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DRIFT AND YIELD MECHANISM BASED SEISMIC DESIGN AND
UPGRADING OF STEEL MOMENT FRAMES
by
Sutat Leelataviwat
A dissertation submitted in partial fulfillment of the requirements for the degree of
Doctor of Philosophy (Civil Engineering)
in The University of Michigan 1998
Doctoral Committee: Professor Subhash C. Goel, Co-chair Assistant Professor Bozidar Stojadinovic, Co-chair Professor William J. Anderson Professor Antoine E. Naaman
ii
Dedicated to my parents and my brothers;
Santi, Surang, Sutee, and Surat Leelataviwat.
iii
ACKNOWLEDGMENTS
The author wishes to express his profound gratitude to Professor Subhash C.
Goel, co-chairman of the doctoral committee, for providing guidance and care both
personally and professionally throughout the course of this study at the University of
Michigan. The author is deeply appreciated for countless hours that he spent mentoring
the author, without which this dissertation could not have been completed. Appreciation
is also extended to Professor Bozidar Stojadinovic, co-chairman of the doctoral
committee, for his invaluable guidance throughout the course of this study. The author
also wishes to express his sincere thanks to his doctoral committee members, Professor
Antoine E. Naaman and Professor William J. Anderson for their helpful suggestions.
The author is most indebted to his parents and his brothers for their love and
encouragement throughout his study, or in fact, throughout his life. The author can not
find any proper words to describe his appreciation. The author also acknowledges the
Rackham predoctoral fellowship from the School of Graduate Studies at the University of
Michigan for their financial support.
This study was greatly facilitated by the generous help from many of the author’s
colleagues in the Department of Civil and Environmental Engineering, who over the
years have become the author’s close friends. The author would like to thank those
friends, notably Dr. Kyoung-Hyeog Lee, Dr. Madhusudan Khuntia, and Arnon
Wongkaew. The help from the technicians at the Structures Laboratory, Robert Spence
and Robert Fischer, is also greatly appreciated.
Last, but not least, the author would like to express special appreciation to
Amornratana Charuratna, Chonawee Supatgiate, and Supana Saivongnual, for their
sincere and wonderful friendship that makes his experience in Ann Arbor a memorable
one.
iv
TABLE OF CONTENTS
DEDICATION................................................................................................................... ii
ACKNOWLEDGMENTS ...............................................................................................iii
LIST OF TABLES ..........................................................................................................vii
LIST OF FIGURES ......................................................................................................... ix
LIST OF APPENDICES................................................................................................. xv
NOTATION .................................................................................................................... xvi
CHAPTER
1. INTRODUCTION............................................................................................. 1
1.1 Background and Motivation................................................................. 1 1.2 Objectives and Organization of the Dissertation.................................. 2
2. A REVIEW OF SEISMIC DESIGN OF STEEL MOMENT
FRAMES............................................................................................................ 6
2.1 Introduction .......................................................................................... 6 2.2 Equivalent Lateral Static Force Procedure (UBC-1994)...................... 7
2.2.1 Design Base Shear............................................................... 7 2.2.2 Distribution of Lateral Forces ............................................. 9 2.2.3 Drift Requirements ............................................................ 10 2.2.4 Beam and Column Strength Requirements for
Controlling the Collapse Mode ......................................... 10 2.3 Equivalent Lateral Static Force Procedure (UBC-1997).................... 11 2.4 Review of Related Research .............................................................. 13
2.4.1 Experimantal Studies......................................................... 13 2.4.2 Analytical Studies ............................................................. 14
2.5 The Study Building ............................................................................ 17 2.6 Nonlinear Analyses of the Study Building ........................................ 20
2.6.1 Methods of Analysis.......................................................... 20 2.6.2 Analytical Modeling of the Study Building ...................... 24 2.6.3 Nonlinear Static Pushover Analysis .................................. 26 2.6.4 Nonlinear Dynamic Analysis ............................................ 29
2.7 Summary and Concluding Remarks .................................................. 35
v
3. DRIFT AND YIELD MECHANISM BASED DESIGN OF MOMENT FRAMES...................................................................................... 38
3.1 Introduction ........................................................................................ 38 3.2 Principle of Energy Conservation ...................................................... 40 3.3 Input Energy in Multi-Degree of Freedom Systems .......................... 42 3.4 Energy-Based Design Base Shear ...................................................... 43
