of the requirements for the degree of
DOCTOR OF PHILOSOPHY
in
Civil and Environmental Engineering
in the
OFFICE OF GRADUATE STUDIES
of the
UNIVERSITY OF CALIFORNIA
DAVIS
Approved:
_____________________________________
_____________________________________
_____________________________________
Committee in Charge
2008
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© Copyright by Benjamin V Fell 2008
All Rights Reserved
Abstract
Special Concentrically Braced Frames (SCBFs) are popular lateral load resisting frames due to
their economy, structural efficiency and stiffness. Following the 1994 Northridge earthquake,
braced-frames became increasingly common after brittle fractures were observed at beam-column
connections in Moment Resistant Frames (MRFs). However, braced-frames are also susceptible
to fracture at the middle plastic hinge of the brace, the brace to gusset plate connection, and the
gusset plate to column or beam connection. This research primarily focuses on fracture at the
middle plastic hinge, where the combined effect of global and local buckling during cyclic
loading amplifies the plastic strain at the brace midpoint and initiates fracture. To develop a better
understanding of the localized mechanisms affecting brace fracture, this work combines a largescale experimental program with an intensive simulation study to investigate brace behavior
across a wide-range of material types and geometries. The simulations employ continuum-based
modeling techniques to accurately reproduce the stress and strain histories during cyclic loading
while a novel micromechanical fracture model is evaluated as a means to predict the fracture
initiation events. The fracture model operates at the continuum-level and captures the
fundamental mechanisms responsible for ductile fracture unique to Ultra Low Cycle Fatigue
(ULCF) conditions which differ from the well-defined High and Low Cycle Fatigue (HCF and
LCF) mechanisms. From the large-scale brace experiments, cross-section width-thickness and
slenderness ratios are shown to influence the brace axial deformation fracture ductility, such that
a larger width-thickness ratio and a smaller slenderness tend to reduce ductility. Furthermore, the
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experiments are used to evaluate the fracture model at the large-scale where small-scale
calibration tests and a multi-scale modeling procedure is used to connect the steel behavior at the
micromechanical level to the finite element simulation results. The fracture predictions are
encouraging considering the high level of complexity in modeling buckling phenomena and
imperfect constitutive model behavior. The model is used to expand the experimental test matrix
through parametric simulation of square and rectangular bracing components which, along with a
synthesis of experimental results over the last twenty years, informs a general relationship
between brace ductility and geometry.
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Acknowledgements
First, I would like to thank my principal advisor, Amit Kanvinde, who has been the single most
influential voice in my life during the last five years. Amit has inspired my development from a
student into an academic and has continually challenged me to critically examine technical and
non-technical topics alike. Over the years we have become great friends and I thank him for all of
the advice, support and friendship that he has provided. I am also very grateful to have worked
with Greg Deierlein through collaborative efforts with Stanford University. His overall
perspective and extensive technical expertise has undoubtedly improved this study and has taught
me the importance of considering the audience of my work. I am also thankful to Sashi Kunnath
for reviewing this dissertation and his general guidance during my tenure at UC Davis.
Mark Rashid and Jeannie Darby have also been very supportive during my time at UC Davis. As
a TA for Professor Rashid, I first began to consider and develop my pedagogical skills and I am
thankful for the interest that he took in my growth as an educator. I would also like to thank him
for chairing my qualifying exam committee and supporting my applications to faculty positions.
As chair of the Civil and Environmental Engineering Department, Jeannie Darby led a
Department which constantly supported me during graduate school and ensured that I receive
every opportunity to have a successful career. I am very thankful to her and the rest of the Civil
and Environmental Engineering Department at Davis.
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My good friend and colleague, Andy Myers, at Stanford University has been instrumental to this
research by conducting the small-scale calibration experiments and dedicating his summer during
2005 to setup the large-scale brace tests at the UC Berkeley Richmond Field Station. The
combination of friendship and dedication that he has brought to this collaborative fracture project
over the last four years has far exceeded anything that I could have hoped for from a colleague. I
am very thankful for his friendship and I wish him the best of luck in his career. Also
instrumental in the large-scale brace tests have been the staff at the Richmond Field Station.
Shakhzod Takhirov, Don Clyde, Wesley Neighbour, Don Patterson, Dave MacLam and Jose
Robles have been incredibly helpful, friendly and professional during the numerous large-scale
test programs that I have been part of during my time at UC Davis. Their dedication was recently
recognized with the Outstanding Service to Researchers award at the 2008 Network for
Earthquake Engineering (NEES) annual meeting, and is certainly well deserved. I am also
grateful for the financial support of from the National Science Foundation (NSF) and NEES as
well as the generous steel donations supplied by the Structural Steel Educational Council (SSEC)
and their financial support through a student fellowship program.
I would also like to thank a wide-range of researchers and practitioners who contributed their
expertise to this research. Professors Helmut Krawinkler and Jack Baker at Stanford University
have been very helpful in providing perspective and technical advice on numerous occasions.
Professor Robert Tremblay, Patxi Uriz, Rafael Sabelli, Walterio López and Mark Saunders have
all provided valuable insight concerning braced-frame performance issues. Especially helpful has
been Patxi Uriz and I appreciate his willingness to share his experimental and simulation results
from his doctoral work. I look forward to more collaboration with all of these people.
I was also fortunate to have participated in a materials science research project at UC Davis. I
greatly appreciate this opportunity and the support of Professor Joanna Groza over the last two
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years. It has been very rewarding to work with Joanna, J.P. Delplanque, Tien Tran and Cory
Guebels and I appreciate their patience as I assimilated into a field outside of my area of
expertise. Tien has been especially helpful and was always willing to answer my questions. I
consider myself lucky to have worked with such an intelligent and friendly colleague who was
able to ease my transition into materials science.
Other than the close friendships that I have formed through my professional collaborations,
several other people have made my time at UC Davis very enjoyable. Ivan Gomez, Marshall
Roberts and Jorge Camacho have provided terrific companionship and I have enjoyed working
with them on other projects over the years. My officemates, Tessa Fojut, Laura Doyle, Harold
Leverenz and Sangam Tiwari are all wonderful people and I am thankful that I was able to share
the small space of 2021 Engineering III with them. The swimming team at Davis Aquatic Masters
(DAM) has provided me with many memorable times and a release from the stress of grad
school. During my short time at Stanford I made several friendships that helped me through a
very rigorous nine month MS program and I am grateful for the support and friendship that Wes,
Tim, Mark, Meris, Paul, Curt, Adam and Molly gave me during that time. I am also thankful for
my relatively new friendships with Yueyue, Dawn, Fabian, Kailash, Spy, Riju, Archis, Abhishek,
and Priya. They are all amazing people and I am fortunate to have them in my life.
Last, but not least, I want to recognize the support from my family and childhood friends. Brian
Middaugh, Stacie Ward, Matt Foreman and Alex Pannone are some of the best friends that I
could ask for and I appreciate their support over the last few years. Perhaps one of the most
influential people in my life has been my high school swimming coach, Dave Mastrantuono, who
gave me guidance at a young age and showed me the benefits of hard work. Through swimming,
and family, we became good friends and I am thankful for everything he has brought into my life.
My extended family, especially my Grandma and Grandpa Reeve and Jim and Barb, have
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provided so much love and support during my entire education. They have been like a second and
third set of parents and have immensely contributed to my personal growth. As far as my
immediate family, it is difficult for words to describe how much they have given to my life, both
personally and academically. I often find myself examining topics in structural engineering the
same way that my Dad would approach projects around the house during my childhood. On the
other hand, my Mom provided such balance in my life and was always able to see events in their
proper perspective. I’ve used a lot of what she has taught me to pursue a healthy and complete
life. Finally, my brother has been a teammate, competitor and best friend for me growing up. His
natural talent at so many things motivated me to set high goals in my life and I am grateful for
everything that he has taught me. This dissertation is a product of their gifts to me and I surely
could not have done it without them.
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Table of Contents
Abstract...........................................................................................................................................ii
Acknowledgements .......................................................................................................................iv
Table of Contents ........................................................................................................................viii
List of Tables ................................................................................................................................xii
List of Figures..............................................................................................................................xiii
Chapter 1
Introduction .............................................................................................................. 1
1.1 Motivation.................................................................................................................................. 1
1.2 Objectives and Scope................................................................................................................. 3
1.3 Organization and Outline........................................................................................................... 5
Chapter 2
Simulating Fracture in SCBF Bracing Components - Background .................. 10
2.1 Performance Evaluation of SCBF Systems ............................................................................. 12
2.1.1 SCBF System Behavior.................................................................................................... 12
2.1.2 Seismic Design Provisions for SCBF Systems................................................................. 15
2.1.3 State-of-the-art in SCBF Performance Assessment.......................................................... 17
2.1.3.1 Studies Examining Seismic Demands in SCBF Systems Through Nonlinear
Dynamic Analysis ................................................................................................................. 19
2.1.3.1.1 Sabelli (2001) Analyses ..................................................................................... 20
2.1.3.1.2 Uriz (2005) Analyses ......................................................................................... 21
2.1.3.1.3 McCormick et al (2007) Analyses ..................................................................... 23
2.1.3.1.4 Redwood et al (1991) Analyses ......................................................................... 23
2.1.3.1.5 Tremblay and Poncet (2005) Analyses .............................................................. 24
2.1.3.1.6 Izvernari et al (2007) Analyses .......................................................................... 25
2.1.3.2 Shake Table Testing .................................................................................................. 25
2.1.3.3 Summary of Demand Assessment of SCBF Systems ............................................... 26
2.1.3.4 Studies Investigating the Capacity of Bracing Members .......................................... 27
2.1.3.4.1 Square and Rectangular Hollow Steel Structural (HSS) Bracing Components . 30
2.1.3.4.2 Round Steel Pipe Bracing Members .................................................................. 32
2.1.3.4.3 Wide-Flanged Bracing Components.................................................................. 33
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2.1.3.4.4 General Ductility Trends of HSS, Pipe, and Wide-Flanged Steel Braces.......... 34
2.1.3.4.5 Full-System Braced Frame Tests ....................................................................... 38
2.2 Modeling Techniques for SCBF Systems................................................................................ 39
2.2.1 Demand Characterization ................................................................................................. 40
2.2.1.1 Techniques for Brace Simulation .............................................................................. 41
2.2.1.1.1 Rule-Based (or Phenomenological) Brace Models ............................................ 41
2.2.1.1.2 Concentrated Hinge Brace Models .................................................................... 42
2.2.1.1.3 Fiber-Element Based Brace Models .................................................................. 42
2.2.1.1.4 Continuum-Based Brace Models ....................................................................... 43
2.2.2 Capacity Characterization ................................................................................................ 45
2.2.2.1 Brace Geometry-Based Fracture Capacity Relationships ......................................... 45
2.2.2.2 Fiber Element-Based Critical Strain Models............................................................. 50
2.2.2.3 Micromechanical Void Growth and Coalescence Models ........................................ 51
2.3 Summary .................................................................................................................................. 53
Chapter 3
Large-Scale Brace Component Tests.................................................................... 71
3.1 Motivation for Large-Scale Brace Tests .................................................................................. 71
3.2 Experimental Setup.................................................................................................................. 73
3.3 Test Program Scope ................................................................................................................. 73
3.3.1 Cyclic Loading Protocols ................................................................................................. 76
3.3.1.1 Standard Cyclic Loading Protocol ............................................................................ 77
3.3.1.2 Near-Field Loading Histories.................................................................................... 79
3.3.1.3 Story Drift and Brace Axial Deformation Relationships .......................................... 80
3.4 Instrumentation and Miscellaneous Testing Results................................................................ 81
3.5 Summary .................................................................................................................................. 84
Chapter 4
Large-Scale Experimental Results and Design Implications ............................. 98
4.1 Qualitative Summary of Experimental Response .................................................................... 98
4.2 Quantitative Summary of Data For All Tests ........................................................................ 100
4.3 Effect of Test Variables on Cyclic Brace Behavior, Limit States and Design Implications.. 104
4.3.1 Effect of Width to Thickness Ratios............................................................................... 104
4.3.2 Effect of Member Slenderness ....................................................................................... 106
4.3.3 Effect of Cross Sectional Shape ..................................................................................... 107
4.3.4 Effect of Grout Filling of HSS Specimens ..................................................................... 107
4.3.5 Effect of Loading Rate ................................................................................................... 108
4.3.6 Effect of Unsymmetrical Buckling................................................................................. 110
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4.4 Brace-Gusset Plate Connection Performance ........................................................................ 110
4.4.1 Experimental Results...................................................................................................... 112
4.5 Comparison of Experimental Data to Commonly Used Formulae for Predicting Strength and
Stiffness of Bracing Members ..................................................................................................... 116
4.5.1 Elastic Stiffness .............................................................................................................. 116
4.5.2 Compressive Strengths ................................................................................................... 117
4.5.3 Maximum Tensile Strength ............................................................................................ 118
4.6 Summary ................................................................................................................................ 119
Chapter 5
Application and Evaluation of the CVGM for Large-Scale Components....... 136
5.1 Development of the Cyclic Void Growth Model (CVGM) ................................................... 139
5.1.1 Material Behavior, Constitutive Models and Calibration............................................... 143
5.1.2 CVGM Calibration ......................................................................................................... 146
5.1.2.1 Calibration of η ....................................................................................................... 147
5.1.2.2 Calibration of λ ....................................................................................................... 147
5.1.3 CVGM Application and Validation in Small-Scale Details ........................................... 148
5.2 Application of CVGM to Large-Scale Bracing Components ................................................ 149
5.2.1 Material Model Calibration ............................................................................................ 150
5.2.2 Large-Scale Brace Simulations ...................................................................................... 153
5.2.2.1 Modeling Global and Local Buckling ..................................................................... 153
5.2.2.1.1 Imperfections and Buckling Simulations......................................................... 154
5.2.2.1.2 Comparison to Experimental Global Buckling................................................ 155
5.2.2.1.3 Comparison to Experimental Local Buckling.................................................. 155
5.2.3 Brace Material CVGM Calibraiton ................................................................................ 156
5.2.3.1 Brace Material Calibration Results: η ..................................................................... 156
5.2.3.2 Brace Material Calibration Results: λ ..................................................................... 157
5.2.4 CVGM Fracture Predictions Applied to Large-Scale Brace Components ..................... 162
5.2.4.1 Application to HSS and Pipe Cross-Sections (Standard Loading).......................... 163
5.2.4.2 Application to Near-Field Loading Histories .......................................................... 166
5.2.4.3 Quasi-Static Versus Earthquake Loading Rates...................................................... 167
5.2.4.4 Application to Unsymmetrical Buckling................................................................. 168
5.2.4.5 Estimating Uncertainty in the CVGM Fracture Predictions.................................... 169
5.3 Summary ................................................................................................................................ 171
Chapter 6
Parametric Simulation of HSS Bracing Members ............................................ 203
6.1 Parametric Simulation Matrix................................................................................................ 204
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6.1.1 Cyclic Loading History .................................................................................................. 205
6.2 CVGM Fracture Predictions .................................................................................................. 205
6.2.1 Effect of Width-Thickness Ratio on Brace Ductility ..................................................... 206
6.2.2 Effect of Slenderness Ratio on Brace Ductility.............................................................. 206
6.3 Synthesis of Experimental and Simulation Results ............................................................... 208
6.3.1 A Simplified Approach for Evaluating the Effect of Brace Parameters on Ductility..... 210
6.3.1.1 Plastic Hinge Length ............................................................................................... 213
6.3.1.1.1 Comparison to Fiber-based Brace Simulations................................................ 216
6.3.2 Evaluating Brace Axial Deformation Capacities Relative to Earthquake Induced
Interstory Drift Demands......................................................................................................... 216
6.4 Summary and Reliability of Parametric Study and Semi-Theoretical Model........................ 222
Appendix A................................................................................................................................. 256
Appendix B ................................................................................................................................. 263
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List of Tables
Table 2.1: Maximum interstory drifts (standard deviation) from nonlinear time-history analyses
on 3 and 6-story SCBF systems. ........................................................................................... 56
Table 2.2: Design level (approximately 10% in 50 years) median (standard deviation) and
maximum interstory drift from Izvernari et al (2007) ........................................................... 56
Table 2.3: Results from 6-story braced-frame shake-table test (Tang, 1987)................................ 56
Table 2.4: Summary of HSS experimental review (63 tests)......................................................... 57
Table 2.5: Summary of calibration constants to predict cyclic fracture life of square and
rectangular HSS bracing members ........................................................................................ 57
Table 3.1: Test parameters and loading histories........................................................................... 85
Table 3.2: Brace material properties .............................................................................................. 85
Table 3.3: Summary of loading protocol modifications ................................................................ 86
Table 4.1: Data for standard (far-field) loading protocol tests..................................................... 123
Table 4.2: Data for near-field (NF) pulse loading protocol tests ................................................. 124
Table 4.3: Experimental results of bracing connections .............................................................. 125
Table 4.4: Comparison of brace strength and stiffness ................................................................ 125
Table 5.1: Measured material properties...................................................................................... 174
Table 5.2: Calibrated kinematic and isotropic hardening law parameters. .................................. 174
Table 5.3: Maximum measured local imperfections.................................................................... 174
Table 5.4: Calibrated CVGM fracture parameters....................................................................... 175
Table 5.5: CVGM fracture prediction summary .......................................................................... 176
Table 6.1: HSS simulation matrix (22 total) ................................................................................ 226
Table 6.2: Brace member lengths across various frame geometries ............................................ 226
Table 6.3: Summary of HSS experimental review (63 tests)....................................................... 226
Table 6.4: Regression parameter values ...................................................................................... 226
Table B.1: Experimental results from square and rectangular HSS tests .................................... 263
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List of Figures
Figure 1.1: Outline of experimental and simulation study............................................................... 9
Figure 2.1: Large-scale Special Concentrically Braced Frame test (Uriz, 2005) and local buckling
induced brace fracture at the middle plastic hinge ................................................................ 58
Figure 2.2: (a) Chevron braced-frame story and typical connection details, (b) Out-of-plane
buckling and tension yielding and (c) Brace plastic hinge formation. .................................. 59
Figure 2.3: (a) Brace gusset-plate net section fracture and (b) Column fracture at base of shear tab
along with beam fracture from prying action of gusset plate. ............................................... 60
Figure 2.4: Plastic hinge formation in beam from bracing force imbalance.................................. 61
Figure 2.5: Influence of brace geometry – in terms of global slenderness and cross-section
compactness – on fracture ductility....................................................................................... 62
Figure 2.6: Interstory drift time history results for the first story of a 3-story SCBF, 2% in 50
years event............................................................................................................................. 63
Figure 2.7: Minimum versus maximum interstory drifts for (a) 3-story and (b) 6-story SCBF
(courtesy Uriz, 2005)............................................................................................................. 64
Figure 2.8: (a) Schematic, one cycle, force-deformation response of a typical brace component
and (b-e) Progression of brace damage ................................................................................. 65
Figure 2.9: (a) Typical brace hysteretic response and definition of Δrange, (b) Influence of Widththickness and (c) Global slenderness ratio on axial deformation range, Δrange/ΔY................. 66
Figure 2.10: Brace modeling techniques........................................................................................ 67
Figure 2.11: Buckled shape, plastic strain contours, and critical fracture node from continuum
HSS4x4x1/4 brace analysis. .................................................................................................. 68
Figure 2.12: Ratio of experimental to predicted fracture deformation for three separate testing
programs................................................................................................................................ 68
Figure 2.13: Micromechanical process of ductile fracture in steel ................................................ 69
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Figure 2.14: (a and b) CVGM fracture prediction and (c) Comparison to experimental fracture
time for the critical node of an HSS4x4x1/4 shown in Figure 2.11 during a standard loading
history.................................................................................................................................... 70
Figure 3.1: (a) Plan and (b) elevation view of brace test setup...................................................... 87
Figure 3.2: Brace drawings for (a) HSS, (b) Pipe and (c) W12x16. .............................................. 88
Figure 3.3: Connection details for (a) HSS and Pipe and (b-c) W12x16....................................... 89
Figure 3.4: (a) Standard cyclic (far-field ground motions), (b) Compression and (c) Tension nearfault pulse loading histories................................................................................................... 90
Figure 3.5: (a) Original and modified SAC loading history and (b) Cumulative plastic drift for
Chevron braced-frame and moment frame under prescribed loading histories..................... 91
Figure 3.6: One-story Chevron braced-frame. ............................................................................... 92
Figure 3.7: Axial transducer measurements for brace end plates. ................................................. 93
Figure 3.8: Intended and measured axial deformations. ................................................................ 94
Figure 3.9: Actuator force measurements versus relative axial deformation for standard cyclic
HSS4x4x1/4 test.................................................................................................................... 94
Figure 3.10: (a) Wire-pot instrumentation plan, (b) Connection detail and (c) Pipe3STD test. .... 95
Figure 3.11: Measured out-of-plane buckling displacements. ....................................................... 96
Figure 3.12: Thermocouple recordings for (a) Quasi-static, near-field test and (b) Earthquake-rate
standard cyclic test. ............................................................................................................... 97
Figure 4.1: Typical progression of brace specimen damage........................................................ 126
Figure 4.2: Typical brace response for (a) Far-field loading, (b) Near-fault compression and (c)
Near-fault tension................................................................................................................ 127
Figure 4.3: Energy dissipated prior to (a) Fracture initiation versus local buckling and (b)
Strength loss versus fracture initiation. ............................................................................... 128
Figure 4.4: Effect of width-thickness ratio on (a) Maximum drift at fracture initiation and (b)
Normalized dissipated energy. ............................................................................................ 129
Figure 4.5: Effect of slenderness ratio on (a) Maximum drift at fracture initiation and (b)
Normalized dissipated energy. ............................................................................................ 130
Figure 4.6: Local buckling shapes. .............................................................................................. 131
Figure 4.7: Symmetric and unsymmetrical buckling. .................................................................. 132
Figure 4.8: Net section details and fracture ................................................................................. 133
Figure 4.9: (a) Experimental connection gage length and (b) Load deformation response of
Pipe3STD and Pipe5STD tests with fracture at the net section. ......................................... 134
Figure 4.10: Maximum experimental forces compared to expected capacities. .......................... 135
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Figure 5.1: Micromechanical process of ductile fracture in steel ................................................ 177
Figure 5.2: Scanning Electron Microscope (SEM) pictures of (a) Monotonic and (b) Cyclic
fracture surfaces (A572, Grade 50). .................................................................................... 178
Figure 5.3: (a) Uniaxial stress-strain behavior and (b) Isotropic yield surface............................ 179
Figure 5.4: Combined isotropic-kinematic yield surface employed in ABAQUS (2004). .......... 180
Figure 5.5: Ideal material calibration flow chart.......................................................................... 181
Figure 5.6: (a) Notched-bar geometry and (b) Typical force-deformation response. .................. 182
Figure 5.7: (a and b) Axisymmetric finite element model of notched-bar specimen and (c) Void
growth demand according to VGM..................................................................................... 183
Figure 5.8: Exponential relationship between ηcyclic/η and damage............................................. 184
Figure 5.9: CVGM fracture prediction for a small-scale notched bar specimen. ........................ 184
Figure 5.10: Predicted versus experimental instance of fracture initiation for small-scale, notchedbar cyclic coupon tests (adapted from Kanvinde and Deierlein, 2007)............................... 185
Figure 5.11: Summary of small-scale specimens for brace material calibration study................ 186
Figure 5.12: Experimental and simulation force-deformation comparisons for small-scale material
calibration tests.................................................................................................................... 187
Figure 5.13: Experimental and simulation force-deformation comparison for large-scale
HSS4x4x1/4 brace (far-field loading). ................................................................................ 188
Figure 5.14: Brace models and meshing schemes. ...................................................................... 189
Figure 5.15: Measured brace cross-section dimensions............................................................... 190
Figure 5.16: (a) Global and (b) Local buckling mode shapes...................................................... 191
Figure 5.17: Cross-section wall imperfection measurements for HSS and Pipe. ........................ 191
Figure 5.18: Experimental and predicted critical buckling loads. ............................................... 192
Figure 5.19: Experimental and predicted local buckling for HSS4x4x1/4 brace (far-field loading).
............................................................................................................................................. 194
Figure 5.20: Triaxiality versus equivalent plastic strain (HSS4x4x3/8 material). ....................... 194
Figure 5.21: Calibration of λNB and λC......................................................................................... 195
Figure 5.22: Constant probability distribution of ε...................................................................... 196
Figure 5.23: Effect of low damage levels on λ variability........................................................... 196
Figure 5.24: CVGM fracture predictions for HSS4x4x1/4 where λNB and λC are used in
calculating two damage functions. ...................................................................................... 197
Figure 5.25: Comparison between experimental and predicted fracture instances (far-field
loading). .............................................................................................................................. 198
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Figure 5.26: Comparison between experimental and predicted fracture instances (near-field
loading). .............................................................................................................................. 199
Figure 5.27: Comparison between quasi-static and earthquake-rate loading tests. ..................... 200
Figure 5.28: Comparison between experimental and predicted fracture instances for HSS4x4x1/4
brace subjected to far-field loading with middle reinforcing plates.................................... 201
Figure 5.29: (a) Deterministic CVGM fracture instances and (b) Cumulative probability function
for HSS4x4x1/4 subjected to far-field loading history........................................................ 202
Figure 6.1: HSS brace geometry .................................................................................................. 227
Figure 6.2: Brace loading history................................................................................................. 227
Figure 6.3: Influence of width-thickness ratio on brace ductility (fracture initiation) in terms of (a)
Axial deformation range and (b) Story drift........................................................................ 228
Figure 6.4: Influence of slenderness ratio on brace ductility (fracture initiation) in terms of (a)
Axial deformation range and (b) Story drift........................................................................ 229
Figure 6.5: Slenderness range for minimum and maximum frame sizes (see Table 6.X) for (a) All
HSS cross-sections and (b) AISC (2005) conforming (i.e., b/t < 16) sections; (c) Influence
of brace slenderness on story drift ductility for fixed b/t (14.2) ratio and brace lengths (119
and 180 in). Also shown are past experimental (b/t=14.2 and L=119 in) results................ 230
Figure 6.6: Experimental brace ductility in terms of drift versus brace parameter for similar (a)
Slenderness (19 tests) and (b) Compactness (25 tests) ratios.............................................. 231
Figure 6.7: Schematic illustration of simplified approach to evaluate effect of brace parameters on
ductility. .............................................................................................................................. 232
Figure 6.8: (a) Fiber-like fracture strain and (b) Deformation range capacity versus b/t ratio for all
experiments and simulations listed in Tables 6.1 and 6.3. .................................................. 233
Figure 6.9: Comparison between maximum experimental and predicted deformation range. .... 234
Figure 6.10: Schematic illustration of plastic hinge length calculation. ...................................... 235
Figure 6.11: Plastic hinge length as a function of increasing drift............................................... 236
Figure 6.12: Continuum finite element simulation of brace specimen showing plastic hinge length
dimension. ........................................................................................................................... 237
Figure 6.13: (a and b) Force-deformation and (c) Plastic hinge length comparison between finite
element and fiber brace models........................................................................................... 238
Figure 6.14: Maximum drift capacity versus width-thickness ratio. ........................................... 239
Figure 6.15: Deformation capacity (in terms of drift), divided by 1.4, versus normalized widththickness ratio...................................................................................................................... 239
Figure A.1: Experimental loading histories and brace hysteretic response. ................................ 256
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1
Chapter 1
Introduction
1.1
MOTIVATION
Structural investigations following the 1994 Northridge earthquake revealed that the combination
of high fracture toughness demands caused by poor detailing of beam-column connections and
low material toughness resulted in widespread fractures in these structural details. Since
Northridge, Special Concentrically Braced Frames (SCBFs) have gained considerable popularity
as a lateral load resisting system in high seismic areas. However, braced frames are also
vulnerable to premature fracture during earthquakes due to the cyclic inelastic buckling of the
bracing elements and the resulting large force and deformation demands on the bracing
connections. In fact, recent studies (Uriz, 2005) have suggested unsatisfactory performance of
bracing systems designed with current codes, leading to fracture in the braces during design-level
earthquakes. However, research regarding SCBFs is relatively less exhaustive when compared to
that regarding moment frame systems. In addition, the mechanisms of earthquake-induced Ultra
Low Cycle Fatigue (ULCF) that could initiate fracture in steel bracing components have only
recently been explained as part of a physics-based approach to predict fracture in small-scale tests
(Kanvinde and Deierlein, 2004 and 2007). Considering the recent concern with fracture in SCBF
systems and a novel physics-based modeling approach to describe ULCF, this study intends to fill
part of the knowledge gap pertaining to brace component behavior, evaluate the ULCF fracture
2
models at a large-scale and present a simulation-based methodology that can be used to
complement experimental programs.
Currently, the tools used by structural engineering researchers to predict fracture are not as
sophisticated as other aspects of structural analysis and common fracture prediction
methodologies are often based on varying degrees of empiricism rather than fundamental
mechanics. Simplistic approaches for fracture prediction in SCBFs involve using the story drift as
an indicator of fracture, while somewhat more advanced approaches may use either critical
longitudinal strain measures, or cycle counting and fatigue life approaches for individual braces
or components (Tang and Goel, 1989). Recent studies (Uriz, 2005) have applied cycle counting
techniques through fiber-based elements to simulate fracture strain demands at a cross section,
instead of along the entire brace. While these represent important advances in the fatigue-fracture
prediction methodology, they fail to directly incorporate the effects of local buckling and the
complex interactions of stress and strain histories that trigger crack initiation in these details. In
part, this is due to the erstwhile lack of computational resources required to model phenomena
such as local buckling and to characterize the stress and strain fields at the location of fracture.
More importantly, however, there has been a lack of suitable stress/strain based fracture
prediction criteria.
In addition, simulation techniques that accurately describe the complex stress and strain states
from global and local buckling events have not been rigorously developed, especially for largescale components and cyclic loading histories. With recent advances in computational power,
three dimensional continuum-based modeling, such as the bracing models illustrated in Figure
1.1, has become an attractive option to model local buckling phenomenon. Continuum-based
modeling presents significant advantages over concentrated hinge or fiber-based formulations as
the buckling induced stress and strain histories that trigger fracture during earthquake loading can
3
be directly modeled. These simulations can be utilized to perform parametric studies, gain
insights into brace performance, and inform general models that characterize member ductility
through geometric properties.
1.2
OBJECTIVES AND SCOPE
Micromechanical based models which act at the continuum level have shown promise in
simulating ductile fracture initiation during cyclic loading for specimens that are relatively small
in scale (on the order of several inches) and free of complex modeling conditions (i.e., buckling)
(Kanvinde, 2004). While the mechanisms of ductile fracture mechanisms in low-carbon steel
under monotonic loading conditions are well established (Rice and Tracy, 1969), the structural
engineering community has not had suitable criteria for assessing fracture demands and capacities
during earthquake-type loading until only recently. In large part, the lack of physics-based
fatigue/fracture models for earthquake-type loading has not been extensively addressed by the
scientific community due to the unique loading conditions of the very large, yet low in number,
earthquake cycles that push a structure well into the inelastic range of material response.
Moreover, large-scale structural analysis methods with the capability of simulating complex
geometric and material nonlinearities have been impractical to employ until recent advances in
computing power. As the earthquake engineering community moves towards a more
performance-based framework of design and analysis, there is a need to develop simulation tools
which can be applied to large-scale details and general loading histories to accurately assess
ductility of steel components. Motivated by this need, the general aim of this study is to reconcile
the modeling gap that exists between current large-scale structural analysis methods that are void
of fundamental fracture prediction capabilities and the validated cyclic fracture initiation
mechanisms at the micromechanical scale. More specifically, the research addresses the
following topics –
4
•
Experimental performance of steel bracing members subjected to earthquake-type
loading conditions. The large-scale tests include square, round and wide-flanged shapes
and investigate the effects of different brace geometries and material types on brace
ductility.
•
The evaluation of a general fracture prediction methodology at the large-scale which can
be applied across diverse loading situations and material types.
•
The application of this methodology to complement the large-scale experimental
program. In the context of this dissertation, the performance of square and rectangular
tube members is investigated through parametric simulations which employ these
advanced simulation techniques.
These objectives are investigated through various experimental, simulation and modeling
components. These components are illustrated schematically in Figure 1.1 and include –
•
Previous experimental results on bracing members and other analytical investigations
aimed at simulating brace behavior and quantifying fracture ductility (not shown in the
figure).
•
Nineteen large-scale tests representative of bracing members in SCBF systems. These
experiments supply a test-bed to evaluate the micromechanical fracture initiation models
as well as provide direct performance-based comparisons for the practicing design
community.
•
A series of small-scale material calibration tests on coupons extracted from the largescale bracing members. These experiments provide the connection between the fracture
mechanisms at the micromechanical scale and the fracture predictions at the continuum
scale in the brace simulations. Accurate and consistent calibration of the multi-axial
5
plasticity model is an important aspect of linking the fracture toughness of the material at
the small-scale to large-scale behavior.
•
Continuum simulations of the large-scale brace tests that accurately capture the complex
stress and strain state induced by global and local buckling during cyclic loading. These
simulations provide insight into localized behavior (i.e., brace cross-section performance)
as well as global performance data (i.e., story drift at local buckling or fracture initiation).
•
A suite of parametric studies on bracing members performed with continuum models and
a set of complementary analyses modeling the braces with fiber elements. The
investigation shows the importance of capturing localized behavior such as cross-section
buckling and describes general models which are informed through the simulation
matrixes.
While the evaluation of the cyclic micromechanical fracture models at the large-scale is an
important scientific aspect of this study, the relationships and conclusions developed from the
parametric study present the greatest opportunity for a more immediate impact on the structural
engineering community. These general models are based on fundamental mechanics and are
informed through results from extensive parametric studies and physics-based fracture models.
By relating brace properties and deformation demands to fiber-type strain measurements, the
models present designers and code writers with a tool to make more informed decisions regarding
steel brace behavior.
1.3
ORGANIZATION AND OUTLINE
Chapter 2 provides a review of large-scale SCBF experimental testing and analytical modeling
results along with a background section looking at the state-of-the-art modeling techniques for
brace behavior and fracture prediction. First, the chapter presents results from past analytical and
experimental studies to quantify earthquake demands on braced-frame components and/or
6
systems. Next, experimental trends in brace component ductility are examined to illustrate the
dependence of ductility on brace global slenderness and cross-section compactness parameters.
After these two sections, the chapter presents the state-of-the-art in modeling techniques that have
been applied to bracing components. The models range from phenomenological models which
describe the force-deformation relationship of an axial strut to sophisticated continuum-based
models which can model global and local buckling events. Following the discussion on brace
modeling techniques, fracture prediction models are introduced that have been specifically
developed for, and applied to, inelastic buckling braces. Finally, a general, physics-based, ULCFfracture model that relies on stress and strain quantities at a continuum point is briefly introduced
with an example of a fracture prediction for a bracing member.
Chapter 3 is a relatively brief chapter that details the experimental setup of the nineteen largescale brace component tests mentioned previously. The chapter reviews the design of the
specimens, test setup, general or special instrumentation for each test and the development of the
loading histories. The primary focus of the chapter is to introduce the test matrix and objectives
of the experimental program.
Chapter 4 presents results from the nineteen large-scale tests of steel bracing members to
examine their inelastic buckling and fracture behavior as related to the seismic design of
concentrically braced frames. The brace specimens include square Hollow Structural Shapes
(HSS), Pipe and Wide-Flange sections. The effect of various parameters, including widththickness and slenderness ratios, cross-section shape, loading history, loading rate and grout fill
on the performance of these braces is investigated. The test data is investigated with respect to
current seismic design limits on maximum width-thickness and slenderness ratios. Also,
measurements of brace stiffness, tensile strength and compressive strength are compared with
design formulae. Future analytical studies to simulate brace buckling and fracture are outlined as
7
a way to generalize the findings of the physical tests. The experimental results are also presented
in the context of net-section limit states in brace-gusset plate connections. Finally, maximum
tensile force measurements of the steel bracing members are presented and compared to codebased provisions that prescribe the expected yield strength (RyFyAg) of the brace for design of the
bracing connections.
Chapter 5 describes a methodology used to predict fracture in large-scale braced frame
components. The micromechanical-based ULCF fracture model (Kanvinde and Deierlein, 2007) –
introduced in Chapter 2 – is used to predict fracture in the brace component tests of Chapters 3
and 4. The physics-based fracture model has been shown to accurately describe the fundamental
mechanisms of void growth, collapse and coalescence in small-scale experiments. This Chapter
evaluates the performance of the physics-based fracture model across the various cross-section
shapes, material types and loading histories of the experimental program presented in Chapter 4.
The methodology demonstrates the importance of modeling the initiation, both the instance and
location, of localized cross-section buckling phenomena. Simulating local buckling is shown to
be linked to cross-section and brace properties (such as width-thickness and global slenderness
ratios), but also to material hardening parameters that are calibrated from small-scale tensile and
cyclic coupon tests. Effects of loading rate are investigated by comparing fracture predictions
from quasi-static loading tests to identical bracing members under fast, earthquake-rate tests.
Chapter 6 utilizes the fracture prediction methodology presented in Chapter 5 to conduct a
parametric study on representative bracing components of SCBF systems. By applying the
micromechanics-based fracture models, the study aims to generalize trends relating to brace
fracture performance. The focus is on Hollow Steel Structural (HSS) members that have recently
been observed to fracture prematurely during experiments. The simulations provide insights into
the failure mechanisms of inelastic buckling braces in the middle plate hinge. Moreover, the finite
8
element analyses provide a means to calculate the plastic hinge length after local buckling
initiation. Results from continuum-based modeling of the HSS braces members are compared to
those from more conventional fiber-based simulations using the OpenSEES platform. Informed
by the findings from past experimental programs and the parametric study, a semi-theoretical
model is proposed to calculate brace deformation capacity, and the associated story drift capacity,
as a function of brace geometry. The past experimental programs provide data from 63 previous
cyclic tests on square and rectangular HSS bracing members. The data is obtained from 10 testing
programs in the USA and Canada, and includes a wide range of brace parameters (slenderness
ratios ranging from 31 to 145, b/t ratios between 8.5 and 31.5).
Chapter 7 summarizes the dissertation by highlighting key observations and conclusions of the
large-scale experimental program as well as the complementary brace simulations and parametric
studies. While the research investigates braced-frame behavior from several different aspects,
from large-scale experimental results to fracture modeling to design provisions, this dissertation
does not seek to provide a “final answer” related to any one of these topics. Rather, the last
chapter emphasizes that the current work 1) Evaluates the use of a sophisticated modeling tool
that can fit into a more general, performance-based methodology, 2) Synthesizes experimental
and modeling results (both from this study and others) to fill parts of the knowledge gap related to
SCBF performance and 3) Highlights possible areas to be examined in future studies.
9
New insights into local
cross-section behavior
(Chapter 4 and 5; future
work)
Model development
(Kanvinde 2004) and
material calibration
(Chapter 5)
Parametric studies
(Chapter 6):
Physics-based
fracture model
evaluation
(Chapter 5)
Design considerations
(Chapter 4 and 6)
Large-scale brace experiments
(Chapters 3 and 4)
Figure 1.1: Outline of experimental and simulation study.
10
Chapter 2
Simulating Fracture in SCBF Bracing Components - Background
Several experimental and analytical investigations (Foutch et al 1987, Uriz 2005) and postearthquake reconnaissance reports (Tremblay et al 1995, Kelly et al 2000) have suggested that
Special Concentrically Braced Frames (SCBFs) may show unsatisfactory performance during
earthquake-type loading. Figure 2.1 illustrates a large-scale SCBF experiment by Uriz (2005) and
component tests conducted as part of the current study which both indicated local bucklinginduced fracture of the bracing members during inelastic buckling and tension yielding. In
addition to the brace fractures shown in the figure, fracture of brace-gusset plate connections
from large force and deformation demands on the connections is also a concern. Since SCBFs
rely on the buckling and yielding action of the brace components to dissipate seismic energy,
sufficient brace ductility must be provided to preclude premature fracture. Assessing the fracture
ductility of large-scale bracing components or systems is expensive, especially considering the
heavy dependence on experiment-based techniques. Moreover, testing of large-scale components
may be infeasible due to the size and strength of structural members. Considering these issues,
reliable fracture simulation and prediction techniques for SCBF systems are highly attractive and
allow for improved practicality in modeling the effects of diverse material properties and loading
conditions at the component, as well as system, level.
11
In this context, this chapter presents a broad overview of various experimental and state-of-the-art
simulation-based investigations that aim to characterize the performance (specifically the fracture
performance) of SCBF systems. Seismic performance assessment of structural systems typically
involves two distinct methodological components – (1) Evaluating deformation or strain demands
at the story, component or continuum level (2) Determining the corresponding capacities through
a combination of experimental and analytical methods. A comparision between the two results in
characterization of structural performance. Although “demands” and “capacities” are abstract
concepts that interactively affect each other in reality, they form a convenient framwork for
characterizing structural performance within the limitations of current simulation techniques.
Thus, the background and literature review presented in this chapter is broadly divided into two
main parts – one focusing on demand characterization, and the other focusing on capacities.
Throughout the chapter the literature is reviewed and synthesized within the context of a
Performance Based Earthquake Engineering (PBEE) methodology.
The first section of the chapter is devoted to a literature review on the performance assessment of
SCBF systems, either through experimental or analytical investigations. These provide
background for subsequent chapters. Next, various modeling techniques are examined to describe
the state-of-the art in structural analysis and fatigue-fracture prediction. With respect to the latter,
these fall into various broad categories including (1) empirical equations to predict fatigue life as
a function of the geometrical parameters of bracing elements (2) fiber based approaches that
employ a critical strain as an indicator of fracture and (3) and micromechanics-based models that
operate at the continuum level and simulate the processes of void growth and coalescense that are
responsible for ductile fracture initiation.
While the major objective of this study is to examine these continuum-based fracture models for
large scale specimens, the chapter also presents work that has advanced the state-of-the art in
12
analyis techniques which have been used on braced-frame components. These are broadly
classified as (1) phenomenological or rule-based models (2) lumped plasticity analysis techniques
(3) fiber-based elements and (4) continuum-based or finite element analyses.
The chapter concludes by summarizing the current state of research, and identifying areas where
advanced simulation and fracture models may benefit structural and earthquake research and
practice.
2.1
PERFORMANCE EVALUATION OF SCBF SYSTEMS
To provide context for the subsequent literature review, this section discusses braced-frame
systems by first describing the desired response and possible fracture events during earthquake
loading, and then reviewing modern design provisions for SCBFs. Afterwards, the literature
review is presented by first focusing on braced-frame demand evaluation and then on the ductility
capacity of these systems.
2.1.1
SCBF SYSTEM BEHAVIOR
A schematic illustration of a one story chevron (inverted-V) braced-frame is shown in Figure 2.2a
where the hollow circles indicate locations of assumed pinned connections. Shear-tabs are used in
the beam-column connections and gusset plates (flexible out of plane) connect bracing members
to beams and columns. While these are more properly classified as partially fixed connections,
they are typically considered to be pinned in design and analysis procedures, although recent
research (e.g. Uriz, 2005) has illustrated that these may be substantially more rigid than often
presumed. As illustrated in Figure 2.2b, frame deformation is accommodated by the rotation of
these connections and axial deformation of the bracing members. Thus, unlike moment frames
where the beam-column connection is fixed and inelastic behavior is concentrated in beam plastic
hinges, braced frames rely on cyclic inelastic buckling and tension yielding of the bracing
13
members to accommodate the inelastic deformations and dissipate seismic energy. The figure
shows a schematic of a chevron braced frame deformed laterally, such that one brace is buckled
in compression and the other brace is in tension. Figure 2.2c shows the frame under larger
deformations, such that the compression brace has now formed a plastic hinge in the middle of
the brace. After several loading cycles – the number depending on cycle amplitude as well as
brace geometry and material type – a local buckle forms at the middle plastic hinge on a
compressive excursion. This localizes the plastic strain accumulation to the region of local
buckling. Fracture initiation closely follows local buckling, with complete cross-section rupture
and strength loss occurring soon thereafter.
Besides local buckling-induced fracture at the middle plastic hinge, other possible fracture events
could occur at the slotted-end brace-gusset connections, beam-column shear tab, or the gusset to
beam-column connections. These are illustrated in Figure 2.3a-b and are described below –
•
Slotted-end brace-gusset (net section) connection fracture is shown schematically in
Figure 2.3a where the slot extending beyond the gusset plate creates a reduced section
susceptible to fracture during severe tensile loading cycles. As the brace is loaded in
tension, inelastic strains accumulate across a short gage length at the reduced area of the
connection, initiating fracture at the net section. Fabrication flaws could promote this
type of failure by introducing surface roughness or imperfections at the slotted section.
Furthermore, the net section is adjacent to the weld between the gusset-plate and brace
such that Heat Affected Zone (HAZ) defects could decrease the ductility of the brace
base metal as a result of material phase transitions or considerable grain growth.
•
Beam-column shear tab fracture is illustrated in Figure 2.3b and was observed during a
large-scale test by Uriz (2005). It was assumed that fracture initiated at fabrication cracks
below the fillet weld of the shear tab to column connection. At large story drifts,
14
considerable rotation demands on the beam-column connection can pry the shear tabs
from the column flange and create localized stress intensities on the weld, and
surrounding base, material. These increased demands, combined with the variability of
base metal properties near welds could cause column fracture at large drift levels.
•
Gusset-plate to beam or column connection fracture shown in Figure 2.3b is also
promoted by large rotation demands at the beam-column connection and the prying
action of the gusset plate on the flanges of the beam or column. Similar to the previous
two fracture events, fracture would also initiate close to the weld in the HAZ-affected
base metal.
In addition to these fracture limit states, inverted-V chevron configurations are susceptible to
concentrated inelastic behavior, and possible plastic hinge formation, at the upper-beam to brace
connection from a large force imbalance between the compression brace (small force) and the
tension brace (large force). The formation of a hinge at the mid-point of the beam could lead to a
story mechanism as illustrated in Figure 2.4. However, braced-frame system geometries with
alternating inverted-V and V bracing members (by story) or a zipper-frame configuration with
vertical axial columns at the beam mid-points (Bruneau et al, 2005) can prevent this type of
behavior by carrying the unbalance force.
Considering these possible failure mechanisms, current American Institute of Steel Construction
(AISC, 2005) Seismic Provisions aim to reduce inelastic effects in beams and columns and ensure
high ductility of the bracing members. While significant brace buckling and yielding is expected
in SCBFs during moderate to large earthquake events, detailing requirements guard against brace
and connection fracture. The design provisions most applicable to SCBF behavior are discussed
in the following section.
15
2.1.2
SEISMIC DESIGN PROVISIONS FOR SCBF SYSTEMS
The aim of modern steel seismic codes, such as the AISC Seismic Provisions (2005), is to ensure
acceptable system behavior through ductile performance of members and connections. Thus, in
the context of SCBFs, ductile system behavior is achieved through proper connection and brace
detailing to account for large rotations and repeated inelastic buckling and tension yield
excursions. The latter (ductile brace behavior) is attained through limits on geometric features
that control buckling mechanisms. While larger brace global slenderness has been shown to
increase brace ductility (Liu and Goel, 1988 and Tremblay, 2002), current seismic provisions
ensure ductile brace performance only through limits on the cross-section width-thickness, or
compactness, ratio (i.e., b/t for HSS). Referring to Figure 2.5, a more compact cross-section will
have a smaller width-thickness ratio and is less susceptible to local buckling during cyclic or
compressive loading. For HSS cross-sections, the Seismic Provisions (AISC, 2005) limit b/t
ratios as follows –
b / t < 0.64 E / Fy = 16 (for Fy = 46ksi )
(2.1.1)
Where b/t is used generically to represent the brace width-thickness ratio – as tabulated in AISC
design charts, such that b/t = B/t – 3 (i.e. (B-3t)/t), where B and t are the overall length of the
buckling face and the design thickness, respectively, as shown in Figure 2.5. Similar to the b/t
limit placed on square and rectangular bracing members, the current AISC Seismic Provisions
(2005) restrict D/t for Pipe cross-sections, such that –
D / t < 0.044 E / Fy = 36.5 (for Fy = 35ksi )
(2.1.2)
Where D/t represents the brace width-thickness ratio shown in Figure 2.5 – as tabulated in AISC
design charts, where D is the nominal outside diameter of the pipe and t is the design thickness.
Current AISC Seismic Provisions (2005) ensure ductile behavior by prescribing limits on WideFlange width-thickness ratios –
b f / 2t f < 0.30 E / Fy = 7.2 (for Fy = 50ksi )
(2.1.3)
16
Where bf and tf are the flange width and thickness dimensions, respectively. Note that Equations
2.1.1-2.1.3 are independent of other parameters, such as the brace slenderness, which is also
presumed to have an impact on brace ductility (Tang and Goel, 1989 and Tremblay, 2002).
Figure 2.5 suggests that a more slender brace will also provide an increase in ductility through the
decreased curvature, and therefore strain, demand at the middle plastic hinge for a longer brace as
compared to a shorter brace for the same axial deformation. However, as the brace becomes
increasingly slender the force imbalance between the tensile yield and compressive buckling
loads will generally increase, negatively affecting the energy dissipation capabilities while
increasing the system overstrength factor and the force demand on the beam in a chevron-type
braced-frame configuration. For these reasons, the current AISC Seismic Provisions (2005) limit
the slenderness of bracing members in SCBF design by –
KL / r < 4 E Fy
(2.1.4)
Thus, KL/r limits are prescribed as 100 (Fy = 46 ksi), 115 (Fy = 35 ksi) and 96 (Fy = 450 ksi) for
HSS A500 Grade B, Pipe A53 Grade B and A992 Wide-Flange bracing members, respectively.
AISC (2005) has recently incorporated an exception to this limit by allowing braces with KL/r <
200 and greater than Equation 2.1.4 if adequate compressive capacity is supplied by the adjoining
columns. This seeks to incorporate the positive affect of large slenderness ratios on brace
ductility.
As mentioned in the previous section, connections in SCBF systems (between the brace, beam
and column) are susceptible to several brittle modes of failure, including weld failure and netsection fracture at the end of the slotted brace. To prevent these types of failure, the Seismic
Provisions (AISC, 2005) require the design of these adjoining connections to be based on the
maximum force that the system can transfer to the connection. Although the Seismic Provisions
17
allow calculation of this force based on pushover or nonlinear time history analyses, they
recognize that (quoting from Section C13.3) – “In most cases, providing the connection with a
capacity large enough to yield the member is needed because of the large inelastic demands
placed on a structure by a major earthquake.” Consequently, these connections are typically
designed for forces corresponding to the expected yield force of the brace as described in
Equation 2.1.5 –
Py = Ry Fy Ag
(2.1.5)
Where Py is the expected yield force of the brace, Fy is the minimum specified yield stress of the
material and Ag is the gross cross-sectional area of the bracing member. The ratio Ry between the
expected yield stress and the minimum specified yield stress of the member material recognizes
that the expected strength will typically be larger than the minimum specified strength. Typical Ry
values are in the range of 1.1 to 1.6 (Liu et al, 2007).
2.1.3
STATE-OF-THE-ART IN SCBF PERFORMANCE ASSESSMENT
To describe the current state-of-the art in SCBF system and component evaluation, the following
sections present experimental and analytical results, as well as fracture predictive approaches
from various investigations focused on the performance of SCBF systems. In general, an
emphasis is placed on brace component behavior, where buckling induced fracture is a concern at
the center of the brace. The literature review is presented by first considering results from SCBF
demand assessment studies, followed by investigations which quantify SBCF ductility capacity.
Given this format, it is useful to first reflect on the motivation, to separate demands from
capacities for structural/earthquake engineering simulations and the resulting dependence on postprocessing techniques for performance evaluation.
18
In the overall context of the theme of “demands” and “capacities”, it is interesting to discuss that
an ideal simulation of structural response should include the direct simulation of each physical
phenomenon (local buckling, fracture initiation, fracture propagation, etc.), leading up to the
complete failure of the system. In this ideal scenario, the simulation would describe the state of
the structure after the loading event and little post-processing (i.e., such as checking the demands
against capacities) effort would be required. The necessity for a post-analysis comparison
between demands and capacities (or limit states) arises from the lack of sophistication of
modeling techniques and their inability to simulate complex phenomena. This lack of
sophistication is an important issue in earthquake engineering where structures undergo large
inelastic deformations, accompanied by various forms of buckling and fracture. While some of
these aspects are routinely incorporated in structural analysis programs, events such as fracture
are typically evaluated by comparing the analysis results (cumulative deformation, maximum
drift, maximum strain, etc.) with experiment-based empirical relationships or other models that
seek to describe the facture ductility of the material or component. This approach is advantageous
in that fracture does not need to be explicitly modeled as part of the analysis, thereby significantly
reducing the computational expense of the analysis model while providing reasonable predictions
of fracture. However, the accuracy of this approach relies on (1) Accurate simulations of
structural behavior leading up to fracture, including effects such as local buckling, etc, that drive
the fracture strains (2) General fracture models that can accurately predict fracture based on these
localized strains and (3) The accurate characterization of local material properties.
Considering the difficulty of simulating complex phenomena leading to fracture events in largescale systems, SCBF performance has been largely characterized through a comparison between
earthquake demands from dynamic analysis techniques and ductility capacities obtained primarily
through experimental techniques. Based on these simulations and experiments, several empirical
and semi-empirical approaches have been suggested to predict the fracture response of braces in
19
these systems. These models represent important advances in brace fracture predictions and they
are reviewed in detail in this section. However, these approaches may be difficult to generalize to
situations different than the experiments used to calibrate them. Braced-frame shake-table tests
are also reviewed in the context of assessing earthquake demands and ductility capacities of the
system, components and connections. Note that shake-table tests experimentally combine demand
and capacity analyses as an assessment of structural behavior and performance is simply the
investigation of the state of the structure following the applied ground motion. Along these same
lines, there has been recent work on incorporating fracture predictions during simulations, thereby
illustrating the “on-the-fly” effect of fracture on structural demands and behavior. However,
suitable analysis and fatigue-fracture prediction models which consider the mechanisms leading
to brace failure are critical elements of such simulation methods.
2.1.3.1 STUDIES EXAMINING SEISMIC DEMANDS IN SCBF SYSTEMS THROUGH NONLINEAR DYNAMIC
ANALYSIS
Several analytical studies have investigated earthquake-imposed demands on SCBF systems
through nonlinear time history dynamic analyses. In this section, three analysis studies on 3 and
6-story SCBF systems, designed according to American design codes (FEMA 1997, AISC 1997),
are reviewed along with three analysis studies on Canadian code (NRCC 1990 and 2005, CSA
1989 and 2005) compliant braced-frames. The analysis results of the US designed frames are
presented first followed by the Canadian designed frames. After the analysis results are presented,
experimental results from a braced-frame shake-table test are presented in the context of
earthquake demands. Finally, a synthesis of these results is provided to compare and contrast the
different studies.
Interestingly, the studies by Sabelli, Uriz and McCormick et al use matching frames and ground
motions with different modeling techniques for the braces. The results from these three analysis
20
programs provide some background for the later discussion on brace modeling approaches
(section 2.2).
2.1.3.1.1
SABELLI (2001) ANALYSES
The design and analysis of the braced-frames for this study is based on a site-specific design for
downtown Los Angeles following the NEHRP Recommends (FEMA, 1997) and the AISC
Seismic Provisions (AISC, 1997). The reader is directed to Sabelli (2001) for the detailed design
of the structures. A suite of 20 horizontal ground motions are used in the time history analysis of
a 3-story and 6-story inverted-V SCBF. The ground motions correspond to the design level
earthquake hazard intensity of 10% chance of exceedance in 50 years, determined according to
the elastic spectral acceleration at the first period of the structure. The reader is referred to
Somerville et al (1997) for more information on the ground motions.
The analyses were performed using the SNAP-2D structural analysis program (Rai et al, 1996),
where the braces were modeled with axial truss members. The axial struts were assigned a
phenomenological (rule-based) uniaxial hysteretic response based on experimental work by Black
et al (1980). The uniaxial force-deformation model depends primarily on the brace slenderness
and other cross-section properties. Fracture was modeled through an empirical cycle-counting
scheme. Upon fracture prediction as per this scheme, the member was removed from the
simulation model.
Referring to Table 2.1 and the 10/50 hazard level, Sabelli (2001) reported mean maximum story
drift values of 3.9 and 1.8% (standard deviations of 3.1 and 0.8) for the 3 and 6-story SCBF,
respectively.
21
2.1.3.1.2
URIZ (2005) ANALYSES
Uriz re-analyzed the previous 3-story and 6-story inverted-V SCBF structures designed by Sabelli
(2001) with the Open System for Earthquake Engineering Simulation (OpenSEES) (McKenna
and Fenves, 2004) analysis platform. The analyses of Uriz are more sophisticated as compared to
those of Sabelli. Nonlinear fiber-based elements were used to model the bracing members where
an initial camber (or member sweep) was assigned to the elements to promote global buckling. In
addition to the design level, 10% chance of exceedance in 50 years, ground motion suite, the
study employed two additional suites of ground motions corresponding to earthquake hazard
intensities of 50 and 2% chance of exceedance in 50 years. While the 10/50 hazard level is a
typical design level event, the 2/50 hazard is the Maximum Considered Earthquake (MCE) level
(IBC, 2006). Table 2.1 reports the results of the nonlinear dynamic analyses which used the sets
of scaled ground motions (10/50 and 2/50 hazard levels) developed for the SAC study (see
Somerville et al, 2007). The Table also distinguishes between analyses which incorporated a
fatigue model to track brace fracture events. The model was applied at the fiber level and once the
fracture limit state was reached, the fiber was removed, effectively diminishing the cross section.
Referring to Table 2.1 and the 10/50 ground motion analyses without the brace fatigue model,
Uriz (2005) reported median maximum story drifts of 1.5 and 1.4% (standard deviations of 0.9
and 0.8) for the 3 and 6-story SCBF, respectively. By modeling brace fracture during the analysis,
the maximum story drift recordings remain approximately the same at 1.6 and 1.1% (0.9 and 0.6)
for the 3 and 6-story SCBF, respectively. At the MCE level (2/50), the median maximum drifts
increase substantially to 5.7 and 5.1% (3.0 and 3.4) without the fatigue model and 5.7 and 4.4%
(2.4 and 2.2) with the fatigue model for the 3 and 6-story SCBF, respectively. Note that several
analyses (3 for the 3-story and 6 for the 6-story) using the 2/50 ground motions revealed brace
22
fracture, as predicted by the built-in fatigue model, and eventual structural collapse. These
analyses are not included in the calculation of the median drifts reported in Table 2.1.
In addition to the maximum story drift data from nonlinear time history analyses, it is also
interesting to note the unsymmetrical response in assessing the demands on SCBF systems and
components. Unlike moment frames, where the response is more symmetrical (under far-field
ground motions), braced frames tend to show a more unsymmetrical response, presumably
because of the unsymmetrical strength and stiffness properties once the compressive brace
buckles. To illustrate this, Figure 2.6 shows a typical story drift time history from the 3-story
SCBF system during a 2/50 ground motion. The important observation is that the response is
highly unsymmetrical, such that the θmax is more than two times θmin, i.e. θmin = 0.4θmax for this
particular story and ground motion. Figure 2.7 illustrate this point further by plotting the
minimum story drift, θmin, versus maximum story drift, θmax, for each ground motion and story of
the 3-story and 6-story SCBF, respectively (Uriz, 2005). On average, the minimum story drift is
shown on the figures as approximately 40% of the maximum story drift for both frames and the
sixty LA-based ground motions. This behavior is notable because braced frame components (such
as braces) are typically subjected to symmetric loading protocols, based on adaptations of
protocols designed to reflect demands in moment frames (Fell et al, 2006, Han et al, 2007,
Shaback and Brown, 2003 and Archambault et al, 1995). Consequently, when the maximum
equivalent drift is reported as a capacity measure, it is calculated as half the range of equivalent
drift applied to the component (e.g. Uriz, 2005). With reference to Figure 2.7, this may
underestimate the capacity of the brace which, in general, will not be subjected to symmetric
cycles under seismic excitation. This issue will be discussed in more detail in Chapter 6 which
looks at the capacity characterization of bracing components.
23
2.1.3.1.3
MCCORMICK ET AL (2007) ANALYSES
McCormick et al (2007) used the same 3 and 6-story SCBF systems as Sabelli (2001) and Uriz
(2005). Similar to Uriz (2005), the analyses were performed using the OpenSEES platform.
Nonlinear fiber-based elements are used for the beams and columns while the braces are assigned
a phenomenological model to describe the hysteretic response. This is similar to the approach by
Sabelli (2001) where the hysteretic model depends on the brace slenderness and other crosssection and material properties. Note that a fatigue-fracture model for the bracing members was
not used in this study.
Referring to Table 2.1, results from nonlinear analyses using the 10/50 and 2/50 ground motions
by Somerville et al (1997) are reported by McCormick et al. For the design-level, 10/50, event
mean maximum story drifts are listed as 3.57 and 1.97 (standard deviation of 1.61 and 0.68) for
the 3 and 6-story, respectively. For the MCE-level, 2/50, event, the corresponding maximum
drifts are 8.13 and 4.67 (standard deviation of 3.03 and 2.70).
2.1.3.1.4
REDWOOD ET AL (1991) ANALYSES
Redwood et al analyzed several 8-story braced-frames designed as per CAN/CSA-S16.1-M89
(CSA, 1989) and the 1990 edition of the National Building Code of Canada (NRCC, 1990). At
the time of this study, three categories of braced-frame systems existed in the Canadian code
corresponding to the expected ductility; these were (i) Ductile Braced Frames (DBF) (ii) Nominal
Ductility Braced Frames (NDBF) and (iii) Braced frames with no special ductility provisions
(SBF). In the context of our discussion on SCBFs, only the DBF results are summarized here.
While the detailed design is provided in Redwood and Channagiri (1991), it should be noted that
a considerably smaller R-factor (or strength reduction factor) is used in the design of the
Canadian DBF as compared to the American designed SCBF from above.
24
The 8-story DBF was analyzed using DRAIN-2D (Kannan and Powell, 1973) where the bracing
elements were modeled with axial truss elements and a phenomenological model (Jain and Goel,
1978) to describe the nonlinear hysteretic behavior. A set of ten earthquake ground motions were
chosen, primarily from recordings in the western United States, and scaled by the Peak Ground
Velocity (PGV) to correspond to a design level event for Vancouver, British Columbia. In terms
of Peak Ground Acceleration (PGA), the scaled earthquake records varied from 0.2g to 0.6g.
Note that from UBC (1997), the design PGA for zone IV, soil profile type SD, is approximately
1.1g. Thus, considering the smaller R factor used for the design and the less intense ground
motions, as compared to the previous 3 studies, these results are not directly comparable to SCBF
behavior but are included here for completeness.
From the nonlinear dynamic analyses, the largest story drift values were recorded on the seventh
story of the 8-story DBF. At this story, the mean maximum story drift across all ten ground
motions was listed as 0.7% (standard deviation of 0.88) by Redwood et al, with a maximum drift
equal to 1.1%. The drifts of the other stories ranged from approximately 0.3 to 0.6%.
2.1.3.1.5
TREMBLAY AND PONCET (2005) ANALYSES
Tremblay and Poncet (2005) analyzed several multi-story concentrically braced frames designed
as per the 2005 edition of the National Building Code of Canada (NRCC, 2005). The primary aim
of the study was to evaluate mass and geometric irregularity affects on behavior. For comparison
with the previous investigations, only the control structure with regular mass and geometric
distributions is considered here. This structure is an 8-story braced-frame with X bracing
members in each story. Similar to the Redwood et al (1991) study, DRAIN-2D (Kannan and
Powell, 1973) is used to analyze the structures where the bracing elements are modeled with axial
truss elements and a phenomenological model (Jain and Goel, 1978).
25
A suite of ten earthquake ground motions, scaled to a design-level event for Vancouver, British
Columbia, was used in the analyses. After scaling, the PGAs ranged from 0.3 to 0.6g across the
ten records. In comparison, PGA values for the Zone IV as per the 1997 UBC design spectra
(corresponding to approximately a 10/50 hazard) disregarding near fault effects are on the order
of 1.1g. The largest drifts were consistently recorded at the 6th story of the structure. The mean
maximum story drift at this story was approximately 1.7% (standard deviation of 0.5). The
maximum drift across all ground motions was listed as 2.7% and also occurred at the 6th story.
2.1.3.1.6
IZVERNARI ET AL (2007) ANALYSES
Izvernari et al (2007) presents results from analyses on several braced-frames designed according
to the 2005 edition of the National Building Code of Canada (NRCC, 2005) as per the Limited
Ductility (LD) and Moderate Ductility (MD) categories. The LD structures are not discussed
here. A total of 5 frames with 2, 4, 6, 8 and 12-stories were analyzed with the OpenSEES
platform where, similar to Uriz (2005), the braces were modeled with fiber-based elements. A
suite of twenty ground motions (ten historical and ten simulated) were used in the nonlinear
analysis of the frames. Referring to Table 2.2, the median drift recordings range from 1.1 to 1.8%
for the five frames, where the shorter 2 and 4-story frames had smaller drifts than the 8 and 12story frames.
2.1.3.2 SHAKE TABLE TESTING
Shake-table experiments can provide valuable earthquake demand data for structural systems,
while at the same time, evaluating the capacity of components and connections. However, it
should be noted that shake-table experiments only provide a single performance data point due to
the single ground motion used to damage the structure. While the experiments are visually
appealing and quite exciting, the results should be viewed objectively as the performance of a
single system during subjected to a single ground motion.
26
Tang (1987), Foutch et al (1987) and Roeder (1989) report findings from a 6-story braced-frame
shake table test conducted in Tsukuba, Japan. The design of the frame was based on both US
(UBC, 1979) and Japanese (Watabe and Ishiyama, 1980) building codes and lacked the detailing
of current design codes (AISC, 2005). For example, gusset plates were not provided at the ends of
the inverted-V bracing members as the braces were welded to the flanges of the beams. Table 2.3
lists maximum story drift measurements from a shake-table test using the Miyagi-Ken-Oki
earthquake scaled to 0.51g, approximately two times that of the “moderate” intensity level
experiment. A corresponding hazard level of the scaled ground motion was not discussed (Tang,
1987). A maximum drift prior to any fracture event was 1.6% in the third story of the frame,
which is within the range of the analytical results of Uriz (2005) and Izvernari et al (2007)
discussed previously with 10/50 simulation-based mean/median maximum drifts between 1.1 and
1.7%. However, while evaluating this comparison it should be noted that U.S. (and Japanese)
design codes have changed considerably between the time of the tests and the nonlinear time
history analyses. After fracture in stories 2-5, the maximum drifts increase significantly due to the
softening affect of fracture. This is most noticeable in story 3 where a maximum drift of 2.5% is
recorded after complete brace fracture and strength loss.
2.1.3.3 SUMMARY OF DEMAND ASSESSMENT FOR SCBF SYSTEMS
Current SCBF design requirements state that “braces could undergo post-buckling axial
deformations 10 to 20 times their yield deformation” (AISC, 2005). Given a system yield level
drift of approximately 0.3% to 0.5% (corresponding to initial brace buckling), the Seismic
Provisions may be interpreted as desiring a deformation capacity of approximately 3% to 5% for
SCBF systems. In this context, the story drift demands reported by Sabelli (2001), Uriz (2005)
and McCormick et al (2007) for the 10/50 (design) event in Table 2.1 range from 1.5 to 3.9% for
the 3-story frame and 1.1 to 2.0% for the 6-story frame. At the MCE, or 2/50, level the drift
demands increased substantially to 5.7 to 8.1% and 4.4 to 5.1% for the 3 and 6-story frame,
27
respectively. Thus, for design-level events, the 10/50 analyses seem to corroborate the expected
demands of the current AISC Provisions. On the other hand, at the MCE level, these results
suggest that SCBF earthquake demands could exceed 5% drift.
The discrepancy between the results of Table 2.1 should be noted, especially considering the
same structures and ground motions were used. This highlights the significance of robust
structural analysis techniques that can accurately characterize response. The different brace
modeling techniques, discussed more in section 2.2, between the three investigations may be
responsible for these differences.
The next section of the literature review discusses experimental studies focused on the cyclic
fracture/fatigue resilience of bracing members.
2.1.3.4 STUDIES INVESTIGATING THE CAPACITY OF BRACING MEMBERS
Where the previous discussion focused on braced-frame demand assessment, this section presents
experimental results which characterize the capacity of SCBF systems and bracing members. Of
primary interest is brace component behavior, where buckling-induced fracture at the middle
hinge could lead to eventual failure. The ductility of bracing components, often expressed in
terms of a maximum, or cumulative axial deformation can be compared to story drift demands
(discussed above) through simple kinematic relationships. While one such relationship is
introduced in Chapter 3, the primary aim of this section is to review experimental work which has
characterized the ductility of inelastic buckling braces. Many of the experimental studies
presented here have sought to generalize the experimental results with fatigue-fracture models of
varying degrees of complexity. However, these will be the focus of a later section. Furthermore,
while experimental investigations on small-scale (for example, 1x1 inch cross-sections) bracing
members have been a popular means to identify important trends in ductility and strength
28
requirements (Khan and Hanson 1976, Jain and Goel 1978, Jain et al 1980), they are not covered
in depth here. Given the highly nonlinear behavior of cyclic brace response, such as local
buckling-induced fracture (where the strains are sensitive to issues such as the length of the local
buckle), similitude might not exist between large and small scale experiments. Furthermore,
material properties and residual stresses within the walls of the cross-section may vary between
small and large-scale sections. Uriz (2005) provides an excellent summary of experimental work
with smaller bracing members.
As discussed previously and illustrated in Figure 2.2, inelastic buckling and yielding of bracing
elements serve as the primary seismic energy dissipation mechanisms in SCBF systems. To
provide background for the current discussion, it is important to consider the events leading to
bracing component failure during earthquake-type cyclic loading. Referring to Figure 2.8, the
first major limit state is brace buckling at point A, which is evident by large lateral deformations
and accompanied by flaking of the whitewash paint due to large strains associated with folding at
the end gusset plates and plastic-hinging at mid-length of the brace. As seen in region A-B,
buckling is followed by a sudden drop in load. However, as the compression in the brace reduces,
the response is mainly driven by bending of the buckled brace, resulting in a more gradual drop in
load after point B. During subsequent cycles, the localized yielding in the gusset plates and midpoint hinge becomes more severe as the amplitude of compressive loading increases. Upon
reversed loading at point C, the stiffness gradually increases (point D) as the out-of-plane
deformations decrease. As the strut straightens, the out-of-plane deformations reduce, and the
brace yields in tension at point E. The subsequent compressive excursion results in a smaller
buckling load (see point F) as compared to the first buckling event due to the Baushinger effect,
increased brace length (from tension yielding), and residual out-of-plane deformations. After
repeated compression-tension loading cycles, a local buckle forms at the middle hinge, similar to
Figure 2.8c, which amplifies the plastic strain and triggers ductile fracture soon thereafter (Figure
29
2.8d). While these events are not evident on the load-deformation curve, the photos of Figure 2.8
are fairly representative of the local buckling and fracture initiation observed in most tests. Upon
further cycling, the rupture propagates in a ductile manner across the section, i.e., for square HSS,
the buckled face ruptures first as shown in Figure 2.8e. Finally, at some point during a subsequent
tensile excursion, the entire cross-section fractures suddenly, severing the brace. This progression
is typical in large-scale braced-frames and has been detailed across numerous studies.
Experimental investigations often compare the cyclic fracture capacity of bracing members to the
brace slenderness ratio, the width-thickness ratio, or a combination of the two. Thus, a general
trend of the following sections will be to discuss the capacity of bracing members with respect to
these two geometric properties. Generally, it is difficult to observe their independent affects as
cross-section compactness is linked with the slenderness ratio through the radius of gyration. This
difficulty provides context for subsequent chapters as 1) there is a lack of general models to
explain brace ductility through geometry properties and 2) continuum-based parametric studies
can be used to expand the experimental date presented in this and future studies.
The current discussion on brace capacity begins with a review of component tests on HSS bracing
members, the most commonly used section in SCBF design. While square and rectangular HSS
have been the most wide-spread shape used in concentrically braced frames, their ability to
provide adequate performance has recently been questioned due to their perceived poor
performance during cyclic loading. Thus, results from Pipe and Wide-Flange experiments are
presented and reviewed as they have gained increased popularity over HSS shapes as bracing
members in SCBF systems. Finally, frame tests – either static or dynamic – are presented to gain
an understanding of system level capacity and performance.
30
2.1.3.4.1
SQUARE AND RECTANGULAR HOLLOW STEEL STRUCTURAL (HSS) BRACING
COMPONENTS
The various experimental programs on square and rectangular HSS bracing members are
summarized in Table 2.4. While a more detailed synthesis of these experiments is the topic of
Chapter 6 (combined with the full-scale tests from this investigation – presented in Chapter 3 and
4), this section provides a general review of experimental research on HSS bracing members.
Given that brace ductility is a function of several parameters, it is difficult to discuss specific
conclusions across all investigations without a common basis that can be used to interpret data
from various test programs, which may have different brace lengths and cross-sectional
dimensions. Thus, the following discussion introduces experimental research on HSS bracing
members by highlighting several important trends that have been reported in the context of HSS
brace ductility.
In general, the findings from all programs listed in Table 2.4 concur that width-thickness and
slenderness ratios have the most significant effect on brace ductility, such that higher widththickness ratios and lower slenderness are detrimental to brace performance. For example, Han et
al (2007) found that for approximately the same slenderness ratio (KL/r ≈ 80), increasing the
width-thickness ratio by 107% (from 13.7 to 28.3) resulted in a decrease in fracture ductility by
75%. In regard to slenderness influence, Tremblay et al (2003) reported that increasing the
slenderness by 60%, while holding the width-thickness ratio constant at 25.7, increased the axial
deformation ductility by 80%.
The investigations by Gugerli, Liu, and Lee with Goel (1982, 1988 and 1988, respectively) and
later Zhao et al (2002) were focused on the relative performance of concrete-filled HSS to
unfilled HSS bracing members. The concrete acts to prevent the inward local buckling observed
31
in most hollow sections, producing a less severe local buckling shape as the cross-section walls
buckle outward. In general, it was concluded that concrete fill has a beneficial effect on ductility
by delaying cross-section local buckling. For example, Lee and Goel (1988) reported a maximum
axial deformation during cyclic loading of 1.7 inches for an unfilled HSS cross-section, whereas
the same concrete-filled brace survived a maximum deformation of 2.6 inches, fracturing
approximately 8 cycles after the unfilled brace.
Archambault et al (1995) and Tremblay et al (2003) used the effective brace length to inform
design provisions on properties such as compressive strength degradation, energy dissipation and
brace fracture resistance. A kinematic relationship was also proposed to predict out-of-plane
buckling deformations. The investigation found that using the brace effective length (KL) resulted
in an accurate prediction of the compressive buckling load and, further, that as the slenderness
increases the energy dissipation capacity of the brace decreases linearly. Lastly, an empirical
fracture resistance equation was proposed, which is proportional to the brace global slenderness
and inversely proportional to the cross-section compactness ratio. Thus, the equation is
descriptive of the behavior that one would expect from the qualitative depiction shown in Figure
2.5, and employs data from all of the studies prior to 1995 (in Table 2.4) to calibrate various
constants of the model. The relationship and its accuracy will be discussed in more detail in
subsequent sections. Shaback and Brown (2003) present similar data and conclusions on HSS
bracing members as those of Tremblay et al (2003) concerning out-of-plane deformations, energy
dissipation and compressive buckling strengths, while employing a different empirical fracture
model that had been developed previously by Lee and Goel (1988) and Tang and Goel (1987).
Similar to the approach of Tremblay et al (2003) the fracture relationship is also a function of
slenderness and cross-section compactness while using a cumulative deformation measure to
assess capacity. This model will also be reviewed in section 2.2.2. While the experimental
investigation on bracing components by Yang and Mahin (2005) provides valuable data on cyclic
32
brace performance, the series of tests were conducted primarily to study the performance of the
slotted brace, gusset-plate end connection.
2.1.3.4.2
ROUND STEEL PIPE BRACING COMPONENTS
As mentioned earlier, there has been a recent shift from using square and rectangular HSS crosssection shapes for bracing members to Pipe and Wide-Flange shapes. In part, the restrictions on
square HSS width-thickness ratios (i.e., Equation 2.1.1) have limited the number of shapes that
are available during design, making Pipe and Wide-Flange shapes a more attractive option.
Moreover as mentioned in the previous section, the cold working of square HSS sections during
fabrication was thought to significantly reduce the fracture toughness of the corner material,
driving fracture initiation at the corners first. Thus, the more gradual bend radii of Pipes and the
rolling process of Wide-Flange sections was thought to provide tougher material as compared to
square or rectangular HSS. This section, and the next, will briefly discuss several experimental
investigations on these alternative shapes.
At the time of the writing of this dissertation, an exhaustive study on steel Pipe bracing members
was being conducted by Tremblay et al at École Polytechnique, Montréal. The study is expected
to carry significant importance as, compared to square HSS members, there are relatively few
studies that have investigated the performance of round Pipes. However, there have been several
other studies that have looked at the cyclic behavior of large-scale Pipe bracing members. One of
the earliest studies on the cyclic behavior of steel Pipe sections was by Popov and Black (1981).
Although the reported experimental results focused more on compressive strength degradation
rather than on fracture events, an important finding was that the more compact Pipe4STD (widththickness ratio = 19, approximately 0.52 times the current maximum ratio recommended by
AISC, 2005) and Pipe4X-Strong (width-thickness ratio of 13.4, 0.37 times the current maximum
limit) braces locally buckled at very large axial deformations.
33
More recently, Elchalakani et al (2003) conducted a series of twenty steel pipe braces under
earthquake-type cyclic loading. The testing matrix for the braces tested to fracture included a
variety of different width-thickness ratios – ranging from a D/t ratio from 9 to 21 – with
slenderness ratios of approximately 24 or 35. Surprisingly, the results suggested that, when
grouped by slenderness, the less slender braces (those with KL/r ratios of approximately 24)
fractured later in the loading history and at larger axial deformations than the braces with a larger
slenderness ratio. Furthermore, the investigation concluded that brace fracture ductility is “more
sensitive” to the cross-section width-thickness ratio as compared to the slenderness ratios. While
the results consistently confirm the less slender braces are more ductile, it should be noted that
the study queried a much larger range of D/t ratios (9 to 21) as compared to brace slenderness
ratios (24 to 35). Nonetheless, the study provides valuable fracture capacity data that can be used
to assess the performance of steel Pipe bracing members.
2.1.3.4.3
WIDE-FLANGED BRACING COMPONENTS
Similar to steel Pipe cross-sections, Wide-Flanged braces have becoming increasingly popular in
recent years due to their perceived superior ductility over HSS members during earthquake
loading. However, there are several challenges with using Wide-Flanged sections in braced-frame
construction. First, novel gusset plate end-connection details, not discussed as a part of this
chapter (see Chapter 3 for an example connection detail), must be devised to ensure out-of-plane
buckling about the member weak axis. These connection details can be quite complex as, unlike
HSS or Pipe sections, traditional slotted-end connections are not feasible for Wide-Flange shapes.
Moreover, the number of Wide-Flanged shapes that can be used as bracing members is somewhat
limited to shapes with small to moderate tensile yield to buckling load ratios. If not selected
correctly, the section properties can create a large force imbalance on the beam (in the case of
chevron-type configurations) and a reduced energy dissipation capacity. While this is accounted
34
for through Equation 2.1.4 (AISC, 2005) it does have the effect of limiting the shapes that can be
used as bracing components.
Popov and Black (1981) and Gugerli and Goel (1982) conducted a series of experiments on
Wide-Flanged bracing members. Popov and Black tested Wide-Flanged members with widththickness ratios ranging from 5 to 8.25 with slenderness ratios from 40 to 120 yet provide little
detail on the relative performance of the members as related to fracture ductility. Gugerli and
Goel present brace fracture capacities for very slender (KL/r from 95 to 175) Wide-Flanged
bracing members with width-thickness ratios between 7.4 and 11.5. As expected, these results
show that ductility improves with increasing slenderness and decreasing compactness.
2.1.3.4.4
GENERAL DUCTILITY TRENDS OF HSS, PIPE, AND WIDE-FLANGED STEEL BRACES
Figure 2.9 compares experimental results from three separate investigations by Shaback and
Brown (2003) on square and rectangular HSS, Elchalakani et al (2003) on round Pipe, and
Gugerli and Goel (1982) on Wide-Flanged cross-sections. The data is presented here for example
purposes to identify issues and questions with respect to assigning brace ductilities based on
slenderness and/or compactness. Furthermore, as each study applies the same loading history to
each brace, the results within each cross-section type can be discussed without considering the
influence of loading history. The shake-table and quasi-static test results from Tang (1987) and
Uriz (2005), respectively, are also shown but will be discussed in the following section. Figures
2.9b and 2.9c illustrate the influence of width-thickness and slenderness ratio, respectively, on the
maximum deformation range in tension and compression (shown in Figure 2.9a) of the brace
prior to failure, Δrange, normalized by the yield deformation, Δy. The brace width-thickness and
slenderness ratios are normalized by the AISC (2005) limits described previously. It is important
to note that the brace component loading protocols varied between the three investigations where
a symmetric history in tension and compression was used for the Pipe tests, a slightly biased
35
compressive loading history for the HSS, and a compressive dominated history for the WideFlanged tests. While a cumulative deformation quantity, such as energy dissipation or cumulative
plastic deformation, may be a more descriptive measure in comparing relative brace capacities,
the maximum range sustained by the brace prior to failure (Figure 2.9a) is used here as this can be
most readily transferred to a simplified maximum story drift. As illustrated in the demand section
(see Table 2.1-2.3), this quantity is most often used when discussing structural system demands
and will set the context for later chapters.
Referring to Figure 2.9b, as the width-thickness ratio (normalized by the respective AISC, 2005
limits) increases within a group of braces with similar slenderness ratios, the maximum
deformation capacity tends to decrease. For example, the Pipe braces with normalized slenderness
ratios of 0.2 (solid circles) show a definite downward trend for increasing width-thickness ratio.
This trend is also observed for HSS and Wide-Flange members when grouped according to
relative slenderness ratios. In general, the experiments selected for this example confirms the
current design methodology of placing upper-bounds on section width-thickness ratios (AISC,
2005).
Figure 2.9c illustrates the influence of slenderness ratio on maximum deformation capacity using
the same data points. The braces are grouped according to relative width-thickness ratios,
facilitating a comparison between ductility capacities as a function of slenderness. The figure
shows an apparent trend between increasing axial deformation ductility and slenderness ratio.
Interestingly, slenderness also appears to mitigate unfavorable width-thickness ratios, as the
compactness of the experiments presented here seems to decrease, on average, as slenderness
increases. However, referring back to Figure 2.9b, the influence of brace slenderness within the
Pipe and HSS sections is questionable. For example, a relative slenderness ratio decrease from
0.62 (solid squares) to 0.53 (hollow squares) for HSS members does not seem to influence
36
ductility, rather it is mostly controlled by the width-thickness ratio. Furthermore, the Pipe crosssections show that increasing slenderness (from 0.2 to 0.3) acts to decrease ductility. However, it
should be mentioned that the HSS and Pipe specimens sample a smaller range of slenderness
ratios as compared to the Wide-Flange braces which have a much larger range of slenderness
ratios (1.0 to 1.8). Thus, the investigations within the HSS and Pipe experiments are somewhat
limited with respect to slenderness affects. Considering the trends presented in Figure 2.9, several
key concerns and knowledge gaps can be raised pertaining to the current state-of-the-art brace
capacity assessment procedures –
1. Given that brace slenderness tends to influence the axial deformation capacity of a brace
during inelastic cyclic loading, should design codes restrict the use of stocky members in
braced-frame construction? For example, in the case of the steel Pipe specimens shown in
Figure 2.9, the axial deformation capacity is severely reduced as compared to the more
slender HSS and Wide-Flanged braces. However, according to AISC Seismic Provisions,
the Pipe members have lower relative width-thickness ratios as compared to the HSS and
Wide-Flanged members. Thus, while the experiments suggest otherwise, the code deems
Pipe members as “more acceptable” compared to the HSS and Wide-Flanged braces
presented here.
Furthermore, should design codes restrict all braces to meet the same compactness
requirement (within cross-section type)? According to the data of Figure 2.9, doing so
may unnecessarily penalize cases where slender braces would perform well, even with a
large width-thickness ratio. For example, the large slenderness ratios (or possibly the
nature of the local buckling deformation) of the Wide-Flanged members presented in
Figure 2.9 seems to mitigate the competing influence of a large width-thickness ratio.
37
2. Does the maximum axial deformation capacity of individual bracing components, used in
Figure 2.9, directly translate to a system level ductility measure? Experimental brace
capacities are often expressed in terms of axial deformation capacity, but few have
translated this into a story drift capacity. This would facilitate a comparison between
earthquake system level demands and component ductility levels.
3. With respect to the first point, can the interactive influence of brace slenderness and
width-thickness ratio be rationally evaluated without relying on a fully empirical,
experiment-based approach? While several relationships have been developed (presented
in the following sections) to describe the ductility capacities for HSS shapes, they are
largely empirical and are not derived from a fundamental mechanics-based approach.
Thus, it could be beneficial to develop a simulation-based methodology that could
investigate the relationship between brace ductility and the governing geometric (or
material) brace properties. Although not explicitly mentioned previously, the brace
length, for example, is often constrained in most experimental due to out-of-plane testing
restraints. Furthermore, actuator force capacity and tests setup, not to mention economics,
can limit the scale and breadth of large-scale brace investigations.
4. Is brace capacity influenced by loading history? With reference to Figure 2.9, the WideFlanged members survived the largest deformation capacity prior to failure, yet were
subjected to a compression dominated loading history (refer Gugerli and Goel, 1982)
with relatively small tensile excursions. Furthermore, the least ductile HSS and Pipe
braces were tested with a slightly biased compressive and standard symmetric loading
history with (i.e., equal compressive and tensile excursions as illustrated in a typical
symmetric hysteretic response in Figure 2.9a), respectively. It could be envisioned that a
symmetric history could be more severe as the tensile action during local buckle
straightening could severely increase the cumulative plastic strain in the middle plastic
hinge region.
38
These issues are specifically addressed in subsequent chapters.
2.1.3.4.5
FULL-SYSTEM BRACED-FRAME TESTS
This section presents results from two experimental investigations on large-scale CBF systems.
The first is the shake-table test on a 6-story CBF that was discussed previously in the demand
section and reported in Tang (1987), Foutch et al (1987), and Roeder (1989). The second is a
static test on a 2-story SCBF system tested as part of the work by Uriz (2005).
Listed in Table 2.3 and illustrated in Figure 2.9 are the approximate brace ductility capacities
from the 6-story shake-table test reported in Tang (1987). Interestingly, the axial deformation
capacities (Δrange/ΔY) prior to brace fracture (stories 2 through 5) are similar to the capacities
discussed in the previous section from brace component tests. This is a valuable comparison as it
provides some justification to apply static, component results to inform large-scale system
behavior. The story drifts recorded prior to the onset of brace fracture range between 1.1 and
1.6%. However, as mentioned previously, this frame lacked modern detailing requirements, such
as gusset plate connections and featured braces with b/t ratios that exceeded current widththickness limits.
Also shown in Figure 2.9 are the brace ductility capacities (in terms of Δrange/ΔY) for two braces in
the lower story of a 2-story SCBF by Uriz (2005). The second story braces did not fracture.
Similar to the shake-table results, these results are in line with the HSS component tests by
Shaback and Brown (2003), thereby supporting the use of component tests to inform system
behavior. One should also note the ability of the Δrange/ΔY to accurately describe the brace
ductility. This will be important in Chapters 6 where the maximum deformation range prior to
brace fracture is used to approximate ductility. The maximum story drift for the first story prior to
any brace fracture for the quasi-static test was approximately 1%. The results illustrated in Figure
39
2.9 lend validity to the numerous component tests described previously. Furthermore, and as
discussed in the following section, if brace component behavior can be simulated correctly, the
extrapolation to large-scale behavior can be relatively straight-forward considering these results.
2.2
MODELING TECHNIQUES FOR SCBF SYSTEMS
Simulating complex physical events has long been a focus of the academic community.
Simulation models reduce the necessity of exhaustive experimentation and extend experimentally
observed phenomena to cases which may be difficult to test. Thus, through the application of
theoretical and analytical models, routine design and construction need not rely on the testing of a
replicate structure. Doing so would be an impractical, and in many cases, an impossible feat. In
this light, the current section presents structural modeling techniques in the context of the bracedframe system. Similar to the previous section, the discussion will be separated by first considering
demand modeling, and second, capacity modeling. The former contains models which aim to
characterize quantities such as deformations, strains and stresses from an applied load, ground
motion, etc. Capacity modeling will be treated as post-processing techniques which are applied
after the analysis and aim to describe the fracture ductility of the steel bracing members.
As discussed in the introduction to this chapter, performance assessment generally takes the form
of, first, developing an analysis model that captures well-known geometric or material behavior
and, second, post-processing the results to gain insights into a variety of different phenomena
which are not included as part of the analysis model. Phenomena investigated in the second stage
are typically more difficult to incorporate into the primary model due to computational expense, a
lack of scientific understanding, or a combination of both. For example, fracture propagation
under certain conditions can be computationally prohibitive if the model is the scale of an actual
building while the elastic modulus of any structural material can be accurately described and is
included in any analysis model. Thus, while the relationship between the elastic strain and stress
40
is contained in the primary model, most fracture assessments are made following the analysis by
comparing, for example, material strains (demands) to critical toughness parameters (capacity).