3.4.1 Design Energy Level ......................................................... 43 3.4.2 Design Base Shear for Ultimate Response........................ 45 3.4.3 Design Base Shear for Serviceability ............................... 52
3.5 Plastic Design of Moment Frames ..................................................... 55 3.5.1 Design of Beams ............................................................... 57 3.5.2 Design of Columns............................................................ 59
3.6 Parametric Study of the Proposed Design Procedure......................... 64 3.6.1 Variation in Number of Stories ......................................... 64 3.6.2 Variation in Design Target Drift ....................................... 72
3.7 Comparison between the Current and the Proposed Design Procedures.......................................................................................... 78
3.7.1 Comparison of Seismic Response ..................................... 78 3.7.2 Comparison of Design Forces ........................................... 83
3.8 Performance-Based Plastic Design .................................................... 84 3.9 Summary and Concluding Remarks................................................... 88
4. SEISMIC UPGRADING OF MOMENT FRAMES USING
DUCTILE WEB OPENINGS........................................................................ 92
4.1 Introduction ........................................................................................ 92 4.2 Concept of Moment Frames with Web Openings .............................. 93 4.3 Testing of Steel Beams with Openings .............................................. 97
4.3.1 Test Set-Up........................................................................ 97 4.3.2 Instrumentation and Test Procedure.................................. 99 4.3.3 Material Properties .......................................................... 100 4.3.4 Specimen 1 ...................................................................... 100 4.3.5 Specimen 2 ...................................................................... 108 4.3.6 Specimen 3 ...................................................................... 111 4.3.7 Specimen 4 ...................................................................... 116 4.3.8 Specimen 5 ...................................................................... 119
4.4 Analysis of Test Data ....................................................................... 124 4.4.1 Overstrength of the Diagonal Members .......................... 125 4.4.2 Overstrength of the Chord Members............................... 126 4.4.3 Ultimate Shear Strength of the Openings........................ 129 4.4.4 Modeling of the Openings under Cyclic Loading ........... 132
4.5 Summary and Concluding Remarks ................................................ 134
vi
5. SEISMIC DESIGN AND BEHAVIOR OF MOMENT FRAMES WITH DUCTILE WEB OPENINGS.......................................................... 137
5.1 Introduction ...................................................................................... 137 5.2 Proposed Design Approach .............................................................. 137
5.2.1 Design of Chord Members .............................................. 139 5.2.2 Design of Diagonal Members ......................................... 140 5.2.3 Design of Vertical Member ............................................. 140 5.2.4 Design of Welds .............................................................. 141 5.2.5 Required Strength of the Opening under Gravity
Loads ............................................................................... 141 5.2.6 Detailing of the Opening ................................................. 142
5.3 The Study Building .......................................................................... 142 5.4 Nonlinear Analyses of the Study Building....................................... 145
5.4.1 Inelastic Static Pushover Analysis .................................. 147 5.4.2 Inelastic Time-History Dynamic Analysis ...................... 149
5.5 Experimental Program...................................................................... 153 5.5.1 Test Set-Up...................................................................... 153 5.5.2 Design of the Girder and the Web Opening .................... 158 5.5.3 Instrumentation and Test Procedure................................ 160 5.5.4 Material Properties .......................................................... 162 5.5.5 Test Results ..................................................................... 162
5.6 Evaluation of the Proposed Design Procedure and the Analytical Modeling ........................................................................ 167
5.7 Summary and Concluding Remarks................................................. 170
6. SUMMARY AND CONCLUSIONS........................................................... 173
6.1 Summary .......................................................................................... 173 6.1.1 Introduction ..................................................................... 173 6.1.2 Conventional Moment Frame Behavior .......................... 174 6.1.3 Drift and Yield Mechanism Based Design...................... 175 6.1.4 Seismic Upgrading with Beam Web Openings ............... 177 6.1.5 Seismic Behavior of Upgraded Frames........................... 179
6.2 Concluding Remarks and Suggested Future Studies ...................... 181 6.2.1 Drift and Yield Mechanism Based Design...................... 181 6.3.2 Moment Frames with Ductile Web Openings ................. 182
APPENDICES ............................................................................................................... 183