In this context, current state-of-the-art modeling approaches are presented by first discussing
simulation methods that aim to characterize earthquake demands on structural systems for steel
braced-frames. Second, models that describe the capacity (usually in terms of a fracture event) of
a braced-frame system or component are presented. Generally, the capacity models are somewhat
empirical in nature and serve the purpose of expanding, or providing an explanation of, trends
from a particular experimental data set. Other models utilize data across a wide-range of
experiments, but could still be classified as empirical in nature and tend not to address the actual
physical mechanisms that control brace failures. The work by Uriz (2005) is highlighted in the
following sections as an example where performance assessment is completed during the analysis
procedure by comparing strain demands to a critical strain measure, thus eliminating a postprocessing-comparison approach for the fracture limit state. While only applied to one material
type (and one brace cross-section type), the work provides context to discuss the advantages of an
“on-the-fly” performance-based framework and areas where the community could benefit from
advanced simulation capabilities.
2.2.1
DEMAND CHARACTERIZATION
This section presents modeling techniques to assess demand quantities (such as deformations,
strains, etc.) for bracing components subjected to earthquake-type loading histories. For purposes
of this discussion, it is assumed that earthquake demand assessment first begins with a sitespecific earthquake hazard analysis, followed by a ground motion scaling procedure (e.g. as
outlined in Haselton, 2006). Once the ground motion(s) are determined, they serve as an input
base excitation to the structure, with the response of the superstructure being determined through
the fundamental equations of dynamic equilibrium. While copious literature is available on the
41
subjects of hazard, risk and dynamic analysis procedures, this section is specifically focused on
determining the response a bracing member once the input deformations (or histories) are known.
These could either be ground motion-induced deformations or some form of a standard loading
history.
2.2.1.1 TECHNIQUES FOR BRACE SIMULATION
Based on the previous discussion describing braced frame response, and Figure 2.8, this section
describes various techniques that have been used to simulate braced-frame response (specifically
the response of the bracing elements themselves). Some of the techniques discussed are aimed
only at providing good simulation of brace load-deformation response, whereas other, more
sophisticated techniques can simulate complex aspects of response including local buckling and
fracture.
2.2.1.1.1
RULE-BASED (OR PHENOMENOLOGICAL) BRACE MODELS
Uniaxial spring elements with rule-based phenomenological, constitutive response are often used
to model bracing elements (Zayas et al 1980, Khatib et al 1988, Sabelli 2001, and McCormick et
al 2007). As illustrated in Figure 2.10a, hysteretic rules are applied to a single degree of freedom
(axial deformation in the case of bracing elements) with the resulting behavior described by
various linear segments, transitioning between different slopes according to empirical calibration
techniques. While convenient, this approach is limited because it does not directly model
buckling events, and thus, cannot provide insight into localized brace behavior. Furthermore,
calibration is often based on empiricism and thus, these models may be more difficult to
generalize to different materials, cross-sectional shapes or loading histories. On the other hand,
phenomenological models are advantageous if the analysis is primarily aimed at assessing
“global” demands, for example, maximum story drift or axial deformations.
42
2.2.1.1.2
CONCENTRATED HINGE BRACE MODELS
Elastic beam-column elements with lumped plasticity end-nodes are often used in structural
analysis programs where the inelastic behavior is known to concentrate at the ends of structural
members (for example, beam elements in moment frame systems). The inelastic behavior at the
end-nodes is simulated through stress-resultant plasticity formulations (Powell and Chen, 1986;
El-Tawil and Deierlein, 1998 and Hajjar and Gourley, 1977) which operate at the cross-sectional
level, i.e., through a P-M interaction curve as shown in Figure 2.10b. For example, Higginbotham
and Hassan (1976), Ikeda and Mahin (1986), and Hassan and Goel (1991) used variations of this
approach to generalize the behavior of buckling braces. It was shown that for general cyclic
loading, the behavior of the bracing elements at the global force deformation level is captured
fairly accurately. This technique is attractive because the key aspects of response (i.e. buckling
and the post-buckling geometric nonlinearities) are explicitly modeled, and calibration is less
subjective.
While these models are generally more robust than the phenomenological models discussed
previously, they are still limited by some degree of empiricism in calibration. Furthermore,
lumped plasticity elements may not accurately capture yielding, such as may occur over the entire
length of the brace during tensile excursions. Refinements to this approach include Jin and ElTawil (2003), based on the prior work of El-Tawil and Deierlein (2001). This research
incorporates the distribution of plasticity along the length and across the cross-section of the
brace.
2.2.1.1.3
FIBER-ELEMENT BASED BRACE MODELS
Recently, the more powerful and versatile fiber-based element has been used to simulate brace
response. In general, a fiber-based model is formulated with integration points along the length of
43
the element where the member cross-section is discretized by a fiber mesh as shown in Figure
2.10c. A constitutive model is defined at the fiber level and allows for strain and stress gradients
through the cross-section.
The fiber-based element formulation can directly model the spread of plasticity along the length
of the member as well as within the depth of the cross-section, whereas a lumped plasticity model
accounts for these effects only indirectly (see Jin and El-Tawil, 2003). Unfortunately, as the fiber
discretization exists at the cross-section level (at integration points along the element), the
element can not explicitly model localized affects along the length of the brace, such as the strain
amplification during local buckling. Thus, in the context of modeling bracing members, which
can develop significant local buckles during cyclic loading, the fiber-element is somewhat
inadequate for capturing localized stress and strain demands. However, Uriz (2005) and others
(Hanbin et al, 2007; Krishnan 2008-in review) have implemented schemes to track a critical
strain measure at the cross-section level of a fiber-element to predict fracture. The approach relies
on empirical calibration to account for the affects of local buckling, but has been shown to
provide reasonably accurate fracture predictions in bracing element, as discussed later.
2.2.1.1.4
CONTINUUM-BASED BRACE MODELS
Continuum finite element models, such as shown in Figure 2.10d and 2.11 constructed through
commercial or research finite element software (e.g., ABAQUS, 2004) can be applied to model
the brace response during cyclic loading. These simulations incorporate continuum (shell or
brick) elements, large displacement and deformation formulations, and continuum cyclic
constitutive response. In addition, by simulating local as well as global imperfections, these
simulations can directly model several complex phenomena (such as local buckling) that lead to
fracture. Although computationally expensive, the resulting simulations provide a high resolution
of the stresses and strains at the local buckle (e.g. Figure 2.11), from which ductile fracture
44
initiation is assessed using micromechanics-based models, such as ones proposed by Kanvinde
and Deierlein (2004 and 2007). Furthermore, the simulations provide interesting insights into
various damage mechanisms. For example, the simulation illustrated in Figure 2.11 indicates that
while the strains in the cross-section due to global buckling and bending alone are on the order of
0.02, the strains induced through the HSS wall thickness by local buckling are more than an order
of magnitude larger (≈ 0.6), thereby underscoring the importance of simulating local buckling
during brace cyclic response.
Continuum-based formulations of bracing elements are advantageous as compared to integration
point fiber-based modeling as the continuum model directly evaluates inelastic brace response at
each point along the length of the member as well as through the depth of the cross-section. Thus,
contrasted with fiber-based elements, a continuum approach allows localized deformation effects,
such as local buckling, to develop along the length the cross-section of the bracing member.
While solid finite element analyses present notable advantages over lumped-plasticity and fiberbased models, there is a heavy computational expense that accompanies these models, especially
considering the complex events of cyclic inelastic compressive buckling and tension yielding of
bracing members during earthquake-type loading. Moreover, currently, fracture initiation and
propagation events are not typically modeled as part of the analysis routine in standard finite
element software packages. In large part, this can be attributed to a lack of fundamental models
that seek to capture the complex micromechanical events that trigger fracture initiation and
propagation. Furthermore, innovative re-meshing schemes often need to be employed at an
advancing crack tip as the fracture propagates through the continuum body (Khoei et al, 2008).
While mechanics-based methodologies have been developed to address these issues (Rao et al,
2007), they are typically applied to relatively simple geometries and loading conditions and, in
general, would add a significant computational expense to large-scale inelastic cyclic brace
analyses.
45
With the advent of general, physics-based initiation and propagation models, continuum analyses
provide the framework to eliminate the separation between demand and capacity evaluation.
However, these fracture models often describe complex micromechanical processes and need to
be rigorously evaluated prior to implementation into FEM analyses. One such initiation model,
which is presented through a post-processing demand versus capacity approach, will be presented
in the following section.
2.2.2
CAPACITY CHARACTERIZATION
The previous section illustrated several modeling techniques that can be used to simulate the
various aspects of inelastic brace behavior during cyclic loading, and the corresponding
deformation or strain demands in the braces. Given these demands, this section provides a brief
review of the various approaches that have been developed to evaluate the fracture ductility of
bracing members, relative to these demands.
2.2.2.1 BRACE GEOMETRY-BASED FRACTURE CAPACITY RELATIONSHIPS
Lee and Goel (1988) described the cyclic ductility capacity of HSS bracing members with (KL/r)
> (KL/r)critical in terms of a normalized (by ΔY) cumulative axial deformation, δf,pred, as per the
Equations below –
δ f , pred
⎡ ⎛ F
Y
= ⎢C1 ⎜
⎢ ⎜⎝ FY , meas
⎣
2
a
a
⎞ 1 ⎛ b ⎞ a2 ⎤ ⎛ B
⎞ 3 ⎛ KL ⎞ 4
⎥
⎟⎟ ⎜ ⎟
⎜ + C2 ⎟ ⎜
⎟
⎠ ⎝ r ⎠
⎠ ⎝ t ⎠ ⎥⎦ ⎝ H
a
(2.2.1)
and,
δ f , pred
⎡ ⎛ F
Y
= ⎢C1 ⎜
⎢ ⎜⎝ FY , meas
⎣
2
a3
a4
⎞ 1 ⎛ b ⎞ a2 ⎤ ⎛ B
⎞ ⎛ KL ⎞
⎥
⎟⎟ ⎜ ⎟
⎜ + C2 ⎟ ⎜
⎟
⎠ ⎝ r ⎠critical
⎠ ⎝ t ⎠ ⎥⎦ ⎝ H
a
(2.2.2)
if (KL/r) ≤ (KL/r)critical, where C1, C2, and a1 – a4 are determined by an empirical fit of the
available HSS data, and a4 = 0 (i.e., slenderness was not considered) in the original formulation
46
by Lee and Goel (1988). The measured and specified yield stresses are included as FY,meas and FY,
respectively. In Equations 2.2.1 and 2.2.2, b/t is used generically to represent the brace widththickness ratio, such that b/t = B/t – 2, where B and t are the overall length of the buckling face
and the design thickness, respectively, as shown in Figure 2.51. The cross-section dimensions B,
H and t are the nominal dimensions of the square or rectangular brace and are also illustrated in
Figure 2.5. The axial displacement fracture prediction, Δf,pred = δf,pred.ΔY, is a cumulative measure
of brace ductility and is determined by summing the compressive and tensile normalized axial
deformations (defined in Figure 2.7a) up to failure of the brace –
Δ f,exp = ΔY
∑ ( 0.1δ
compression
+ δ tension )
(2.2.3)
Several other experimental investigations on HSS bracing members have used the empirical
relationship proposed by Lee and Goel (1988) to calibrate the constants in Equations 2.2.1 and
2.2.2. Table 2.5 lists the results of the various calibration studies conducted on HSS bracing
members tested by Tang and Goel (1989), Archambault et al. (1995), and Shaback and Brown
(2003). It is important to note that Tang and Goel expressed ductility in terms of a standardized
number of cycles to fracture (Nf) instead of the normalized axial deformation, δf,pred, used by the
others (see Tang and Goel, 1989, for more details). Also, Shaback and Brown used both
Σ(0.1δcompression + δtension) and Σ(δcompression + δtension) to calibrate Equations 2.2.1 and 2.2.2 as they
argued that the “penalty” of 0.1 on the normalized cumulative compressive excursions is not
representative of the full deformation demands on the brace, especially considering the largely
compressive dominant loading histories applied by the Goel et al investigations. Due to these
discrepancies, Table 2.5 does not intend to suggest average calibration values for Equations 2.2.1
and 2.2.2; rather, the constants are supplied to provide a synthesis of the fracture predictions for
HSS bracing member which take the form of the above model. The accuracy of each model
compared to three separate testing programs will be discussed later.
1
Note: Since the original formulation of Equations 2.2.1 and 2.2.2, b/t is now generally listed as B/t-3.
47
Tremblay (2002) and Tremblay et al (2003) developed similar empirical relationships for the
cyclic fracture life of HSS bracing components. The first (Tremblay, 2002) provides a rigorous
synthesis of past experimental work on the cyclic performance of HSS bracing members. Thus,
the fracture model is calibrated across six separate experimental studies with a variety of loading
histories and section properties. Interestingly, the total fracture ductility, measured in terms of the
maximum compressive and tensile ductility, μcompression and μtension, respectively, was found to be
function only of the global brace slenderness ratio, λ, defined as –
λ=
KL FY ,meas
r
π 2E
(2.2.4)
Assuming a linear relationship, the expected ductility of an HSS brace component was found to
be –
μ f , pred = μcompression + μtension = 2.4 + 8.3λ
(2.2.5)
or –
Δ f , pred = ΔY ,meas ( 2.4 + 8.3λ ) =
FY , meas
E
L ( 2.4 + 8.3λ )
(2.2.6)
Note that, Δf,pred in Equation 2.2.6 differs from the previous empirical models as it simply
describes the sum of the maximum compressive and tensile deformations rather than the
cumulative compressive and tensile deformations of each cycle. A later publication by Tremblay
et al (2003) suggested a relationship for the maximum rotation prior to brace failure that is more
similar in form to the original model by Lee and Goel (1988) –
Θ f , pred
⎛b d ⎞
= 0.091⎜
⎟
⎝t t ⎠
−0.1
⎛ KL ⎞
⎜
⎟
⎝ r ⎠
0.3
(2.2.7)
Where b/t and d/t are used generically to represent the brace width-thickness ratio, such that b/t =
B/t – 4 (or d/t = D/t – 4). Note these ratios differ from the B/t – 2 and B/t – 3 used in the previous
empirical relationships and current AISC standards (2005), respectively. Considering the
48
kinematic relationship between the rotation and axial deformation of a buckling brace with plastic
hinges at both ends and at the midpoint, Equation 2.2.7 can also be expressed as –
Δ f , pred
0.3 2 ⎫
−0.1
⎧F
⎪ Y ,meas ⎡
⎛ b d ⎞ ⎛ KL ⎞ ⎤ ⎪
= 2L ⎨
+ ⎢0.046 ⎜
⎟ ⎜
⎟ ⎥ ⎬
⎝ t t ⎠ ⎝ r ⎠ ⎥⎦ ⎪
⎢⎣
⎪⎩ 2 E
⎭
(2.2.8)
Where Δf,pred is the sum of the maximum tensile and compressive axial deformations.
To show the accuracy of these empirical fracture predictions, Figure 2.12 illustrates the ratio of
experimental fracture measurements, Δf,exp, for three separate testing programs to predicted
fracture capacities, Δf,pred, versus the model number used to calculate the capacity according to the
following key –
1. Lee and Goel (1988) model described by the general Equation 2.2.1 (see Table 2.5 for
calibration constants), where the cumulative experimental deformation, Δf,exp, is
calculated with Equation 2.2.3. Note this model does not consider global slenderness
affects (a4 = 0 in Equation 2.2.1).
2. Archambault et al (1995) model; similar in form to Lee and Goel (1988) model, but
considering affects of brace slenderness (i.e., a4 ≠ 0).
3. Shaback and Brown (2003); Archambault et al (1995) model with different calibration
coefficients (see Table 2.5).
4. Tremblay (2002) model; linear relationship between the summed maximum compressive
and tensile deformations before fracture (Δf) and the global slenderness parameter
(Equation 2.2.6).
5. Tremblay et al (2003) model described by Equation 2.2.8; empirical relationship similar
in form to models 1–3, but with Δf, taken as the sum of the maximum compressive and
tensile deformations before fracture.
49
Referring to Table 2.5 and Figure 2.12, the fracture predictions of model 1 are the farthest from
the experimental fracture deformations (average Δf,exp to Δf,pred ratio of 2.6 with a COV of 0.42),
most likely from the omission of a slenderness dependent quantity. Models 2 and 3 include a
global slenderness term (KL/r), resulting in more accurate fracture capacity predictions (average
Δf,exp/Δf,pred = 1.3 and 1.2, respectively) with model 3 showing less scatter (COV = 0.30) than
model 2 (0.41). Interestingly, model 4 is one of the more accurate models (Δf,exp/Δf,pred = 0.8, COV
= 0.41) despite the absence of a b/t term in the formulation (refer Figure 2.5). The last model (5),
is the most accurate empirical-based model (Δf,exp/Δf,pred = 0.9, COV = 0.37). Interestingly,
whereas the first three models used a cumulative deformation measure to assess fracture capacity,
model 5 relies only on the summed maximum compressive and tensile deformations (or
deformation range). This is an important basis for the work presented in Chapter 6 which relies
on the assumption that brace capacity can be accurately characterized through the maximum
deformation range prior to fracture.
While empirical-based relationships are useful, they tend to be quite sensitive to variations in
loading history (Shaback and Brown, 2003) and brace material property. Moreover, the results in
Table 2.5 and Figure 2.12 are calibrated for only HSS bracing sections, under a limited set of
loading histories. To extend the methodology to other brace cross-sections, more experimental
data would be needed to calibrate accurate relationships. While the form of additional equations
would be similar to the relationships presented above, the confidence of empirical-based models
relies solely on a comprehensive data set with tests encompassing a wide range of b/t and KL/r
ratios and loading histories. Thus, one could argue that there is a need to develop general models
that implicitly reflect the relationships described above while not relying on numerous large-scale
tests data sets to calibrate brace ductility.
50
2.2.2.2 FIBER ELEMENT-BASED CRITICAL STRAIN MODELS
Uriz (2005) recently implemented an “on-the-fly” rain-flow counting scheme to monitor the
accumulated cyclic damage of an HSS6x6x3/8 bracing member for several different loading
histories. The model was used together with a fiber-based brace model (described previously) and
was shown to predict fracture quite accurately. The damage is expressed in terms of an equivalent
cycle, ni, for a cycle with a strain amplitude of, εi, at any point and is compared to a critical strain
measure according to –
ε crit < ε i nim
(2.2.9)
Where m and εcrit are calibrated according to standardized fatigue testing procedures. For the case
of monotonic loading, ni = 1 and Equation 2.2.9 simplifies to a comparison between the strain at
any point, εi, and the critical monotonic fracture strain, εcrit. During cyclic loading (i.e., ni > 1) the
strain demand needed to induce fracture at a point is reduced by the assumption that the material
is undergoing a fatigue capacity reduction. Once the critical strain is exceeded, the material is
assumed to have fractured at that point. Referring to the previous discussion on the fiber-based
modeling approach, and considering that stress and strain is defined at a fiber level, a fiber can be
“removed” (strength and stiffness set equal to zero) once the strain reaches the critical strain
according to Equation 2.2.9. This allows the model to mimic the behavior of fracture initiation
through to propagation using a simple fiber-based model. In the context of performance-based
earthquake engineering, this approach is highly attractive as fiber-based models are
computationally cheap compared to more detailed continuum models, yet are more fundamental
in nature as compared to other simplified models (i.e., phenomenological models).
While the work by Uriz (2005) suggests a critical strain of εcrit = 0.095 with m = 0.5 for an
HSS6x6x3/8 bracing member, one may argue that these strains, being monitored at the fiber
level, are not the “true” strains responsible for fracture, and thus may be configuration dependent.
51
However, continuum analyses combined with novel fracture initiation models (discussed next)
could calibrate these material parameters for a wider-range of bracing member types and
geometries. A continuum-based calibration procedure combined with the computationally
inexpensive fiber models could provide an advantageous approach to simulate fracture in fullscale steel structures.
2.2.2.3 MICROMECHANICAL VOID GROWTH AND COALESCENCE MODELS
This section presents a brief discussion of micromechanics-based models to predict fracture in
bracing elements. These models, operating at the continuum level can be used in conjunction with
continuum finite element simulations discussed earlier. Thus, rather than relying on global or
abstracted strain or deformation definitions, they rely on direct estimates of stress and strain
histories simulated through sophisticated finite element simulations. The resulting predictions
offer the advantages of being general as well as accurate, as compared to the less sophisticated
approaches, which are limited by their assumptions and oversight of critical aspects of response.
Ductile fracture and fatigue in steel is caused by the processes of void nucleation, growth, and
coalescence as illustrated in Figure 2.13 (Anderson, 1995). As the steel material experiences a
state of triaxial stress, voids tend to nucleate and grow around inclusions (mostly sulfides or
carbides in mild steels) in the material matrix and coalesce until a macroscopic crack is formed in
the material. Previous research (Rice and Tracey, 1969) has shown that void growth is highly
dependent on the equivalent plastic strain, εp, and stress triaxiality, T = σm/σe, where σm is the
mean or hydrostatic stress and σe is the von Mises stress. Assuming that voids grow when the
localized triaxiality is positive and shrink when negative, Kanvinde and Deierlein (2007)
quantified cyclic void growth – described by the ratio of the current void size, R, to the original
void size, R0 – with a modified version of the Rice and Tracy model for monotonic loading
(Equation 2.2.10) –
52
ηcyclic
⎛R ⎞ ⎛
= ln ⎜ n ⎟ = ⎜
⎝ R0 ⎠ ⎜⎝
⎞
⎛
exp ( 1.5T ) d ε p ⎟ − β ⎜
⎟
⎜
ε np−1
⎠T >0
⎝
p
ε np ε n
∑∫
0
⎞
exp ( 1.5T ) d ε p ⎟ ≥ 0 (2.2.10)
⎟
ε np−1
⎠T <0
p
ε np ε n
∑∫
0
For fracture to occur, the void growth demand, ηcyclic, should exceed the void growth capacity or
critical void size. Under cyclic loading, the monotonic ductility measure, η, decays according to a
damage law, which depends on another material parameter, λ. Thus, the fracture criterion
according to the CVGM can be expressed as –
⎛ Rcrit
⎝ R0
ηcyclic ≥ ln ⎜
⎞
p*
⎟ = exp −λε η
⎠
(
)
(2.2.11)
The ratio of ηcyclic at any time in the loading history to the critical void size expresses how close
the material is to ductile crack initiation. Thus, a low ratio would suggest that the material is safe
from fracture, while a ratio closer to unity indicates a higher probability of ductile fracture at that
point. For details of the expressions in Equations 2.2.10 and 2.2.11, refer Kanvinde et al (2004
and 2007).
Unlike traditional fracture mechanics, the above fracture imitation model is not limited by the
assumption of a preexisting crack or imperfection and is valid in the presence of large-scale
yielding - often the case in regions of plastic hinge formation. Moreover, the proposed fracture
index is not directly dependent on geometry or rain-flow counting schemes to simplify an
earthquake loading history into standard cycles. Of course, if modeled correctly, the resulting
stresses and strains of the structure will be indirectly related to the geometry and loading history.
Most importantly, this fundamental approach offers important insights into localized effects that
cause fracture, which greatly improves design intuition and extends the results of this study to
situations beyond those directly tested.
53
As Chapter 5 presents this methodology in greater more detail, the use of the model will only be
highlighted in the current discussion. As an example, Figure 2.11 and 2.14 will be used to
illustrate the fracture prediction at a critical node for an HSS4x4x1/4 bracing member under a farfield loading history (symmetric tension and compression excursions). Illustrated in Figure 2.14a
and 2.14b are Equations 2.2.10 and 2.2.11 at the critical node (continuum point which is predicted
to fracture first) shown in Figure 2.11 of the brace during the loading history shown in Figure
2.14c. Referring to the figure, it is apparent that elastic behavior is observed prior to cycle 24
after which point the bracing member buckles globally. While the brace is far from fracture
initiation, cycle 24 is the first sign of inelastic behavior both experimentally and analytically.
Local buckling is observed at approximately the same axial deformation during the first large
compressive pulse for both the experiment and ABAQUS simulation. As the plastic strain is
significantly amplified during local buckling, the critical parameter that drives fractures is the
plastic strain, and less so the triaxiality. Local buckling induced damage significantly decreases
the capacity in Figure 2.14a (or critical void size) and leads to a sharp increase in the demand to
capacity ratio (Figure 2.14b). Several cycles after local buckling initiation, the demand/capacity
ratio reaches unity and is predicted to fracture at that point in the loading history. From a
comparison with the experimental results in Figure 2.14c, it can be seen that the prediction is
quite close to the actual instance of fracture.
2.3
SUMMARY
The experimental results presented in this chapter (and later in Chapter 4) suggest that bracing
members are prone to fracture at the middle plastic hinge region from the increased plastic strain
accumulation during local buckling. Relative to current AISC (2005) limits, braces with low
slenderness (KL/r) and high width-thickness (b/t, D/t, or bf/2tf) ratios tend to fracture earlier in a
cyclic loading history compared to more slender and compact members which delay local
buckling. A large majority of research has been conducted to synthesize the combined affects of
54
slenderness, compactness and material type on brace fracture ductility; but most of the proposed
relationships have been of an empirical nature. These relationships often assign a cumulative or
maximum axial deformation capacity, which corresponds well to the scale of past analysis
techniques primarily aimed at characterizing force-deformation response. Only recently have
advances in computational power allowed researchers to simulate full-scale structures and
components directly incorporating response at the cross-section and even continuum-level.
Modeling approaches that leverage these advances in computational power may offer improved
and general fracture predictive methods. Thus, in addition to these modeling techniques (i.e.
sophisticated continuum FEM), there is a need for suitable stress/strain based fracture criteria to
accurately evaluate the complex interactions of stresses and strains at the continuum level
resulting in fracture. Recently, novel micromechanics-based model to predict Ultra Low Cycle
Fatigue (ULCF) have been developed by Kanvinde and Deierlein (2007) to address this need.
The work described in this dissertation develops a methodology that uses sophisticated FEM
simulation of analyze large-scale bracing components (incorporating complex events such as
local buckling) in conjunction with the ULCF model to predict fracture. The ULCF fracture
model operates at the continuum level and is derived from monotonic void growth and
coalescence fracture models for ductile metallic materials. The model relies on accurate
simulation of the stress and strain histories for small-scale calibration and large-scale brace
experiments. Especially important in the context of brace modeling is simulating local bucklingwhich leads to significant plastic strain accumulation at the critical cross-section. Thus, the
dissertation will discuss the simulation of local buckling in detail by considering cross-section
geometry, member and section imperfections, as well as the strain hardening properties of the
material.
55
While the large-scale brace component tests and complementary continuum analyses provide a
rigorous test bed to evaluate the general, physics-based fracture prediction methodology, the
simulations also provide insights into localized brace behavior, which may be invaluable in the
context of calibrating or developing “macro” models such as described in preceding sections.
Finally, it is hoped that these advances in modeling will reduce the reliance on experiment-based
research and provide a useful research tool for studying design requirements of fracture-critical
structures. As discussed earlier, one of the most important advantages offered by these ULCF
models is the insight into localized effects, and their relation to global parameters that are used to
inform design and detailing considerations. For example, numerical parametric studies may be
used to develop insights into the combined influence of slenderness and width-thickness ratios on
brace ductility. The continuum-based models can extend and generalize experimentally observed
trends to untested parameter sets.
56
Table 2.1: Maximum story drifts (standard deviation) from nonlinear time-history analyses on 3
and 6-story SCBF systems.
Drift
Value
10% in 50 years
2% in 50 years
3-story 6-story 3-story 6-story
3.9
1.8
Sabelli
NA
Mean
(3.1)
(0.8)
(2001)
1.5
1.4
5.7
5.1
(0.9)
(0.8)
(3.0)
(3.4)
Uriz (2005)
Median
1.6
1.1
5.7
4.4
(0.9)
(0.6)
(2.4)
(2.2)
3.6
2.0
8.1
4.7
McCormick
Mean
(1.6)
(0.7)
(3.0)
(2.7)
et al (2007)
*Incorporated fatigue-fracture law in bracing modeling
Investigation
Software/Brace Model
SNAP-2D/Rulebased*
OpenSEES/Fiberbased
OpenSEES/Fiberbased*
OpenSEES/Rulebased
Table 2.2: Design level (approximately 10% in 50 years) median (standard deviation) and
maximum story drift from Izvernari et al (2007)
Investigation
Izvernari et al
(2007)
Number
of
Stories
2
4
8
12
16
Median Story
Drift
1.1 (0.4)
1.1 (0.6)
1.4 (0.8)
1.6 (0.9)
1.8 (0.4)
Maximum
Story
Drift
3.5
3.4
4.2
4.6
2.7
Software/Brace Model
OpenSEES/Fiber-based
Table 2.3: Results from 6-story braced-frame shake-table test (Tang, 1987)
HSS
Shape
4x4x3/16
5x5x3/16
5x5x1/4
6x6x1/4
Story
(b-3t)/t
KL/r
6
5
4
3
20
26
19
23
79
62
63
60
Max
Drift* (%)
0.6
1.7
1.6
2.5
6x6x1/4
2
23
60
2.1
Fracture Drift (Def)**
(%, Δrange/ΔY)
NA
1.1 (7.9)
1.2 (9.0)
1.6 (9.0)
1.4
(8.6N, 7.7S)
NA
Damage
Minor buckling
Fracture initiation (N brace)
Fracture initiation (N brace)
Strength loss (N brace)
Fracture initiation (both)
6x6x1/2
1
10
64
1.0
Buckling
*Maximum drift recorded during earthquake record
**Maximum drift and maximum deformation range recorded prior to fracture initiation
57
Table 2.4: Summary of HSS experimental review (63 tests)
Test Program [No.]
Gugerli and Goel [1]
Liu and Goel [2]
Lee and Goel [3]
Archambault, [4]
Tremblay, Filiatrault
Walpole [5]
Shaback and Brown [6]
Yang and Mahin [7]
Fell, Kanvinde, and
Deierlein [8]
Han et al [9]
Yoo, Lehman, and
Roeder [10]
Year
Published
1982
1987
1988
Average
Fy,meas
60
54
67
1995
57
1996
2003
56
64
2005
60
2006
69
2007
2008
59
No. of
Tests
4
3
6
10
4
3
8
3
1
4
1
4
General Description of
Cyclic Loading History
Unsymmetric compressive
Unsymmetric compressive
Unsymmetric compressive
Standard symmetric
Unsymmetric
Standard symmetric
Standard symmetric
Standard symmetric
Unsymmetric tensile
Standard symmetric
Unsymmetric compressive
Standard symmetric
67
12
Standard symmetric
Table 2.5: Summary of calibration constants to predict cyclic fracture life of square and
rectangular HSS bracing members
Model
1
2
3
Investigation
C1
a1
a2
a3
a4
C2
(KL/r)crit FY
Lee and Goel
32.68 0.6
-0.8
1.0
0
0.25
NA
46
Archambault et al
0.124 0.6
-0.25 0.8
2.0 0.25
70
46
Shaback and Brown
0.240 -1.75 -0.6
0.55 2.0 -0.125 70
51
Tang and Goel*
16.19 0
-1.0
1.0
1.0 0
60
NA
Shaback and Brown**
5.11
0.51
-1.25 0.55 2.0 -0.125 70
51
*Used number of cycles to failure, Nf, instead of (typical) cumulative normalized deformation, Δf,pred
**Used Δf,pred = Σ(Δcompression + Δtension) instead of Δf,pred = Σ(0.1Δcompression + Δtension)
58
Figure 2.1: Large-scale Special Concentrically Braced Frame test (Uriz, 2005) and local buckling
induced brace fracture at the middle plastic hinge.
59
(a)
Compression (outof-plane buckling)
Tension
(b)
Compression (outof-plane buckling
and plastic hinge
kinking)
Tension
(c)
Figure 2.2: (a) Chevron braced-frame story and typical connection details, (b) Out-of-plane
buckling and tension yielding and (c) Brace plastic hinge formation.
60
Net section
fracture
(a)
Gusset-beam
fracture
Shear tabcolumn fracture
(b)
Figure 2.3: (a) Brace gusset-plate net section fracture and (b) Column fracture at base of shear tab
along with beam fracture from prying action of gusset plate.
61
Figure 2.4: Plastic hinge formation in beam from bracing force imbalance.
62
Plastic
Hinge
Δ
See below
Δ
Plastic
Hinge
L2 > L1
L1
Increasing global slenderness (KL/r)
Increasing brace ductility
Increasing compactness (b/t, D/t, bf/2tf)
H
b B − 3t HSS
=
t
t HSS
t2-HSS > t1-HSS
t1-HSS
B
D
D
t Pipe
t2-Pipe > t1-Pipe
t1-Pipe
bf
bf
t1-f
2t f
t2-f > t1-f
Figure 2.5: Influence of brace geometry – in terms of global slenderness and cross-section
compactness – on fracture ductility.
63
Interstory Drift, θ (rad)
0.06
θmax = 0.048
0.03
0
θmin = 0.019 (0.4θmax)
-0.03
Figure 2.6: Story drift time history results for the first story of a 3-story SCBF, 2% in 50 years
event.
64
0.06
Symmetric
Behavior
0.04
θmin
θmin = 0.37θmax
0.02
0
0
0.02
0.04
0.06
0.08
θmax
(a)
0.06
Symmetric
Behavior
θmin
0.04
θmin = 0.42θmax
0.02
0
0
0.02
0.04
0.06
0.08
θmax
(b)
Figure 2.7: Minimum versus maximum story drifts for (a) 3-story and (b) 6-story SCBF (courtesy
Uriz, 2005).
65
P/PY
(b)
E
D
1/3
Δ/ΔY
C
(c)
F
B
A
δtension
δcompression
(d)
(a)
(e)
Figure 2.8: (a) Schematic, one cycle, force-deformation response of a typical brace component.
Typical progression of brace damage: (b) Global buckling (point A), (c) Local buckling, (d)
Fracture initiation and (e) Loss of tensile strength.
66
300
Force (k)
150
0
-150
Δrange
(a)
-300
-2
25
Δrange/ΔY
0
1
Axial Displacement (in)
2
Pipe (0.2)
Pipe (0.3)
HSS (0.53)
HSS (0.62)
WF (1.0-1.8)
HSS (0.6)*
HSS (0.54)**
20
15
10
5
-1
**Static
SCBF test
(b)
0
0.0
*Shake-table
data
1.0
2.0
Width-thickness Ratio/AISC Limit
25
Δrange/ΔY
20
**Static
SCBF test
15
10
5
(c)
*Shake-table
data
0
0.0
Pipe (0.3)
Pipe (0.4)
Pipe (0.5-0.6)
HSS (0.9)
HSS (1.1)
WF (0.9-1.6)
HSS (1.2-1.6)*
HSS (0.9)**
1.0
2.0
Slenderness Ratio/AISC Limit
Figure 2.9: (a) Typical brace hysteretic response and definition of Δrange, (b) Influence of Widththickness and (c) Global slenderness ratio on normalized axial deformation range, Δrange/ΔY, for
HSS (experiments from Shaback and Brown, 2003), Pipe (Elchalakani et al, 2003) and Wideflanged (Gugerli and Goel, 1982) members. Also shown are large-scale shake-table and quasistatic brace ductilities from Tang (1988) and Uriz (2005), respectively. Note: legends for (b) and
(c) list the corresponding slenderness and width-thickness ratios, respectively.
67
Elastic Element
P
P
F
M
Δ
(a)
(b)
Integration Points
σ
60
40
20
-4.00E-03
-2.00E-03
0
0.00E+00
-20
2.00E-03
4.00E-03
ε
6.00E-03
-40
-60
(c)
(d)
Figure 2.10: Brace modeling techniques in order of increasing localized simulation abilities: (a)
Phenomenological (or rule-based) model, (c) Lumped plasticity element, (c) Fiber-based model
and (d) Continuum model.
68
Critical Node
Figure 2.11: Buckled shape, plastic strain contours and critical fracture node (as determined by
section 2.2.2.3 and Figure 2.14) from continuum HSS4x4x1/4 brace analysis.
6
Δf,exp/Δf,pred
Archambault et al (1995)
5
Shaback and Brown (2003)
4
Han and Kim (2007)
3
2
1
0
0
1
2
3
4
Model Number
5
6
Figure 2.12: Ratio of experimental to predicted fracture deformation for three separate testing
programs. The five empirical-based fracture models are summarized in the text.
69
Void Nucleation
Void Growth and Strain
Localization
Necking Between Voids
Void Coalescence and
Macroscopic Crack Initiation
Figure 2.13: Micromechanical process of ductile fracture in steel
6
Monotonic
Capacity (η)
CVGM
Prediction
Cyclic Capacity
(Eqn. 2.2.11)
Fracture Index
Void Size and Critical Void Size
70
4
CVGM
Prediction
2
1
Demand, ηcyclic
(Eqn. 2.2.10)
(b)
(a)
0
0
23
25
23
27
25
27
Cycle Number
Cycle Number
4
Experimental Fracture
Predicted Fracture
Δ (in)
2
0
-2
(c)
-4
20
22
24
26
28
30
Cycle Number
Figure 2.14: (a and b) CVGM fracture prediction and (c) Comparison to experimental fracture
time for the critical node of an HSS4x4x1/4 shown in Figure 2.11 during a standard loading
history.
71
Chapter 3
Large-Scale Brace Component Tests
3.1
MOTIVATION FOR LARGE-SCALE BRACE TESTS
Concentrically braced steel frames (CBFs) are attractive lateral load resisting steel-framed
systems due to their economy, structural efficiency and high stiffness. However, as indicated by
several recent studies (outlined in the previous chapter), braced frames are vulnerable to
premature fracture during earthquakes due to the interactive effects of overall flexural buckling
combined with concentrated local buckling in the plastic hinge that forms near the midpoint of
the brace. Connections between the braces and frame are also prone to fracture, prompting
proposed provisions (AISC 2005, and Yang and Mahin 2005) to mitigate this through connection
detailing that accommodates brace end rotations and provides reinforcement to avoid net section
fracture. Nevertheless, because braced systems rely on cyclic inelastic buckling of braces for
energy dissipation, buckling-induced brace fracture has a significant effect on overall system
ductility.
As discussed in Chapter 2, previous studies (e.g. Jain et al. 1978, Popov and Black 1981,
Tremblay 2000 and 2002, Shaback and Brown 2003, Lee and Bruneau 2005, Han et al 2007)
have examined the effect of various parameters, such as brace slenderness, compactness, and
72
cross-section shape, on fracture ductility and energy dissipation in braces. Qualitatively, the
studies concur that cross-sectional shapes, slenderness ratios and width-thickness ratios most
strongly affect the fracture ductility of bracing elements. Tremblay (2002) summarizes most of
these studies and found local buckling to be more severe with smaller brace slenderness, even
with compact cross-sections. Studies have also shown that more compact cross-sections tend to
have higher ductility than less compact sections. Furthermore, Tremblay’s review (2002) suggests
an important loading history effect, such that braces loaded with asymmetric compression cycles
are more prone to fracture as compared to those subjected to symmetric cycles of
tension/compression loading. However, data on these effects is sparse and specific quantitative
criteria to relate these parameters to brace performance is lacking, particularly in the case for Pipe
and Wide-Flange brace members.
With the goal of developing improved understanding of brace buckling and fracture, this study
involves tests of nineteen large-scale braces conducted as part of a Network for Earthquake
Engineering Simulation and Research (NEESR) project. The test specimens were subjected to
reversed-cyclic loading histories with large deformation amplitudes. The specimens are
approximately two-thirds of full-scale, as compared to braces used in typical buildings, with end
connections that represent the flexibility of commonly used gusset plate connections. The cross
sections investigated include square HSS4x4x1/4 and HSS4x4x3/8 sections, standard pipe
sections – Pipe3STD and Pipe5STD, and a Wide-Flange section – W12x16. These tests examine
the effects of section compactness, section geometry, loading histories, loading rates, and grout
fill in the HSS members.
The tests presented in this chapter and Chapter 4 are dual purpose tests that provide valuable data
regarding the seismic performance of braces, while serving as an evaluation test-bed for the
micromechanics-based fracture modeling methodology introduced in Chapter 2. This chapter
73
presents the test program and setup, whereas Chapter 4 focuses on the practical implications of
the tests. Based on the test results, subsequent chapters will address the fracture modeling aspects
of the study. Moreover, the results presented in this chapter (and Chapter 4) will be revisited in
Chapter 6, in the context of a numerical parametric study based on the validated fracture models.
3.2
EXPERIMENTAL SETUP
The tests were conducted at the UC Berkeley NEES facility located at the Richmond Field
Station. The facility offers state-of-the-art testing resources and versatility with respect to the
application of boundary conditions, forces, and loading rates.
As shown in Figure 3.1, the brace test rig provides a fixed-fixed boundary condition, where one
end of the brace specimen is bolted directly to a large reaction block and the other end is attached
to a moving cross-beam. The entire setup was attached to the strong floor and stood
approximately three feet high. The connection gusset plates are oriented in the vertical plane, thus
permitting buckling in the horizontal plane with an effective buckling length roughly equal to
brace length. Load is applied through two servo-hydraulic actuators, each with a 220 kip force
capacity and a +/- 10 inch stroke capacity. The tests were performed in displacement control with
the actuators set in a master-slave feedback-control to minimize end rotations and maintain a
fixed boundary condition at the translating end. The axial brace deformation, Δa, is measured as
the relative deformation between the two specimen end plates.
3.3
TEST PROGRAM SCOPE
Current seismic design standards (AISC 2005) distinguish between Ordinary Concentrically
Braced Frames (OCBF) and Special Concentrically Braced Frames (SCBF), where the latter have
slightly more stringent requirements. With regard to the bracing member provisions, the AISC
requirements for HSS and Pipe sections are similar for OCBFs and SCBFs. Both employ the
74
same
section
compactness
( b / t − 3 < 0.64 E / Fy ),
round
limits
Pipe
for
square
and
rectangular
( D / t < 0.044 E Fy ),
and
HSS
W-shape
members
braces
( b f / 2t f < 0.3 E Fy ). Likewise, the overall brace slenderness limits for both system
designations are similar ( LB / r < 4. E Fy ), excepting more slender braces that are permitted in
certain SCBF and OCBF systems. While the results in this dissertation are generally applicable
for both types of braced frames, the design comparisons in this paper are made in the context of
SCBF systems since these systems are prevalent in regions with high seismic activity where
design level events are expected to induce inelastic brace buckling. Moreover, design provisions
anticipate that OCBF systems will experience smaller deformation demands as compared to
SCBFs, and thus the large inelastic deformations reported in this study may not reflect expected
demands in OCBF systems.
Summarized in Table 3.1 is the test matrix for the nineteen specimens, including information on
the brace cross sections, details, and loading variables. To reflect common design practice, the
test matrix and various component details were developed in consultation with the Structural
Steel Educational Council and practicing engineers at Rutherford and Chekene Structural
Engineers. The test plan is organized to provide insights into a variety of design parameters that
affect brace buckling and fracture. The testing program includes eight square HSS specimens,
eight pipe specimens, and three wide-flange specimens. The member sizes were selected to
investigate the effect of slenderness (LB/r) and width-thickness (b/t and D/t) ratios. For example,
the two HSS sections have similar overall slenderness ratios and thus provide a comparative
assessment of the influence of width-thickness ratios. The alternative Pipe sections and W-shape
allow for an assessment of slenderness effects combined with section properties.
75
The section compactness and member slenderness ratios are summarized in the fourth and fifth
columns of Table 3.1. The numbers in parentheses are ratios of the specimen properties to the
AISC (2005) limits (listed previously) for SCBF systems. The section compactness and member
slenderness ratios for HSS and Pipe sections are all well within the AISC limits. On the other
hand, the flange compactness of the W-shape is close to the AISC limit and the weak-axis
(governing) slenderness ratio exceeds the AISC limit by about 60%. The tests were conducted
using one of three alternative loading protocols listed in Table 3.1 (distinguished by the crossed
cells for each test) and these are summarized later. Loading rates were quasi-static except for two
tests (HSS1-3 and HSS2-2), which were tested under faster earthquake loading rates.
The dimensions of the test specimens are shown in Figures 3.2 and 3.3 with the corresponding
measured material properties summarized in Table 3.2. All specimens are 10’-3” long, measured
from the outside of each end plate, including a clearance of 1.5” between the end of the brace and
bolted end plate to accommodate end rotations associated with brace buckling. The gusset plates
are designed to resist buckling in compression (Astaneh-Asl 1998) and yielding in tension
(Whitmore 1950). Design forces, equal to the expected brace yield strength (RyFyAg), are used to
design all tension critical components including welds. The AISC (2005) specified material yield
and ultimate strengths (Fy, Fu ) and expected strength factors (Ry, Rt) are also listed in Table 3.2.
As shown in Figure 4.2 and Table 3.1 (with crossed cells in the “Reinf.” column), most of the
HSS and Pipe specimens had reinforcement to preclude net-section fractures at the connection.
Reinforcement was omitted from four pipe tests (P1-2, P1-4, P2-2 and P2-4) to investigate its
relative importance for different loading histories. Two of the HSS braces (HSS1-4 and HSS1-5;
designated with crossed cells in the “Fill” column) were filled with grout to assess whether the
fill would delay local buckling and subsequent fracture initiation. One of the specimens (HSS1-6)
featured 18x2x0.5 inch reinforcing plates welded at the center of its non-buckling faces (see
76
Figure 4.7). These plates were intended to encourage unsymmetric buckling of the brace (with
respect to the length), as would be produced due to dissimilar end conditions.
3.3.1
Cyclic Loading Protocols
In contrast to moment frame systems, where peak seismic story drift demands are fairly stable
with respect to design variables (Gupta and Krawinkler 1999), drift demands for braced frames
are more sensitive to variations in bracing configurations and highly nonlinear brace buckling
behavior. For example, Tremblay (2000) showed that the slenderness ratio of the bracing
elements can have a significant influence on drift demands. The loading protocols for this study
are based on an adaptation of protocols for moment frame systems that are adjusted for
earthquake loading demands in braced-frames.
As shown in Figure 3.4a, the standard cyclic protocol follows a symmetric loading history that
represents demands imposed by far-field (non near-fault) earthquake ground motions. The two
other loading protocols, shown in Figure 3.4b-3.4c, represent non-symmetric pulse type demands
as might be imposed by near-fault ground motions. These are distinguished between pulses
dominated by compression or tension loading of the brace. The loading histories are expressed in
terms of building story drift demands, where the drift ratio is assumed to be 0.2% at brace
buckling and 4% at the Maximum Considered Earthquake (MCE) demand. These displacement
demands are based on analyses of chevron configuration braced frames by the authors and others
(Izvernari et al 2007, McCormick et al 2007, Uriz and Mahin 2004, and Sabelli 2001, Uriz 2005
,McCormick et al 2007). Referring to the previous chapter, some of these investigations (Uriz
2005 and McCormick et al 2007) have shown that drift demands may exceed 4% in SCBF
systems for MCE-type events. For example, Uriz (2005) reports median drifts of 5.7 and 5.1% for
3 and 6-story SCBF frames, respectively, during ground motions with a 2% probably of
exceedance in 50 years (referred to as 2/50 ground motions). This type of MCE demand is also
77
consistent with the capacity of SCBFs as implied by the commentary to the AISC (2005) seismic
provisions, which state that “braces could undergo post-buckling axial deformations 10 to 20
times their yield deformation”. Assuming a system yield level drift of approximately 0.3% to
0.5%, the AISC statement may be conservatively interpreted as desiring a drift capacity of
approximately 3% to 5%.
3.3.1.1 Standard Cyclic Loading Protocol
The far-field (or general) loading history was developed by adapting one from ATC-24 (ATC,
1992) to represent SCBF behavior. This protocol is based on nonlinear time history investigations
by Gupta and Krawinkler (1999), who demonstrated that the dissipated energy demands that
result from the testing protocol are consistent (under reasonable assumptions) with realistic
seismic demands in ductile moment frames. The authors modified the moment frame loading
protocol to braced frames using concepts outlined in Krawinkler et al (2000).
Table 3.3 outlines the original ATC/SAC loading protocol. The protocol is defined in terms of
cycles of story drift angles of successively increasing magnitudes. As shown in the figure the
loading history consists of three increasing sets of six cycles (θ = 0.00375, θ = 0.005, and θ =
0.0075) followed by four cycles at the approximate yield drift of a moment frame (θY = 0.01), and
four progressively increasing sets of two cycles each with the fourth set corresponding to the
Maximum Considered Earthquake – MCE level (θ = 0.015, θ = 0.02, θ = 0.03, and θMCE = 0.04).
The modified ATC/SAC far-field protocol used in the current study for SCBF systems is also
listed in Table 3.3 and illustrated in Figure 3.4a. Referring to Table 3.3, the four cycles at the
MRF yield level (1% drift – inelastic cycle group 0 in Table 3.3) are scaled to coincide with the
onset of inelasticity in an SCBF system, typically the buckling of the brace. Load steps 1-3 (θ =
78
0.00375, θ = 0.005, and θ = 0.0075 from the original history) are scaled using the same factor.
The intent of this modification is to ensure a relatively consistent number of inelastic damaging
cycles between the modified and original ATC/SAC protocols. The justification for maintaining a
similar number of inelastic cycles for the modified history is based on the observations that (1)
once a structure begins to yield, the period elongates so that the demands are more ground motion
dependent rather than structure (initial stiffness) dependent and (2) recent research (Uriz and
Mahin, 2004 and McCormick et al, 2007) suggests that the MCE story drift level for SCBFs is in
the 3-5% range, which is comparable to that for MRFs. Based on this reasoning, scale factors
were developed that allowed the inelastic cycle set to increase such that (1) the number of
inelastic cycles would be preserved between the ATC/SAC and the new protocol and (2) the
largest cycles would reflect a drift level consistent with the θMCE ATC/SAC protocol.
The scaling method for each group of cycles is based on a rationale similar to that used by
Krawinkler et al (2000), which assumes the cumulative drift as a measure of system damage.
Following this rationale, the deformation amplitudes and numbers of cycles in the modified
loading protocol were adjusted such that the relationship between peak drift and cumulative drift
would roughly approximate the trends (for moment frames) outlined by Krawinkler et al (2000),
and then applied to the SAC protocol. Adjusting the protocol in this way enables a convenient
interpretation of the test results, such that at any drift amplitude, the damage to the brace may be
assumed consistent with the damage induced by a ground motion that produces a similar drift
amplitude in the system.
This scaling procedure is schematically illustrated in Figure 3.5a, and involves two steps. First,
the inelastic cycle groups 0 and 4 are fixed at 0.2 and 4.0%, respectively. Next, the slope of the
79
line connecting groups 0 to 3 is modified (through trial and error) to produce the effect described
in the previous discussion.
It is important to discuss that the protocol is based on the assumption that story deformation
histories in moment frames and braced frames are similar. However, nonlinear time history data
from some studies (Uriz, 2005; refer Chapter 2) on braced frames indicate that unlike moment
frames, the unsymmetric response of the braces may result in a ratcheting response of the braced
frames. Thus, the peak drift may be significantly larger than half the drift amplitude, which
contradicts the implicit assumption of the symmetric loading protocol. A systematic consideration
of such issues will require detailed analysis of several nonlinear time history simulation histories
(in an exercise similar to that performed by Krawinkler et al, 2000 for moment frames), and is
outside the scope of this dissertation. As discussed subsequently in Chapter 6, the test results
themselves may be interpreted in a more appropriate manner, if such an analysis becomes
available at a later time.
3.3.1.2 Near-Field Loading Histories
To reflect demands imposed by near-fault ground motions, two loading protocols – asymmetric
compression and asymmetric tension – were used for several of the brace tests. In contrast to
moment frames where the component response is fairly symmetric, for braced frames, the pulse
like near-fault protocol must be distinguished between cases dominated by either tension or
compression response. As with the general protocol (described previously) the near-field protocol
is based on a similar one developed in the SAC project for moment frames. These loading
protocols are illustrated in Figures 3.4b and 3.4c.
As shown in Figure 3.4b, the compression dominated history is identical to the ATC/SAC nearfault protocol. However, following completion of the near-field protocol, the far-field loading
80
protocol (of Figure 3.4a) is appended so as to extract additional information from the test in the
event that the brace survives the near-fault loading. Aside from providing data for validating the
ductile fracture models, this subsequent loading is envisioned to represent an aftershock
earthquake that follows the first large pulse of the main earthquake fault rupture.
The tension dominated history (Figure 3.4c) consists of a large monotonic pull followed by
subsequent cycles. For tension dominant loading, where one of the goals is to apply large tension
demands on the brace and connections prior to buckling, the initial negative (compression) pulse
is omitted and the positive (tension) pulse is increased to +8% drift, to impose the largest tensile
demands on the specimens given the limitations of the test setup. In anticipation of instances
where the brace may not fail during the near-fault pulse loading, the standard cyclic history is
appended to the pulse protocol.
3.3.1.3 Story Drift and Brace Axial Deformation Relationships
The drift demands in the loading protocols are converted to corresponding axial deformations (for
application in the experiments) through the following relationship:
(
)
Δ a = cos 2 45D LBθ = 0.5 LBθ
(3.1)
Where θ is the story drift angle (expressed in radians), Δa is the corresponding axial deformation
and LB is equal to the distance between the fold lines of the gusset plates (9’-9 ½” for HSS; 9’-10
½” for Pipe and W specimens). This relationship is based on a chevron brace configuration
shown in Figure 3.6, assuming center-line dimensions with the braces inclined at 45 degrees and
ignoring flexural deformations in the beams and columns. Given that axial deformations are
applied to the experimental brace specimens, Equation 3.1 provides an approximate measure to
compare the various brace capacities to earthquake-induced story drifts and does not take into
account all affects. For instance, Equation 3.1 is based on frame geometry that considers center-
81
line dimensions of the chevron frame, neglecting the joint size. This effect may be included in
Equation 3.1 by assuming a rigid-link distance between the gusset-plate fold lines and the
working point of the beam-column connection. Following this reasoning, a modification factor
(1+C) may be introduced in Equation 3.1, resulting in Equation 3.2, where C is the ratio of the
rigid-link length (on both ends of the brace) to the brace length LB.
(
)
Δ a = (1 + C ) cos 2 β LBθ
(3.2)
Owing to the wide-variety of brace, gusset-plate configurations in SCBF construction, it is
difficult to prescribe a consistent or precise value for the modification factor C. Recognizing the
uncertainty in other aspects of this kinematic relationship (such as the brace angle β), and
moreover the subjectivity in the characterization of the drift demands themselves (refer earlier
discussion), this chapter and the next relies on Equation 3.1 to relate the brace axial deformation
to a corresponding drift level. In the presence of these uncertainties, relationships such as the one
presented in Equation 3.2 may be used to interpret the data presented in this paper in the context
of specific frame designs, or for examining the sensitivity of the findings to other geometrical
parameters such as the connection size or brace angle.
3.4
Instrumentation and Miscellaneous Testing Results
This section describes the instrumentation used for the brace experiments. In addition to the
primary channels of load and deformation (which are discussed in detail in Chapter 4), this
section describes various ancillary measurements.
The primary measurements for each test include the total axial force resistance of the brace,
obtained from the readings of both actuators shown in Figure 3.1a, and the axial deformation
measured across the length of each brace. The latter was inferred using four displacement
transducers, one each on the top and bottom of each brace end plate. The transducers on the west-
82
end (sliding end) of the experimental setup in Figure 3.1 had a range of +/- 20 inches while the
fixed-end (east) transducers had a range of +/- 1 inch. Representative recordings for each of these
instruments are shown in Figure 3.7a-3.7b for an HSS4x4x1/4 bracing member subjected to the
standard cyclic loading history (HSS1-1). As can be seen from the figure, the deformations on the
east end are negligible compared to the sliding end. However, the figure indicates a slight
discrepancy between the top and bottom recordings at each end, suggesting a slight rotation
during large tensile excursions. Attributed to construction imperfections, these deformations are
small – a maximum difference of approximately a tenth of an inch on the west end and a
hundredth of an inch on the east – and are not considered to have a substantial affect on the
testing results. Lateral slip of the brace end plates (North-South movement of end plates in Figure
3.1a) is also measured with +/- 1 inch position transducers. These measurements are shown to be
small (on the order of 0.02 inch) in Figure 3.7c with the maximum lateral slippage recorded at the
inelastic tension peaks on the sliding end of the experimental test setup.
The axial brace deformation was determined from the difference between the average (of the top
and bottom) of the east and west position transducers. Thus calculated, the relative deformation
was used to trigger actuator reversal during the experiment. In contrast to direct control of the
actuator displacement (which also includes deformation of the test-setup, this procedure resulted
in a highly accurate application of the loading history as illustrated in Figure 3.8. From the figure,
it is observed that the intended and measured deformation histories are identical.
The total axial force was calculated by adding the two actuator force measurements. Figure 3.9
illustrates the axial deformation-force relationship for each actuator during the standard cyclic
loading test on the HSS4x4x1/4. As shown in the figure, the force measurements from the two
actuators are similar throughout the experiment and demonstrate the symmetry of the loading
frame (and boundary conditions) and high-quality fabrication of the experimental specimens.
83
A series of cable-extension position transducers (string pots) were used to measure the large outof-plane buckling displacements of the bracing members. The string pots were mounted on the
floor of the lab and attached at the brace third points as depicted in Figure 3.10. At the midpoint,
two string pots were needed to track the position of the brace point in space as the axial
deformation, from symmetry, is approximately half of the end displacement at this point. A
schematic illustrates this configuration in Figure 3.10b. At the other points, three pots were
required to track the position of the brace, where the additional device is mounted along the axis
of the bracing member. Before each test, the initial locations of the brace third points are
measured with respect to each string pot connected at that point. With these measurements, a
fairly simple program can track the position of each point given the recordings of the string pots.
Figure 3.12 shows the out-of-plane measurements at each third-point versus the brace axial force
for an HSS4x4x1/4 during a standard loading history (HSS1-1). As expected, the mid-point outof-plane deformations are larger than the third-points.
Thermocouples were used on several of the experiments to ascertain the influence of rate affects
on steel fracture behavior. It was found that rate affects did not significantly alter the behavior or
performance of the bracing members as compared to the slower quasi-static experiments. The
results of the faster rate experiments and a discussion on rate-induced temperature increases in the
context of ductility and fracture predictions are presented in more depth in Chapters 4 and 5,
respectively. Figure 3.12a illustrates the temperature increase due to inelastic buckling and
tension yielding for an HSS4x4x1/4 brace during the quasi-static near-field compression loading
history. The maximum recorded temperature occurs on the compression side of the brace during
the first inelastic excursion to -6.0% drift. Note the cooling period during the elastic cycles
between the end of the near-field history and the appended far-field history. Figure 3.12b shows a
significantly larger temperature increase, reaching a maximum of nearly 200°F, recorded for an
HSS4x4x3/8 brace during an earthquake-rate.
84
3.5
Summary
To provide background for the upcoming chapters, this chapter introduced the test matrix, setup
and loading histories applied to nineteen large-scale bracing component members, representative
of members in Special Concentrically Braced Frame (SCBF) systems. The tests, designed to
represent typical conditions in steel braced frames, complement previous studies by investigating
three cross section types (HSS, Pipe and Wide flanged sections) of varying section compactness
and brace slenderness. The test program also investigates the effects of loading histories, loading
rates, connection reinforcement, and grout filling of HSS sections. Results presented in the
subsequent chapters will heavily reference the background information presented in this chapter.
85
Table 3.1: Test parameters and loading histories
Test
No.
HSS1-1
HSS1-2
HSS1-3*
HSS1-4
HSS1-5
HSS1-6#
HSS2-1
HSS2-2*
P1-1
P1-2
P1-3
P1-4
P2-1
P2-2
P2-3
P2-4
W1
W2
W3
Bracing
Member
Test Specimen Parameters
b/t or D/t
KLB/r
LB
(AISC**) (AISC**)
HSS4x4x1/4
14.2
(0.89)
77
(0.77)
HSS4x4x3/8
8.5
(0.53)
80
(0.80)
Pipe5STD
21.6
(0.59)
63
(0.55)
16.2
(0.44)
102
(0.89)
7.5
(1.04)
153
(1.59)
Pipe3STD
9’9½”
9’10½”
W12x16
Reinf.
Loading Protocol # #
Std. Pulse Pulse
FF NF-C NF-T
Fill
*fast (earthquake) loading rate
**the numbers in parentheses are the section width-thickness or brace slenderness ratio
normalized by the criteria specified by the AISC Seismic Provisions (2005)
#HSS1-6 was reinforced at the midpoint (top and bottom) with 18” long plates (see Figure 4.7)
# #FF: Standard far-field loading (Figure 3.4a); NF-C: Near-field compression loading (Figure
3.4b); NF-T Near-field tension history (Figure 3.4c)
Table 3.2: Brace material properties
Brace
CrossSection
HSS1*
HSS2*
P1
P2
W
Measured
Properties
Fu,meas
Fy,meas
(ksi)
(ksi)
Corner: Corner:
74
80
Center:
Center:
67
71
Corner: Corner:
73
78
Center:
Center:
72
79
47
61
54
67
60
69
Fy
(ksi)
Specified
Properties
Fu
Ry
(ksi)
Rt
46
58
1.4
1.3
35
60
1.6
1.2
50
65
1.1
1.1
Fy ,meas
Fu , meas
Fy
Fu
Corner:
1.6
Center:
1.5
Corner:
1.6
Center:
1.6
1.4
1.5
1.2
Corner:
1.4
Center:
1.2
Corner:
1.4
Center:
1.4
1.0
1.1
1.2
*center material properties are from ASTM specified rectangular tensile coupons. All other data
are from non-ASTM cylindrical coupons
86
Table 3.3: Summary of loading protocol modifications
Inelastic
Group
Number
elastic
loading
0
1
2
3
4
5
Number
of
Cycles
6
6
6
4
2
2
2
2
2+n
Original SAC
Loading History
Drift (%)
0.375
0.50
0.75
1.00 – Y
1.50
2.00
3.00
4.00 - MCE
5.00
Modified SAC
Loading History
Drift (%)
Δa (in)
0.08
0.04
0.10
0.06
0.15
0.09
0.20 – B
0.12
1.03
0.61
1.85
1.10
2.68
1.59
4.00 - MCE
2.38
5.00
2.99
Y: Approximate Moment Frame yield; B: Approximate braced-frame inelastic buckling; MCE:
Maximum Considered Earthquake
87
Sliding Beam
Constraint Frame
N
Actuator 2 (220 kip, +/- 10 in)
10’-3”
Cyclic
5’-2”
Loading
Specimen
Brace buckling
Reaction
Wall
Actuator 1 (220 kip, +/- 10 in)
(a)
Sliding
Beam
Constraint
Frame
Cyclic
Loading
Actuator
3’-3”
Reaction
Wall
(b)
Figure 3.1: (a) Plan and (b) elevation view of brace test setup.
88
HSS1: 10”
HSS2: 1’-3”
Reinf. Plates:
HSS1: 8x2x1/4”
HSS2: 8x2x3/8”
¼”
2”
HSS-section
½” Gusset Plate (typ)
1 ½” (typ)
HSS1: 1’-1 ½”
HSS2: 1’-6 ½”
(a)
1” (typ)
P1: 1’-0”
P2: 6 ½”
1’-8”
(typ)
Reinf. Plates:
P1: 11x3x¼”
P2: 6x2x¼”
¼”
1 ½”
Pipe-section
P1: 1’-3”
P2: 9 ½”
10’-3” (typ)
(b)
6”
8x5x3/8”
Slotted Plate
¼”
W12x16
1 ½”
CJP
See Figure 3.3
7.5”
(c)
Figure 3.2: Brace drawings for (a) HSS, (b) Pipe and (c) W12x16.
89
¼”
½” Gusset (typ.)
9/16”
≈ ¼”
Reinforcing PL
(a)
CJP
8x5x3/8”
Plate
¼”
45°
6”
¼”
6”
¼”
6”
½”
R=3/8”
2 13/16”
W12x16
Web
½” Gusset
CJP
¼”
45°
(b)
¼”
6”
¼”
6”
8x5x3/8”
Plate
9/16” slot
5”
W12x16
Flange
2”
6”
½” Gusset
(typ.)
(c)
Figure 3.3: Connection details for (a) HSS and Pipe and (b-c) W12x16.
90
Maximum Considered (4)
Drift (%)
4
2
2.36
1.18
Expected Buckling (0.2)
0
0
-2
-1.18
-4
-2.36
Axial Deformation (in)
3.54
6
-3.54
-6
0
10
20
30
Cycle Number
(a)
4
Compression dominated
near-field loading history
2
1.18
0
Drift (%)
0
-1.18
-2
-2.36
-4
Far-field loading history
appended to near-field
-6
-3.54
Axial Deformation (in)
2.36
-4.72
-8
0
10
20
30
Cycle Number
(b)
Far-field loading history
appended to near-field
Drift
Drift (%)
(%)
6
3.54
4
2.36
2
1.18
Tension dominated
near-field loading history
0
0
-1.18
-2
Axial Deformation (in)
4.72
8
-2.36
-4
0
10
20
30
Cycle Number
(c)
Figure 3.4: (a) Standard cyclic (far-field ground motions), (b) Compression and (c) Tension nearfault pulse loading histories.
91
Peak Drift (%)
6
Original SAC
Modified SAC
4
Fixed at 4%
--MCE
Fixed at 0.2%
--Buckling (B)
2
Variable Slope
0
0
2
4
6
Inelastic Cycle Group Number
(a)
ΣθP (rads)
1.5
HSS4x4x1/4
HSS4x4x3/8
Pipe3STD
Pipe5STD
W12x16
Moment Frame
1
0.5
0
0
2
4
6
Inelastic Cycle Group Number
(b)
Figure 3.5: (a) Original and modified SAC loading history and (b) Cumulative plastic drift for
Chevron braced-frames during modified history and a moment frame under original SAC history.
92
hinge
Δa
θ
LB
β
Figure 3.6: One-story Chevron braced-frame.
93
3
Top
Bottom
Displacement (in)
2
1
0
-1
(a)
-2
Top
Bottom
2
-2
Displacement x 10 (in)
3
1
0
-1
-2
(b)
-2
Displacement x 10 (in)
2
East
West
1
0
(c)
-1
Figure 3.7: Axial transducer measurements for (a) West and (b) East top and bottom brace end
plates and (c) Lateral (North-South) displacements of end plates. For (a) and (b) positive
displacement corresponds to tension of the bracing member.
Imposed Axial Deformation (in)
94
2
1
0
Elastic
cycles
-1
-2
-2
-1
0
1
2
Measured Axial Deformation (in)
Figure 3.8: Intended and measured axial deformations.
150
Actuator 1
Actuator 2
Axial Force (kip)
100
50
0
-50
-100
-2
-1
0
1
2
Axial Deformation (in)
Figure 3.9: Actuator force measurements versus relative axial deformation (between east and
west end plates) for standard cyclic HSS4x4x1/4 test.
95
N
(a)
Brace
Washer
Cable
String pot device
(b)
Unistrut connection
to lab floor
(c)
Figure 3.10: (a) Wire-pot instrumentation plan, (b) Connection detail and (c) Pipe3STD test.
96
300
West
East
Force (kip)
200
100
0
-100
(a)
-200
0
4
8
Lateral Displacement (in)
300
Force (kip)
200
100
0
-100
(b)
-200
0
4
8
12
16
Lateral Displacement (in)
Figure 3.11: Measured out-of-plane buckling displacements at (a) East and west (refer Figure
3.1a) brace third-points and (b) Brace mid point.
97
95
Local Buckling Face
o
Temperature ( F)
Opposite Face
85
75
65
(a)
225
Local Buckling Face-MidPoint
185
Opposite Face-Offset
o
Temperature ( F)
Opposite Face-MidPoint
145
105
65
(b)
Figure 3.12: Thermocouple recordings for (a) Quasi-static, near-field test and (b) Earthquake-rate
standard cyclic test.
98
Chapter 4
Large-Scale Experimental Results and Design Implications
This chapter summarizes results from the tests described previously in Chapter 3, in the context
of the performance of SCBF systems and current design provisions. The chapter begins with a
qualitative description of brace response and the observed damage states. This is followed by a
discussion of brace performance for all nineteen tests with the aim to examine the influence of
various test parameters. Two distinct types of failure are observed in the tests; one involves
fracture at the center of the brace due to local buckling at the middle plastic hinge region, whereas
the second involves failure of the brace-gusset plate connections. To address these issues
individually, the discussion of test results includes two separate sections. Based on the
experimental data, the last section of the chapter examines the applicability of strength equations
for bracing members commonly used for the capacity design of connections.
4.1
QUALITATIVE SUMMARY OF EXPERIMENTAL RESPONSE
The typical sequence of events leading up to fracture of an HSS brace is illustrated in Figures
4.1a-d for test HSS1-1, with the corresponding load versus deformation response shown in Figure
4.2a. The initial elastic cycles do not induce any visually observable deformation in the specimen.
The first major limit state is global brace buckling (Figure 4.1a) at a drift ratios of approximately
99
0.3%1, accompanied by large lateral deformations and flaking of the whitewash paint at the end
gusset plates and near the mid-point of the brace. Upon further loading, a plastic hinge develops
at the mid-point of the brace, which experiences local buckling (Figure 4.1b) at a drift ratio of
about 2%. Subsequently, cyclic loading triggers ductile fracture initiation (Figure 4.1c), which for
HSS1-1 occurred after the first reversed cycle to 2.7%. Soon after initiation, the fracture
propagates by ductile tearing through the section (Figure 4.1d) leading to a noticeable loss of
force capacity in the hysteretic response. In the square HSS, the buckled face ruptures first at the
corners and then propagates up the sides, leading to complete severance of the brace and loss of
strength. As the imposed story drifts increase up to 4%, the lateral deformations of the brace
become quite large - on the order of LB/8 (15-18 inches). It should be noted that unlike global
buckling and strength loss, which can be observed accurately through sudden drops in the loaddeformation plot, the precise instants of local buckling and fracture initiation are somewhat more
subjective to ascertain, since they are inferred through visual and photographic observations.
However, this has a relatively minor impact on overall performance assessment, since the
catastrophic event of final fracture and strength loss of the brace almost immediately follows
local buckling and fracture initiation.
Shown in Figure 4.2b is the load-deformation response of HSS1-2 subjected to the near-fault
compression dominated loading history. In this case, global buckling occurs at a smaller
compressive load (119 kips for HSS1-2 as opposed to 157 kips for HSS1-1), due to the tensile
elongation during the initial excursion to 2.0% drift. Local buckling occurred during the large
compressive pulse at a drift of 2.6%. Subsequent cycling of the already buckled brace about its
residual drift (refer to the near-field loading histories in Chapter 3) did not produce appreciable
1
This and other similar drift ratios reported in this section are specific to the tests described in this section,
which are used only for a qualitative illustration of brace response. These drift ratios may vary significantly
for other tests depending on test parameters. Table 4.1 and 4.2 summarizes these for all the specimens.
100
straining in the plastic hinge region. The brace ultimately survived twelve cycles of the appended
standard cyclic loading history before fracture initiation at a drift of -1.1%.
As compared to the other loading protocols, the tension dominated near-fault pulse presents a
more critical test of net section fracture at the brace end. Three of the five tension dominated tests
were reinforced at the net-section (P1-1, P2-3 and W3) and survived the large tension pull,
whereas the two pipe sections that were not reinforced (P1-4 and P2-4) fractured at the netsection during the initial tension pull. The results of test P1-1, shown in Figure 4.2c, is typical of
the tension dominated response where net-section fracture does not occur. The loading history
begins with an initial tensile excursion to 8% drift, followed by a few cycles of compression and
tension loading. The appended standard cyclic loading ultimately leads to local buckling and
fracture similar to that described previously in Figures 4.1a-d. Being of a different section type,
the pipe section data in Figure 4.2c are not directly comparable to the HSS results in Figures 4.2a
and 4.2b. However, compared to other pipe section tests (P1-1 and P1-2), the initial tension
excursion causes a reduction of about 30% in the compression buckling strength (summarized in
Tables 4.1 and 4.2). This reduction is larger than the 25% reduction observed between the two
HSS tests (HSS1-1 and HSS1-2), shown in Figures 4.2a and 4.2b. Strain hardening during the
large tension pulse increases the maximum tensile strength of test P1-3 by about 20%, relative to
other pipe tests (P1-1 and P1-2).
4.2
QUANTITATIVE SUMMARY OF DATA FOR ALL TESTS
Data for all nineteen tests are summarized in Tables 4.1 and 4.2 (for standard cyclic and nearfault pulse loading, respectively), including the maximum measured forces, deformation and drift
levels corresponding to key damage states, and total dissipated energy up to failure. Referring to
Table 4.1, the following data are reported for the limit states of standard cyclic loading tests: (1)
Δ a −GB - axial deformation at global buckling, (2) the maximum deformations sustained prior to
101
the occurrence of Δ a − LB - local buckling, Δ a − FI - fracture initiation (both observed visually), and
Δ a − SL - strength loss
(3) the drift levels as per Equation (1) for each of these events
( θ GB , θ LB , θ FI , θ SL ), (4) ΣΔ aP− SL - the cumulative plastic deformation and the corresponding
P
cumulative plastic drift ( Σθ SL
) sustained prior to strength loss and (5) ΣESL / E y - the normalized
dissipated energy, i.e., the summation of energy dissipated up to the point of strength loss
normalized by the tension yield energy (calculated as the product of the measured yield strength
and the yield displacement). For example, referring to Figure 4.2a and Table 4.1, HSS1-1
experiences global buckling at θ GB = 0.3%, local buckling at the peak drift of 1.9% during the
second compression cycle to 1.9%, fracture initiation at a drift of 1.7% during the second tension
cycle to 2.7%, and strength loss at a drift of 2.5% during the second cycle to 2.7%. Note that in
Table 4.1 the maximum drifts for θ FI and θ SL are both reported as 2.7%, since this is the
maximum drift that is sustained prior to fracture initiation and strength loss. Local buckling
occurred on the first compressive cycle to 1.9%, thus in this case, the drift at the instant of
buckling is the maximum drift sustained by the brace to that point. In cases where the specimen
survives the standard protocol (Figure 3.4a), the cycles at 5% drift were repeated. For example,
the W-shape specimen (W1) sustained 14 cycles at 5% drift prior to fracture initiation. These
maximum values by themselves may not provide a complete description of the brace
performance, since they do not incorporate information regarding the loading history. In this
context, an examination of the cumulative plastic deformation or energy dissipation provides a
better relative performance assessment, although it is difficult to compare these quantities directly
to imposed seismic demands that are typically expressed as peak drifts. Thus, the maximum
values of sustained drift (and axial deformation) are used for discussion, mainly due to their
simplicity and convenience of interpretation with respect to system level drift demands. Refer to
102
Appendix A for detailed documentation regarding the instants (during the loading histories) of
local buckling, fracture initiation and strength loss limit states.
Referring to the near-fault loading data of Table 4.2, all of the specimens except for P1-4 and P24 survived the near-fault loading without fracture, and only HSS1-2 and W2 experienced local
buckling during the large compression pulse. The drift at global buckling is not reported for the
near fault cases, since the buckling drift is not unique and depends on the prior loading history
(e.g., see Figures 4.2b and 4.2c). Drifts for damage states reached during the appended standard
cyclic loading protocol are reported as relative values, with the datum being the residual drift at
the end of the near-fault loading. Referring to Figure 3.4b-c, the near-fault residual drift was 3%
in either the positive (tension) or negative (compression) sense. Referring to HSS1-2 (Figure 4.2b
and Table 4.2), local buckling occurred during the large near-fault compression pulse at -2.6%
drift. Fracture initiation and strength loss did not occur until well into the subsequent standard
cyclic loading, where fracture initiation occurred at 1.9% drift during the first tension cycle to
1.9% drift and strength loss occurred at 2.7% drift during the first tension cycle to 4% drift.
Having already survived the first compression cycle to 4% drift, the maximum value for θ SL is
listed as 4%.
Referring to Table 4.1, lateral brace buckling (global buckling) occurs at 0.3% for most of the
specimens, with slenderness values ranging from KLB/r = 63 to 80. The buckling drift is lower
(0.2%) for the more slender W-shape brace with KLB/r = 153 and larger (0.4%) for the grout
filled HSS brace. These values are generally consistent with previous studies (e.g. Tremblay
2000).
103
Compared to global buckling, larger variations were observed among drifts at local buckling,
which indicates the sensitivity of the local buckling limit state to the cross sectional shape, widththickness ratio, and grout fill. For the standard loading (Table 4.1), the maximum drifts at local
buckling ranged from 1.9% to 5%, where the larger resistance was observed in the more stocky
HSS and Pipe sections and the W-section. Local buckling occurred at larger drifts for the nearfault compressive dominated cases (HSS1-2, HSS1-5, and W2 in Table 2). For example,
comparing HSS1-1 and HSS1-2, the initiation of local buckling occurred at 1.9% drift under
standard loading and 2.6% under the near-fault pulse. Similarly, comparing HSS1-4 and HSS1-5,
local buckling for the grout filled HSS initiated at 2.7% drift under standard loading and survived
a 6% drift pulse during the near-fault loading.
In general, fracture initiation and strength loss closely follow local buckling, where fracture
initiation and loss of strength occurred between 2.7% and 5.0% drifts. In this context, Figure 4.3
further illustrates the dependence of brace fracture on local buckling (tests with connection
failure, i.e., P1-4 and P2-4, are not shown). Figure 4.3a plots the normalized the energy dissipated
prior to local buckling versus a similar quantity for fracture initiation. From the figure (and the
associated linear regression fit between the local buckling and fracture energies), it is interesting
to observe that when all brace specimens and loading histories are considered, fracture initiation
succeeds local buckling after a relatively constant interval = 10.5 (indicated as the y-intercept on
Fig 4.3a), measured in terms of normalized dissipated energy. A similar trend is observed
between fracture initiation and strength loss (see intercept indicated on Figure 4.3b) such that
strength loss succeeds fracture by a constant interval = 3.8 (again in terms of normalized
dissipated energy). Thus the key observation is that once local buckling occurs, fracture initiation
and strength loss occur soon after. While the “lag” between any two of these limit states is
relatively constant, strength loss appears to follow fracture initiation quite quickly, in contrast to a
slightly larger separation between local buckling and fracture initiation.
104
As observed for local buckling, the near-fault pulse loading was not as damaging as the standard
cyclic loading, where all of the tests sustained the pulse loading without fracture initiation. The
fracture endurance (especially dissipated energy) for members subjected to the standard cyclic
loading after the compression pulse was similar to that observed under the standard cyclic loading
alone. The endurance of specimens that were subjected to the tension dominated pulse was
improved compared to their respective standard cyclic tests, largely due to cycling the specimen
about a residual tension elongation, thereby delaying local buckling.
4.3
EFFECT OF TEST VARIABLES ON CYCLIC BRACE BEHAVIOR, LIMIT STATES AND
DESIGN IMPLICATIONS
While the previous section summarized general trends and observations with respect to all the
experiments, this section investigates the effects of cross section geometry, width-thickness ratio,
slenderness ratio, loading rates and histories on brace performance. To provide a more
meaningful discussion of brace capacity in the context of expected demands, the experimental
results are compared to an assumed 2% design drift and a 4% Maximum Considered Earthquake
(MCE) event level drift. Recall the prior discussion (Chapter 3) that presented the rationale for
using 4% as an approximate measure of MCE drift (2/50 ground motions). On similar lines (and
based on data from Sabelli, 2001; Uriz, 2005; and McCormick, 2007) a value of 2% drift is
considered indicative of the mean demands expected in 10/50 ground motions. However, it
should be recognized that these values are largely subjective, and a rigorous system performance
assessment is needed to accurately establish acceptability criteria for braces. Such an assessment
is outside the scope of this dissertation.
4.3.1
Effect of Width to Thickness Ratios
Referring again to Figure 4.1b, brace fracture is driven by the amplified local strains induced by
the interactive effects of global and buckling during reversed cyclic loading. The drifts at fracture
105
initiation and the normalized energy dissipation capacity are plotted versus the normalized widththickness ratios in Figure 4.4 and versus global slenderness in Figure 4.5. The horizontal lines
drawn at 4% and 2% drift (Figures 4.4a and 4.5a) are considered to represent the minimum
required and design drift capacities of SCBF systems, respectively. As the fracture ductility is
known to be controlled by a combination of slenderness and width-thickness ratios (Tang and
Goel 1989), one should be mindful that the trends in the plots of Figures 4.4 and 4.5 are interrelated.
Within each cross section type, the tendency for local buckling and fracture initiation increases
with increasing width-thickness ratios, resulting in reduced drift capacity at fracture. The two
HSS sections with similar slenderness ratios and varying width-thickness ratios (HSS1-1, HSS13, HSS2-1, and HSS2-2, shown by the hollow squares in Figure 4.4) provide the most direct
evidence of the relationship between section compactness and fracture ductility. In this case, the
reduction in width-thickness ratio by 40% resulted in about a 65% increase in fracture drift
capacity and a 90% increase in energy dissipation capacity. Similar trends are observed when
comparing the pipe sections (P2-1, P2-2, P1-2, and P1-3, shown by circles in Figure 4.4), where a
25% reduction in the diameter to thickness ratio increased the fracture drift capacity by 50% and
energy dissipation by about 60%.
While all of the HSS and Pipe braces are well below the compactness limits of the AISC SCBF
provisions, the drift capacities of the less compact cross-sections do not meet the acceptance
criteria of 4% drift. The average drift at fracture initiation is equal to 2.9% for the HSS1 section
and 3.4% for the P1 (pipe) section. These results suggest that the current AISC SCBF
compactness limits for HSS and pipe sections may be unconservative and, in the least, warrant
further review. The data in Figures 4.4a and 4.4b suggest that a reduction to about three-fourths
of the current compactness limits for HSS and Pipe sections would achieve a drift capacity of 4%.
106
On the other hand, if one considers the 2% story drift (corresponding to design events), all but
one (HSS1 during the near-fault compression history) of the nineteen braces survive this drift
without fracture.
The W-shape braces exhibit high ductility despite having a flange width-thickness ratio that
exceeds the AISC compactness limit by about 5%. This can be attributed in part to the high
slenderness of the specimen which limits plastic strains in the central plastic hinge. Perhaps
equally significant is that the local buckling shape in the W-section induces less severe material
strains as compared to the HSS or Pipe sections.
4.3.2
Effect of Member Slenderness
As the LB/r slenderness increases, the brace buckling is more elastic and the smaller cross section
dimensions (relative to the brace length) lead to smaller strain demands at the central plastic
hinge. Previous studies (e.g. Jain et al. 1978, Tang and Goel 1989) have documented the
beneficial effects of increased slenderness on brace performance. In fact, some studies have
determined slenderness ratio to be the most important parameter controlling brace response (e.g.
Lee and Bruneau 2005).
Referring to the plot of slenderness ratio versus fracture drift and dissipated energy in Figures
4.5a and 4.5b, the data (particularly of the pipe braces) suggest that fracture ductility increases
slightly with member slenderness. However, since the member slenderness between tests of a
given cross section is not varied as much as the section compactness, it is difficult to draw clear
conclusions about LB/r slenderness from the data. The large capacities obtained for the W-shape
underscore the effect of overly large slenderness on ductility. Despite having a large widththickness ratio, the high member slenderness, as well as the local buckling shape, of the W12x16
appears to contribute to its large fracture resistance.
107
4.3.3
Effect of Cross Sectional Shape
Representative inelastic local buckling mode shapes are shown in Figure 4.6 for the three types of
cross sections, plus the filled section. As the local buckling shapes are quite different in form, the
resulting strain concentrations that trigger fracture are different as well. The HSS shapes exhibit
severe local buckling and crimping at the corners that greatly amplifies the local strain (Figure
4.6a) leading to fracture at that location. On the other hand, the local buckling deformations in
Pipes and W-shapes are more gradual, leading to a less severe strain gradient at the critical
location (Figures 4.6b and 4.6c). Thus, the Pipe and W-sections are inherently more resilient to
local buckling induced fracture. However, a suitable combination of slenderness and widththickness ratios can mitigate the effects of cross section geometry. As suggested by the data in
Figure 4.4a and 4.4b, the HSS section performance improves with reduced width-thickness ratios.
Conversely, in spite of their favorable shape, Pipes with large diameter-thickness ratios and lower
slenderness may not provide the required ductility.
4.3.4
Effect of Grout Filling of HSS Specimens
Previous experimental investigations (Liu and Goel 1988) indicate that concrete filled sections
may exhibit higher ductility and withstand more cycles of reversed loading as compared to
hollow sections. The concrete fill delays local buckling and minimizes the severity of strain that
drives fracture initiation. When local buckling occurs in concrete-filled tubes, the tubes tend to
buckle outward (Figure 4.6d), such that the cyclic strain demands are reduced compared to the
unfilled section (Figure 4.6a). This increased ductility is account for by AISC (2005) by
specifying relaxed b/t limits on filled-HSS braces –
b / t < 1.4 E / Fy = 35.5 (for Fy = 46ksi )
(4.3.1)
Note that the hollow b/t limit of 16 is significantly less than the value obtained by Equation 4.3.1.
108
The effect of grout fill can be seen by comparing HSS1-1 to HSS1-4 and HSS1-2 to HSS1-5,
where HSS1-4 and 5 have high strength grout fill ( f c' = 6-8 ksi). Referring to Table 4.1, the drift
at fracture initiation for the filled HSS1-4, θ FI = 4.1%, is about 50% larger than for the unfilled
HSS1-1; the dissipated energy for the HSS1-4 is about 20% larger. For the near-fault
compression loading (Table 4.2), the filled HSS1-5 exhibits a large (160%) increase in drift
capacity and (170%) dissipated energy compared to the unfilled HSS1-2. While these tests
confirm the beneficial effects of the fill, the degree of improvement is highly variable and
warrants further study, particularly considering constructability costs and larger cross sections
where size effects associated with cracking of the grout or concrete fill may play a role.
Furthermore, after normalizing the b/t ratio of the HSS4x4x1/4 (14.2) by the AISC (2005) limits
for composite HSS members (35.5) results in a ratio of approximately 0.4. However, comparing
the ductilities of HSS1-4 and HSS2-1 (with θ FI = 5.0%; normalized b/t limit = 0.53) suggests that
Equation 4.3.1 may be too relaxed and warrants further investigation in the context of SCBF
bracing components.
4.3.5
Effect of Loading Rate
Two experiments (HSS1-3 and HSS2-2) were conducted at higher loading rates, comparable to
rates that would occur during earthquakes, as in contrast to the quasi-static rates used in the other
tests. The earthquake loading rate was determined using the approximate secant stiffness at the
design drift and corresponding elongated period (0.8 seconds) for a typical SCBF frame. The
resulting peak loading excursion rate of 6 in/sec is about 360 times faster than the slow rate of
0.017 inch/sec (1 inch/min) used in the other tests.
High loading rates and the associated high strain rates can induce elevated stresses, due to ratedependent yielding and strain hardening behavior (Anderson 1995), which may increase the
109
tendency for brittle cleavage fracture. Conversely, the higher loading rates may cause a
temperature rise in the regions of high localized strain, which will tend to improve fracture
resistance as is evident in the Charpy energy curve (e.g. Barsom and Rolfe 1987). Thus, the
higher loading rates can have competing adverse and beneficial impacts on fracture ductility,
depending on the structural component geometry, stress constraint, presence of cracks, ambient
temperature, and material properties.
Referring to Table 1, the differences in fracture ductility between the high and low rate test
(HSS1-1 versus HSS1-3 and HSS2-1 and HSS2-2) are not significant. It should be noted that a
direct comparison between the tests is difficult because the loading histories for the high and low
rate tests are not identical. This variation results from the reduced ability to accurately control the
actuators at the high-rate tests, such that the imposed displacements are somewhat larger than the
specified target limits. The apparent insensitivity of fracture resistance to loading rate is
consistent with the fact that the brace fractures occur due to ductile tearing in regions of relatively
low-constraint, such that the modest effects of loading rate and temperature change (for the
ranges considered) do not significantly alter the fracture mechanisms. In addition to the observed
fracture response, thermo-couples installed on the surface of the braces provide a comparison of
temperature rise in the high-rate tests. The maximum recorded brace surface temperature for the
high-rate test, HSS2-2, was nearly 200°F, compared to the peak 90°F reading during a quasistatic test, HSS1-2. Since the typical brittle-ductile transition temperatures for mild structural
steels range between -100°F to 70°F (Koteski et al 2005), the temperature increase from 90°F to
200°F does little to affect the ductility of fracture in the brace.
110
4.3.6
Effect of Unsymmetrical Buckling
One of the experiments – HSS1-6 had two 18x2x0.5” reinforcing plates welded at the center to
the top and bottom (non-buckling faces) of the brace to deliberately induce unsymmetric buckling
(see Figure 4.7). The objective of this detail was to encourage unsymmetric buckling, such as
may be caused in field details due to dissimilar end conditions or imperfections. As indicated by
kinematics-based calculations, this type of unsymmetric response may increase the brace plastic
hinge rotations for a given axial deformation, relative to the idealized situation in which the brace
plastic hinge forms at the center of the brace. In all other respects, the specimen was similar to
HSS1-1. As expected, the ductility decreased by 43% from a maximum sustainable drift before
fracture of 1.57% in HSS1-1 (symmetric) to 1.1% in HSS1-6. The welded attachment itself is not
believed to influence fracture substantially (other than by causing the non-symmetric buckling
pattern) since fracture initiation occurred at a distance of approximately two inches from the end
of the reinforcing plate (see Figure 4.7).
These observations highlight the extent to which non-ideal conditions may affect the response,
especially when interpreting test results from specimens that are highly idealized representations
of field construction. The degree to which these non-ideal conditions, or unloaded attachments
(such as the plates in HSS1-6), will influence the response is uncertain. However, the above
discussion supports the requirement of a protected zone for design of SCBF systems which
guards the lateral load resisting elements against nonstructural attachments that negatively impact
the desired response (AISC, 2005).
4.4
BRACE-GUSSET PLATE CONNECTION PERFORMANCE
Fracture often controls the strength and ductility of critical structural members and connections.