BIBLIOGRAPHY ......................................................................................................... 201
vii
LIST OF TABLES
Table
2.1. Floor Masses of the Study Building ............................................................... 17
2.2. UBC Design Lateral Forces for the Original frame ....................................... 20
2.3. Design Story Shears and Story Drifts ............................................................ 20
2.4. Characteristics of Earthquake Records .......................................................... 24
3.1. Design Parameters (2% Drift Limit) .............................................................. 65
3.2. Design Lateral Forces (in kips) ...................................................................... 66
3.3. Design Parameters .......................................................................................... 72
3.4. Design Lateral Forces (in kips) ...................................................................... 72
3.5. Performance Criteria ...................................................................................... 86
3.6. Earthquake Design Levels.............................................................................. 86
4.1. Average Yield Stress of Key Members ........................................................ 100
4.2. Shear Force Contributed by Chord Members .............................................. 131
4.3. Shear Force Contributed by Diagonal Members .......................................... 131
4.4. Comparison between Expected and Experimental Ultimate Shear Strength ........................................................................................................ 131
5.1. Design of Web Openings ............................................................................. 145
5.2. Member Sizes of the Modified Frame with Web Openings......................... 145
5.3. Comparison Between Design and Attained Overstrength Values ............... 149
5.4. Average Yield Stress of Key Members ........................................................ 162
A1. Distribution of Beam Strength ..................................................................... 187
B1. Weights of the Equivalent One-Bay Frame.................................................. 192
B2. Design Lateral Forces ................................................................................... 193
B3. Calculation of Beam Proportioning Factors ................................................. 193
viii
B4. Minimum Weight Beam Sections................................................................. 194
B5. Lateral Forces at Ultimate Drift Level.......................................................... 195
B6. Axial Forces in an Exterior Column (kips) .................................................. 196
ix
x
LIST OF FIGURES
Figure
1.1. Organization of the Dissertation..................................................................... 3
2.1. Plan View of the Study Building.................................................................. 18
2.2. A Typical Three-Bay Moment Frame in the N-S Direction ........................ 19
2.3. Scaled Pseudo-Velocity Spectra of the Earthquakes Used in This Study (5% Damping).................................................................................... 22
2.4. Four Selected Earthquakes Used in this Study............................................. 23
2.5. The Original Frame and the Equivalent One-Bay Idealized Model............. 25
2.6. Base Shear - Roof Drift Response from Pushover Analysis ........................ 27
2.7. Sequence of Inelastic Activity from Pushover Analysis .............................. 27
2.8. Distribution of Beam Moment in Columns at the Second Floor Joint......... 29
2.9. Maximum Floor Displacements due to the Four Selected Earthquakes ...... 30
2.10. Maximum Story Drifts due to the Four Selected Earthquakes..................... 31
2.11. Location of Inelastic Activity and Rotational Ductility Demands due to the Four Selected Earthquakes ..................................................................... 31
2.12. Roof Displacement Time Histories under the Four Selected Earthquakes. ................................................................................................. 33
2.13. Distribution of Column Strength along the Height ...................................... 34
2.14. Maximum Column Moments Due to the Four Selected Earthquakes.......... 35
3.1. Typical Response of Structures .................................................................... 39
3.2. Design Pseudo-Acceleration and Pseudo-Velocity Spectra (UBC-94)........ 44
3.3. Equivalent One-Bay Frame at Mechanism State ......................................... 46
3.4. Drift and Yield Mechanism Based Design Base Shear Coefficients ........... 51
3.5. Expected Response of a Structure Designed to Satisfy Serviceability ........ 54
3.6. Frame with Global Mechanism .................................................................... 56
xi
3.7. Frame with Soft-Story Mechanism .............................................................. 58
3.8. Free Body Diagram of the Column in the Equivalent One-Bay Frame ....... 60
3.9. Typical Story of the Study Frames............................................................... 64
3.10. Member Sizes of the 2-, 6-, and 10-Story Frame with 2% Target Drift ...... 66
3.11. Base Shear versus Roof Drift Response of the Study Frames ..................... 68
3.12. Location of Inelastic Activity in the Three Frames at 3% Roof Drift.......... 69
3.13. Maximum Story Drifts of the 2-, 6-, and 10-Story Frames .......................... 70