The 1994 Northridge earthquake revealed the vulnerability of steel beam-column connections to
111
brittle fracture. Research since Northridge (e.g. FEMA, 2000) has resulted in significant
improvements in design practices regarding beam-column details in moment resisting frames.
However, earthquake-critical details commonly used in other lateral load resisting systems such
as Special Concentrically Braced Frames (SCBFs) have received relatively lesser attention in
research. A consequence of this is the limited understanding of the fracture resistance of these
details.
Brace-gusset plate connections in SCBF systems are an example of practical relevance where
earthquake-induced net section fracture may be a governing limit state. SCBF systems rely on
cyclic yielding and buckling of bracing members, subjecting the brace-gusset connection to large
tensile loads. Figure 4.8 illustrates commonly used brace connections for tubular HSS, Pipe and
Wide-Flanged sections. Figures 4.8a and 4.8b show the brace-gusset connection for these sections
where the tubular brace is slotted to accommodate the gusset plate. As shown in the figure, the
slot, extending beyond the gusset plate is typical of these connections and creates a reduced
section susceptible to fracture during severe tensile loading cycles. Fractures at these sections,
illustrated in Figure 4.8d, have been observed in earthquakes (Tremblay et al, 1995) and simple
calculations illustrate that the nominal strength of the net section may often be smaller than the
demands imposed by the yielding brace indicating that fracture in these details may well be a
likely mode of failure. Moreover, experiments by Tremblay et al (2003), Uriz (2005) and this
investigation have confirmed the likelihood of such fractures. Figure 4.8c shows a similar detail
for a Wide-Flanged brace where weld access holes create a reduced section.
In view of these developments, an objective of this chapter is to present experimental data from
the nineteen large-scale cyclic tests of brace specimens to examine the fracture resistance and
strength of brace-gusset connections, considering the effect of net-section reinforcement, loading
history and cross-section type. Recent tests (Liu et al, 2006, Yang and Mahin, 2005, Korol (1996)
112
and Cheng et al, 1998) have confirmed the likelihood of such failures in realistically sized
members under earthquake type loading. Moreover, the Seismic Provisions (AISC, 2005) require
that the design strength of CBF or SCBF braces based on net-section rupture (φRtFuUAn) should
be at least equal to the expected tensile strength of the brace (RyFyAg) to avoid net-section
fracture. Typically, Ry is 1.4 for HSS tubes, 1.6 for Pipe sections and 1.1 for wide-flanged shapes
(Liu et al, 2007). Assuming U = 0.85, and given that An must be smaller than Ag, a simple
calculation shows that the net section strength of the member will be smaller than the tensile axial
strength, unless the net section is reinforced. The concerns are even more important as thicker
gusset-plates result in smaller net areas. However, the net section is often not reinforced in
practice, mainly because of the lack of guiding experimental data. This investigation
complements previous research by examining additional cross-sections and shapes (e.g. Pipes and
Wide-Flanged sections).
4.4.1
Experimental Results
Qualitatively, the specimens exhibited two main types of response. As discussed in the previous
section, a large majority (17 of 19) of the braces showed global buckling (Figures 4.1a) during the
compressive cycles, followed by local buckling (Figures 4.1b) of the cross section at the central
plastic hinge. The local buckles resulted in severe strain amplifications, eventually leading to
fracture at the center of the brace (Figures 4.1c and 4.1d). However, in the context of this section,
these specimens serve as examples of brace-gusset connections that survived earthquake-type
loading histories where both the strength and ductility are limited by brace, rather than
connection, response. For typical earthquake loading histories, once buckling occurs, damage
localizes at the center of the brace, essentially protecting the connection region from further
damage or fracture. Of the nineteen tests, only two (Pipe sections without reinforcement)
exhibited net-section fracture at the brace-gusset connection. Figure 4.8d shows a photo of the
fracture at the connection (for Pipe3STD), whereas Figure 4.9b shows the load deformation curve
113
of these experiments where the connection deformation is measured over each end region (onethird length – refer to Figure 4.9a) of the brace to distinguish between response observed at each
end connection. In both of these experiments, the specimens were subjected to a near fault tension
dominated history (refer Chapter 3, Figure 3.4c) where a large tensile pulse was applied before
any cyclic loading could concentrate buckling-induced damage at the center. In both cases, the
braces fractured on the first tensile pulse before buckling-induced damage could localize at the
center.
Deformation data for the brace specimens was recovered in two formats. The first measure is the
overall deformation of the brace, measured over the entire brace length (see Figure 3.1a), which
can be converted to a corresponding drift ratio as discussed earlier. While this enables a
convenient comparison with expected story drift demands, it does not capture local effects, e.g. if
deformations localize at one end before the other. Thus, additional deformation measurements
were made over each end region (one-third length – refer Chapter 3 and Figure 3.1a) of the brace
to distinguish between response observed at each end connection.
Table 4.3 summarizes deformation capacities (in terms of drift) corresponding to fracture
initiation in the specimens. Also indicated in the Table are connection deformations for the braces
which failed at the net section (tests P1-4 and P2-4) prior to any damage localization at the center
of the brace from global and local buckling. The drift capacities correspond to the maximum
drifts sustained by the member before fracture was observed. For example, if a brace fractures
during a tensile cycle at a specific drift, but encountered a larger drift during a prior compressive
excursion, the larger drift is reported. The Table also includes a comparison of the maximum
experimental tensile strength capacities with nominal capacities (based on a minimum of gross
yield and net-section strengths). Several observations regarding brace and connection behavior
can be made based on the data presented in Table 4.3.
114
•
When HSS or Pipe braces are reinforced at the net-section with reinforcing plates none of the
specimens (total 12) exhibited net section fracture. Irrespective of cross section type or
loading history, the damage localized at the center due to buckling and low cycle fatigue. In
fact, several of the specimens sustained drifts as large as 5-8% prior to failure, well in
exceedance of the expected seismic drift demands in SCBF systems. This indicates that
reinforcement effectively prevents net section fracture in these details.
•
Four Pipe specimens (P1-2, P1-4, P2-2, and P2-4) were not provided with reinforcement
plates at the net-section. Of these, two (P1-4 and P2-4) fractured during the first tensile pull
of the near-fault tension dominated history. This loading history was applied as a worst-case
scenario for tensile fracture at the connection, because it applied large deformation demands
before buckling damage could localize at the center of the brace. Despite the severity of the
loading history, the details fractured at deformations corresponding to drift levels of 6.4 and
5%, still fairly large as compared to expected demands. The two other unreinforced pipe
specimens (P1-2 and P2-2), subjected to far-field loading histories, did not fracture and
survived drifts as large as 2.68 and 5.0%, before fracturing at the center.
•
Three tests (W1, W2 and W3) featured W-section braces, subjected to compression and
tension dominated near-fault, as well as a regular far-field loading history. All these
specimens survived drifts between 5 and 8%, fracturing at the center due to buckling induced
damage. For the W-section braces, even the tension dominated near-fault history failed to
produce net section fracture, despite imposing tensile drift demands as large as 8%. This large
ductility can be attributed to the connection detail (see Figure 4.8c), where the weld access
hole produces a long reduced section resulting in smaller strains at the connection.
115
•
Shown in bold-face in Table 4.3, the specimens P2-3 and P2-4 (Pipe3STD) are controlled by
net section failure as opposed to brace yielding, i.e., Pn = Rt FuUAn < Py = Ry Fy Ag . Note, the
φ-factor is removed in the comparison of the two capacity quantities. Of these, P2-3 failed at
the center (under a far field loading history) whereas P2-4 failed at the connection due to net
section
fracture.
Also
interesting
to
note
is
that
in
test
P1-4,
where
Pn = Rt FuUAn > Py = Ry Fy Ag (suggesting that gross yielding would govern), the specimen
failed by net section fracture. This can be attributed to strain hardening in the brace that
increases the brace yielding force resulting in connection failure.
•
In context of the previous point, refer the last column of Table 4.3 that summarizes the ratio
between the experimental and predicted brace force Pmax/min(Py, Pn). Recall that Pn governed
only for P2-3 and P2-4. For all the other tests, is apparent that the measured maximum tensile
strengths (Pmax) are up to 25% larger than the expected yield strength ( Pu = Ry Fy Ag ). This
effect is most significant for the HSS specimens where the maximum tensile forces were 13
to 21% larger than the expected strength, Py. This suggests that the current practice of
designing bracing connections based on the expected yield force may be underestimate force
demands in connections, mainly because it ignores the effect of strain hardening that
amplifies the brace forces. Others have noted this too, e.g. Tremblay et al (2002) who
suggested amplifying brace forces by appropriate factors (α), such that Pbrace = α Ry Fy Ag
incorporates the effects of strain hardening. This is discussed in the following section.
Thus, the experimental data suggests that (1) net-section reinforcement in brace-gusset
connections effectively prevents fracture, (2) in far-field type earthquake loading histories, braces
typically exhibit damage localization at the center thereby protecting the net-section from failure
and (3) fracture at the net-section occurs only during the most severe loading histories where a
116
large tension pulse precedes cyclic loading. Even in these cases, the deformation capacity of the
member exceeds the expected seismic drift demands in SCBF systems (4) maximum tensile
forces observed in the large-scale brace tests may be somewhat larger than the expected yield
force (RyFyAg) typically used to design the bracing connections. This final point will be discussed
in the following section.
4.5
COMPARISON OF EXPERIMENTAL DATA TO COMMONLY USED FORMULAE FOR
PREDICTING STRENGTH AND STIFFNESS OF BRACING MEMBERS
This section compares the experimental brace stiffness and maximum strength (compressive and
tensile yield) data to commonly used analytical values. Table 4.4 lists the calculated and
measured strengths and elastic stiffness for each of the tests along with statistics for each
comparison. For example, where the first column lists the expected critical buckling loads of each
brace, Pc,exp, the second column, RPc, is the ratio of the experimental to predicted buckling load.
Referring to the table, the elastic stiffness, compressive buckling load, and expected tensile yield
and ultimate forces will be discussed in this section.
4.5.1
Elastic Stiffness
The initial elastic stiffness is calculated as Kel = EAg/L, where E = 29,000 ksi, Ag is the nominal
brace cross-sectional area, and L is the brace length (measured from end-to-end of the gusset
plates). For the grout filled braces, the stiffness of the fill is calculated similarly and is assumed to
act in parallel with the brace. The average test-prediction ratio for the stiffnesses is 1.04, with a
coefficient of variation of 0.11. This may be attributed to errors in estimating the grout stiffness
and other simplifications, such as errors in stiffness estimation of the gusset plates regions.
117
4.5.2
Compressive Strengths
The expected compressive strengths, Pc,exp, are calculated based on the AISC Specification
(2005), except that the expected yield strengths are used and the effective buckling length is
assumed equal to the distance between the gusset plate fold lines (see Figure 3.2; taken as the LB
listed in Table 3.1). For example, HSS1-1 had a measured compressive resistance of 157 kips,
whereas the corresponding values calculated as per AISC are:
Pc,exp = Fcr A g = (30.7 ksi)(3.37 in 2 ) = 103kips
Fe =
π 2E
⎛ KL ⎞
⎜
⎟
⎝ r ⎠
2
=
π 2 (29,000 ksi)
⎡ (1.0 ) (118") ⎤
⎢
⎥
⎣ 1.52" ⎦
KL
E
≤ 4.71
Since
r
Fy
Fy
⎡
Fcr = ⎢ 0.658 Fe
⎢
⎣
2
= 47.5 ksi
( 77.6<118 )
(4.3.2)
46
⎤
⎡
⎤
⎥ Fy = ⎢0.658 47.5 ⎥ 46 ksi = 30.7 ksi
⎥
⎢⎣
⎥⎦
⎦
The estimated critical buckling loads presented in Table 4.4 use Ry to account for the increase in
yield stress from the minimum specified value to the expected value:
Pc ,exp = Fcr − Ry Ag = (36.5 ksi)(3.37 in 2 ) = 123kips
Fe =
π 2E
⎛ KL ⎞
⎜
⎟
⎝ r ⎠
2
=
π 2 (29,000 ksi)
⎡ (1.0 ) (118") ⎤
⎢
⎥
⎣ 1.52" ⎦
KL
E
≤ 4.71
Since
r
R y Fy
Fcr − Ry
Ry Fy
⎡
⎢
= 0.658 Fe
⎢
⎣
2
= 47.5 ksi
( 77.6<100 )
(4.3.3)
(1.4 )( 46 ) ⎤
⎤
⎡
⎥ Ry Fy = ⎢0.658 47.5 ⎥ (1.4 ) ( 46 ksi ) = 36.5 ksi
⎥
⎢⎣
⎥⎦
⎦
For the grout filled braces, HSS1-4 and HSS1-5, the fill strength was assumed equal to its
nominal specified value of f’c = 7 ksi. The strengths for these specimens are calculated as per
AISC (2005), which involves adding the contributions from the strengths of the grout and steel,
as described by Equations 4.3 through 4.6 below –
118
Po = Ag Ry Fy + 0.85 Ac f c '
(
Pe = π 2 EI eff
) ( KL )
2
B
(4.3.4)
(4.3.5)
Where EI eff = Es I s + C3 Ec I c and C3 = 0.9 for this investigation.
(a) When Pe ≥ 0.44 Po
Pc ,exp
⎛ Po ⎞ ⎤
⎡
⎜ ⎟
P
= Po ⎢0.658⎝ e ⎠ ⎥
⎢
⎥
⎣
⎦
(4.3.6)
(b) When Pe < 0.44 Po
Pc ,exp = 0.877 Pe
(4.3.7)
Where Ag, Fy, Is and Ac, fc’, Ic are the properties of steel and grout, respectively.
Overall, for all tests, the ratio of measured to calculated expected compressive strengths is 1.24
with a standard deviation of 0.30, where the expected compressive strengths are, in general,
smaller than the experimental buckling loads. The smaller expected compressive strengths may
be a result of assigning an effective length factor, K = 1.0, in the above calculations. Assuming
the gusset plates do provide some fixity and assigning K = 0.9 decreases the measured to
calculated ratio to 1.08 with a standard deviation of 0.22. The remaining error may be attributed
to the larger measured yield strengths, Fy,meas, above RyFy and the varying effective length factors
from the different brace and gusset-plate geometries.
4.5.3
Maximum Tensile Strength
The expected tensile yield, Py,exp = RyFyAg, and ultimate, Pu,exp = RtFuAg, brace strengths are
calculated as the product of the expected material strengths and the gross cross-sectional area
(this was introduced previously in the context of brace-gusset connections). Figure 4.10a
compares the maximum measured experimental tensile forces with the expected yield and
119
ultimate strengths. The average ratio of the maximum measured brace strength to the expected
brace yield strength is 1.14 with a standard deviation of 0.07. In some cases (e.g. test W3), the
measured strength is as large as 1.25 times the expected yield strength, which while conservative
from the perspective of member design, can be unconservative for determining capacity design
requirements for connections. This is especially true for tension near-field loading for the Pipe
and Wide-Flanged sections. The average ratio of the measured strength to the expected ultimate
strength is 0.92 with a standard deviation of 0.08. On average, an improved estimate of the
expected brace strength would be to use an average of the expected yield and ultimate strengths,
i.e., Pmax = 0.5(RyFy+ RyFy)Ag. This is illustrated in Figure 4.10b. Using this measure, the ratio of
the maximum experimental brace forces to this capacity measure is 1.01 with a standard deviation
of 0.08).
4.6
SUMMARY
This chapter presents findings and design implications based on nineteen large scale tests of
concentrically-loaded steel braces subjected to earthquake type cyclic loading. The tests are part
of a larger project that has the dual aims of validating and applying new micromechanics based
models to simulate ductile fracture under cyclic loading and to develop practical behavioral
information and design guidance of steel braced frames.
The tests, designed to represent typical conditions encountered in steel braced frames,
complement previous studies by investigating three cross section types (HSS, Pipe and Wide
flanged sections) of varying section compactness and brace slenderness. The test program also
investigates the effects of loading histories, loading rates, connection reinforcement, and grout
filling of HSS sections.
120
Qualitatively, the tests all followed a similar sequence of events leading to failure. Global
buckling of the brace (at displacements corresponding to 0.2-0.4% story drift) leads to the
formation of a plastic hinge at the midpoint of the brace. Subsequently, local buckles form in the
hinge region (at 2% to 5% story drift) that amplify the strains and trigger fracture initiation (at
2%-8% story drift). Soon after this, the fracture propagates through the entire cross section,
severing the brace. Brace buckling is accompanied by large out of plane displacements that pose
threats to surrounding architectural enclosures.
One of the main conclusions of this study is that brace fracture ductility is primarily a function of
section compactness and to a lesser extent member slenderness and loading history. Specifically,
fracture ductility increases with more compact cross sections and more slender members. Further,
the standard loading protocols (modeled to represent general or far-field ground motions) are
more damaging than loading protocols developed to represent pulse-like near-field ground
motions. The tests further demonstrate that the local buckles in HSS sections result in more
severe straining of the steel material, leading to fracture initiation near the corners of the brace.
This is in contrast to Pipe and Wide-Flange sections that exhibit more gradual local buckling
modes. This is not to imply that Pipes and Wide-Flange braces are naturally more ductile than
HSS. In fact, the Pipe5STD (large D/t) brace fractured at a smaller drift ratio than the
HSS4x4x3/8 (small b/t).
The tests suggest that the section width-thickness ratios in the AISC Seismic Provisions (2005)
for HSS and Pipe sections may not result in adequate deformation capacities for seismic design.
HSS members with width-thickness ratios equal to about 90% of the limiting compactness
criteria, and subjected to the general loading protocol, fractured at drift ratios in the range of 2.7%
to 3.0%. Pipe members with diameter to thickness ratios equal to 60% of the limit fracture at drift
ratios of 2.7%. While the drifts achieved by these members are larger than the approximate
121
design level drift of 2%, they are smaller relative to the 4% drift demand criteria implied by
several previous investigations and current design requirements. On the other hand, W-shape
braces, which slightly violated the compactness criteria, sustained drift ratios of up to 5%. These
results are sensitive to loading history, since the endurance for all of the braces increased
considerably (up to two or three times) when subjected to the near-fault loading protocol that
subjected the braces to fewer reverse loading cycles. Tests to investigate the effect of loading rate
on fracture performance demonstrated essentially no difference in response between quasi-static
and earthquake loading rates.
Tests of braces filled with high-strength grout fill did not increase the fracture capacity as much
as expected. As expected, the grout fill postponed local buckling and increased the fracture
resistance by about 160% in the near-fault tests while only a modest increase in drift ratio (50%)
was observed in the far-field tests. Thus, the effectiveness of grout fill to improve braced frame
performance warrants further study. One of the braces, fitted with reinforcing plates to induce
unsymmetric buckling showed significantly reduced ductility. This indicates that experimental
data from idealized specimens must be interpreted with caution, since imperfections in field
details may negatively impact response.
Comparison of measured and calculated strengths for brace strength and stiffness generally
confirm expectations and the legitimacy of standard assumptions. In particular, ratios of measured
compressive buckling strengths to calculated strengths (using the standard AISC column curve
equation and expected yield strengths with Ry factors specified by AISC) have a mean value of
1.24 and a standard deviation of 0.30. Ratios of measured tensile strengths are estimated fairly
well by the average of the expected yield and ultimate brace strengths (calculated using RyFy and
RtFu values specified by AISC) with a mean value of 1.01 and a standard deviation of 0.08.
122
While the brace test data provides valuable insights into the brace buckling and fracture behavior,
it is difficult to generalize the experimental findings since (a) the tests cover only a limited range
of parameters and configurations, and (b) in and of themselves, the tests provide limited data to
quantify the localized stress and strain combinations that trigger fracture and fatigue. Thus,
analytical simulations are necessary to generalize the data; for example, to develop quantitative
relationships between various parameters (such as slenderness and width-thickness ratios) and
fracture ductility. These topics are addressed in the following chapters.
123
Table 4.1: Data for standard (far-field) loading protocol tests
Test
No.
(kip)
Pc,max
(kip)
HSS1-1
247
157
HSS1-3*
255
161
HSS1-4
(filled)
257
194
HSS1-6#
249
163
HSS2-1
348
186
HSS2-2*
362
184
P1-1
241
181
P1-2
243
177
P2-1
132
80
P2-2
130
84
W1
286
93
Pt,max
Δ a −GB , in
Δ a − LB , in
Δ a − FI , in
Δ a − SL , in
( θGB , %)
( θ LB , %)**
( θ FI , %)**
( θ SL , %)**
0.18
(0.3)
0.20
(0.3)
0.21
(0.4)
0.21
(0.35)
0.17
(0.3)
0.20
(0.3)
0.18
(0.3)
0.18
(0.3)
0.16
(0.3)
0.16
(0.3)
0.09
(0.2)
1.10
(1.9)
1.26
(2.1)
1.57
(2.7)
1.10
(1.9)
2.99
(5.0)
2.64
(4.5)
1.57
(2.7)
1.57
(2.7)
2.99
(5.0)
2.99
(5.0)
2.99
(5.0)
1.57
(2.7)
1.73
(3.0)
2.40
(4.1)
1.10
(1.9)
2.99
(5.0)
2.64
(4.5)
1.57
(2.7)
2.36
(4.0)
2.99
(5.0)
2.99
(5.0)
2.99
(5.0)
1.57
(2.7)
1.73
(3.0)
2.40
(4.1)
1.57
(2.7)
2.99
(5.0)
2.64
(4.5)
2.40
(4.0)
2.36
(4.0)
2.99
(5.0)
2.99
(5.0)
2.99
(5.0)
ΣΔ aP− SL , in
( Σθ aP− SL ,
rad)
22.3
(0.4)
23.1
(0.4)
31.0
(0.5)
17.7
(0.3)
77.9
(1.3)
78.2
(1.3)
30.5
(0.5)
39.8
(0.7)
101.5
(1.7)
91.3
(1.5)
222.5
(3.8)
ΣESL
Ey
29.6
41.1
39.3
25.0
62.5
73.5
35.3
41.8
65.3
60.9
81.7
*fast (earthquake) loading rate
**values for the limit states of local buckling (LB), fracture initiation (FI) and strength loss (SL) are reported as the maximum value of axial
deformation (Δa, in) or drift (θ, %) that was sustained by the specimen prior to the limit state
#HSS1-6 was reinforced at the midpoint (top and bottom) with 18” long plates (see Figure 4.7)
124
Table 4.2: Data for near-field (NF) pulse loading protocol tests
NF Loading #
Test
No.
Pt,max
(kip)
Pc,max
(kip)
HSS1-2
249
119
HSS1-5
(filled)
263
136
W2
287
82
P1-3
292
127
P2-3
149
57
W3
323
75
P1-4
279
--
P2-4
144
--
Event
Δ a , in
( θ , %)**
Appended Standard
Loading #
Event
Δ a , in
( θ , %)**
Near Fault Compression Pulse Loading
-1.50
FI
1.10 (1.9)
LB
(-2.6)
SL
2.36 (4.0)
LB
2.99 (5.0)
None,
-3.54
FI
2.99 (5.0)
Survived
(-6.0)
SL
2.99 (5.0)
Pulse of:
-2.68
FI
2.99 (5.0)
LB
(-4.5)
SL
3.66 (6.2)
Near Fault Tension Pulse Loading
1.61 (2.7)
2.40 (4.0)
2.40 (4.0)
LB
None,
2.87 (4.9)
+4.72
FI
Survived
2.87 (4.9)
(+8.0)
SL
Pulse of:
2.87 (4.9)
2.32 (3.9)
2.32 (3.9)
3.07 (5.2)
+3.86
--FI/SL at
(+6.5)
net
+2.99
section
--(+5.1)
ΣΔ aP− SL , in
( Σθ aP− SL ,
rad)
ΣESL
Ey
62.2
(1.1)
19.3
119.7
(2.0)
52.2
140.6
(2.4)
36.4
92.2
(1.6)
52.7
136.6
(2.3)
67.1
475.5
(8.0)
95.0
3.6
(0.1)
2.8
(0.05)
14.7
14.5
#limit states observed during the standard, far-field loading protocol that is appended to the near-field loading are reported in terms of drift ratios
measured relative to the residual drift of +/- 3% that existed at the end of the near-field loading (see Figure 3.4b and 3.4c). Otherwise, the values
are described in the same way as for those reported in Table 4.1 for the standard (far-field) loading.
125
Table 4.3: Experimental results of bracing connections
Test
Cross
Section
Detail Type
Failure Type
P1-3
P1-4
P2-3
P2-4
W2
Pipe5STD
Pipe5STD*
Pipe3STD
Pipe3STD*
W12x16
Reinforced
Unreinforced
Reinforced
Unreinforced
NA
Fracture in middle of brace
Net section Fracture at end
Fracture in middle of brace
Net section Fracture at end
Fracture in middle of brace
Fracture/
Maximum Drift
(%)
8.0#
6.5 (3.9 in)
8.0#
5.1 (3.0 in)
8.0#
Pt, max
min(Py , Pn )
**
1.21
1.16
1.05
1.17
1.25
*Failure at net section. Connection deformations are listed in parentheses.
** Py = R y Fy Ag ; Pn = Rt FuUAn = 0.85 ⎡( Rt Fu Ag )
⎢⎣
plate
+ ( Rt Fu An )brace ⎤ . Number in bold are controlled
⎥⎦
by net section failure (i.e. Pn = Rt FuUAn < Py = Ry Fy Ag )
#Denotes maximum drift sustained without fracture at net section. Failure occurred during the
subsequent cyclic loading (refer Table 4.2 for details).
Table 4.4: Comparison of brace strength and stiffness
Test
No.
HSS1-1
HSS1-2
HSS1-3
HSS1-6
HSS1-4
HSS1-5
HSS2-1
HSS2-2
P1-1
P1-2
P1-3
P1-4
P2-1
P2-2
P2-3
P2-4
W1
W2
W3
average
std.dev.
Pc,exp
(kip)*
123
156
169
174
53
50
*Assumes K = 1.0
RPc
(K = 1.0)
1.27
0.96
1.30
1.32
1.24
0.87
1.10
1.09
1.04
1.02
0.73
N/A
1.51
1.58
1.07
N/A
1.85
1.63
1.49
1.24
0.30
RPc
(K = 0.9)
1.14
0.86
1.17
1.10
0.77
1.18
0.98
0.97
0.98
0.96
0.69
N/A
1.28
1.34
0.91
N/A
1.50
1.32
1.21
1.08
0.22
Py,exp
(kip)
217
217
308
241
125
259
RPy
1.14
1.15
1.18
1.15
1.18
1.21
1.13
1.18
1.00
1.01
1.21
1.16
1.06
1.04
1.19
1.15
1.10
1.11
1.25
1.14
0.07
Pu,exp
(kip)
254
254
360
310
161
337
RPu
0.97
0.98
1.00
0.98
1.01
1.04
0.97
1.01
0.78
0.78
0.94
0.90
0.82
0.81
0.93
0.89
0.85
0.85
0.96
0.92
0.08
Kel
(kip/in)
825
1216
1165
1039
537
1136
RKel
1.13
1.13
1.11
1.14
0.77
0.78
1.06
0.9
1.07
1.04
1.12
1.08
1.09
1.07
1.12
1.12
1.00
1.08
1.04
1.04
0.11
126
(a)
(b)
(c)
(d)
Figure 4.1: Typical progression of brace specimen damage (a) Global buckling, (b) Local
buckling, (c) Fracture initiation and (d) Loss of tensile strength.
127
1200
Max.
Force
1099 kN
(a)
Force (kN)
800
Max.
Drift
2.7%
Fracture
Initiation
1.7%
400
0
-400
Local
Buckling
1.9%
Global Buckling
θ GB =0.3%, 698 kN
Loss of
Tensile
Strength
2.5%
-800
-4
-2
0
2
4
Drift (%)
1200
Max.
Force
1108 kN
(b)
Fracture
Initiation
-1.1% (1.9%)
Force (kN)
800
Loss of
Tensile
Strength
-0.3%
(2.7%)
400
0
Max.Drift
-7.0%
(-4.0%)
-400
Local
Buckling
-2.6%
Global Buckling
θ GB =1.0%, 529 kN
-800
-8
-6
-4
-2
0
2
4
Drift (%)
1600
(c)
Max.
Force
1299 kN
Force (kN)
1200
Fracture
Initiation
5.9% (2.9%)
800
400
Max. Drift
-1.0%
(-4.0%)
Loss of
Tensile
Strength
6.4%
(3.4%)
0
Local
Buckling
0.3% (-2.7%)
-400
-800
-2
0
2
Global Buckling
θ GB =6.8%, 565 kN
4
6
8
10
Drift (%)
Figure 4.2: Typical brace response for (a) Far-field loading (HSS1-1 shown), (b) Near-fault
compression (HSS1-2) and (c) Near-fault tension (P1-3). Drifts in parenthesis are relative to
residual drift after near fault loading. Drifts underlined are reported in Tables 4.1 and 4.2.
128
100
(a)
ΣEFI/EY
ΣEFI/EY = 0.99ΣELB/EY + 10.5
HSS1 (Grout Fill)
50
HSS1
HSS2
P1
P2
10.5
W
0
0
50
ΣELB/EY
100
100
(b)
ΣESL/EY
ΣESL/EY = 1.01ΣEFI/EY + 3.8
HSS1 (Grout Fill)
50
HSS1
HSS2
P1
P2
3.8
W
0
0
50
ΣEFI/EY
100
Figure 4.3: Energy dissipated prior to (a) Fracture initiation (FI) versus local buckling (LB) and
(b) Strength loss (SL) versus fracture initiation.
129
6
Fracture Drift (%)
HSS1 (Grout Fill)
HSS1
4
HSS2
P1
P2
2
W
(a)
0
0
1
2
Width-Thickness Ratio/AISC Limit
100
HSS1 (Grout Fill)
HSS1
ΣE/EY
HSS2
P1
50
P2
W
Connection
Failure
(b)
0
0
1
2
Width-Thickness Ratio/AISC Limit
Figure 4.4: Effect of width-thickness ratio on (a) Maximum drift at fracture initiation and (b)
Normalized dissipated energy.
130
6
Fracture Drift (%)
HSS1 (Grout Fill)
HSS1
4
HSS2
P1
P2
2
W
(a)
0
0
1
2
3
Slenderness Ratio/AISC Limit
100
HSS1 (Grout Fill)
HSS1
ΣE/EY
HSS2
P1
50
P2
W
Connection
Failure
(b)
0
0
1
2
3
Slenderness Ratio/AISC Limit
Figure 4.5: Effect of slenderness ratio on (a) Maximum drift at fracture initiation and (b)
Normalized dissipated energy.
131
(a)
(b)
(c)
(d)
Figure 4.6: Local buckling shapes for (a) HSS, (b) Pipe, (c) Wide Flanged and (d) Grout-filled
HSS cross sections.
132
CL
(a)
Reinforcing Plates
CL
(b)
Figure 4.7: (a) Symmetric local buckling at brace midpoint and (b) Middle reinforcement plate
(top and bottom)-induced unsymmetrical buckling.
133
Net
Section
Reinforcement plate
Net
Section
(a)
(b)
Weld access hole
(c)
(d)
Figure 4.8: Net section details for (a) Reinforced HSS4x4x1/4, (b) Pipe3STD, (c) W12x16; (d)
Net section failure of Pipe3STD.
134
Sliding Beam
Constraint Frame
N
Actuator (220 kip, +/- 10 in)
Monotonic
10’-3”
Specimen
Reaction
Wall
Loading
Connection Gage Length (3’-3”)
(a)
Force (kip)
300
Pipe5STD
Pipe3STD
200
100
(b)
0
0
1
2
Connection Deformation (in)
Figure 4.9: (a) Experimental connection gage length (refer Figure 3.10a for locations of wirepots) and (b) Load deformation response of Pipe3STD and Pipe5STD tests with fracture at the net
section.
135
Design Capacity (kips)
400
R
RyFyAg
yFyAg
R
RtFuAg
tFuAg
Overestimated
200
Underestimated
(a)
0
0
200
400
Pmax (kips)
1/2[RyFyAg + RtFuAg] (kips)
400
200
(b)
0
0
200
400
Pmax (kips)
Figure 4.10: Maximum experimental forces compared to (a) Py,exp=RyFyAg and Pu,exp=RtFuAg
capacities and (b) Average of Py,exp and Pu,exp.
136
Chapter 5
Application and Evaluation of the CVGM for Large-Scale
Components
Prevailing approaches to characterize the fatigue and fracture performance of braced frame and
other structural components are based mostly on empirical or semi-empirical methods. For
braces, previous research has relied on critical longitudinal strain measures at the material or
cross sectional level, or cycle counting and fatigue-life approaches at the component level (Tang
and Goel, 1989). In fact, recent studies (e.g. Uriz and Mahin, 2004) have combined these
approaches by applying fatigue-life approaches to a fiber strain at a brace cross-section. While
these approaches represent important advances in the fatigue-fracture prediction methodology for
structures, they do not directly incorporate the effects of local buckling or the complex
interactions of stress and strain histories that trigger crack initiation in bracing components.
Consequently, large-scale testing is still required to characterize the fracture performance of these
details (Lehman et al., 2008).
In part, the dependence on experiment-based approaches can be attributed to the lack of
computational resources required to simulate phenomena such as local buckling that create
localized stress and strain gradients that cause fracture. However, where fracture is of concern,
137
the reliance on simplistic models is primarily due to the lack of suitable stress/strain based
fracture criteria to accurately evaluate the complex interactions of stresses and strains. This is
particularly the case when fracture occurs in structural components subjected to large-scale
yielding and cyclic loading where traditional fracture mechanics approaches are not accurate.
Moreover, many of these situations (especially those found in braced-frames) do not contain a
sharp crack or flaw, which is another necessary assumption for the use of traditional fracture
mechanics. Finally, earthquakes produce Ultra Low Cycle Fatigue (ULCF) in structures where
very few (typically less than 10) cycles of extremely large magnitude (several times yield) are
typical during the dynamic response of a building. This ULCF behavior is quite different from
low or high cycle fatigue, which occurs in bridges and mechanical components. Consequently,
continuum-based models that capture the fundamental physics of the ULCF/fracture phenomena
are required to capture the complex stress-strain interactions leading to fracture. The models
presented in this chapter simulate the micromechanical processes of ULCF to predict fracture
from a fundamental physics-based perspective. They are fairly general, can be applied to a wide
variety of situations as they work at the continuum level, and are relatively free from assumptions
regarding geometry and other factors. Finally, these models require inexpensive tension coupon
type tests for calibration.
Kanvinde and Deierlein (2007) suggested an explanation for the processes which may govern
micromechanical ULCF behavior during earthquake-type cyclic loading. The resulting
continuum-based, Cyclic Void Growth Model (CVGM) has shown promise in simulating the
physical void growth, shrinkage and damage events that lead to ductile fracture initiation during
cyclic loading in small-scale experiments. The model has been rigorously examined by Kanvinde
and Deierlein (2007) at the small-scale across a variety of steel types, geometries and loading
histories. In this chapter, the CVGM is applied to the brace specimens described in Chapters 3
and 4 to extend the work of Kanvinde and Deierlein and assess the accuracy and limitations of the
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model at a larger scale. Referring to the previous discussion, the brace fracture at the middle
plastic hinge presents a case where large-scale yielding and smooth geometries limit the
applicability of traditional fracture mechanics. Furthermore, the testing matrix presented in
Chapter 3 provides a rich collection of different steel types, cross-sectional shapes and loading
histories to assess the capabilities of the CVGM at the large-scale. However, the global and local
buckling-induced behavior of the braces in the presence of cyclic inelastic response under
multiaxial stresses presents challenges in simulating these components which must be resolved to
accurately predict the fracture.
A successful large-scale evaluation of the physics-based fracture model provides a powerful
framework which can be used to investigate structural component or system behavior without
experimentation. For example, while the results from the nineteen brace tests, presented in
Chapter 4, suggested trends between brace ductility and geometric and material properties, the
experimental results alone do not form a basis for providing conclusive recommendations in the
context of brace performance and detailing guidelines. In fact, large-scale testing results are
rarely exhaustive due to insufficient lab capabilities and high material and laboratory costs that
accompany large-scale testing. Thus, the development of simulation methodologies provides
researchers with powerful tools to generalize experimental data with the ultimate goal of an
improved assessment of fracture susceptibility in steel systems.
To provide support for such a methodology for large-scale structural components, this chapter
reviews a finite-element simulation study and the performance of micromechanical-based fracture
models applied to the brace experiments of Chapters 3 and 4. The chapter begins by reviewing
the formulation of the Cyclic Void Growth Model (CVGM) and the calibration procedure for the
CVGM parameters. Next, the small-scale CVGM validation results from a prior study (Kanvinde
and Deierlein, 2007) are briefly introduced to illustrate the accuracy of the CVGM fracture
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predictions across multiple small-scale experimental specimens, seven different steels, and
diverse loading histories. The fracture prediction methodology is then developed for the largescale brace component tests presented as part of this study. Here, the results of finite element
simulations, and corresponding CVGM fracture predictions, of each bracing member are
presented to examine the performance of the model across various material types and loading
histories. The fracture predictions are discussed in a deterministic as well as a probabilistic
framework, where the latter incorporates material uncertainty. Finally, the implications of this
study are discussed in the broader context of the state of the art in fracture modeling in
earthquake engineering.
Since the micromechanical-based fracture model is based on continuum stress and strain
quantities, the model is sensitive to the accurate description of the material constitutive response
as well physical phenomena which interactively affect the continuum stresses, especially for postbuckling behavior. Thus, issues associated with finite-element modeling, such as proper
calibration of material constitutive models, and modeling of initial brace imperfections and crosssection local buckling are given special attention when developing the fracture prediction
methodology.
5.1
DEVELOPMENT OF THE CYCLIC VOID GROWTH MODEL (CVGM)
This section presents the fracture criterion for the CVGM model. The model is expressed in terms
of continuum stresses and strains which intend to reflect void growth and coalescence during
inelastic cyclic loading. Next, material constitutive model calibration is discussed as the accuracy
of the CVGM depends on correctly simulating phenomena such as necking and buckling which
are highly sensitive to material constitutive response. Then, the calibration procedure for the
CVGM fracture parameters is reviewed. Finally, the accuracy of the CVGM fracture predictions
for small-scale (on the order of several inches) details is discussed.
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The CVGM is derived from the findings of McClintock (1968) and Rice and Tracey (1969) that
the monotonic growth rate of a single spherical void, dr/r, in an elastic-perfectly plastic
continuum is exponentially related to the hydrostatic stress state, σm –
⎛ 1.5σ m
dr
= 0.283exp ⎜
r
⎝ σY
⎞ p
⎟ dε
⎠
(5.1.1)
Where σY is the yield stress, and dεp is the incremental equivalent plastic strain defined by –
dε p =
2 p
d ε ij .d ε ijp
3
(5.1.2)
For an incremental plastic loading excursion, from εpn-1 to εpn, Equation 5.1.1 can be integrated to
determine the new void size in relation to the void size at the previous step –
ε np
∫
ε np−1
εp
n
⎛ 1.5σ m
dr
= C exp ⎜
r εp
⎝ σY
∫
n −1
ε np
ln ( r ) ε p =
n−1
ε np
⎞ p
⎟ dε
⎠
⎛ 1.5σ m ⎞ p
⎟ dε
σY ⎠
∫ C exp ⎜⎝
ε np−1
(5.1.3)
(5.1.4)
If the new void size at εpn is Rn, and the original size at εpn-1 is Rn-1, the above expression can be
used to describe the ratio of the new void size to the previous void size –
εp
⎛ R ⎞ n
⎛ 1.5σ m ⎞ p
ln ⎜ n ⎟ = C exp ⎜
⎟ dε
⎝ σY ⎠
⎝ Rn −1 ⎠ ε np−1
∫
(5.1.5)
Subsequently, D’Escata and Devaux (1979) suggested replacing the yield stress by the effective,
or von Mises, stress, σe, to provide a better description of void growth in the presence of strain
hardening. Following this refinement, the ratio σm/σe referred to as the stress triaxiality, T, is a
scalar quantity which affects the rate of void growth in ductile metals during inelastic loading.
Equation 5.1.5 is based on analytical derivations of the growth of a single void and does not
explicitly account for the interaction effects of neighboring voids. However, several researchers
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including Hancock and Mackenzie (1976), Panontin and Sheppard (1995), Chi and Deierlein
(2000) and recently Kanvinde (2004) have demonstrated the effectiveness of this type of criterion
to predict fracture during monotonic tensile loading. To facilitate calibration, and thus the
prediction methodology itself, Equation 5.1.5 is best expressed in terms of the void ratio at any
plastic loading increment, Rn, with respect to the original void size, R0 –
⎛R
ln ⎜ n
⎝ R0
⎞
⎟=
⎠
ε np
∫ C exp (1.5T ) d ε
(5.1.6)
p
0
With this Void Growth Model (VGM), as it will be referred to henceforth, simple notched-bar (to
generate the necessary triaxial stress state) tensile tests and complementary finite-element
simulations are used to calibrate the material dependent, critical void size, Rcrit. From the
equivalent plastic strain and triaxiality evolution at the critical fracture point, it is assumed that at
fracture, Equation 5.1.6 describes a fracture parameter, η –
ln ⎛⎜
η= ⎝
Rcrit
C
⎞
R0 ⎟⎠
p
ε crit
=
∫ exp (1.5T ) d ε
p
(5.1.7)
0
A monotonic fracture criterion can then be proposed as –
ε np
∫ exp (1.5T ) d ε
p
≥η
(5.1.8)
0
While Equation 5.1.8 is developed by considering the growth of a single void, the calibration of η
implicitly accounts for void interaction. By evaluating η at the first occurrence of a macroscopic
crack, the parameter describes void-cluster coalescence rather than a critical size of a single void.
Thus, the criterion tracks the critical void size that will lead to necking instabilities between voids
as illustrated in Figure 5.1. In general, when the stress and strain gradients are relatively shallow,
the fracture criterion is satisfied over a large volume of material. If the criterion is evaluated in
the presence of sharp stress and strain gradients (for example, at a crack tip), then it must be
satisfied over a characteristic volume of the material. This volume is considered a material
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property and is usually presented as the characteristic length, l*. In the context of the shallow
stress and strain gradients from the notched-bar and brace geometries discussed in this chapter, it
will not be necessary to consider the characteristic length effect.
While the intent of the VGM is to model void growth under positive triaxiality (T > 0), i.e. tensile
hydrostatic stress, it cannot simulate void collapse under a negative (or compressive) triaxial
stress state during cyclic loading such as observed during earthquakes. To account for void
collapse during negative triaxialities, Kanvinde (2004) proposed the following modification to the
VGM –
ηcyclic
⎛R ⎞ ⎛
= ln ⎜ n ⎟ = ⎜
⎝ R0 ⎠ ⎜⎝
⎞
⎛
exp ( 1.5T ) d ε ⎟ − β ⎜
⎟
⎜
ε np−1
⎠T >0
⎝
p
ε np ε n
∑∫
0
p
p
ε np ε n
∑ ∫ exp ( 1.5T ) d ε
0 εp
n−1
p
⎞
⎟ ≥ 0 (5.1.9)
⎟
⎠T <0
This expression will be referred to as the CVGM and introduces a second term to the original
VGM to account for loading excursions with negative triaxialities (T < 0). Subtracting this term
simulates the effect of void collapse mechanisms. Following Kanvinde (2004), different loading
rates for growth and collapse are not considered. Therefore, β is assumed to be equal to 1 for this
study. In addition, ηcyclic, or ln(Rn/R0), is restricted to values greater than or equal to zero. This
may be interpreted to imply that Rn/R0 remains greater than or equal to one, or Rn ≥ R0, i.e., the
void size is assumed to remain larger than or equal to the unity. However, this cannot be
independently verified, but as indicated by Kanvinde and Deierlein (2007) and discussed
subsequently in this chapter, this assumption provides results that are in good agreement with
experimental observations.
Similar to the VGM, a fracture criterion is also specified for cyclic loading. However, unlike
monotonic loading, the fracture toughness is assumed to reduce from the effects of cyclic loading.
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The formulation by Kanvinde and Deierlein (2007) account for this damage by specifying the
CVGM fracture criterion as –
ηcyclic ≥ f ( D )η
(5.1.10)
Where f(D) is a damage function which reduces (i.e., f(D) ≤ 1) the monotonic fracture parameter,
η. The damage is assumed to occur primarily during compressive loading when voids collapse
into a more oblate shape, introducing damage at the corners of the void. This damage is verified
in Figure 5.2 by examining the fractured surfaces of a steel material after monotonic and cyclicinduced fracture. Referring to the figure, Scanning Electron Microscope (SEM) images of
fractured surfaces from Kanvinde and Deierlein (2007) are shown for monotonic and cyclic
loading (Figure 5.2a and 5.2b, respectively). A contrast can be seen between the smaller,
somewhat squished dimples formed prior to cyclic fracture initiation as compared to the larger,
more spherical dimples from monotonic fracture. To reflect the damage to the fracture toughness,
an exponential damage function is specified by Kanvinde and Deierlein (2007) as –
(
f ( D ) = exp −λε p*
)
(5.1.11)
Where λ is a model parameter and the damage is determined as the equivalent plastic strain at any
load reversal point (εp*) determined by a switch from a negative to positive triaxiality. Thus, if
plotted versus cycle number, the damage takes the form of a decreasing step function. This will
be illustrated in the following sections.
5.1.1
MATERIAL BEHAVIOR, CONSTITUTIVE MODELS AND CALIBRATION
Considering Equations 5.1.9 and 5.1.10 above, the accuracy of the fracture predictions depends
on an accurate finite element simulation of the material constitutive (stress and strain) response at
the critical fracture locations. Since the simulations are conducted at a continuum, rather than a
micromechanical scale, the material constitutive models cannot directly simulate the
micromechanical processes responsible for material plasticity (such as dislocation motion). This
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results in two issues in the context of this study (1) The constitutive models themselves may not
be sophisticated enough to capture complex hardening behavior, especially under cyclic nonproportional loading, resulting in situations where it is impossible to develop calibrated parameter
sets that simulate response accurately for all experiments and (2) The calibration is based on
phenomenological, rather than physical procedures, resulting in a high degree of subjectivity and
tedium, in the calibration process.
When developing constitutive relationships for steel and other common alloys, several
simplifying assumptions are typically made to describe the average, macroscopic, material
behavior. First, it is common to assume that steel is both homogeneous and isotropic. Material
homogeneity is best described through the statement that, for any loading condition, every
material point in the body will show an identical response. Isotropy is the assumption that the
material has identical properties in all directions and is not orientation dependent. For small
deformations, the material response can be classified as “linear elastic” in that the stress can be
uniquely related to the strain, and vice versa, through a material dependent constant referred to as
the elastic (or Young’s) modulus. However, at larger deformations, ductile materials yield and
deform inelastically as illustrated in the uniaxial stress versus strain response of Figure 5.3a.
Here, the elasticity assumption no longer holds as the work done on the body can not be fully
recovered during load reversal. This is primarily due to permanent deformations that accompany
dislocation (i.e., line imperfections in the material matrix) motion. For implementation into an
analysis program such as ABAQUS (2004), uniaxial, inelastic stress/strain data (along with the
elastic properties) can be used with an isotropy assumption to describe the material response
along any loading plane. Following this assumption, this type of material model is typically
referred to as an isotropic hardening model and is usually represented by the growth of a (yield)
surface in multi-dimensional stress space such as Figure 5.3b instead of the uniaxial stress
increase of Figure 5.3a.
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During cyclic loading, cold-working anisotropic effects are introduced through dislocation
multiplication and pile-up. The movement of dislocations in one direction generates local back
stresses which tend to assist movement in the opposite direction upon load reversal. This
produces the well-known Bauschinger effect, decreasing the yield stress in compression, for
example, if the material was first loaded in tension. Since loading direction determines the
evolution of the stress state, the isotropic assumption is no longer valid. To account for this
anisotropy, the yield surface (i.e., consider the surface shown in Figure 5.3b) is allowed to move
as well as grow, thus introducing a combined kinematic-isotropic hardening rule. The kinematic
hardening rule is often described in terms of a back stress (typically designated as α), named from
the effect of dislocation pile-up behavior described previously. For example, the ArmstrongFrederick (1966) model is often used to decribe the incremental back stress vector, dαij –
dα ij =
C
σ0
(σ
ij
− α ij ) d ε p − γα ij ε p
(5.1.12)
Where C and γ are material parameters, dεp and εp are the incremental and current equivalent
plastic strain quantities, respectively, defined by equation 5.1.2. The growth of the yield surface,
or the isotropic behavior, is described by σ0 –
(
σ 0 = σ Y + Q∞ 1 − e− bε
p
)
(5.1.13)
Where Q∞ and b characterize the maximum size of the yield surface and the saturation rate,
respectively, and σY is the value of the yield stress as before. With the back stress tensor fully
defined, the yield surface, f, can be written as –
f =
3
( Sij − α ij )( Sij − α ij ) − σ Y 2 = 0
2
(5.1.14)
Where Sij is the deviatoric stress tensor. Figure 5.4 illustrates this combined hardening model in
principal stress space (ABAQUS, 2004). As illustrated in Figure 5.4, the ratio of C/γ is
proportional to the radius of the yield surface and describes the saturation limit of the back stress
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component, α. Considering the saturation value of Equation 5.1.14 at large plastic strains, the
model is bounded by a limit surface of the size proportional to σY + Q∞ + C/γ. Armstrong and
Frederick (1966), and later, Lemaitré and Chaboche (1990), along with the ABAQUS Theory
Manual, version 5.5 (ABAQUS, 2004) can provide further details.
From Equations 5.1.13 and 5.1.14, the isotropic-kinematic hardening law necessitates the
calibration of five material parameters, σY, Q∞, b, C and γ, to fully describe the model. The yield
stress, σY, is relatively easy to obtain from a standard tension coupon experiment. The isotropic
hardening parameters, Q∞ and b, are calculated from a uniaxial cyclic test with multiple
increasing symmetric plastic excursions. From this data, a first estimate can be made on the
maximum size and the rate of growth of the elastic envelope. Once the yield stress and isotropic
hardening parameters are specified, the kinematic hardening components can be calibrated to
provide the best fit across cyclic and monotonic tests. Inevitably, this process (outlined in Figure
5.5 and discussed more in subsequent sections) requires some iteration to properly tune the model
parameters.
5.1.2
CVGM CALIBRATION
This section describes the calibration procedure for the two CVGM parameters, η and λ. Unlike
the material constitutive model which is calibrated with smooth uniaxial coupon tests, the CVGM
parameters are calibrated with circumferentially notched-bar specimens. The notch, such as
shown in Figure 5.6a, produces a high-triaxiality with a very shallow gradient over the central
region of the notched cross section. Since the VGM reflects the processes of void growth and
coalescence that typically occur under high triaxiality values (T > 0.4; Bao and Wierzbicki,
2005), the notched bar geometries are fabricated to generate relatively high triaxiality values.
Referring to Figure 5.6a, the magnitude of the triaxiality is controlled by the ratio of the neck
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diameter, D0, to the notch radius, r*. A smaller r*/D0 suggests a larger constraint, and therefore a
higher triaxiality in the notch region, as compared to a larger r*/D0 which provides less
constraint.
5.1.2.1 CALIBRATION OF η
To calibrate η, the notched bar tests are complemented by finite element simulations, such as
described in detail by Kanvinde (2004) and shown in Figure 5.7. These feature an axisymmetric
finite element model, incorporating the material constitutive relationships described in the
previous section. Figure 5.6b shows good agreement between the simulated and the experimental
load-deformation response, including the point of experimental fracture. The stress-strain
histories leading up to this point are integrated according to Equation 5.1.8, described earlier, to
generate an estimate of η for each experiment. The notch geometry ensures that the highest
triaxiality is observed at the center of the notched cross section, requiring this integration only at
one node, i.e. the central node of the notched cross section (Refer Figure 5.7). Repeating this
procedure for each experiment provides statistical information regarding the parameter η.
The parameter, η may also be indirectly inferred if the upper shelf Charpy V Notch Energy is
known or specified. For this purpose, an empirical relationship (Kanvinde and Deierlein, 2006)
between the Charpy V-notch (CVN) upper-shelf energy, ECVN (ft-lbs) may be used.
η = 0.018 ECVN − 1.30
(5.1.15)
5.1.2.2 CALIBRATION OF λ
Once η is calibrated, cyclic notched-bar experiments and complementary finite element
simulations are used to calibrate λ. Referring to Equation 5.1.10 and Figure 5.8, λ is obtained
from an exponential regression analysis through the average monotonic fracture parameter (0, η)
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and multiple cyclic data points (εp*, ηcyclic) such that the x-coordinate indicates the accumulated
damage at the beginning of the failure cycle, whereas the y-axis coordinate plots the ηcyclic/η at
the instant of fracture observed in the experiment. Referring to a previous discussion, the damage
is assumed to be the equivalent plastic strain at the beginning of the tensile excursion of
experimental fracture. Also indicated on Figure 5.8, an exponential function of the form
f(εp*)=exp(-λ.εp*) is fit to this data through a regression fit, resulting in a calibrated value of the
CVGM parameter, λ . The regression fit is constrained to pass through the point (0, η ) to
appropriately reflect monotonic fracture.
Multiple cyclic tests are used to query a range of damage values prior to fracture. The damage
prior to fracture can be controlled by varying the loading history and/or the specimen triaxiality.
In general, specimens with lower triaxiality will accumulate greater damage prior to fracture as
compared to specimens with high triaxiality. The calibration range can be determined from the
expected magnitude of damage where fracture is predicted. This may necessitate an approximate,
a priori simulation of the component of interest. The notched-bar tests used to predict large-scale
brace fracture in this study highlight the possible shortcomings of using cyclic material
calibration experiments which do not sample appropriate damage levels. This will be addressed in
a subsequent section. In addition, the cyclic loading histories for the notched-bar calibration tests
are designed according to the applicable range of ULCF where a relatively small number (< 5-10
cycles) of very large inelastic cycles are applied prior to fracture.
5.1.3
CVGM APPLICATION AND VALIDATION IN SMALL-SCALE DETAILS
Once the model parameters (η and λ) are calibrated, along with the material constitutive model,
the CVGM can be used to predict fracture. Figure 5.9 illustrates the fracture prediction for a
cyclically loaded notched bar. A detailed finite element analysis is used to determine the stress
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and strain histories at the center of the notched-bar – the location of experimental fracture
initiation. The increasing solid line is calculated according to Equation 5.1.9 and represents the
demand, or ηcyclic, at the critical material point, while the dashed line is the fracture capacity
described by Equation 5.1.11. Note that the capacity, following Equation 5.1.11 and prior
discussions, degrades in a stepwise fashion. Fracture is predicted to occur when the demand
exceeds the capacity, as indicated on Figure 5.9. The instant in the simulation corresponding to
this event can be compared to the experimental fracture instance to evaluate accuracy of the
CVGM. Note that the prediction shown in Figure 5.9 is based on mean η and λ values.
Probabilistic aspects of these predictions will be discussed in a subsequent section.
Kanvinde and Deierlein (2007), demonstrated the effectiveness of the CVGM in predicting
fracture in various material types and notch geometries, exposed to different loading histories.
Referring to Figure 5.10 (adapted from Kanvinde and Deierlein, 2007), the accuracy of the
CVGM is demonstrated by comparing the equivalent plastic strain at the predicted fracture
instance, εpanalytical, to the equivalent plastic strain at experimental fracture initiation, εpexperimental.
The monotonically increasing equivalent plastic strain is used in the comparison for the
convenience it allows in identifying a specific instance within a cyclic loading history. Thus, the
solid line corresponds to a perfect prediction while the dashed lines indicate a 25% error margin
between the prediction and the experiment. In general, the study concluded that the CVGM
performs well for the material types, loading histories and scale of the 46 notched-bar
experiments.
5.2
APPLICATION OF CVGM TO LARGE-SCALE BRACING COMPONENTS
Considering the accuracy of the CVGM predictions applied to small-scale notched bar
experiments, this section evaluates the methodology for large-scale details. The focus of the
discussion will be validating the CVGM using the representative SCBF brace component tests of
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Chapters 3 and 4. Similar to the previous discussion, the methodology will consist of material
model and CVGM calibration, as well as large-scale finite element brace simulations. Analogous
to the notched-bar analyses, the accuracy of the brace fracture predictions depend on accurately
simulating the stress and strain histories at the critical point of fracture. In the context of the
buckling-induced fracture processes in brace component behavior, the accurate simulation of
complex phenomena such as local buckling is shown to control the fracture prediction accuracy.
5.2.1
MATERIAL MODEL CALIBRATION
To reintroduce the large-scale testing matrix (review Chapter 3 and 4 for more detail), Figure
5.11 illustrates the three cross-section types used in the experimental program where two HSS
(HSS4x4x1/4 and HSS4x4x3/8), two Pipe (Pipe3STD and Pipe5STD) and one Wide-Flanged
shape (W12x16) comprise the nineteen experiments. Also shown in the figure are locations along
the cross-section where material coupons were extracted for constitutive model calibration as well
as CVGM parameter calibration. The dimensions for the circumferentially notched and smooth
tensile coupons are shown along with the number of specimens extracted from each location. The
large-notched (LN) bar is subjected to a cyclic loading history (illustrated in Figure 5.12e) while
the other specimens are monotonic tension experiments. Also shown in Figure 5.11, are typical
“flat” tension coupons from both HSS cross-sections and fabricated according to ASTM Standard
E8 (2005): Standard Test Method for Tension Testing of Metallic Materials.
Using the experimental data from these specimens, Figure 5.5 illustrates the calibration procedure
for the combined isotropic-kinematic hardening model described previously. First, an average
yield stress (σY) is calculated from the smooth and ASTM tension coupons using a 2%-strain
offset rule. These measurements are listed in Table 5.1 along with other material properties. Next,
the circumferentially large-notched cyclic experiments are used to approximate the isotropic
hardening law (i.e., Q∞ and b). The specimen is modeled in ABAQUS using the axisymmetric
151
element formulation and loaded with the cyclic history shown in Figure 5.12e (cycles of +0.15/0.05 inches across a 1 inch gage length). From experience, the initial kinematic hardening
coefficients are taken as C = 300 and γ = 50 with Q∞, and b set to 15 and 5, respectively. The
isotropic parameters are adjusted using 2-4 analyses to obtain a first approximation. An iterative
approach is used as it is difficult to rigorously characterize the growth of the yield surface with
the relatively large, and few numbers of, plastic excursions during the cyclic loading of the
notched-bar test. Ideally, a uniaxial specimen with a symmetric loading history comprised of a
large cycle count would be used to measure the growth of the yield surface. However, as
mentioned previously, a large cycle count combined with a smooth geometry does not facilitate
the CVGM calibration. Therefore, to reduce fabrication and testing costs, the approximate cyclic
material behavior is obtained from the notched-bar experiments with the final check being the
large-scale brace force-deformation behavior (see Figure 5.5).
After the isotropic hardening parameters, Q∞ and b, are approximated from the large-notch
experiments, the smooth and small-notched monotonic tests, along with the first pull of the largenotched cyclic test, are used to calibrate the kinematic hardening rule. The isotropic parameters
can be slightly adjusted during this stage, keeping in mind the approximate values obtained from
the large-notch experiments. The new kinematic and isotropic parameters are then applied to the
large-notch analysis to ensure the cyclic behavior is acceptable. Figure 5.12 compares the
simulation results after this calibration procedure with the experimental data from the corner
material specimens of the HSS4x4x1/4 cross-section. A single set of parameters (σY = 73, Q∞ =
850, b = 160, C = 14.5 and γ = 5.27) for the combined hardening law is able to describe the
behavior of the various specimens fairly well. This list of parameters is recorded in Table 5.2
along with the other material types. The same procedure is used to calibrate the material for the
walls of the HSS brace. After both material types are defined, a large-scale brace continuum
152
model (discussed more in a later section) is analyzed with the small-scale parameters to verify the
constitutive response at the large-scale. Figure 5.13 compares the simulated and experimental
force-deformation response for an HSS4x4x1/4 bracing member subjected to a far-field loading
history. Also of importance is the instance of local buckling in the simulation as compared to the
experiment. Buckling is addressed more in a later section.
While the good agreement between the experiments and simulations presented in Figures 5.12
and 5.13 is representative of all material types and bracing members, only the HSS4x4x1/4 and
Pipe5STD brace/material types are calibrated with a single set of hardening parameters (i.e., σY,
Q∞, b, C and γ). The HSS4x4x3/8 and Pipe3STD brace/material types required different
parameter sets to describe the behavior of the small and large-scale specimens. Ideally, a unique
constitutive model parameter set could predict material behavior across the variety of scales,
geometries and loading histories. However, this is somewhat of an unrealistic aim as the
constitutive model should be considered imperfect, and whose parameters may not be applicable
across a wide range of inelastic strain levels (ABAQUS, 2004). In fact, in the case of the
HSS4x4x3/8 and Pipe3STD braces which, in general, survived several more inelastic cycles as
compared to the less compact HSS4x4x1/4 and Pipe5STD (see Chapter 4) the repeated loadings
could be highlighting an inherent cumulative error effect of the combined hardening model.
Considering that the objective of the current study is to apply and evaluate the CVGM at the
large-scale, the finite element simulations, and therefore the constitutive model, are regarded as a
tool which can reproduce the stress and strain histories of the experimental specimens. In the case
where the calibration procedure outlined in Figure 5.5 is not applicable across all experiments, the
parameters are adjusted accordingly so as to capture the observed experimental behavior. For
example, for the small-scale specimens, the load-deformation behavior was used to determine
accurate material modeling, while for the large-scale brace experiments, it was considered
paramount to predict the instance of local buckling. While local buckling is dependent on brace
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geometry through brace slenderness and cross-section compactness, material hardening also
influences behavior. The modeling of the large-scale brace simulations and buckling phenomena
is discussed in the following section.
5.2.2
LARGE-SCALE BRACE SIMULATIONS
Figure 5.14 illustrates a representative finite element model for a brace specimen constructed
using the ABAQUS simulation software (ABAQUS, 2004). The simulations featured threedimensional, twenty-node brick elements with reduced integration (type C2D20R in the
ABAQUS element library), which were determined to simulate through-thickness bending of the
specimen walls with a high degree of accuracy. The simulations were based on the measured,
rather than the nominal dimensions of the braces (see Figure 5.15). To reflect experimental
conditions accurately, a 1.5 inch clear distance between the end of the brace and the gusset plate
was incorporated in all brace models (Figure 5.14). Symmetry in the boundary conditions,
specimen and the deformation mode are leveraged to construct the quarter model as shown in the
figure.
Typical meshes for various cross sections are shown in Figure 5.14. Referring to the figure, the
mesh is finer at the center of the brace where severe plastic deformations are expected. The mesh
gradually transitions to a coarse mesh at the third point of the brace, such that the element lengths
increase in a geometric progression. A total of 25 elements are used along the middle-third of the
half-brace, with the elements in the hinge region one-fourth the size of the elements at the ends of
this region. A detailed mesh-refinement study verified the accuracy of this mesh.
5.2.2.1 MODELING GLOBAL AND LOCAL BUCKLING
Since the strains responsible for fracture are controlled by global and local buckling of the brace,
it is important to accurately simulate these events in the brace simulations. To achieve this, global
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and local imperfections are introduced to the brace model by scaling eigenvectors from elastic
buckling analyses. The following subsections discuss these aspects of modeling, and examine the
accuracy of the adopted approach.
5.2.2.1.1
IMPERFECTIONS AND BUCKLING SIMULATIONS
Elastic buckling analyses are performed with ABAQUS to determine the buckling mode shapes
of each brace specimen. The “global” buckling analysis involves the application of a compressive
axial load at the gusset plate while a separate “local” buckling involves the application of a
compressive load at the partition between the fine and coarse mesh region, 20 inches from the
brace mid-point. Figure 5.16 shows the buckling mode shapes obtained from the elastic buckling
analyses for the HSS4x4x1/4. These buckling modes are scaled appropriately and incorporated as
deformation imperfections into the inelastic cyclic loading simulations discussed subsequently.
The scaling procedure is discussed next.
The global imperfections were introduced by scaling the global buckling mode shape, such that
the peak deflection at the center was L/1000, which is equal to the maximum sweep of
compression members assumed by AISC (2005). Thus, the elastic global buckling eigenvector is
scaled such that the maximum out-of-plane imperfection is 0.12 inches. Following the
measurement procedures by Schafer and Peköz (1998), the local imperfection magnitudes are
determined from digital end-mill measurements along the walls of four-foot sections of each
brace cross-section type (with the exception of the Wide-Flange section). These measurements
are illustrated in Figure 5.17 and listed in Table 5.3 for the HSS and Pipe cross-sections. The
maximum values from each brace cross-section listed in Table 5.3 were chosen as the local
imperfection magnitudes in the brace analyses. Interestingly, these magnitudes do not influence
the final solution. In fact, virtually identical stress and strain histories are obtained without
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incorporating local imperfections. Considering the relatively small imperfections, the solution
tends to be dominated more by the plastic strains accumulated during repeated cyclic loading.
5.2.2.1.2
COMPARISON TO EXPERIMENTAL GLOBAL BUCKLING
The first inelastic event during the far-field cyclic loading experiments is brace buckling, while
the near-field loading histories yield the brace in tension prior to buckling (refer to Chapters 3 and
4 for loading history descriptions). Buckling, often described by the maximum, or critical,
compressive load is dependent on member geometry, material properties, and imperfection
magnitudes. Figure 5.18 compares the experimental buckling loads (presented earlier in Chapter
4) to the simulated buckling loads from the finite element models. On average, the ratios of the
predicted to experimental buckling load are 1.06 and 1.35 for the far-field and near-field
simulations, respectively. The HSS4x4x1/4 simulations are used as an example to compare the
larger predicted buckling loads for the near-field loading history (Figure 5.18b) to the far-field
loading (Figure 5.18a). These observations from the near-field buckling analyses may be
explained by the lack of axial residual stresses in the model as well as the tension loading prior to
buckling (see Chapter 4) which has the effect of removing the initial global imperfections.
However, in the context of predicting fracture, it is not critical to accurately predict the buckling
load as the plastic strain accumulation from local buckling overwhelms global buckling strains.
Local buckling simulation is discussed in the next section.
5.2.2.1.3
COMPARISON TO EXPERIMENTAL LOCAL BUCKLING
Referring to the qualitative description of brace failure from Chapter 4, the amplified inelastic
strain demand from local buckling is primarily responsible for fracture initiation in the middle
plastic hinge of the brace. For accurate application of the CVGM, the instant of local buckling
initiation must be correctly modeled to accurately reproduce the continuum stress and strain
histories at the critical fracture point. Figure 5.19 illustrates the progression of global and local
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buckling for an HSS4x4x1/4 bracing member at the compressive peaks of a far-field loading
history. The figure compares the recorded experimental events to the continuum-based simulation
events. The solid dot on the second compressive excursion to 1.1 inches indicates the first visible
local buckling event recorded during the experiment. This point is accurately predicted by the
simulation as well. This is representative of all brace analyses presented in this chapter.
5.2.3
BRACE MATERIAL CVGM CALIBRATION
This section presents the results of the small-scale material calibration study for the various brace
material types. While a previous section discussed the calibration of the material constitutive law,
here the focus is on calibrating the CVGM parameters, η and λ.
5.2.3.1 BRACE MATERIAL CALIBRATION RESULTS: η
Monotonic tension notched bar specimens, extracted from various locations within each brace
cross-section, were used to calibrate the fracture parameter η. Figure 5.11 shows the locations
where material coupons were sampled from each brace type. Referring to the HSS experimental
results in Chapter 4, fracture initiates at the corner of the tubes rather than on the face. Therefore,
more calibration tests are performed on the HSS corner material as compared to the wall material.
Referring to the Pipe cross-section material, the welded seam material is different as compared to
the base metal. For the Wide-Flange, only the flange material is investigated as local buckling
initiates in and remains restricted to the flanges.
Table 5.4 presents the calibration results for each material type investigated as part of the smallscale study. The materials are listed by cross-section type and the location within the crosssection. For each monotonic notched-bar experiment, the fracture deformation, Δf, is presented
along with the corresponding fracture parameter, η (the λ calibration results are presented in the
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next section). Axisymmetric finite element simulations are used to determine the stress and strain
histories at the fracture location of the notched specimen and Equation 5.1.7 is used to estimate η.
In this way, each experiment/simulation combination provides a single estimate of η, while
multiple experiments provide statistical data on η. Table 5.4 provides average values, along with
the Coefficient of Variation (COV), for each material type.
Referring to Table 5.4, the HSS4x4x1/4 corner material is the toughest material tested with an
average η of 5.98. It is interesting to note the low toughness of the same (nominal) material type
(A500 Grade B specified by AISC, 2005) at the corner of the HSS4x4x3/8 where the average η is
found to be 3.62. This may be due to different plate materials and thicknesses used to form the
two cross-sections. A comparison between the mean toughness values of the HSS wall and corner
materials illustrates the effect of cold working on the steel ductility. The HSS4x4x1/4 corner
material was found to be 12.4% less tough as compared to the wall material, while a 3.7%
decrease was observed in the HSS4x4x3/8. However, it is relevant to note that despite the
apparent trends observed in the mean toughness values, the variations in the wall toughness
measurement are large (COV = 22 and 35% for the HSS4x4x1/4 and HSS4x4x3/8, respectively).
Referring to the toughness values of the two Pipe materials, the 3 inch Pipe toughness of 4.40 is
considerably larger than the less ductile 5 inch Pipe with an average η equal to 2.46.
5.2.3.2 BRACE MATERIAL CALIBRATION RESULTS: λ
Referring to the discussion in the first section of this chapter λ is calibrated through an
exponential regression fit. The regression analysis uses experimental data and finite element
simulations of cyclic notched-bar tests. The notched-bar tests measure the toughness of the
material as a function of inelastic damage during cyclic loading. Since the small-scale calibration
informs the large-scale fracture prediction, the calibration tests should ideally sample material
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behavior over the range of damage expected in the large-scale specimen. This recommendation is
offered after studying the CVGM fracture predictions of the large-scale braces. While the
predictions are discussed in the next section, the different damage levels between the calibration
and brace experiments lead to inconsistent fracture predictions in some cases. As illustrated in
Figure 5.20, the lower triaxiality in the braces (0.4-0.5) also results in the higher damage levels as
compared to those observed in the notched bars, where the triaxiality is significantly higher (in
the range of 1.2-1.3). Furthermore, only three cyclic tests were conducted for each material type
(four for the HSS4x4x1/4 corner material) and the loading history was not varied between
specimens. Thus, the cyclic data is somewhat limited from the small-scale study. The λ values
calibrated for each material type are presented in Table 5.4. At the conclusion of this section a
more detailed discussion is provided on the effect of insufficient small-scale cyclic data on
fracture prediction in the large-scale components.
Figure 5.21 illustrates the degradation of fracture capacity in the small-scale cyclic calibration
tests. From section 5.1, the x-coordinate indicates the accumulated damage at the beginning of the
failure cycle and the y-axis coordinate plots the ηcyclic/η at the instant of fracture observed in the
experiment. Following the arguments of Kanvinde and Deierlein (2007), the damage is
considered equal to the equivalent plastic strain (εp*) at the beginning of the final tensile pull to
fracture. Also shown in Figure 5.21 is an exponential function which is fit to this scatter data
through a regression fit, resulting in a calibrated value of the second CVGM parameter λ. The
regression fit is constrained to pass through the point (0, η) to appropriately reflect monotonic
fracture.
The HSS wall and Pipe seam material are not included in Figure 5.21 since fracture did not
initiate in these materials during the large-scale tests. The Wide-Flange braces will not be
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included in the large-scale fracture predictions, so this material is also excluded. Following the
procedure outlined in the preceding paragraph, λ-values are generated for each material type, and
are summarized in Figure 5.21, and listed in Table 5.4. The subscript NB in the figure and the
Table indicates that the trend fit is done using only the notched-bar (NB) experiments (in contrast
to a more expanded data set, discussed next).
It is also useful to compare the damage incurred to the large-scale braces with the damage to the
notched-bar experiments. Introduced previously, Figure 5.21 also includes scatter points for the
braces in a manner similar to the scatter points for the notched bars described previously. It is
interesting to note that in general, these points are below the trend fit based only on the notched
bar scatter points. The dashed line on the figure shows a similar trend fit; however, this fit is
based on the entire data set (including the brace scatter points). Correspondingly, the subscript C
on the calibrated λ-values indicates that it is based on a “combined” data set. The λC values, for
each of the HSS and Pipe steels are higher than the corresponding λNB values. This indicates that
calibrating the λ-value based only on a data set that queries a limited range of damage may be
inaccurate. Moreover, for the specific situation discussed here, the larger damage observed in the
brace specimens implies that using only the notched bar specimens, will result in a lower bound
estimate of the λ-value. Since lower λ-values shifts the trend fit upwards, brace fracture
predictions based on the λNB values, in general, correspond to a later instance in the loading
history as compared to the experimental observations.
While the next section describes the prediction results in more detail, here the likelihood of
obtaining the results from the notched-bar experiments is examined assuming that the true
behavior of the material is the exponential function described by the λC-value. In other words, if
the small-scale experiments were repeated, what is the probability that the characteristics of the
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notched-bar data points (i.e., mean and standard deviation), relative to the exponential function
described be the λC-value, will be repeated? This is interesting to consider because, for each
material type, it quantifies the likelihood that the notched-bar experiments follow the material
behavior described by the λC exponential function. Therefore, the combined damage function
represents the population while the notched-bar experiments represent the sample. To investigate
this, an error quantity, ε, is first introduced to describe the vertical difference from any
experimental point to the damage law. From the notched-bar and brace experiments, a normal
probability distribution is generated where the mean at ε = 0 is, by definition, located on the
exponential damage law defined by λC. This is illustrated in Figure 5.22. Second, a null
hypothesis is proposed which states the mean epsilon of the population is equal to zero (H0: ε =
0) while an alternative hypothesis states that the mean is not equal to zero (HA: ε ≠ 0). Finally,
utilizing the Student’s t-distribution and assuming the population error (ε) is normally distributed,
a P-value can be computed to assess the probability that the null hypothesis is true. A small Pvalue suggests the null hypothesis may not be true and leads to the conclusion that the notchedbar results are unlikely assuming the behavior of the population is governed by the λC exponential
function from the notched-bar and brace experiments. For the purposes of this discussion, a small
P-value is defined as less than 5% (Montgomery et al, 2001).
Following the hypothesis stated above, the P-values for the HSS4x4x1/4, HSS4x4x3/8,
Pipe3STD and Pipe5STD materials are calculated as 30.0, 3.4, 0.1 and 87.6%, respectively.
Considering the large P-values for the HSS4x4x1/4 and Pipe5STD materials, the null hypothesis
can not be rejected. From this, it can not be stated that with 95% certainty the mean error of the
notched-bar data is inconsistent with the exponential damage function defined by λC. While the
HSS4x4x3/8 fails the hypothesis test, the P-value is relatively close to the predefined degree of
confidence (0.95). On the other hand, the null hypothesis is certainly rejected for the Pipe3STD
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material due to the low sample standard deviation in ε. A small standard deviation increases the
value of the test statistic since it is unlikely (with respect to the null hypothesis) that repeated
experimentation will consistently produce data points which are equal distance from the
exponential function. From this hypothesis testing we can conclude that for two of the brace
materials, the null hypothesis can not be rejected at a high degree of confidence, while the null
hypothesis for the Pipe3STD and HSS4x4x3/8 notched-bar samples is rejected with 95%
confidence. In summary, two of the four brace material types fail the t-test, concluding that for
these materials we can say with 95% confidence that the notched-bar data is inconsistent with the
true behavior of the material.
While hypothesis testing describes the likelihood that the notched-bar observations are not
consistent with the assumed true behavior, it does not reconcile the difference between the λNB
and λC calibration values for the two exponential functions. However, referring to Figure 5.21, it
is interesting to note that for the material types which failed the hypothesis testing, the brace
points are consistently below the dashed line representing the combined fit while the notched-bars
are above. Discussed in the next section, these notched-bar tests also provide the worst fracture
predictions. This may highlight a fault in CVGM such that different mechanisms are active at
lower triaxialities.
Furthermore, calibrating λ at small or large damage levels can influence the variation in λ. As
discussed previously and explained through the low triaxiality (relative to notched-bars) of the
brace experiments shown in Figure 5.20, the damage ranges for the two data sets are quite
different, such that the notched-bars incur less damage prior to fracture initiation. Assuming the
constant ε distribution (Myers, 2009) shown in Figure 5.22, the variation in the different λ values
can be explained by the behavior illustrated in Figure 5.23. Referring to the figure, the bounds
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through plus and minus one standard deviation of the error distribution are tighter if the
calibration includes large damage values as compared to calibrating the function with samples
which have smaller damage values. As expected, this behavior is verified through Monte-Carlo
simulations. Thus, in lieu of matching the expected level of damage at the fracture location with
the calibration tests, the small-scale experiments could be designed to produce large damage
values to reduce the variation in λ.
5.2.4
CVGM FRACTURE PREDICTIONS APPLIED TO LARGE-SCALE BRACE COMPONENTS
This section presents the fracture predictions for the large-scale bracing members where the stress
and strain histories from the brace simulations and the fracture parameters (η and λ) are used to
determine the instant when the CVGM fracture criterion (Equation 5.1.10) is satisfied. The
predictions are compared to the experimental instances of fracture presented in Appendix A and
discussed in Chapter 4. The accuracy and limitations of the CVGM are investigated through the
wide-range of brace cross-section and material types as well as the various loading histories and
rates of the large-scale testing matrix.
Deterministic fracture predictions based on the mean η and λ values are first presented for the
far-field, general loading history experiments of the HSS4x4x1/4, HSS4x4x3/8, Pipe3STD and
Pipe5STD. Next, the CVGM is applied to the bracing members subjected to asymmetric, nearfield loading histories, to investigate the influence of loading history on the accuracy of the
predictions. Then, the fracture predictions are presented for the two HSS bracing members
subjected to earthquake loading rates. Finally, an unsymmetrical deformation shape, produced by
reinforcing plates at the middle of an HSS4x4x1/4 brace (HSS1-6), further examines the
performance of the CVGM with different stress and strain histories.
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Following the deterministic results, a probabilistic approach is developed to account for the
uncertainty introduced into the predictions by the regression analysis used to fit the exponential
function. The error quantity (i.e., ε) distributions, discussed previously, are used to generate
probabilities of failure at each instance during the loading history.
5.2.4.1 APPLICATION TO HSS AND PIPE CROSS-SECTIONS (STANDARD LOADING)
Figure 5.24 illustrates the CVGM prediction for an HSS4x4x1/4 bracing member (HSS1-1)
subjected to the standard loading history. Using the finite element analysis results and the
calibrated CVGM material parameters listed in Table 5.4, the critical node is determined to be at
the corner of the square cross-section. This point is identified on the finite element mesh in Figure
5.24 and coincides with the experimental location of fracture initiation. Also shown in the figure
is the increasing demand, or void growth (ηcyclic), and the decreasing capacity, or critical void
size, at the critical point of the brace cross-section. At the intersection of ηcyclic and the decreasing
capacity function, the fracture criterion (Equation 5.1.10) is satisfied and fracture initiation is
predicted to occur. The figure compares the expected fracture instances for two different capacity
functions according to the λNB and λC parameters calculated previously. Referring to the figure,
ηcyclic exceeds the fracture capacity calculated with the larger λ-value (λC) one cycle before the
prediction with the smaller λNB. This is expected considering that a larger λ-value decreases the
fracture capacity at a faster rate as compared to a smaller λ.
Figure 5.25a shows the CVGM fracture instances from Figure 5.24 as deterministic points on the
standard loading history of the HSS4x4x1/4 test where the predicted (i.e., using λNB) and
experimental instances of fracture are approximately the same. Also shown in Figure 5.25a is the
expected fracture instance according to the larger λC-value used to define the capacity of the
brace material. Referring to the figure, using the λC-value, calibrated with both the notched-bar
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and brace experiments, does not result in a good comparison to the experimental instance of
fracture. However, on average (across all HSS4x4x1/4 specimens), it is expected that the CVGM
damage function (Equation 5.1.11) will be more accurately characterized by the λC-value because
it is based on the experimental points from the large-scale brace tests. The probabilistic
methodology presented in the following section will provide a better framework to assess the
performance of the CVGM with the two different λ-values.
Figure 5.25 also illustrates the fracture predictions for the HSS4x4x3/8, Pipe3STD and Pipe5STD
cross-section types subjected to the standard loading history. Referring to Figure 5.25d, the
CVGM estimates the instance of fracture initiation accurately for the Pipe5STD bracing member,
while the predictions for the HSS4x4x3/8 and Pipe3STD (Figures 5.25b and 5.25c, respectively)
are several cycles after the experimental fracture instances. These predictions are the result of the
small λNB-values calibrated with the notched-bar tests. The smaller λ-values incorrectly estimate
the experimental damage of the Pipe3STD and HSS4x4x3/8 bracing members. However, the
larger λC-values, calibrated with both the notched-bar and brace experiments, provide better
estimates of the damage to the Pipe3STD and HSS4x4x3/8 brace materials.
These results indicate that either 1) the CVGM is not an appropriate criterion to predict fracture in
large-scale bracing members or 2) calibrating the λ-value based only on a data set that queries a
limited range of damage may be inaccurate. Referring to the first point, the model may be
inappropriate to model fracture under the brace loading conditions due to several factors
including, but not limited to, –
•
The relatively low triaxiality range at the critical location of fracture (0.4-0.5) in the brace
experiments. Considering the small-scale validation study by Kanvinde and Deierlein
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(2007) included specimens with relatively large constraint (producing triaxialities in the
range of 1.0-2.0), the CVGM has not been tested in situations with low-triaxiality. This
could trigger somewhat different micromechanical failure mechanisms other than those
assumed by the CVGM.
•
The exponential damage function employed by the CVGM may be fundamentally flawed,
suggesting there is another function which better characterizes damage. While the
exponential degradation was shown to work well at the small-scale, the stress and strain
histories of the brace experiments are unlike the histories from the notched-bar tests. This
may alter the rate of damage accumulation in the braces relative to the previously
observed behavior in the small scale validation tests by Kanvinde and Deierlein (2007).
While investigating alternative functions is outside the scope of the current study, it is a
recommended topic for future work.
However, the expected fracture instances using the λC-values illustrate that even if the functional
form of the damage law is inaccurate, and/or the low triaxiality alters the assumed void growth
mechanisms, the CVGM is still able to describe the behavior of the brace specimens.
Referring to the second point, the inaccurate λ-values could be a product of inappropriate smallscale calibration tests such that the damage levels are inconsistent with the brace experiments.
Furthermore, referring to Figure 5.23, calibrating λ with the notched-bar data set may produce
larger variations in λ as compared to a data set with larger damage levels. Thus, if small-scale
experiments were designed to sample larger damage levels, the calibration may be more
consistent with the brace points. Nonetheless, the predictions from the HSS4x4x3/8 and
Pipe3STD brace types expose some faults of the CVGM if λ is not calibrated properly.
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5.2.4.2 APPLICATION TO NEAR-FIELD LOADING HISTORIES
To examine the capabilities and limitations of the CVGM further, two near-field loading histories
(see Chapter 3) are used to generate stress and strain histories in the bracing members which are
unlike those observed in the specimens tested with the standard loading history. Considering the
CVGM operates at the continuum level, it is interesting to investigate the accuracy of the model
in the presence of different loading cases. For example, while the predictions from standard
loading may be accurate, a more compressive (or tension) dominated stress and strain history may
influence the accuracy of the model. Investigating various loading histories is also of practical
importance considering the unsymmetrical behavior of braced-frames during earthquake loading
(see Chapter 2).
In this section, the near-field loading histories presented in Chapters 3 and 4 are used to examine
the accuracy of the CVGM during asymmetric loading histories. Referring to the discussion in
Chapter 3, the asymmetric compression history is marked by a large pulse in compression
followed by deformation cycles about a residual compressive offset. This is applied to an
HSS4x4x1/4 bracing member. An asymmetric tension history is applied to both the Pipe3STD
and Pipe5STD bracing members and contains a large initial tensile pulse followed by deformation
cycles applied at a residual tensile offset.
Referring to Figure 5.26a, using the λNB-value to describe the decreasing fracture capacity for the
HSS4x4x1/4 bracing member subjected to the compressive near-field history results in a
deterministic fracture prediction which is approximately four cycles after the experimentally
observed fracture instance. However, the λC-value shifts the expected fracture instance closer to
the experimental. Figure 5.26b illustrates the prediction (i.e., using the λNB-value) for the
Pipe3STD bracing member subjected to the tension near-field history. The figure shows the same
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effect which was observed during the standard loading history of the Pipe3STD brace where the
low λNB-value pushes the prediction several cycles past the instance of experimental fracture. On
the other hand, the fracture prediction instance for the Pipe5STD bracing member during the
asymmetric tension loading history is only one cycle after the experimental fracture instance.
Similar to the HSS4x4x1/4, the λC-value provides a more accurate fracture initiation instance for
both Pipe specimens.
5.2.4.3 QUASI-STATIC VERSUS EARTHQUAKE LOADING RATE
Chapters 3 and 4 describe two earthquake-rate loading experiments, where the standard loading
history is applied to each of the HSS brace cross-sections (HSS4x4x1/4 and HSS4x4x3/8) at 360
times the loading rate of the quasi-static experiments. The justification for this rate is discussed
more in the previous chapters. Referring to the experimental observations discussed in Chapter 4,
the earthquake loading rates did not significantly alter the behavior of the bracing members as
compared to the quasi-static experiments. Thus, the accuracy of the CVGM fracture predictions
for these fast-rate loading experiments is not expected to be notably changed from the quasi-static
tests.
Inconsistent displacement limits (from actuator over-shooting described in Chapter 4) between
the quasi-static and fast-rate experiments make it somewhat difficult to judge if the increased
strain rate or temperature in the region of fracture had a substantial effect on the performance of
the brace. However, after both experimental histories are input into the finite element model of
the brace and fracture initiation is predicted according to the CVGM, the resulting deviations
between the prediction and the experiment are essentially the same for both tests. Therefore,
within the precision of the models, rate effects do not seem to affect fracture significantly. This is
illustrated in Figure 5.27 where (a) and (b) are replicates of the quasi-static results from Figure
5.25 and Figures 5.27c-d correspond to the earthquake-rate tests.
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5.2.4.4 APPLICATION TO UNSYMMETRICAL BUCKLING
To further examine the capabilities of the CVGM, two eighteen inch reinforcing plates were
welded at the center to the top and bottom (non-buckling faces) of an HSS4x4x1/4 bracing
member to deliberately induce unsymmetric buckling. In all other respects, the specimen was
similar to the HSS4x4x1/4 experiment during far-field loading discussed previously. The welded
attachment is not believed to influence fracture substantially (other than by causing the nonsymmetric buckling pattern) since fracture initiation occurred approximately two inches from the
end of the reinforcing plate (see Figure 5.28a).
As expected, and shown in Figure 5.28c, the experimental ductility is less than that recorded for
the symmetric experiment shown in Figure 5.25a. This is due to the larger strains that develop
when the plastic hinge is not at the center of the brace and can be explained by considering the
kinematics of a buckling brace with varying plastic hinge locations. This behavior is modeled by
incorporating reinforcing plates into the brace finite-element model where a tie constraint is used
to replicate the welded connection between the plates and the brace. To simulate the
unsymmetrical buckling effect, the plates are offset by 0.1 inches from the brace mid-point.
Figure 5.28a illustrates the results of the continuum-based analysis which predicts local buckling
on one side of the reinforcing plates. The predicted fracture location is determined to be at the
same location as the experimental specimen, i.e. one inch from the edge of the reinforcing plate.
Referring to Figure 5.28c, the CVGM fracture prediction (using λNB) is shown to be one cycle
after the experimental instance of fracture. Again, the prediction using the λC-value shows an
improved accuracy over the damage function calibrated with just the notched-bar experiments.
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5.2.4.5 ESTIMATING UNCERTAINTY IN THE CVGM FRACTURE PREDICTIONS
While the deterministic fracture predictions in the previous section describe the mean behavior of
CVGM, they do not provide the likelihood of fracture at any point in the loading history. This
may be useful for determining conservative lower bounds on predictions, investigating the
influence of material uncertainty, or providing a tool to examine the influence of loading history
on the CVGM predictions in a probabilistic sense. While Myers (2009) provides a more detailed
explanation of the probabilistic approach presented here, a simplified version of his method is
applied to the large-scale brace simulations.
Referring to Figure 5.22, an error quantity (ε) with a normal distribution is assigned with a mean
of zero along the exponential trend-line. Furthermore, the distribution is assumed constant for
each damage point along the exponential function. Thus, the CVGM fracture criterion presented
in the first section of this chapter can be expressed as –
ηcyclic
≤ f ( D ) = exp ( −λε p* ) + ε
η
(5.2.1)
Where ε is a random variable with a normal probability distribution. Solving for ε at any time (ti)
in the loading history, the difference between the CVGM demand (ηcyclic/η) and the fracture
toughness can be calculated with –
εi =
ηcyclic
− exp ( −λε p* )
η
(5.2.2)
The expression for εi provides a bound on the integral of the probability density function of ε,
such that on one tensile excursion –
εi
∫ f (ε | B ) d ε
ε
c
Pr[T f < ti | B c ] =
0
1−
ε0
∫ f (ε | B ) d ε
c
−∞
(5.2.3)
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Where Tf is a random variable corresponding to the instance of fracture, ti is any instance during
the loading history, f(ε) is the probability density function and Bc is the complement of the event
that the brace fractured at the peak of the previous tensile excursion. Thus, Equation 5.2.3 is
conditioned on the event that the brace did not fail on the previous cycle. Note that the expression
is also normalized to account for ε0 > -∞ . The probability of fracture at any analysis step is the
joint probability, determined by “scaling” Equation 5.2.3 by the maximum likelihood that fracture
did not occur on the previous cycle –
Pr[T f < ti , B c ] = Pr[ B] + Pr[T f < ti | B c ]Pr[ B c ]
(5.2.4)
Where Pr[Bc] is 1-Pr[B]. Furthermore, during compressive loading (T < 0) the probability is
assumed to be equal to the maximum probability from the previous tensile excursion. This
prevents the cumulative probability function from decreasing. As an example, the probability of
failure during the far-field loading history of an HSS4x4x1/4 bracing member is shown in Figure
5.29b. The figure shows two cumulative distribution functions (CDFs) for each of the λ-values
calibrated previously as well as a vertical line at the observed experimental fracture location. As
expected from the deterministic fracture prediction in Figure 5.25a, the prediction using the λNBvalue from the notched-bar calibration tests intersects the experimental fracture location at
approximately 36%. Note that an exact deterministic prediction translates to a 50% likelihood of
fracture at the experimental fracture instance. If the λC-value is used to describe the damage
function, the expected (i.e., at the 50th percentile) instance of fracture initiation is on the previous
tensile cycle while the probability at the experimental fracture instance is approximately 85%.