3.14. Distribution of Maximum Story Shears from Dynamic Analyses ............... 71
3.15. Three Six-Story Frames with 1.5%, 2.5%, and 3% Target Drifts ................ 73
3.16. Base Shear versus Roof Drift Response of the Study Frames ..................... 74
3.17. Location of Inelastic Activity in the Three Study Frames at 3% Roof Drift .............................................................................................................. 74
3.18. Maximum Story Drifts under the Four Selected Earthquakes ..................... 75
3.19. Comparison between Design Target and Attained Maximum Drifts........... 76
3.20. Distribution of Story Shears from Dynamic Analyses ................................. 77
3.21. Member Sizes of the Original Frame and the Redesigned Frame................ 78
3.22. Base Shear versus Roof Drift of the Original and the Redesigned Frames .......................................................................................................... 80
3.23. Sequences of Inelastic Activity under Increasing Lateral Forces ................ 81
3.24. Maximum Story Drifts of the Original and the Redesigned Frames............ 82
3.25. Location of Inelastic Activity under the Four Selected Earthquakes ........... 82
3.26. Comparison of Design Base Shear Coefficients .......................................... 83
3.27. Recommended Performance Objectives, Adapted from [SEAOC 1995] ............................................................................................. 85
3.28. A Possible Quantification of the Performance-Based Design Space ........... 87
3.29. Design Base Shear for Different Performance Objectives ........................... 88
xii
4.1. Yield Mechanism of Special Truss Moment Frame and Moment Frame with Girder Web Opening ............................................................................ 96
4.2. Schematic Diagram of a Typical Test Set-Up.............................................. 98
4.3. Typical Test Set-Up ..................................................................................... 99
4.4. Test Specimen 1 ......................................................................................... 101
4.5. Loading History 1 of Specimen 1 .............................................................. 102
4.6. Loading History 2 of Specimen 1 .............................................................. 102
4.7. Specimen 1 before Removal of Diagonal Members .................................. 103
4.8. Specimen 1 after Removal of Diagonal Members ..................................... 103
4.9. Hysteretic Loops of Specimen 1 with Diagonal Members ........................ 104
4.10. Hysteretic Loops of Specimen 1 without Diagonal Members ................... 104
4.11. Yielding and Buckling in Specimen 1 with Diagonal Members................ 106
4.12. Yielding in Specimen 1 without Diagonal Members ................................. 107
4.13. Cracking of the Chord Member ................................................................. 107
4.14. Test Specimen 2 ......................................................................................... 108
4.15. Loading History for Specimen 2 ................................................................ 109
4.16. Hysteretic Loops of Specimen 2 ................................................................ 109
4.17. Yielding and Buckling in Specimen 2........................................................ 110
4.18. Cracking in the Chord Member of Specimen 2.......................................... 111
4.19. Test Specimen 3 ......................................................................................... 112
4.20. The Opening in Specimen 3 ....................................................................... 112
4.21. Loading History for Specimen 3 ................................................................ 113
4.22. Hysteretic Loops of Specimen 3 ................................................................ 113
4.23. Deformation of the Test Specimen 3 (Positive Direction) ......................... 114
4.24. Deformation of the Test Specimen 3 (Negative Direction) ....................... 115
4.25. Local Buckling of Chord Members............................................................ 115
xiii
4.26. Test Specimen 4 ......................................................................................... 117
4.27. A Close-Up View of Specimen 4 ............................................................... 117
4.28. Loading History of Specimen 4 ................................................................. 118
4.29. Hysteretic Loops of Specimen 4 ................................................................ 118
4.30. Local Buckling and Fracture of Specimen 4 .............................................. 119
4.31. Test Specimen 5 ......................................................................................... 120
4.32. Close-Up View of Specimen 5 ................................................................... 121
4.33. Loading History of Specimen 5 ................................................................. 121
4.34. Hysteretic Loops of Specimen 5 ................................................................ 122
4.35. Deformation of the Test Specimen (Negative Direction) .......................... 122
4.36. Deformation of the Test Specimen (Positive Direction) ............................ 123
4.37. Comparison of Strain Hardening Values ................................................... 127
4.38. Comparison of Yield Stresses .................................................................... 128
4.39. Equilibrium of Internal Forces in the Opening .......................................... 130