Also interesting to note is the sharp increase of the CDFs across these two tensile excursions of
the far-field loading history. This can be mostly attributed to the influence of local buckling on
the stress and strain state at the critical cross-section and infers that a probabilistic analysis is not
necessary for the specific case of predicting large-scale brace fracture. Moreover, the sharp nature
of the CDF further underscores the importance of modeling local buckling phenomena accurately.
171
The probabilistic analysis is applied to each brace simulation from the previous section. Table 5.5
lists the probability of fracture initiation at each experimental fracture instance using both λvalues. Referring to the Table, an approximate 0% probability of fracture initiation is calculated
at the experimental fracture instance for the HSS4x4x3/8 and Pipe3STD bracing members if λNB
is used to describe the fracture capacity. This is consistent with the deterministic fracture
predictions of these two cross-sections and can be attributed to the λ calibration issues as well as
the sharp nature of the CDF discussed previously. The analysis of the HSS4x4x1/4 and Pipe5STD
bracing members calculates non-zero probabilities where the CVGM predictions are shown to be
the most accurate for the Pipe5STD brace type. For each cross-section type, using the λC-values
provides probabilities of fracture which are closer to 50%. Similar to the deterministic
predictions, the probabilistic analyses demonstrate that if calibration specimens are designed
according to an expected damage range then the CVGM can accurately simulate cyclic fracture
behavior.
5.3
SUMMARY
This chapter introduced a micromechanical-based fatigue/fracture initiation model developed and
validated at the small-scale by Kanvinde (2004). The monotonic Void Growth Model (VGM) and
the corresponding Cyclic-VGM (CVGM) operate at the continuum level and utilize stress and
strain histories from finite element simulations to predict fracture during monotonic and cyclic
loading. Therefore, the accuracy of the models relies on accurate material constitutive simulation
and, related to this, correct simulation of localized phenomena which lead to fracture. Calibration
procedures for two CVGM fracture parameters, η and λ, are outlined in the chapter. While the
monotonic fracture parameter, η, is calibrated using notched-bar tensile tests, the degradation
parameter, λ, requires more judicious cyclic calibration experiments. It is found to be important
for the damage range of the calibration experiments to approximately match the expected damage
172
of the component (large-scale or otherwise) where the fracture prediction is being made. In lieu of
this, Figure 5.23 illustrates that calibrating the exponential function with large damage levels will
reduce the variation in λ. The chapter highlights the shortcomings of the CVGM fracture
predictions applied to large-scale bracing members if this condition is not observed. However,
using the brace experiments and simulations as calibration points, it is shown that the form of the
damage model (exponential) accurately tracks the damage processes leading to eventual fracture
in the bracing elements. To illustrate this, a combined damage law is calibrated based on the
notched-bar experiments as well as the large-scale brace experiments. Statistical P-values are
used to characterize the likelihood of observing the notched-bar experimental results if the
combined damage law is the “true” behavior of the material. Finally, a probabilistic framework is
introduced utilizing the inherent error, or uncertainty, in fitting a trend-line to the cyclic damage
points. While the predictions are shown to be somewhat inaccurate for the HSS4x4x3/8 and
Pipe3STD specimens, the HSS4x4x1/4 and Pipe5STD fracture predictions are quite accurate
across a variety of loading histories. Moreover, the unsatisfactory performance of the CVGM
when applied to the HSS4x4x3/8 and Pipe3STD brace types is suspected to be a result of
insufficient small-scale calibration tests and not the formulation of the model itself.
In light of the evaluation of the CVGM performance discussed in this chapter, the methodology
can be applied to several practical situations. First, the large-scale brace simulations provide a
tool to inform inelastic buckling behavior during earthquake-type loading. For example, the
ability to accurately simulate the localized inelastic strains at the middle plastic hinge region
during local buckling provides valuable insights into the mechanisms which eventually produce
brace fracture. Furthermore, as design codes incorporate more performance-based techniques, the
sole reliance on large-scale testing results to evaluate ductility limits for steel components may
not provide a comprehensive description of all factors controlling fracture limit states. Through
the CVGM-based fracture methodology presented in this chapter, parametric simulation studies
173
can expand testing programs to provide improved insights into the behavior of large-scale brace
components. This is the topic of the next chapter.
174
Table 5.1: Measured material properties (refer to Figure 5.12 for locations of each specimen).
Material (Specimen) Type
No. Tests
σY, ksi
A500 Gr. B (ASTM)
HSS4x4x1/4 (Wall)
3
67 (1.8)
HSS4x4x3/8 (Wall)
3
72 (5.2)
A500 Gr. B (UN)
HSS4x4x1/4 (Corner)
2
74 (2.2)
HSS4x4x1/4 (Wall)
2
70 (3.4)
HSS4x4x3/8 (Corner)
2
73 (1.8)
HSS4x4x3/8 (Wall)
2
80 (6.0)
A53 Gr. B (UN)
Pipe3STD (Wall)
2
54 (1.0)
Pipe5STD (Wall)
3
47 (9.2)
A992
W12x16 (Flange)
2
60 (0.1)
*Nearly identical measured fracture diameters
σU, ksi
εF, in/in
μ = εF/εY
71 (1.5)
79 (4.1)
0.98 (2.2)
0.90 (1.8)
249 (3.9)
182 (4.3)
80 (2.1)
74 (1.9)
78 (1.6)
84 (7.7)
1.14 (6.6)
1.02 (0.0*)
1.04 (2.3)
0.89 (5.0)
242 (8.4)
216 (0.05*)
224 (2.9)
184 (10.8)
67 (1.3)
61 (2.2)
0.96 (0.0*)
0.86 (3.6)
259 (0.6*)
251 (12.3)
79 (0.3)
0.89 (4.9)
192 (5.0)
Table 5.2: Calibrated kinematic and isotropic hardening law parameters.
Brace Material Type
HSS4x4x1/4 (Corner)
HSS4x4x1/4 (Wall)
HSS4x4x3/8 (Corner)
HSS4x4x3/8 (Wall)
Pipe3STD (Wall)
Pipe3STD (Seam)
Pipe5STD (Wall)
Pipe5STD (Seam)
All Gusset Plates
σY, ksi
73
68
64
64
55
55
51
51
50
C, ksi
850
300
1450
1450
500
500
489
489
500
γ
160
25
70
70
35
35
26
26
38
Q∞, ksi
14.5
10
18.5
18.5
52
52
13
13
17
b
5.25
6
6
6
2
2
7
7
5
Table 5.3: Maximum measured local imperfections (x 10-3 inches) along 3-foot section of each
brace cross-section.
Seam (HSS-location
within wall; Pipe-along
seam)
Adjacent to Seam (HSSlocation within wall;
Pipe-180° from seam)
HSS4x4x1/4
0.8(Left)
1.2(Right)
1.1 (Middle)
2.3(Left)
2.7 (Right)
2.8 (Middle)
*Global camber assumed as L/1000
HSS4x4x3/8
1.5 (Left)
1.0 (Right)
0.6 (Middle)
1.4(Left)
0.5 (Right)
1.3 (Middle)
Pipe3STD
Pipe5STD
1.9
4.1
3.6
6.4
175
Table 5.4: Calibrated CVGM fracture parameters.
Brace Material Type
HSS4x4x1/4 (Corner)
HSS4x4x1/4 (Wall)
HSS4x4x3/8 (Corner)
HSS4x4x3/8 (Wall)
Pipe3STD (Wall)
Pipe3STD (Seam)
Pipe5STD (Wall)
Pipe5STD (Seam)
W12x16 (Flange)
Sample
No.
1
2
3
Avg.
COV
1
2
Avg.
COV
1
2
3
Avg.
COV
1
2
Avg.
COV
1
2
3
Avg.
COV
1
2
3
Avg.
COV
1
2
3
Avg.
COV
1
2
3
Avg.
COV
1
2
3
Avg.
COV
Δfexp (in)
η
0.0150
0.0153
0.0156
0.0153
1.96%
0.0156
0.0192
0.0174
14.63%
0.0102
0.0112
0.0116
0.0110
6.56%
0.0093
0.0137
0.0115
27.05%
0.0196
0.0213
0.0224
0.0211
6.69%
0.0197
0.0203
0.0204
0.0201
1.88%
0.0099
0.0106
0.0129
0.0111
14.10%
0.0048
0.0078
0.0092
0.0073
30.94%
0.0186
0.0198
0.0198
0.0194
3.57%
5.83
5.98
6.13
5.98
2.51%
5.78
7.88
6.83
21.74%
3.29
3.70
3.87
3.62
8.33%
2.82
4.69
3.76
35.21%
4.03
4.44
4.72
4.40
7.89%
4.05
4.19
4.22
4.15
2.18%
1.98
2.24
3.17
2.46
25.40%
0.59
1.30
1.74
1.21
47.96%
2.98
3.30
3.30
3.19
5.79%
λNB
λC
0.09
0.17
0.04
--
0.05
0.13
0.09
--
0.15
0.25
--
0.16
0.21
--
0.34
--
176
Table 5.5: CVGM fracture prediction summary
Test No.
Bracing
Member
HSS1-1
HSS1-2
HSS1-3*
HSS1-6#
HSS2-1
HSS2-2*
P1-1
P1-2
P1-3
P2-1
P2-2
P2-3
HSS4x4x1/4
HSS4x4x1/4
HSS4x4x1/4
HSS4x4x1/4
HSS4x4x3/8
HSS4x4x3/8
Pipe5STD
Pipe5STD
Pipe5STD
Pipe3STD
Pipe3STD
Pipe3STD
Loading History
Std.
FF
Pulse
NF-C
Pulse
NF-T
Deterministic
Figure No.
5.25a
5.26a
5.27c
5.28
5.25b
5.27d
5.25d
5.26c
5.25c
5.26b
Probability of Fracture**
(%)
λNB Damage λC Damage
Law
Law
35
85
13
42
3
26
1
13
Approx. 0
47
Approx. 0
43
37
43
23
26
57
74
Approx. 0
45
Approx. 0
26
Approx. 0
17
*fast (earthquake) loading rate
#HSS1-6 was reinforced at the midpoint (top and bottom) with 18” long plates (see Figure 5.28)
**Calculated at experimental fracture instance
177
Void Nucleation
Necking Between Voids
Void Growth and Strain
Localization
Void Coalescence and
Macroscopic Crack Initiation
Figure 5.1: Micromechanical process of ductile fracture in steel
178
(a)
(b)
Figure 5.2: Scanning Electron Microscope (SEM) pictures of (a) Monotonic and (b) Cyclic
fracture surfaces (A572, Grade 50). Note the larger dimples of the monotonic as compared to the
cyclic surface.
179
100
n
σTrue
σ = σY + Kε
50
Model
Experiment
0
0
0.025
0.05
εTrue
(a)
σ1
σ1 = σ2 = σ3
σ3
σ2
(b)
Figure 5.3: (a) Uniaxial stress-strain behavior and (b) Isotropic yield surface.
180
S33
(
2
σ Y + Q∞ + C γ
3
limit surface
2C
3γ
Limiting
location
of α
∂F
∂σ ij
α ijdev
S11
)
2
σY
3
S22
yield surface
Figure 5.4: Combined isotropic-kinematic yield surface employed in ABAQUS (2004).
181
Yield stress, σY
(uniaxial
tension coupon)
Isotropic hardening
parameters, Q∞ and b
(notched-bar cyclic test;
initial values: C=300,
γ=30, Q∞=15, and b=5)
Kinematic hardening
parameters, C and γ
(notched-bar
monotonic test)
Verify that parameters work
well for uniaxial tension and
notched-bar cyclic tests
Yes; use parameters
in large-scale model
No; repeat small-scale
calibration loop
Verify large-scale model
properly simulates localized
phenomena that lead to
fracture (i.e., local buckling)
Yes: done
No; adjust parameters
to track local buckling
and check against
small-scale tests
Figure 5.5: Ideal material calibration flow chart.
182
D0
r*
Force
Experimental
fracture, Δfexp
Model
Experiment
Deformation
(b)
(a)
Figure 5.6: (a) Notched-bar geometry and (b) Typical force-deformation response. See next figure
for notched-bar continuum model.
183
Node 1
Node 2
(b)
(a)
η
⎛R⎞
ln ⎜ ⎟
⎝ R0 ⎠
Node 1
Node 2
exp
Δf
Deformation (in)
(c)
Figure 5.7: (a and b) Axisymmetric finite element model of notched-bar specimen. The contours
show (a) Equivalent plastic strain and (b) Triaxiality in the notched region. The large triaxiality at
Node 1 is primarily responsible for the discrepancy in (c) the Void growth demand between the
two nodes.
184
1
ηcyclic/η
f (D) = exp(-λε p* )
1 cyclic test
ε p*
Figure 5.8: Exponential relationship between ηcyclic/η and damage (equivalent plastic strain at the
beginning of the tensile excursion to fracture, εp*).
1.5
Void Growth and
Critical Void Size
η
1
Predicted
Fracture Time
0.5
0
Analysis Time
Figure 5.9: CVGM fracture prediction for a small-scale notched bar specimen.s
185
4
AP50
AP110
AP70HP
JP50
AW50
JP50HP
JW50
2
ε
p
Analytical
3
1
0
0
1
2
ε
p
3
4
Experimental
Figure 5.10: Predicted versus experimental instance of fracture initiation for small-scale, notchedbar cyclic coupon tests (adapted from Kanvinde and Deierlein, 2007).
186
Corner:
2S
3 SN
3 LN
Wall:
2 UN
3 Flats
2 SN
2 LN
HSS – HSS4x4x1/4,
HSS4x4x3/8
Seam:
2S
3 SN
1 LN
Wall:
2 UN
3 SN
3 LN
Pipe – Pipe3STD,
Pipe5STD
1.0”
3.0”
0.5”
3.0”
0.1”
1.5”
r = 0.5”
1.0”
3.0”
0.75”
HSS Flats
(thickness: 0.25”)
D0 = 0.1”
Smooth (S)
monotonic tension
D0 = 0.1”
r* = 0.05”
r* = 0.025”
3.5”
(typ)
Flange:
2S
3 SN
3 LN
Wide-Flanged –
W12x16
0.2”
(typ)
Small-notched (SN)
monotonic tension
Large-notched (LN)
cyclic
Figure 5.11: Summary of small-scale specimens for brace material calibration study.
187
700
Load (lbs)
Load (lbs)
700
350
Model
1
2
(a)
0
0.00
350
0.04
0.08
0
0.05
0.12
(b)
0.075
Deformation (in)
0.1
Neck Diameter (in)
1000
1000
500
Model
1
2
3
(c)
0
0
0.01
Deformation (in)
Load (lbs)
Load (lbs)
Model
Experiments
500
Model
1
2
3
(d)
0
0.02
0
0.02
0.01
Deformation (in)
0.02
1500
Fracture
0.01
Load (lbs)
Deformation (in)
1000
0
(e)
-0.01
500
0
Model
1
2
3
-500
-1000
-1500
-0.01
(f)
0
0.01
Deformation (in)
0.02
Figure 5.12: Experimental and simulation force-deformation comparisons for small-scale material
calibration study on HSS4x4x1/4 corner material. Shown above is (a) Smooth bar forcedeformation behavior and (b) Fracture load versus fracture diameter; (c) SN force-deformation
behavior and (d) First tension excursion for the cyclic LN experiments; (e) Cycle loading history
and (f) LN cyclic force-deformation relationship.
188
300
Experiment
Model
Force (k)
200
100
0
-100
-200
-2
-1
0
1
2
Axial Displacement (in)
Figure 5.13: Experimental and simulation force-deformation comparison for large-scale
HSS4x4x1/4 brace (far-field loading).
189
Δ1=0
Δ2=0
Δ3→ loading
n=10, p=2
8
15
14
16
20”
n=25, p=3
20
13
5
6
4
Δ3=0
17
2
Δ2=0
3
7
19
11
3
18
12
10
1
1
9
2
HSS4x4x3/8
HSS4x4x1/4
Pipe3STD
Pipe5STD
W12x16
Figure 5.14: Brace models and meshing schemes.
190
Outside corner
radius = 0.625”
Outside corner
radius = 0.800”
4”
4”
4”
4”
0.254”
0.355”
(b)
(a)
5.56”
3.5”
0.254”
0.216”
(c)
(d)
4.08”
0.270”
0.20”
12.0”
(e)
Figure 5.15: Measured brace cross-section dimensions for (a) HSS4x4x1/4, (b) HSS4x4x3/8, (c)
Pipe3STD, (d) Pipe5STD and (e) W12x16.
191
(a)
(b)
Imperfection Magnitude, in (10-3)
Figure 5.16: (a) Global and (b) Local buckling mode shapes.
8
Pipe3
Pipe5
HSS4x4x1/4
HSS4x4x3/8
4
Along Pipe Seam
Along HSS
Seam
0
Figure 5.17: Cross-section wall imperfection measurements for HSS and Pipe.
192
300
100
0
-100
Experiment
Model
200
Force (k)
200
100
0
-100
(a)
-200
(b)
-200
-2
-1
0
1
2
-6
-4
Axial Displacement (in)
Experimental Buckling Load (k)
Force (k)
300
Experiment
Model
250
150
0
Axial Displacement (in)
HSS4x4x1/4
HSS4x4x3/8
Pipe3STD
Pipe5STD
200
-2
10% margin
lines
100
50
(c)
Near-field
loading
0
0
50
100
150
200
250
Predicted Buckling Load (k)
Figure 5.18: Predicted buckling loads for (a) Far-field loading as compared to (b) Near-field
loading for HSS4x4x1/4 brace. (c) Experimental and predicted critical buckling loads.
2
193
Axial Deformation (in)
2
1
0
0.61" 0.61"
(1) (2) 1.1" 1.1"
(1) (2)
-1
-2
20
22
24
26
28
Cycle Number
0.61” (1)
0.61” (2)
1.1” (1)
1.1” (2)
Figure 5.19: Experimental and predicted local buckling for HSS4x4x1/4 brace (far-field loading).
.
194
Triaxiality
2
Brace
Notched-Bar
1
0
-1
-2
0
2
4
6
eP
Figure 5.20: Triaxiality versus equivalent plastic strain (HSS4x4x3/8 material).
195
1
1
λΝΒ = 0.05
0.5
ηcyclic/η
ηcyclic/η
λΝΒ = 0.09
λC = 0.17
Notched-Bar
Brace
λC = 0.13
0.5
Notched Bar
Brace
(a)
0
0
0
2
4
6
8
0
2
4
P*
ε
6
8
εP*
1
1
λΝΒ = 0.15
0.5
λΝΒ = 0.16
ηcyclic/η
ηcyclic/η
(b)
λC = 0.25
Notched Bar
Brace
0
2
4
P*
ε
λC = 0.21
Notched-Bar
Brace
(c)
0
0.5
(d)
0
6
0
2
4
6
P*
ε
Figure 5.21: Calibration of λNB and λC using the notched-bar (NB) and combined (C) data points,
respectively, for (a) HSS4x4x1/4, (b) HSS4x4x3/8 corner material and (c) Pipe3STD and (d)
Pipe5STD base metal.
196
1
f (D)
Probability
Distribution of ε
0.5
λC = 0.17
ε
ε=0
Notched-Bar
Brace
0
0
2
4
6
P*
ε
Figure 5.22: Constant probability distribution of ε.
1
ηcyclic/η
Damage
Figure 5.23: Effect of low damage levels on λ variability.
8
197
6
Experimental
Fracture
4
Void Size and Critical Void Size
Void Size and Critical Void Size
6
CVGM
Prediction: λ NB
2
4
CVGM
Prediction: λ C
2
0
0
23
24
25
26
Cycle Number
27
28
23
24
25
26
27
Cycle Number
Figure 5.24: CVGM fracture predictions for HSS4x4x1/4 where λNB and λC are used in
calculating two damage functions (circle and triangle, respectively).
28
198
4
2
2
Combined Fit
Δ (in)
Δ (in)
4
Experimental Fracture
Notched-Bar Fit
0
-2
0
-2
(b)
(a)
-4
20
-4
22
24
26
28
30
20
24
32
36
Cycle Number
4
4
2
2
Δ (in)
Δ (in)
Cycle Number
28
0
0
-2
-2
(d)
(c)
-4
20
-4
24
28
32
Cycle Number
36
20
22
24
26
28
30
32
Cycle Number
Figure 5.25: Comparison between experimental and predicted fracture instances for (a)
HSS4x4x1/4, (b) HSS4x4x3/8, (c) Pipe3STD and (d) Pipe5STD. Note both Pipe sections have
two experimental points.
2
6
0
4
Δ (in)
Δ (in)
199
-2
-4
0
Experimental Fracture
Notched-Bar Fit
Combined Fit
-6
0
6
12
2
(a)
18
24
30
(b)
-2
36
0
6
12 18 24 30 36 42 48
Cycle Number
Cycle Number
6
Δ (in)
4
2
0
(c)
-2
0
6
12
18
24
30
36
Cycle Number
Figure 5.26: Comparison between experimental and predicted fracture instances for (a)
HSS4x4x1/4 brace subjected to the asymmetric compressive near-field loading history, (b)
Pipe3STD and (c) Pipe5STD subjected to the asymmetric tension near-field loading history.
200
4
2
2
Combined Fit
Δ (in)
Δ (in)
4
Experimental Fracture
Notched-Bar Fit
0
-2
0
-2
(b)
(a)
-4
20
-4
22
24
26
28
30
20
24
32
36
Cycle Number
4
4
2
2
Δ (in)
Δ (in)
Cycle Number
28
0
-2
0
-2
(d)
(c)
-4
-4
20
22
24
Cycle Number
26
28
20
24
28
32
36
Cycle Number
Figure 5.27: Comparison between HSS4x4x1/4 and HSS4x4x3/8 (a and b) Quasi-static (shown
previously in Figure 5.26) and (c and d) Earthquake-rate tests.
201
(a)
(b)
2
Experimental Fracture
Notched-Bar Fit
Combined Fit
Δ (in)
1
0
-1
-2
(c)
20
22
24
26
28
Cycle Number
Figure 5.28: Comparison between experimental and predicted fracture instances for HSS4x4x1/4
brace subjected to far-field loading with middle reinforcing plates.
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4
Experimental Fracture
Notched-Bar Fit
Combined Fit
Δ (in)
2
0
-2
(a)
-4
Probability of Fracture
26
27
Cycle Number
28
Experimental Fracture
1
Notched-Bar Fit
Combined Fit
0.5
(b)
0
26
27
Cycle Number
28
Figure 5.29: (a) Deterministic CVGM fracture instances and (b) Cumulative probability function
for HSS4x4x1/4 subjected to far-field loading history.
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Chapter 6
Parametric Simulation of HSS Bracing Members
To generalize experimental findings using the CVGM methodology, this chapter presents a
parametric simulation study on square and rectangular HSS bracing members subjected to
earthquake-type cyclic loading. The braces are modeled with continuum and fiber elements to
compare the advantages and disadvantages of the modeling techniques in the context of
simulating brace fracture. The continuum-based Cyclic Void Growth Model (CVGM), evaluated
in Chapter 5, is used to predict the fracture initiation in the brace simulations. These predictions
expand the experimental testing matrix presented in Chapter 4 by examining a wider range of
geometric properties which may influence brace ductility.
The parametric study results are combined with rectangular and square HSS experimental data
from the last twenty years to examine the factors which control brace ductility. Based on this
combined experimental and simulation data set, a semi-theoretical model is proposed, which
relates the brace axial deformation capacity, and associated story-drift capacity to the bracing
member geometry (i.e. slenderness and cross sectional geometry). An illustrative example is
provided wherein the semi-theoretical model is used to suggest a limiting cross-section widththickness, or b/t, ratio for square and rectangular bracing members, to provide acceptable
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performance. Referring to Chapter 2, limits on cross-section b/t ratios are used by the AISC
Seismic Provisions (2005) to ensure acceptable ductility with respect to brace local buckling and
fracture. Thus, the results presented in this chapter provide an example where the advanced
simulation tools presented in Chapter 5 can be applied to develop insights into localized behavior
affecting fracture in large-scale steel structures, and subsequently improve design provisions.
6.1
PARAMETRIC SIMULATION MATRIX
Table 6.1 summarizes the simulation matrix, which includes 22 brace simulations with varying
cross-section sizes and width-thickness, aspect and slenderness ratios. The cross-section sizes
range from 4 to 8 inches square with width-thickness ratios between 8.5 and 14.2 for the square
cross-sections. Three slenderness ratios of 40, 80 and 120 are selected to complement the testing
matrix presented in Chapter 3 where the experimental HSS4x4x1/4 and HSS4x4x3/8 both had
slenderness ratios of approximately 80. Four rectangular brace simulations are conducted to
investigate the influence of aspect ratio, or B/H, on brace ductility where B is the overall length of
the buckling face and H is the perpendicular depth of the cross-section as shown in Figure 6.1.
Corner bend radii are assumed to be approximately 2t where t is the thickness of the crosssection, according to the minimum specified fabrication limits for HSS bend radii from AISC
(2005). The brace cross-sections are attached to end gusset plates to replicate realistic boundary
conditions. The gusset plates are designed according to the procedure discussed in Chapter 3.
The HSS4x4x1/4 material type, presented in Chapter 5, is used for the parametric study discussed
in this chapter, given that it resulted in excellent predictions of fracture. Similar to the finite
element models presented in Chapter 5, a fine mesh is used in the middle third of the half-brace,
transitioning to a coarse mesh at the ends such that the element lengths increase in a geometric
progression. This allows for a fine mesh at the middle plastic hinge region where local buckling is
expected and a coarse mesh at the end of the brace where minimal inelastic strains develop. For
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the larger sections, the HSS4x4 meshes developed in Chapter 5 are used as guidelines to
determine the number of elements along the dimensions of the cross-section.
6.1.1
CYCLIC LOADING HISTORY
Figure 6.2 illustrates the cyclic loading history for the parametric brace simulations expressed in
terms of story drift. The peak drifts for each loading cycle are identical to the experimental farfield history presented in Chapter 3. The story drift is converted to axial deformations according
to the relationship –
(
)
Δ a = cos 2 45D LBθ = 0.5 LBθ
(6.1.1)
Where θ is the story drift angle (expressed in radians), Δa is the corresponding axial deformation
and LB is the length of the bracing member between gusset plate fold lines. Refer to Chapter 3 for
more details.
6.2
CVGM FRACTURE PREDICTIONS
This section presents the results of the parametric study where the CVGM is used to predict the
instance of fracture in each of the brace simulations of Table 6.1. The CVGM fracture parameters
are assumed to be equal to the calibrated values from the HSS4x4x1/4 brace material where η is
equal to 5.98 and λ is 0.17. Note that λ is calibrated from the combined data set of the notched
bar and brace experiments because it characterizes the cyclic damage of the experiments more
accurately. The fracture predictions expand the experimental testing results from Chapter 4 and
inform relationships which govern brace ductility. From the discussion in Chapters 2 and 4,
width-thickness (b/t) and brace slenderness ratio (KL/r) were shown to affect brace ductility.
Thus, the fracture predictions are discussed in the context of these parameters.
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6.2.1
EFFECT OF WIDTH-THICKNESS RATIO ON BRACE DUCTILITY
In this section, the results of the simulation study are used to determine the effect of widththickness ratio on brace ductility. The maximum axial deformation range, Δrange=Δmax+|Δmin| (See
Figure 2.9 or 6.7), prior to fracture initiation is used to describe the capacity of the bracing
components. While cumulative ductility capacities (i.e., cumulative plastic deformation) are, in
general, more comprehensive, the consistent loading history diminishes the usefulness of a
cumulative measure. Furthermore, the semi-theoretical model in Section 6.3 will use the
deformation range because it may be evaluated relative to maximum story drift demands.
Referring to Figure 6.4a, the influence of brace width-thickness ratio on axial deformation
ductility is difficult to ascertain due to the effect of the brace length on the ductility. If the brace
simulations had been the same length, the ductility trend would be similar in appearance to Figure
2.9. Thus, converting the axial deformation range to the story drift range, θrange, allows the effect
of the width-thickness ratio to be more easily identified. Figure 6.4b illustrates the expected
decreasing ductility for increasing width-thickness ratio after the deformation range is normalized
by the brace length. Referring to the figure, for a slenderness ratio of KL/r = 80, a 40% reduction
in width-thickness ratio resulted in about a 100% increase in fracture drift ductility. This is
consistent with the other slenderness ratios as well as the experimental observations presented in
Chapters 2 and 4.
6.2.2
EFFECT OF SLENDERNESS RATIO ON BRACE DUCTILITY
This section uses the results of the parametric study to investigate the influence of brace
slenderness ratio on ductility. Referring to Figure 6.4a, brace slenderness has a dramatic effect on
the axial deformation ductility, such that increasing the slenderness ratio of an HSS6x6x3/8
bracing member from 40 to 120 increases the ductility by nearly 330%. This trend was also
207
shown in Chapter 2 across a variety of cross-section types. However, as illustrated in Figure 6.4b,
expressing ductility in terms of the maximum story drift range considerably lessens the effect of
brace slenderness ratio. In terms of drift, increasing the HSS6x6x3/8 slenderness ratio from 40 to
120 increases the ductility by 43%. Note that for the group of simulations with b/t < 14, the
ductility increases for a given slenderness because of the decreasing width-thickness ratios
included in that group (i.e., 10.8, 9.9 and 8.5, refer Table 6.1). As expected, this is not observed
for the group of simulations where b/t = 14.
Referring to Figure 6.4, the vertical line represents the AISC (2002 and 2005) upper-limit on
brace slenderness (100). However, it should be noted that AISC Seismic Provisions (2005) has
adopted an exception which allows slenderness ratios greater than 100 (for A500 Gr. B HSS
braces), but less than 200, if the expected yield force of the brace is less than the compressive
capacity of the adjoining columns. While the intent of the new provision is to improve brace
ductility through slenderness ratio, slenderness ratios above 100 may be somewhat unlikely
considering the larger HSS shapes used in high seismic regions (i.e., HSS8x8 and above). For
example, Table 6.2 lists the corresponding brace lengths (assuming a 45° Chevron orientation) for
story heights equal to 13 and 20 feet and frame bay widths equal to 20, 30 and 40 feet. Referring
to the table, assuming typical braced-frame dimensions, the range of brace lengths is between
16’-5 and 28’-3 (197 and 339 in). Using these lengths, and an effective length factor (K) equal to
0.9, Figure 6.4a illustrates the resulting slenderness ratios for all available square HSS shapes.
Horizontal lines are also drawn at the AISC (2005) slenderness limits of 100 and 200, where the
200 limit can only be used if the adjoining columns have adequate compressive capacity. Figure
6.4b illustrates the available square HSS shapes with b/t ratios less than the maximum b/t limit of
16 prescribed by the AISC Seismic Provisions (2005). Referring to these Figures, for shapes
which are relatively common in SCBF design (larger than 5x5), the largest slenderness ratio is
approximately 120. Thus, the parametric study (and the following synthesis of experimental data)
208
appropriately queries slenderness ratios which can be expected in typical braced-frame
construction.
In the context of the previous discussion, it may be more appropriate to investigate the influence
of slenderness ratio on brace ductility through the radius of gyration, r, by fixing the overall brace
length and b/t ratio. This eliminates the compactness (discussed previously) and length effects on
brace ductility. Using the results from the parametric study, Figure 6.4c illustrates story drift
capacities for fixed brace lengths equal to 119 and 180 inches (b/t =14.2) with varying r.
Referring to the figure, the ductility does not increase for the 119 inch length (slenderness range
from 40 to 80) while a negligible increase is observed for the 180 inch length (80 to 120). Also
shown in Figure 6.4 are results from past experimental testing programs (discussed next) where,
similar to the parametric study results, a fixed brace length of 119 inches and a constant b/t limit
equal to 14.2 examines the influence of increasing r on story drift ductility. The experimental
trends seem to confirm the results of the simulations such that an increasing slenderness ratio (50
to 145) has negligible effects on ductility.
6.3
SYNTHESIS OF EXPERIMENTAL AND SIMULATION RESULTS
This section uses the results from the parametric study simulations along with HSS brace tests
conducted over the last twenty years to develop a semi-theoretical model to predict ductility as a
function of brace geometry. The relationship can inform design procedures of HSS braces in
SCBF systems and evaluate their performance relative to expected seismic demands. Specifically,
the methodology generalizes the experimental and simulation results and proposes a relationship
that may be used to determine limiting b/t ratios considering various aspects of brace response.
A review of ten experimental studies (a total of 63 experiments) on HSS bracing members is
summarized in Table 6.3. Important results from all the experiments within each program are
209
summarized in Appendix B. Referring to Appendix B, these programs have examined the effect
of various parameters, such as brace slenderness, compactness, and cross-section shape, on
fracture ductility and energy dissipation in braces. The yield stress for all the HSS steel materials
listed in Table 6.3 and Appendix B is fairly consistent (Mean = 62 ksi, COV = 0.09).
In general, the findings from all programs concur that the width-thickness ratios and slenderness
ratios control brace ductility, such that higher width-thickness ratios and lower slenderness are
detrimental to brace performance. However, synthesizing the data from diverse experimental
programs presents several challenges. Since the different programs have dissimilar test setups,
brace geometries (length and cross-sections) and loading protocols, it is difficult to directly
interpret results from these various programs. To illustrate this point, Figures 6.6a and 6.6b
include example plots of the observed maximum drift (expressed as half the maximum drift
range, θmax) for selected experimental data points from various test programs. Figure 6.6a plots
the drift against the b/t ratio for 19 braces of approximately similar slenderness (≈ 80). On the
other hand, Figure 6.6b plots the drift capacity versus brace slenderness for 25 braces of
approximately equal b/t ratios (≈ 14.2). Referring to the figures, two observations may be made –
1. Although the b/t ratios appear to negatively influence the brace ductility, the experiments
exhibit maximum drift ductilities (expressed as half of the total range, as is often done for
symmetric cyclic loading protocols) that are smaller (average 2.6%) than the expected
4% during MCE events, suggesting that almost all HSS braces are deficient, many even
under design level events (that correspond to 2% drift). Referring to Figures 6.3 and 6.4,
this is also true for the simulation results where the maximum drift ductilities (again,
calculated as half θrange) are, on average, 3.4%.
210
2. Similar to the parametric study results presented in Figure 6.4b and 6.5c, Figure 6.6b
does not indicate a strong positive effect of the brace slenderness on ductility (for
constant b/t).
Both these observations are somewhat surprising and raise two key issues –
1. Given that brace ductility is a function of several parameters, a common basis is required
for interpreting data from various test programs, which may often have different brace
lengths and cross-sectional dimensions. This common basis can then be used to
generalize results of the various programs. This is the topic of the next section.
2. A consistent approach to evaluate the brace drift capacities relative to drift demands is
desirable, especially given the variability in loading protocols. This is the topic of Section
6.3.2.
6.3.1
A SIMPLIFIED APPROACH FOR EVALUATING THE EFFECT OF BRACE PARAMETERS ON
DUCTILITY
A simplified approach is proposed to examine the effect of various parameters such as the brace
buckling length and cross sectional dimensions on brace ductility. The main assumptions of this
approach are summarized schematically in Figure 6.7 and listed below –
1. Neglecting elastic deformations in the brace, the range of axial brace deformations
Δrange=Δmax+|Δmin| can be kinematically related to a gross strain quantity corresponding to
fracture (similar to a fiber strain) at the center of the brace. Referring to Figure 6.7 and
Equation 6.3.1, LB is the brace buckling length, and is distinct from the effective length,
KL –
211
εF =
H
Lh ,est
Δ range
LB
=
2 ( Δ max + Δ min
2H
H +B
LB
)
(6.3.1)
2. The plastic hinge length, Lh,est, at the center of the brace is equal to the average of the
width, B, of the cross-section over which the local buckle forms and the depth of the
cross-section, H. Refer to the following discussion for an explanation of Lh,est.
Thus, the relationship presented in Equation 6.3.1 can be used to convert the brace axial
deformation range to the local (fiber-level) strains corresponding to fracture. This relationship is
advantageous because it provides a common variable that can be examined across various
simulations, experiments and testing programs and eliminates the effect of brace length,
dimensions, and loading protocol. However, these gross strains do not incorporate the
amplification due local buckling of the cross-section wall. To include the local buckling effect,
Figure 6.8a plots the fiber-level strain, εF, determined according to Equation 6.3.1 versus the b/t
ratio for all the 63 tests. Also shown are the 22 simulation points from the parametric study.
Referring to the figure, a strong relationship is observed between the b/t ratio and εF. This may be
expressed through a regression fit (shown as dashed lines on Figure 6.8a) as shown below –
⎛b⎞
εF = a⎜ ⎟
⎝t⎠
b
(6.3.2)
Where a and b are constants which are calibrated with a least squares fit. Referring to Table 6.4
and Figure 6.8a, the above regression fit is determined for the experiments (63 points),
simulations (22) and a combined data set of the experiments and simulations (85). The relatively
close values of the regression parameters, a and b, across the three data sets is encouraging in that
the simulation results are corroborated by the experimental observations. For comparison, Figure
6.8b directly plots Δrange against b/t for all the experiments. Referring to the figure, εF results in a
clearer dependence on b/t.
212
The regressed relationship from Equation 6.3.2 may be used in conjunction with Equation 6.3.1 to
develop a predictive relationship for the brace axial deformation capacity such that –
Δ
p r edicted
range
b
L ⎡ a ( H + B) ⎛ b ⎞ ⎤
= B⎢
⎜ ⎟ ⎥
2 ⎣⎢ 2 H
⎝ t ⎠ ⎦⎥
2
(6.3.3)
Figure 6.9 plots the ratio of the measured brace axial deformation capacity to the predicted
deformation capacity (Δrange,exp/Δrange,pred) for all the experiments. Given the variability between
various test programs, the results presented in Figure 6.9 are encouraging (mean
Δrange,exp/Δrange,pred=1.03, and COV of 26%).
It is important to note that the relationships presented in this section are based on the brace
buckling length, LB, instead of the brace effective length (with respect to elastic buckling). This
can be explained by considering that once the brace buckles and forms a central plastic hinge, the
strains are driven by a kinematic relationship which is a function of the buckled brace geometry
(rather than the slenderness ratio, which corresponds to the elastic buckling curve). Thus, if the
brace buckles elastically, i.e. KL/r > 118 (for Fy = 46 ksi), the mechanisms discussed in this
section may not be active. Out of the 63 experiments listed in Table 6.3, only 4 had KL/r values
greater than 118 while 6 simulations are approximately equal to the elastic buckling limit (120).
Moreover, the relationships assume loading history independence and quantify the capacity in
terms of a single parameter, Δrange. While this is a simplified estimate, it provides good agreement
with test data and can be conveniently interpreted. This may be refined through a more detailed
analysis of test data.
In summary, this section presents an approach to determine brace axial deformation capacities
given brace parameters. However, these predictions (as well as data from other experiments) must
be evaluated relative to expected story drift demands. This is discussed in the Section 6.3.2.
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6.3.1.1 PLASTIC HINGE LENGTH
The spread of plasticity along the length of structural components during inelastic loading may be
approximately characterized with the plastic hinge length. Although it is subjectively defined,
(given the irregular shape of the plastic zone) it often serves as a convenient measure of the size
of the zone of concentrated plasticity. Previous research has expressed plastic length as a function
cross-section geometry. For example, in circular bridge piers, the plastic hinge length is often
considered equal to half the diameter of the reinforced concrete column (Priestley and Park,
1987). Similarly, the plastic hinge length of rolled steel members is often assumed to be equal to
the depth of the cross-section across the neutral axis. Although the plastic hinge is commonly
visualized as a zone of concentrated plastic deformation within a member, the plastic hinge length
itself is somewhat difficult to characterize in a precise manner for a variety of reasons. For
example, weak moment gradients within a member may extend the zone of plasticity well outside
the commonly visualized hinge region. Thus, in the context of this discussion, the plastic hinge
length is interpreted as a parameter which enables the convenient determination of fiber-like
strains based on global deformations, rather than a precise
physical quantity that may be
objectively characterized.
In this section, the plastic hinge lengths of the brace simulations are examined. The hinge length
is shown to be dependent on local buckling such that the buckling wavelength of the HSS wall
controls the distribution of inelastic strains. To calculate the plastic hinge length in the brace
simulations, the curvature along the length of the bracing member is integrated to determine the
total rotation between the gusset end plates. Assuming this rotation is uniformly concentrated in a
concentrated plastic hinge at the brace mid-point, an approximate hinge length can be interpreted
such that –
214
LB
Lhφmax = ∫ φ ( x)dx
(6.3.4)
0
Where Lh is the plastic hinge length, φmax is the maximum curvature at the center of the brace,
and φ(x) is the curvature along the length of the brace, LB. The curvature along the length of the
brae is determined through differentiation of the brace profile on the face which does not locally
buckle. The plastic hinge calculation is illustrated schematically in Figure 6.10.
Referring to Figures 6.11a and 6.11b, the plastic hinge lengths at the compressive peaks of the
standard loading history are calculated according to Equation 6.3.4 for the HSS4x4x1/4 and
HSS6x6x3/8 bracing member simulations, respectively. Figures 6.11c and 6.11d illustrate the
plastic hinge lengths for the HSS4x2x1/4 and HSS4x2x3/16 with B/H ratios of 2 and 0.5,
respectively. The vertical dashed lines show the location of local buckling while the horizontal
dashed lines correspond to an estimated plastic hinge length once adequate deformation occurs to
clearly form the plastic hinge. This length will be discussed below. Two vertical lines are used in
Figures 6.11a and 6.11b because local buckling occurred at a later cycle for the braces with large
slenderness ratios (L/r=120).
To associate the plastic hinge length results from Equation 6.3.4 to the brace geometry (and
explain the horizontal dashed lines in Figure 6.11), Figure 6.12 illustrates the deformed shape of
an HSS4x4x1/4 during local buckling. Connecting the plastic hinge length with the brace
dimensions is also convenient in light of the semi-theoretical model described above. Thus, an
approximate plastic hinge length is developed considering the spread of plasticity with respect to
the post-buckling amplified strains. Referring to Figure 6.12, the hinge length is mostly influence
by local buckling, such that the buckling wavelength is B and the effective buckling wavelength
is approximately B/2. Furthermore, considering the plastic hinge length is used in the semitheoretical model as a parameter to estimate fiber-like strains, the hinge length is determined
215
along the center-line of the brace at a distance of H/2 from the buckling face. From this, the
plastic hinge length is estimated as –
Lh ,est =
B
+ mH
2
(6.3.5)
Where m is the slope of the plane section which extends from the local buckling face to the
center-line of the brace. Equation 6.3.5 can be explained physically by the fact that while the
plastic hinge length is dependent on the local buckling wavelength of the compression face, the
depth provides rotational constraint such that a larger depth develops a larger hinge length (for
equal B). The slope m can be considered proportional to the out-of-plane buckling deformation of
the brace divided by half the brace length. However, realizing the complexity of the inelastic
behavior during global and local buckling, Equation 6.3.5 is simplified by assuming m = 0.5
Thus, while the expression captures the physical behavior of the plastic hinge length during local
buckling, it does not necessitate a dependence on the global deformation shape. In light of the
semi-theoretical model proposed in the next section, as well as the complex nature of inelastic
buckling phenomena, Equation 6.3.5 with m = 0.5 serves as a reasonable estimate for the plastic
hinge length.
Referring back to Figures 6.11a-b, the hinge lengths from the square cross-section simulations
tend to converge to Lh,est, i.e., the wall dimension of the square tubes, regardless of slenderness
ratio and cross-section size. Moreover, referring to Figures 6.11c and 6.11d, the brace simulations
with aspect ratios not equal to 1 (HSS4x2x1/4 and HSS4x2x3/16, respectively) illustrate that the
finite-element hinge lengths also approach Lh,est after local buckling initiates. Therefore, square
and rectangular simulations suggest that the hinge length after local buckling can be
approximated as the average of the cross-section dimensions, B and H. In addition, these
simulation results further emphasize the influence of local buckling on the strain amplification at
the middle plastic hinge of bracing members.
216
6.3.1.1.1
COMPARISON TO FIBER-BASED BRACE SIMULATIONS
Modeling bracing components with line models, such as the fiber element, is frequently used in
analyses of braced-frame behavior (Uriz et al, 2008 and Izvernari and Tremblay, 2007). The
popularity of fiber-based modeling can be attributed to the accuracy of the model in predicting
system and component level behavior such as story drifts and deformation demands in the
presence of material and geometric nonlinearities. Furthermore, the computational expense of
continuum models for large-scale structural modeling renders fiber-models a more efficient
choice in the context of modeling global response. However, fiber-models cannot simulate
phenomena such as local buckling. Thus, while the fiber-model can accurately predict global
behavior and provides computational advantages over continuum models, the model may not be
suitable in assessing localized behavior which can lead to fracture events. Figure 6.13 compares
the results from a continuum and fiber model of an HSS4x4x1/4 bracing member during the farfield loading history shown in Figure 6.2. The fiber-based model uses approximately 20 elements
along the length of the brace where, similar to the continuum model, more elements are used
along the middle-third of the half-brace. Referring to the figure, while the load-deformation
behavior from the fiber model matches the experimental results well, the plastic hinge length
remains relatively constant during the loading. Compare this to the continuum-based predictions,
which show the sudden decrease in plastic hinge length after local buckling occurs. Thus, the
amplified strain demands from local buckling are not predicted accurately using the fiber model.
6.3.2
EVALUATING BRACE AXIAL DEFORMATION CAPACITIES RELATIVE TO EARTHQUAKE
INDUCED STORY DRIFT DEMANDS
To provide limits on brace geometries, the brace axial deformation capacities, Δrange, determined
either directly from experiments or through an approach such as the one suggested in the previous
section, must be evaluated relative to expected story drift demands in SCBF buildings. For this
217
discussion, the drift demands will be discussed in terms of the maximum story drift, θmax. Several
components, already presented in this chapter or previous chapters, are combined to develop a
relationship between the axial deformation capacity and the expected maximum drift. These are –
1. Estimating the maximum story drifts, θmax, that may be encountered in SCBF buildings
during earthquakes. This was discussed in Chapter 2.
2. Examining the relationship between the maximum story drift demands θmax and the range
of drift, θrange = θmax +|θmin|, expected within a story. This is also discussed in Chapter 2.
3. Developing a relationship between θrange and the brace axial deformation, Δrange. This was
discussed in Chapter 3.
Referring to the first point, current design requirements for SCBFs (AISC, 2005) state that
“braces could undergo post-buckling axial deformations 10 to 20 times their yield deformation”.
Given a system yield level drift of approximately 0.3 to 0.5%, the Seismic Provisions could be
interpreted as desiring a deformation capacity of approximately 3 to 5% for SCBF systems.
Referring to the nonlinear time history simulations in Chapter 2, mean story drifts were
determined to be between 4.4 (6-story from by Uriz) and 8.1% (3-story frame by McCormick et
al) at the Maximum Considered Earthquake (MCE) demand level (2% chance of exceedance in
50 years). However, for the ground motions and buildings used as part of these studies, large
coefficients of variations (COV) are found to accompany these results (approximately 50 and
37%, respectively). Considering the discrepancy between the analysis results of Chapter 2 and the
interpretation of the AISC Seismic Provisions (2005), it is difficult to select an expected
maximum story drift demand. While it is outsides the scope of the current work, for the purposes
of this chapter, θmax is selected as 4%. Additional simulations and further analysis of existing
simulation data are suggested to refine this value.
218
With reference to the second point, Chapter 2 (Figure 2.6) illustrates the highly unsymmetrical
drift time history of SCBF systems under a 2/50 ground motion (Uriz, 2005) such that the
maximum drift is approximately two and a half times the maximum drift in the other direction of
loading, i.e., θmin=0.4θmax. Unlike moment frames, where the response is more symmetrical
(under far-field ground motions), braced frames tend to show a more unsymmetrical response,
presumably because of the unsymmetrical strength and stiffness properties once the compressive
brace buckles. However, braced frame components (such as braces) are typically subjected to
symmetric loading protocols, based on adaptations of protocols designed to reflect demands in
moment frames (Fell et al, 2006, Han et al, 2007, Shaback and Brown, 2001 and Archambault et
al, 1995). Consequently, when the maximum equivalent drift is reported as a capacity measure, it
is calculated as half the range of equivalent drift applied to the component (e.g., Chapter 4). This
may underestimate the capacity of the brace which, in general, will not be subjected to symmetric
cycles under seismic excitation. In fact, one may argue that this is one of the main reasons why an
examination of all the experimental data (as illustrated previously in Figure 6.6) indicates an
unusually low capacity for HSS braces. For a better interpretation of test data with respect to
realistic seismic demands, Equation 6.3.6 below incorporates the unsymmetrical nature of the
brace deformation discussed in Chapter 2 –
θ range = θ max + θ min = θ max (1 + α )
θ max =
1
θ range
1+α
(6.3.6)
Where α = 0.4 is chosen from Figure 2.7. While more investigations are required to characterize
this parameter accurately, these will likely reveal that α is smaller than one which is implicitly
assumed while interpreting results of symmetric loading protocols. Thus, based on Equation
6.3.6, using the estimate of α = 0.4 < 1 to interpret the test data increases the calculated drift
capacity, such that for the test data discussed earlier and shown in Figure 6.6, the average drift
219
capacities are 3.6% for the experiments (0.71/0.5 = 1.4 times 2.6% from above) and 4.8% for the
simulations described in the beginning of this section.
Finally, referring to the final point introduced at the beginning of this section, the story drift
range, θrange, can be related to the brace axial deformation, Δrange, with the kinematic relationship
presented in Chapter 3. Assuming that the story drift is accommodated by the deformation of the
braces (a conservative assumption from the standpoint of brace capacity), the kinematic
relationship was presented as –
(
)
Δ range = (1 + C ) cos 2 β LBθ range
(6.3.7)
Where θrange is the story drift angle (expressed in radians), Δrange is the corresponding axial
deformation, β is the brace angle, and C is the ratio of the rigid-link length (on both ends of the
brace) to the brace length, LB.
Thus, based on the ideas presented in this section, the brace axial deformation capacity, Δrange,
may be converted to a corresponding drift capacity by combining Equations 6.3.6 and 6.3.7 such
that –
⎛ 1 ⎞⎛ 1 ⎞ ⎛
1
⎞Δ
range
θ max = ⎜
⎟
⎟⎜
⎟⎜
2
C
L
+
+
1
1
cos
α
β
⎝
⎠⎝
⎠⎝
⎠ B
(6.3.8)
As discussed earlier, this relationship may be used to directly characterize brace capacities based
on experimental estimates of Δrange, or alternatively, it might be used in conjunction with
Equation 6.3.3 to develop a relationship between the critical width-thickness ratio and the brace
capacity, expressed in terms of maximum drift demand –
⎡ 2H
⎤
⎛b⎞
2 (1 + C )(1 + α ) ( cos 2 β )θ max ⎥
⎜ ⎟ =⎢
⎝ t ⎠crit ⎢⎣ a ( B + H )
⎥⎦
1
b
(6.3.9)
220
By substituting α = 0.4, a = 0.98 and b = -0.56 from Table 6.4, and C = 0 and β = 45° from the
assumptions discussed in Chapter 3, Equation 6.3.9 may be simplified to –
⎛b⎞
⎛ 2H ⎞
⎜ ⎟ = 0.71⎜
⎟
⎝ t ⎠crit
⎝B+H ⎠
−1.78
−0.89
θ max
(6.3.10)
By substituting B = H in Equation 6.3.10, i.e. for square cross-sections –
⎛b⎞
−0.89
⎜ ⎟ = 0.71θ max
⎝ t ⎠crit
(6.3.11)
Figure 6.14 plots Equation 6.3.11 in the (b/t)-θmax space, indicating that lower (b/t) ratios will
result in higher θmax capacities. If an expected drift demand of θmax = 4% is substituted into
Equation 6.3.11, the limiting (b/t) ratio is calculated as 12.5. Shown by the dashed line on Figure
6.13, this is approximately 22% lower than the AISC limiting (b/t) = 16. On the other hand,
substituting the AISC value (b/t) = 16 into Equation 6.3.11 results in θmax = 3.0%, indicating that
although the current design provisions may not guarantee the 4% drift capacity implied by the
code, they may not be as deficient as have been recently suggested (especially when interpreting
results of experiments subjected to symmetric loading protocols).
Also illustrated in Figure 6.14 is an interesting dependence on a cross-section aspect ratio term,
2H/(B+H), for rectangular sections in Equation 6.3.10, where B is the overall length of the
buckling face and H is the distance between the compression (buckling) face and the tension face
(see Figure 6.12). If the cross-section is oriented to buckle about the weak-axis, 2H/(B+H) is less
than one and Equation 6.3.10 implies a more ductile configuration through an amplified (b/t)critical.
For the uncommon condition when the brace buckles about the strong axis, the 2H/(B+H) ratio is
greater than unity and the critical with-thickness ratio is reduced. Referring to Figure 6.12, this
effect can be explained by considering the strain gradient through the depth of the cross section,
H. For equal curvature, a larger H acts to increase the strain demand at the extreme fiber of the
cross-section and promotes a less ductile configuration.
221
Another interesting observation is that, according to Equations 6.3.9-6.3.11, the width-thickness
limit is not dependent on the length or the slenderness of the brace. Since this is somewhat
counterintuitive, it merits additional discussion. Referring to Equation 6.3.3, the brace length LB
affects the axial deformation capacity Δrange in a positive manner. However, when the capacity is
expressed in terms of the story drift (i.e., combining Equation 6.3.3 with Equation 6.3.8), the LB
term cancels out. Physically, this may be explained by considering brace elements of identical
cross-sections included in a large and small braced-frame. The brace in the larger frame will have
a larger LB, and consequently a proportionally larger axial deformation capacity Δrange. If both
these frames are subjected to an equal story drift, the larger brace will be subjected to
proportionally larger deformations, such that both will encounter similar strains in the plastic
hinge region. Furthermore, referring to Figure 6.5c, for identical cross-section b/t ratios and frame
geometries (i.e., equal bracing member lengths), the drift capacity was shown to remain constant
for varying radii of gyration. This can be explained through Equation 6.3.1 such that with an
increase in r for square cross-sections, H and B will be proportionally larger, thereby increasing
the length of the plastic hinge. Thus, this acts to offset the increase in strain which accompanies a
larger cross-section depth H.
While the model presented in this chapter is primarily focused on inelastic buckling behaviors,
future work is recommended to develop expressions for brace lengths which may produce large
slenderness ratios. In the context of the AISC Seismic Provisions (2005), this could allow for
larger (b/t)critical ratios for unusually slender braces. Also, the estimates of the (b/t) limits
presented in Figure 6.14 should be used with caution when applied to slenderness ratios that are
significantly outside the range (less than 31 and greater than 120) of the experimental and
simulation data presented here. For example, as discussed earlier, for KL/r > 118, i.e. elastic
buckling, the mechanisms discussed in this chapter might not be active. Similarly, for very low
KL/r, other mechanisms, such as local buckling under pure axial compression without global
222
buckling, may be active. Additionally, the relationships may not be valid for braces with different
material properties or cross-section types such as Pipe or Wide-Flanged sections. This
experiments synthesized in this investigation were comprised of square and rectangular brace
experiments with average yield strengths of 62 ksi and a coefficient of variation equal to 9%.
Moreover, the simulation study assumes the material yield strength to be approximately 60 ksi
and a fracture toughness of η=5.98 (HSS4x4x1/4 material type from Chapter 5). Using Equation
5.1.15, this fracture toughness corresponds to a very large CVN value of approximately 404.
Thus, in the absence of further study, the results of this chapter are applicable only within the
experimental and simulation parameters investigated. The next section discusses the reliability of
the presented approach.
6.4
SUMMARY AND RELIABILITY OF PARAMETRIC STUDY AND SEMI-THEORETICAL
MODEL
This chapter synthesizes the results of a parametric study and experimental data from ten
independent testing programs to develop improved insights into the fracture capacity of HSS
bracing members. The parametric study suggests trends in ductility as a function of widththickness and slenderness ratio while expanding the results of the experimental program in
Chapter 4 to investigate a wider-range of brace parameters. Applying the same loading history
across all simulations allows for clear relationships to be obtained between ductility and brace
geometry. The simulations also establish the length of the plastic hinge after cross-section local
buckling as the average of the width, B , of the cross-section over which the local buckle forms of
the buckling face and the depth of the brace cross-section, H. This is a critical component in the
development of a semi-theoretical model used to synthesize the results of the experimental and
simulation studies. Perhaps of more interest, however, is the validation of using advanced
simulation techniques to expand the set of experimental data. Referring to Table 6.4 and Figure
6.8, separate least-squares fits on the experimental (63 tests) and simulation (22) data sets
223
provides very similar regression parameters. Even considering the experimental programs of
Table 6.3 with varying loading histories and geometries, the simulation results (obtained with
CVGM fracture predictions) seem quite similar plotted in the εF versus (b/t) space in Figure 6.8.
While the simulations and experimental results suggest certain trends, it is somewhat challenging
to reconcile experimental data from diverse test programs. Therefore, to eliminate the effects of
geometric variables such as brace length and cross-sectional dimensions, a kinematic relationship
is proposed to relate the brace axial deformation capacity, Δrange, to a “fiber” strain-like quantity,
εF, in the central plastic hinge region of the buckling brace. A relatively strong trend is observed
between this strain and the (b/t) ratio and a regressed power-law model is used to express this
relationship. This relationship is then used to determine the brace axial deformation capacity,
Δrange, as a function of the width-thickness ratio, (b/t).
Once the brace axial deformation capacity is determined, either directly from experimental data
or based on the proposed relationship, it is useful to express it in terms of equivalent drift capacity
for meaningful comparison with seismic demands. This includes developing a kinematic
relationship between the brace axial deformation capacity, Δrange, and the equivalent drift θrange.
An important issue associated with this is the unsymmetrical nature of story drift time histories.
Therefore, determining the maximum drift capacity using half of the total range results in
conservative estimates, but is typically done when reporting testing results. Based on observations
from various nonlinear dynamic simulations, a preliminary estimate of θmax = 0.71θrange is
proposed. This is combined with the relationship between Δrange and (b/t), and results in a limiting
function for the maximum (b/t) ratio given a desired level of drift capacity. To examine the
efficacy of this approach, Figure 6.15 plots all the drift capacities from the 63 experiments and 22
simulations against their (b/t) ratios normalized by the critical limit calculated using Equation
224
6.3.10. Following the reasoning outlined previously, the drift capacities are calculated as θmax =
0.71θrange, where θrange = θmax +|θmin| is the range of equivalent drift experienced by the test
specimen.
Plotted on the figure is the horizontal line corresponding to the 4% drift capacity expected from
SCBF systems. Also drawn on the figure is the vertical line corresponding to (b/t)/(b/t)critical = 1,
where (b/t)critical is calculated according to Equation 6.3.10 and is the critical (b/t) ratio required to
meet the drift demand of 4%. While this value is constant for square braces ((b/t)critical = 12.5),
results are plotted in this way because the data points also include those corresponding to nonsquare cross-sections. These two limits divide the space into four quadrants, where the Roman
numerals are followed by the count of data points in that particular quadrant –
•
Quadrant I (22%): Experiments which are “safe” (i.e., do not fail before 4%) and
predicted to be safe (i.e., less than the b/t limit of 1.0)
•
Quadrant II (16%): Experiments which are safe, but predicted to fail
•
Quadrant III (12%): Experiments which fail, but are predicted to be safe
•
Quadrant IV(50%): Experiments which fail, and are predicted to fail
A perfect model would result in 100% of the data points in either Quadrant I or IV where the
experimental data matches model predictions. As shown in Figure 6.15 for the proposed
relationship, approximately 72% of the data points lie in these two Quadrants. In addition, only a
small percentage of data points (12%) lie in Quadrant III where the relationship
(unconservatively) predicts safety. Thus, given the variability amongst the test programs and the
subjectivity in characterizing demands in SCBF systems, the proposed approach is fairly reliable
when evaluated against a large data set.
225
The results presented in this chapter are based on calibrated empirical relationships, and therefore
should be used with caution when applied to brace parameters (especially slenderness ratios
smaller than 60 or greater than 120 and materials dissimilar to those listed in Table 6.3). In
addition, the approach makes several simplifying assumptions, for example, it neglects the effect
of loading history on brace capacity. While these assumptions may not affect the efficacy of the
approach (in an average sense) with respect to a large sample of test data, they may result in
significant errors when applied to individual components. Finally, further study is needed to
accurately quantify both the maximum drift demands, the nature of the deformation histories in
SCBF systems and brace behavior at larger slenderness ratios.
226
Table 6.1: HSS simulation matrix (22 total)
Shape
HSS4x4x1/4
HSS4x4x3/8
HSS4x2x1/4
HSS4x2x3/16
HSS6x6x3/8
HSS6x6x1/2
HSS6x2x3/8
HSS6x2x3/16
HSS8x8x1/2
HSS8x8x5/8
B
4
4
4
2
6
6
6
2
8
8
H
4
4
2
4
6
6
2
6
8
8
B/H
1
1
2
0.5
1
1
3
0.3
1
1
b/t
14.2
8.5
14.2
8.5
14.2
9.9
14.2
8.5
14.2
10.8
LB
60,120,179
60,120,179
62
111
90,180,271,410
90,180,271
61
161
120,241,362,547
120,241,362
Approx. LB/r
40,80,120
40,80,120
80
80
40,80,120
40,80,120
80
80
40,80,120
40,80,120
Table 6.2: Brace member lengths across various frame geometries
Frame Bay Width (feet)
Story Height (feet)
20
16’-5
22’-4
13
20
30
19’-10
25’-0
40
23’-10
28’-3
Table 6.3: Summary of HSS experimental review (63 tests)
Test Program [No.]
Gugerli and Goel [1]
Liu and Goel [2]
Lee and Goel [3]
Archambault, [4]
Tremblay, Filiatrault
Walpole [5]
Shaback and Brown [6]
Yang and Mahin [7]
Fell, Kanvinde, and
Deierlein [8]
Han et al [9]
Lehman et al [10]
Year
Published
1982
1987
1988
Average
Fy,meas
60
54
67
1995
57
1996
2003
56
64
2005
60
2006
69
2007
2008
59
67
Table 6.4: Regression parameter values
Data Set
Experimental
Simulation
Combined
a
0.90
1.13
0.98
b
-0.53
-0.62
-0.56
No. of
Tests
4
3
6
10
4
3
8
3
1
4
1
4
12
General Description of
Cyclic Loading History
Unsymmetric compressive
Unsymmetric compressive
Unsymmetric compressive
Standard symmetric
Unsymmetric
Standard symmetric
Standard symmetric
Standard symmetric
Unsymmetric tensile
Standard symmetric
Unsymmetric compressive
Standard symmetric
Standard symmetric
227
Outside corner
radius = 2t
H
Buckling
face
B
t
Figure 6.1: HSS brace geometry
6
Maximum Considered (4)
Drift (%)
4
2
Expected Buckling (0.2)
0
-2
-4
#cycles
6
6
6
4
2
2
2
2
2+n
-6
0
10
20
30
Cycle Number
Figure 6.2: Brace loading history
Drift (%)
0.08
0.10
0.15
0.20 – B
1.03
1.85
2.68
4.00 - MCE
5.00
228
20
40
80
120
Δrange (in)
15
10
5
(a)
0
5
10
15
20
Width-thickness (b/t)
0.12
40
θrange (rad)
80
120
0.08
0.04
(b)
0
5
10
15
20
Width-thickness (b/t)
Figure 6.3: Influence of width-thickness ratio on brace ductility (fracture initiation) in terms of (a)
Axial deformation range and (b) Story drift.
229
20
Small b/t (<14)
Large b/t (14)
Δrange (in)
15
10
5
(a)
0
0
50
100
150
Slenderness (L/r)
θrange (rad)
0.12
0.08
0.04
Small b/t (<14)
(b)
Large b/t (14)
0
0
50
100
150
Slenderness (L/r)
Figure 6.4: Influence of slenderness ratio on brace ductility (fracture initiation) in terms of (a)
Axial deformation range and (b) Story drift.
230
500
300
(KL/r)max
200
100
300
(KL/r)max
200
100
(a)
0
L=204in
L=281in
400
Slenderness
400
Slenderness
500
L=204in
L=281in
(b)
0
0
5
10
15
0
5
Cross-section size (in)
10
15
Cross-section size (in)
θrange (rad)
0.08
0.04
L=180 in
L=119 in
L=119 (Experimental)
(c)
0
0
50
100
150
200
Slenderness (L/r)
Figure 6.5: Slenderness range for minimum and maximum frame sizes (see Table 6.2) for (a) All
HSS cross-sections and (b) AISC (2005) conforming (i.e., b/t < 16) sections; (c) Influence of
brace slenderness on story drift ductility for fixed b/t (14.2) ratio and brace lengths (119 and 180
in). Also shown are past experimental (b/t=14.2 and L=119 in) results.
231
0.06
0.5 x θrange (rad)
KL/r ~ 80
0.03
(a)
0.00
0
10
20
30
40
Width-thickness, b/t
0.06
0.5 x θrange (rad)
b/t ~ 14.2
0.03
(b)
0.00
0
50
100
150
200
Slenderness, KL/r
Figure 6.6: Experimental brace ductility in terms of drift versus brace parameter for similar (a)
Slenderness (19 tests) and (b) Compactness (25 tests) ratios.
232
300
Force (k)
200
100
0
-100
-200
-1.5
Δrange
-1
-0.5
0
0.5
1
Axial Displacement (in)
Strong relationship
From experiments
fracture
Φ hinge
= Lh ,pred .
Δrange
H
. ε F → f (b / t )
2
From regression
δ
Ф/2
(= 0.5(H+B), based on
FEM simulations)
LB
Figure 6.7: Schematic illustration of simplified approach to evaluate effect of brace parameters on
ductility.
233
εF
0.4
Experiments
Simulations
Experiment Fit
Simulation Fit
Combined Fit
0.2
(a)
0
0
10
20
30
40
b/t
Experiments
Simulations
0.5 x θrange (rad)
0.06
0.03
(b)
0.00
0
10
20
30
40
b/t
Figure 6.8: (a) Fiber-like fracture strain and (b) Deformation range capacity versus b/t ratio for all
experiments and simulations listed in Tables 6.1 and 6.3.
234
Δ range,exp/Δ range,pred
2
Experiments
Simulations
1
0
0
2
4
6
8
10
S
12
Test Program
Figure 6.9: Comparison between maximum experimental and predicted deformation range.
235
φ(x)
LB
= ∫ φ ( x)dx
φmax
0
x
LB
Lh
(a)
φ(x)
φmax
LB
= ∫ φ ( x)dx
0
x
LB
Lh
(b)
Figure 6.10: Schematic illustration of plastic hinge length calculation where the total rotation is
equated to an equivalent area defined by φmax and Lh. Shown above is the curvature profile (a)
Before and (b) After local buckling.
236
6
4
40
80
120
Lh/Lh,est
Lh/Lh,est
4
40
80
120
2
2
(b)
(a)
0
0
0
0.02
0.04
0.06
0
0.02
θ (rad)
0.06
0.04
0.06
θ (rad)
4
4
Lh/Lh,est
Lh/Lh,est
0.04
2
(c)
0
0
2
(d)
0
0.02
0.04
θ (rad)
0.06
0
0.02
θ (rad)
Figure 6.11: Plastic hinge length as a function of increasing drift for (a) HSS4x4x1/4, (b)
HSS6x6x3/8, (c) HSS4x2x1/4 and (d) HSS4x2x3/16 analyses.
237
H
B
Lh
(a)
m
B
Lh
B/2
H/2
(b)
Figure 6.12: Continuum finite element simulation of brace specimen showing plastic hinge length
dimension in (a) Isometric and (b) Top view.
238
300
Experiment
Continuum Model
200
Force (k)
100
0
-100
(a)
-200
-2
-1
0
1
2
Axial Deformation (in)
300
Experiment
Fiber Model
Force (k)
200
100
0
-100
(b)
-200
-2
-1
0
1
2
Axial Deformation (in)
8
Continuum Model
Fiber Model
Lh/Lh,est
6
4
2
(c)
0
0
0.02
0.04
0.06
θ (rad)
Figure 6.13: (a and b) Force-deformation and (c) Plastic hinge length comparison between finite
element and fiber brace models.
239
0.10
θmax (rad)
B=H
B/H=2
B/H=0.5
0.05
0.03
12.5
0.00
0
5
10
15
20
25
b/t
Figure 6.14: Maximum drift capacity versus width-thickness ratio.
0.71θrange (rad)
0.09
I: 19
safe/predicted
safe
II: 14
safe/predicted
to fail
III: 10
IV: 43
fail/predicted
to fail
0.06
0.03 fail/predicted
safe
0
0
0.5
1
1.5
2
2.5
(b/t)/(b/t)critical
Figure 6.15: Deformation capacity (in terms of drift), divided by 1.4, versus normalized widththickness ratio.
240
Chapter 7
Summary, Conclusions and Future Work
This chapter summarizes various findings and conclusions from the study, while discussing areas
that need further examination. Within the broad context of simulating structural fracture using
advanced modeling methods, this dissertation had three specific goals (1) Investigation of the
cyclic inelastic buckling and fracture of steel braces in concentrically braced frames (2)
Development of a methodology wherein novel physics-based models are evaluated for full-scale
structural components (3) Application of this methodology for generalizing experimental data
sets to inform design considerations, while demonstrating the effectiveness of micromechanicsbased fracture modeling.
The bracing members investigated in this study are representative of those in Special
Concentrically Braced Frames (SCBFs) and are subjected to loading histories which impose
realistic seismic demands across a wide-range of ground motion intensities. Square Hollow Steel
Sections (HSS), steel Pipe and Wide-Flanged cross-sections are investigated in a rigorous testing
matrix which investigates a range of width-thickness and slenderness ratios along with unique
conditions such as grout-filled HSS, earthquake-rate loading and unsymmetrical buckling. In
addition to the practical data generated by the large-scale tests, the experimental specimens also
241
serve as a test-bed to validate a micromechanics-based modeling approach in large-scale details.
The Cyclic Void Growth Model (or CVGM), previously developed by Kanvinde and Deierlein
(2007), is evaluated using the experimental observations from the brace tests during earthquaketype cyclic loading. Since a continuum-based model is used, the accuracy of the modeling
approach is contingent on accurate characterization of stress and strain histories in the presence of
phenomena such as local and global buckling.
This chapter reviews the conclusions of the large-scale experimental program along with a
summary of the model validation and parametric simulation studies. Limitations of the results and
method are discussed, leading to recommendations for future work.
7.1
SUMMARY AND CONCLUSIONS
The following three sections present a summary of the previous chapters. First, general
conclusions from the large-scale experimental study are presented. This is followed by a section
focusing on the fracture prediction methodology for the large-scale brace specimens. Finally, the
parametric simulation study on square and rectangular HSS sections, as well as the semitheoretical model used to generalize the experimental and simulation results, is reviewed.
7.1.1
Large-Scale Brace Experiments
Results of the large-scale testing program on nineteen bracing elements subjected to earthquaketype cyclic loading are reported in Chapters 3 and 4. The experiments featured brace specimens
detailed according to current codes, and were subjected to various types of cyclic loading
histories designed to reflect realistic seismic demands. The testing matrix included a diverse
blend of parameters including cross section width-thickness, slenderness, type of cross section,
loading history, loading rate and special details such as grout filled braces. Various limit states,
242
such as local buckling, fracture initiation and loss of strength were monitored, and related to
system drift levels.
The braces subjected to cyclic loading failed due to fracture at the center, which was triggered by
strains highly amplified due to local buckling. Consequently, cross section width-thickness ratios
were found to strongly influence brace ductility for all cross sections, and higher width-thickness
ratios resulted in a severe decrease in ductility. Importantly, in some experiments, current AISC
limits for width-thickness ratios could not ensure acceptable performance, resulting in fracture at
unacceptably low story drift levels.
In addition to width-thickness, slenderness was determined to be another important factor
affecting brace fracture, in that more slender braces suffered relatively lower levels of inelasticity,
delaying fracture. In fact, the axial deformation at fracture was determined to be governed by a
combination of slenderness and width-thickness. For example, the wide-flange section with an
undesirable width-thickness ratio exhibited excellent ductility, due to its very high slenderness
(above the elastic limit) and the less severe nature of the local buckling shape as compared to
HSS sections. Since brace slenderness is a system level design variable, it might not be feasible to
provide large slenderness with the sole intent to prevent fracture. Moreover, large slenderness can
reduce energy dissipation in the brace, and place excessive tensile demands on connections (due
to overstrength). In addition, when the axial deformation capacity is expressed in terms of
equivalent story drift, the positive effects of slenderness tend to be diminished for slenderness
ratios below 120 (see Chapter 6 discussion of HSS simulation study).
In addition to slenderness and width-thickness, various other factors were considered. Of these,
the nature of the local buckling shape was found to differ in severity across the various crosssection types (HSS, Pipe and Wide-Flanged). The square HSS local buckling shapes were found
243
to be particularly severe while the Pipe and Wide-Flange shapes developed more gradual
buckling shapes. However, this does not imply that Pipe and Wide-Flange behavior are naturally
superior to HSS (as illustrated with the Pipe5STD tests described in Chapter 4). Filling the braces
with concrete resulted in a somewhat larger ductility in one of two tests, but given the logistical
challenges to this, it may be more economical to achieve similar levels of ductility by using either
a more compact shape or an alternate cross section. Rate effects were examined and determined
to be relatively unimportant.
Connection performance regarding net section fracture at slotted brace-ends was investigated by
subjecting these to tension dominated near-fault loading histories with a large initial tensile pulse.
These tests, conducted for pipe sections and one wide-flange section, confirmed previous findings
that net section reinforcement increases ductility substantially and prevents fracture at the
connection. In fact, for the pipe specimens, the large difference between yield and ultimate
strengths resulted in large ductilities even for unreinforced connections. Overall, the variations in
the expected versus nominally specified material properties demonstrate the degree to which the
net section fracture response may differ between different structures. The test data did confirm
that the expected yield strength (RyFyAg) and the expected ultimate strength (RtFuAg) tend to
bracket the maximum measured strength fairly well. Furthermore, an accurate prediction of the
maximum brace tensile force was found to be the average of the expected and ultimate strengths.
7.1.2
CVGM Evaluation in Large-Scale Details
Referring to Chapter 5, the Cyclic Void Growth Model (CVGM) developed by Kanvinde and
Deierlein (2007) is used to predict the instant of fracture initiation in the large-scale experimental
brace specimens. Smooth (and ASTM-specified flat for HSS wall material) tensile coupons along
with circumferentially notched bars are tested to calibrate the material constitutive model
parameters as well as the CVGM fracture parameters (η and λ). Large-scale finite-element brace
244
simulations are employed to investigate the stress and strain histories at the critical fracture
location during global and local buckling mechanisms. The simulations incorporate global and
local imperfections, as well as multiaxial von-Mises plasticity with isotropic and kinematic
hardening.
The CVGM fracture criterion is expressed as ηcyclic ≥ exp(λ.D)η, where ηcyclic is a void growth
function and D is the damage, assumed to be the equivalent plastic strain at a load reversal point
marked my a switch from negative to positive triaxiality. While the fracture parameter η can be
calibrated conveniently and accurately through monotonic notched bar experiments, calibrating
the λ-parameter is more challenging. In fact, for two out of the four steels examined in the current
study, fracture predictions of the braces based on a λ-value calibrated from notched bar tests were
highly inaccurate. A closer inspection revealed that the notched bar specimens failed at low
damage levels (owing to the high triaxiality present in the notched bars), relative to the braces,
which had lower triaxialities. While the low triaxialities in the braces may trigger entirely distinct
fracture mechanisms (such as void shearing), an analysis of the data indicates that that the source
of the error is most likely in the selection of calibration tests themselves, rather than the damage
model. Thus, for future use of the CVGM it is recommended that the cyclic tests sample large
damage levels, or appropriately sample the magnitude of damage in the specimen where the
CVGM will be applied.
The CVGM fracture predictions are also presented in a probabilistic framework incorporating the
effect of material uncertainty. The analysis indicates that, for all specimens, the probability of
failure increases sharply 2-3 cycles after local buckling is observed in the simulations (for the
standard loading histories). This illustrates the strong influence of local buckling on the CVGM
predictions and the importance of simulating it accurately. In fact, the quick succession of
245
fracture following local buckling indicates that it may not be as important to simulate fracture, if
local buckling is accurately modeled. However, the sensitivity of local buckling initiation to
material model parameters is not explicitly investigated in this study.
7.1.3
Parametric Simulation of Bracing Members
A parametric simulation study on square and rectangular bracing members (total of 22 analyses)
is performed to expand the experimental data set presented in Chapter 4 and demonstrate the
applicability of advanced simulations methods. The simulations, which are similar to the ones
used for the CVGM evaluation, incorporate large deformations, multi-axial plasticity and brick
elements. The Cyclic Void Growth Model (CVGM) is used to predict the fracture initiation
instances of the 22 simulations and investigate a wide-range of parameters which affects brace
failure. The finite-element simulations are also used to determine the length of the plastic hinge
after cross-section local buckling. Referring to Chapter 6, the plastic hinge length is found to be
approximately equal to the average of the width of the cross-section, B, over which the local
buckle forms of the buckling face and the depth of the brace cross-section, H.
Experimental and simulation data from this study is synthesized with experimental data from 10
previous testing programs to present a semi-theoretical relationship that may be used to predict
the fracture deformation of HSS braces directly from brace properties such as cross-sectional
dimensions and slenderness ratios. The relationship is shown to reconcile fracture data from
several diverse experimental programs and simulation studies, which encompass a wide range of
test variables. The use of this relationship to develop design considerations is presented as an
illustrative example. Specifically, for the range of b/t and KL/r ratios investigated in this study,
the semi-theoretical relationship reveals 1) Overall slenderness has a minimal effect on ductility
when expressed in terms of a story drift and 2) Weak axis bending (H < B) leads to a more ductile
configuration.
246
7.2
FUTURE WORK
This dissertation presents a large and small-scale experimental testing program, nonlinear finite
element modeling, the evaluation of a physics-based fracture criterion at the large-scale, and the
development of a semi-theoretical model to estimate brace ductility. The results raise some
interesting scientific questions regarding the feasibility of using micromechanics-based models to
predict fracture in full-scale components. In addition to the modeling aspects, the study also
motivates further examination of several practical issues. All of these issues are now summarized-
1. Development of appropriate loading protocols for braced frames: While this study
presented a loading history based on a modified SAC loading protocol, it was not
developed through rigorous nonlinear dynamic analyses of braced-frame systems
(comparable to the SAC loading history for moment frames). Furthermore, analyses by
Uriz (2005) and others (McCormick et al 2007) suggest that a symmetric loading history
for braced-frame components may not be appropriate due the unsymmetrical response of
braced-frames during cyclic loading. Thus, the experimental results presented in Chapter
4 with respect to brace fracture ductility may be over-conservative in light of the
symmetric loading history used in this (and previous) studies.
2. Characterizing braced-frame demands: Considering the previous discussion, with
advances in structural modeling techniques and increasing computational power, there is
an opportunity to better characterize braced-frame demands across varying building
geometries, such as frame and story height, as well as brace properties (i.e., slenderness
ratios) and system configurations (Chevron, cross-bracing, etc.). This will assist in
performance comparisons between brace ductilites and expected seismic demands.
3. Experimental investigation of alternate cross-sections: Considering the recent popularity
of Pipe and Wide-Flanged sections over HSS, a comprehensive experimental or
247
simulation study (similar to Chapter 6) is warranted for these shapes. While the
experiments conducted as part of this investigation expanded the database of Pipe and
Wide-Flanged tests, there is a need to better characterize the ductility of bracing members
with these cross-sections.
4. Using the CVGM to inform fiber-based fracture criterion: Referring to Chapter 2, the
efficiency of fiber-based elements for earthquake engineering analyses has contributed to
their popularity in modeling bracing, and other structural, components. Following the
approach developed by Uriz (2005), an empirical-based fracture criterion could be
calibrated with the CVGM and then used in the analyses of full-scale braced-frames. This
would allow for “on-the-fly” analysis procedures by combining demand and capacity
characterizations which are typically separate in the context of fracture events.
Referring to the previous section and Chapter 5, several limitations of the CVGM have been
highlighted by applying the model in the context of brace fracture –
1. Low triaxiality: The low triaxiality ranges observed in the brace tests may invalidate the
underlying assumptions of the CVGM. Considering the development of the CVGM (at
the small-scale) did not investigate these low triaxiality ranges (Kanvinde and Deierlein,
2007), the micromechanical fracture behavior of the braces may not be governed by the
void growth and cyclic damage mechanisms assumed by the CVGM. Thus, it is
important to investigate the fatigue and fracture mechanisms which are triggered in the
presence of a low triaxiality stress state. This may motivate the need for alternative
damage models other than the exponential function presented in Chapter 5.
2. Crack Propagation: Although not explicitly mentioned in Chapter 4, the brace tests with
near-field loading histories demonstrated that cross-section tensile strength loss may not
occur immediately after fracture initiation (refer Appendix A). This is mostly attributed
248
to the loading history offset at 3% drift for these histories such that the stress and strain
state needed to propagate the crack does not develop as rapidly during near-field loading
as compared to the more severe tension histories in the far-field tests. Thus, the
simulation of ductile propagation events may provide a more reliable measure of brace
ductility. Considering the unsymmetrical behavior of braced-frames, this may prove
useful in characterizing the various performance limit states of bracing members.
249
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256
Appendix A
Figure A.1 illustrates the loading histories and load-deformation plots for the nineteen brace
experiments. The limit states of global and local buckling, fracture initiation, and strength loss are
reported on each figure in terms of story drift, while the stiffness and maximum tensile and
compressive forces are shown on the hysteretic plots. The test numbers correspond to Table 3.1
and titles are also provided to distinguish the specimens, loading histories, and other attributes.
Figure A.1: Experimental loading histories and brace hysteretic response (continued on next
page)
257
Figure A.1: Experimental loading histories and brace hysteretic response (continued on next
page)
258
Figure A.1: Experimental loading histories and brace hysteretic response (continued on next
page)
259
Figure A.1: Experimental loading histories and brace hysteretic response (continued on next
page)
260
Figure A.1: Experimental loading histories and brace hysteretic response (continued on next
page)
261
Figure A.1: Experimental loading histories and brace hysteretic response (concluded on next
page)
262
Figure A.1: Experimental loading histories and brace hysteretic response
263
Appendix B
This appendix presents experimental results from past tests on large-scale bracing members. The
table summarizes the brace geometries and deformation capacities for each experiment used in
the development of the semi-theoretical model presented in Chapter 6. It should be noted that the
deformation capacities may differ slightly from the true experimental results owing to the
difficulty of assimilating data from diverse testing programs.
Table B.1: Experimental results from square and rectangular HSS tests (continued on next page)
Test Program (Year)
[Material]
Gugerli and Goel (1982)
[ASTM-A500, Gr. B]
Liu and Goel
(1987)
[ASTM-A500, Gr. B]
Lee and Goel
(1988)
[ASTM-A500, Gr. B]
Archambault, Tremblay,
Filiatrault
(1995)
[G40.21-350W]
Test
I.D.
TW2
TW3
TW4
TW6
T633H
T424H
T422H
1
2
4
5
6
7
S1A
S1B
S2A
S2B
S3A
S3B
S4A
S4B
S5A
S5B
S1QA
S1QB
S4QA
S4QB
Brace Properties
Shape
HSS5x3x1/4
HSS4x2x1/4
HSS7x5x1/4
HSS6x3x3/16
HSS6x3x3/16
HSS4x2x1/4
HSS4x2x1/8
HSS5x5x3/16
HSS5x5x3/16
HSS4x4x1/8
HSS4x4x1/4
HSS4x4x1/4
HSS4x4x1/4
HSS5x3x3/16
HSS5x3x3/16
HSS4x3x3/16
HSS4x3x3/16
HSS3x3x3/16
HSS3x3x3/16
HSS5x2.5x3/16
HSS5x2.5x3/16
HSS4x3x1/4
HSS4x3x1/4
HSS5x3x3/16
HSS5x3x3/16
HSS5x2.5x3/16
HSS5x2.5x3/16
B
5
4
7
6
6
4
4
5
5
5
4
4
4
3
3
3
3
3
3
2.5
2.5
3
3
3
3
2.5
2.5
H
3
2
5
3
3
2
2
5
5
5
4
4
4
5
5
4
4
3
3
5
5
4
4
5
5
5
5
LB
136
138
132
136
113
120
120
116
126
126
122
130
130
181
181
182
182
182
182
182
182
182
182
181
181
182
182
tgusset
Not
Used
0.63
0.63
0.50
0.63
0.63
0.50
0.63
0.63
0.63
0.44
0.44
0.44
0.44
0.38
0.38
0.44
0.44
0.50
0.50
0.44
0.44
0.43
0.43
Deformation
Capacity
Δmax
Δmin
0.57
-1.7
0.78
-2.55
0.57
-1.7
0.57
-1.7
0.43
-1.70
0.85
-2.55
0.68
-1.70
0.33
-0.85
0.32
-0.85
0.21
-2.13
0.24
-3.40
0.21
-2.13
0.32
-1.70
1.70
-1.70
1.70
-1.70
2.52
-2.52
2.39
-2.39
3.31
-3.31
3.32
-3.32
1.52
-1.52
1.80
-1.80
3.28
-3.28
2.70
-2.70
1.69
-1.69
1.81
-1.81
1.52
-1.52
1.80
-1.80
264
Table B.1: Experimental results from square and rectangular HSS tests
Walpole
(1996)
[AS1163-C350]
Shaback and Brown
(2001)
[G40.21-350W]
Yang and Mahin
(2006)
[ASTM-A500, Gr. B]
Han and Foutch
(2007)
[ASTM-A500, Gr. B]
Fell, Kanvinde, Deierlein
(2007)
[ASTM-A500, Gr. B]
Lehman, Roeder, Herman,
Johnson, Kotulka
(2008)
[ASTM-A500, Gr. B]
RHS1
RHS2
RHS3
1B
2A
2B
3A
3B
3C
4A
4B
1
3
4
5
85-14A
70-18
82-19
77-28
HSS1-1
HSS1-2
HSS1-3
HSS2-1
HSS2-2
HSS-2
HSS-3
HSS-4
HSS-5
HSS-6
HSS-7
HSS-8
HSS-9
HSS-10
HSS-11
HSS-12
HSS-13
HSS6x4x1/4
HSS6x4x1/4
HSS6x4x1/4
HSS5x5x5/16
HSS6x6x5/16
HSS6x6x3/8
HSS5x5x1/4
HSS5x5x5/16
HSS5x5x3/8
HSS6x6x5/16
HSS6x6x3/8
HSS 6x6x3/8
HSS 6x6x3/8
HSS 6x6x3/8
HSS 6x6x3/8
HSS4x4x1/4
HSS5x5x1/4
HSS4x4x3/16
HSS4x4x1/8
HSS4x4x1/4
HSS4x4x1/4
HSS4x4x1/4
HSS4x4x3/8
HSS4x4x3/8
HSS5x5x3/8
HSS5x5x3/8
HSS5x5x3/8
HSS5x5x3/8
HSS5x5x3/8
HSS5x5x3/8
HSS5x5x3/8
HSS5x5x3/8
HSS5x5x3/8
HSS5x5x3/8
HSS5x5x3/8
HSS5x5x3/8
6
6
6
5
6
6
5
5
5
6
6
6
6
6
6
4
5
4
4
4
4
4
4
4
5
5
5
5
5
5
5
5
5
5
5
5
4
4
4
5
6
6
5
5
5
6
6
6
6
6
6
4
5
4
4
4
4
4
4
4
5
5
5
5
5
5
5
5
5
5
5
5
106
80
99
134
157
157
173
173
173
193
192
115
115
115
115
133
133
135
132
118
118
118
118
118
162
162
158
162
162
153
166
162
161
153
139
153
Not
Used
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
0.63
0.63
0.63
0.63
0.50
0.50
0.50
0.50
0.50
0.50
0.50
0.50
0.38
0.38
0.88
0.38
0.50
0.50
0.88
0.50
0.50
1.39
0.97
0.49
1.36
1.25
1.31
1.02
1.47
1.34
1.16
1.18
1.55
1.25
2.90
1.93
1.33
1.00
0.67
0.33
1.59
1.18
1.24
3.00
2.50
1.53
1.53
1.53
1.73
1.73
1.33
2.35
1.33
1.94
1.12
1.43
1.94
-1.39
-0.97
-0.49
-2.67
-2.03
-2.58
-2.91
-2.49
-3.78
-2.95
-3.56
-1.90
-1.01
-0.70
-1.51
-1.33
-1.00
-0.67
-0.33
-1.59
-3.54
-1.77
-3.00
-2.50
-2.14
-3.06
-2.96
-3.16
-3.06
-2.86
-2.65
-2.45
-2.55
-1.53
-2.14
-2.14