4.40. Axial Hysteretic Model for Diagonal Members [ Jain et al. 1978]............ 133
4.41. Analytical Modeling of Specimen 3........................................................... 134
5.1. Equilibrium of Forces at the Middle Joint ................................................. 141
5.2. The Modified Frame with Beam Web Openings ....................................... 144
5.3. The Modified Frame and its Analytical Model .......................................... 146
5.4. Base Shear – Roof Drift Response of the Original and the Modified Frames(Based on Expected Yield Strength) .............................................. 148
5.5. Sequences of Inelastic Activity of the Modified Frames .......................... 149
5.6. Maximum Floor Displacements of the Modified and the Original Frames ........................................................................................................ 150
5.7. Maximum Interstory Drifts of the Modified and the Original Frames ...... 151
xiv
5.8. Location of Inelastic Activity in the Modified Frame under the Four Selected Records ........................................................................................ 151
5.9. Maximum Overstrength Values Under the Four Selected Records ........... 152
5.10. Overall View of the Test Set-Up ................................................................ 154
5.11. Close-Up View of the Test Specimen ........................................................ 155
5.12. Lateral Bracing of the Test Specimen ........................................................ 155
5.13. Beam-to-Column Connection of the Test Specimen.................................. 156
5.14. Dimensions of the Test Specimen .............................................................. 157
5.15. Dimensions of the Web Opening in the Test Specimen............................. 158
5.16. Close-Up View of the Special Opening ..................................................... 159
5.17. Diagonal-to-Chord Junction ....................................................................... 159
5.18. Vertical-to-Chord Junction......................................................................... 160
5.19. First Loading History ................................................................................. 161
5.20. Second Loading History. ............................................................................ 161
5.21. Hysteretic Loops from the First Loading History ...................................... 163
5.22. Hysteretic Loops from the Second Loading History.................................. 164
5.23. Deformation of the Test Frame (Positive Displacement)........................... 164
5.24. Deformation of the Test Frame (Negative Displacement) ......................... 165
5.25. Inelastic Activity in the Opening ............................................................... 165
5.26. Yielding of the Chord and the Diagonal Members .................................... 166
5.27. Fracture in the Chord Member ................................................................... 166
5.28. Analytical Model of the Test Specimen ..................................................... 168
5.29. Analytical Simulation of the Experiment with the First Loading History ........................................................................................................ 169
5.30. Analytical Simulation of the Experiment with the Second Loading History ........................................................................................................ 169
xv
A1. Typical Story of the Six-Story Frame Used to Calibrate iβ ...................... 185
A2. Four Six-Story Frames Used to Calibrate iβ ............................................. 187
A3. Distribution of Maximum Story Shears under the Four Selected Records ....................................................................................................... 188
A4. Variation of Error Function X .................................................................. 189
A5. Comparison between 50.0)/( nii VV=β and Relative Shear
Distributions from Dynamic Analyses ....................................................... 189
B1. Drift and Yield Mechanism Based Design Procedure Flowchart .............. 191
B2. Internal Forces in the Roof Beam .............................................................. 195
B3. Distribution of Moment in an Exterior Column (Units in kips and ft.) ..... 196
B4. Member Sizes of the Redesigned Frame .................................................... 198
xvi
LIST OF APPENDICES
Appendix
A. CALIBRATION OF BEAM PROPORTIONING FACTOR ................. 184
B. DESIGN EXAMPLE................................................................................... 190
C. ABSTRACT ................................................................................................. 199
xvii
NOTATION
a Normalized design pseudo-acceleration (with g )
ea Base shear coefficient for serviceability (elastic) level
oa Mass-proportioning damping coefficient
)(τga Ground acceleration at time τ
ga Ground acceleration
A Design pseudo-acceleration
eA Design pseudo-acceleration for serviceability
b Numerical factor for beam proportioning factor
fb Flange width of beam
1B , 2B Amplification Factors used to determining uxM for combined
bending and axial force design
c Viscous damping coefficient
][C Damping matrix
C Seismic coefficient (UBC-94)
aC , vC Seismic coefficients (UBC-97)
bd Depth of beam
cd Depth of column
E Input energy form earthquake
E Young’s Modulus
eE Elastic vibrational energy, the sum of kinetic energy and elastic
strain energy
esE Elastic strain energy
pE Cumulative hysteretic energy
kE Kinetic energy
xviii
dE Damping energy
af Axial compressive stress in column
sf Restoring force
abF Actual yield stress of beam
acF Actual yield stress of column
crF Critical stress
iF Equivalent inertia force applied at level i of the structure
iuF Equivalent inertia force at level i at ultimate response
tF Concentrated force applied at the top floor of the structure
ybF Nominal yield stress of beam
ycF Nominal yield stress of column
g Acceleration due to gravity
G Shear Modulus
aG , bG Ratio of column stiffness to beam stiffness for column design
h Height
h Total height of structure
1h Height of the first story
ih , jh Height of floor level i (or level j ) of the structure above the
ground
sh Story height
H Horizontal force in the story used to calculate 2B
I Importance factor (UBC-94, UBC-97)
cI Moment of inertia of chord member
eI Earthquake intensity
k Effective length factor
xk , yk Effective length factor for buckling about x-axis (or y-axis)
l Unbraced length of column
xix
xl Unbraced length of diagonal member
L Span Length
0L Length of special segment, Length of opening
m Mass of single degree of freedom system
][M Mass matrix
M Total mass of the system
)(hM c Moment in the column at a height h above the ground
chM Plastic moment of chord member
ltM Required flexural strength in member due to lateral translation
nM Nominal Flexural Strength
ntM Required flexural strength in member assuming no lateral
translation
pM Plastic moment
ipbM , jpbM Plastic moment of beam at level i (or level j )
rpbM Reference plastic moment of beams
pcM Plastic moment of columns at the base of the equivalent one-bay
frame
pzM Beam moment when panel zone shear strength reaches the value
specified in the UBC
yM Yield moment of beam
uxM Required flexural strength for x-axis bending
n Number of stories
aN Near source acceleration factor (UBC-97)
vN Near source velocity factor (UBC-97)
p Fraction of cumulative plastic energy dissipated at peak response
)(hPc Total axial force in column at a height h above the ground
xx
)(hPcg Axial force in column due to gravity loads at a height h above the
ground
nP Nominal compressive strength
uP Required axial strength
vP Axial force in the vertical member
xyP Tensile yield force of the diagonal member
xcP Buckling force of the diagonal member
xr Radius of gyration about x-axis
yr Radius of gyration about y-axis
R Structural system coefficient (UBC-97)
bR Reaction force from cross beam
wR Response modification factor (UBC-94)
S Site coefficient (UBC-94)
vS Pseudo-velocity
t Time
ft Flange thickness of beam
wct Web thickness of column
wt Web thickness of beam
T Fundamental period of the structure
V Design base shear
cV Ultimate shear provided by the chord members
eiV Maximum earthquake-induced story shear in story level i
eijV Maximum earthquake-induced story shear in story level i in
case j
enV Maximum earthquake-induced story shear in the top story
(level n )
xxi
enjV Maximum earthquake-induced story shears in the top story (level
n ) in case j
iV Static story shear at level i due to the equivalent inertia forces
nV Static story shear at the top story (level n ) due to the equivalent
inertia forces
oV Ultimate shear strength of opening
pV Shear in the panel zone
uV Base shear at ultimate
xV Ultimate shear provided by the diagonal members
iw , jw Weight of the structure at level i (or level j )
W Total weight of the structure
x Displacement in the x direction
x� Velocity in the x direction
x�� Acceleration in the x direction
X Error function use to calibrate beam proportioning factor
Z Seismic zone factor (UBC-94, UBC-97)
cZ Plastic modulus of column
bZ Plastic modulus of beam
α Design base shear parameter
iβ Beam proportioning factor at level i
δ Story displacement
eδ Serviceability drift level
iδ Step function for calculation of column moment and axial force
plδ Inelastic story drift
pδ Story displacement due to panel zone deformation
yδ Yield story drift
m∆ Expected maximum inelastic drift (UBC-97)
xxii
s∆ Elastic drift due to design level forces (UBC-97)
φ Resistance factor for bending
cφ Resistance factor for axial compression
η Strain-hardening factor
µ Rotational ductility
max,pθ Maximum plastic rotation
pθ Plastic rotation, Inelastic drift
xθ Angle between the diagonal and the chord members
yθ Yield rotation
τ Time instant
ω , nω Natural circular frequency
cξ Overstrength factor for the chord members
iξ Overstrength of the beam at level i
sξ Overstrength due to strain hardening
xξ Overstrength factor for the diagonal members
ζ Damping as a fraction of the critical value
1
CHAPTER 1
INTRODUCTION
1.1 BACKGROUND AND MOTIVATION
Moment-resisting steel frames have long been regarded as one of the best
structural systems to resist seismic forces. The load-carrying mechanism of these frames
depends on the capability of their moment-resisting joints to transfer the applied forces
between members. Therefore, the strength and ductility of these joints play a crucial role
in the seismic response of these frames. Unfortunately, an unprecedented number of
beam-to-column connections and other failures were reported in the aftermath of the
1994 Northridge and the 1995 Kobe earthquakes [SAC 1995c, Nakashima et al. 1998].
These incidents clearly show that our knowledge about seismic behavior of moment-
resisting frames at present is not adequate. It creates a profound impact that can be felt by
everyone involved in the design and construction of moment-resisting frames. After three
years of intensive research, the engineering community remains shrouded in doubts. It is
not apparent how safe the existing moment-resisting frames are, how existing moment-
resisting frames should be retrofitted, or how new moment-resisting frames should be
designed.
At a glance, the design of moment-resisting frames involves only fundamentals of
structural analysis and simplified structural dynamics. After a closer look, however, the
design of moment-resisting frames requires a clear understanding of earthquake-structure
interaction and inelastic distribution of stresses, both at the member and at the system
levels.
2
At the member level, recent studies at the University of Michigan [Goel et al.
1997, Lee et al. 1998] have shown that the stress distribution at beam-to-column
connections in moment-resisting frames defies the classical beam theory. The design of
these connections requires a clear understanding of the stress paths and the boundary
effects. At the system level, studies [Goel and Leelataviwat 1998] have shown that
moment resisting frames designed by the elastic method using equivalent static forces
may undergo inelastic deformations in a rather uncontrolled manner, resulting in uneven
and widespread formation of plastic hinges. Thus, combined lack of ductility of the
connections and the use of unrealistic design approaches could hold a major key in
explaining the recently observed poor performance of steel moment frames.
The research work presented herein focuses on answering two imminent
questions: how a new moment frame should be designed and how an existing moment
frame could be retrofitted. The behavior of a moment-resisting frame designed by the
conventional method was studied using extensive nonlinear static and nonlinear dynamic
analyses. Guided by the performance of this conventionally designed frame, a new design
concept was proposed based on the principle of energy conservation and theory of
plasticity. This study was then extended to include seismic upgrading of existing steel
moment frames for future earthquakes.
1.2 OBJECTIVES AND ORGANIZATION OF THE DISSERTATION
The objectives of this study were: 1) To investigate the behavior of moment-
resisting frames designed by conventional methods; 2) To propose a new design
procedure that addresses explicitly the ultimate drift and the yield mechanism of moment
frames; 3) To propose a new upgrading scheme for existing moment frames. The
organization of the dissertation can be best summarized by the chart presented in Figure
1.1. The results of this study are presented in the following five chapters and two
appendices:
3
DRIFT AND YIELD MECHANISM BASED SEISMIC DESIGN OF STEEL MOMENT FRAMES
CHAPTER 1:INTRODUCTION
CHAPTER 2: A REVIEW OF SEISMIC DESIGN OF STEEL MOMENT FRAMES
NEW MOMENT FRAMES
CHAPTER 3: DRIFT AND YIELD MECHANISM BASED DESIGN OF
MOMENT FRAMES
CHAPTER 6: SUMMARY AND CONCLUDING REMARKS
Figure 1.1. Organization of the Dissertation.
EXISTING MOMENT FRAMES
CHAPTER 4: SEISMIC UPGRADING OF MOMENT FRAMES USING
DUCTILE WEB OPENINGS
CHAPTER 5: SEISMIC DESIGN AND BEHAVIOR OF MOMENT FRAMES WITH DUCTILE WEB OPENINGS
4
• Chapter 2 focuses on the behavior of conventionally designed moment frames.
This chapter presents a review of the underlying concepts behind current design
procedures for steel moment frames based on the Uniform Building Code [UBC 1994,
UBC 1997]. The implications of the current design philosophy for steel moment frames
are discussed. An existing moment frame structure designed by the conventional design
method was taken as a study case. Nonlinear static and nonlinear dynamic analyses were
used to identify potential problems. The results of these analyses are presented and
discussed. The findings in Chapter 2 led to the development of a new design procedure
presented in Chapter 3.
• Chapter 3 presents a new drift and yield mechanism based (DYMB) seismic
design procedure for steel moment frames. In this procedure, the structure is designed at
the ultimate level. The ultimate design base shear for plastic analysis is derived by using
the input energy from the design pseudo-velocity spectrum, a pre-selected yield
mechanism, and a target drift. The procedure also includes a step to determine the design
forces in order to meet specified target drifts in the elastic stage under moderate ground
motions. The results of nonlinear static and nonlinear dynamic analyses of an example
steel moment frame designed by the proposed method are presented and discussed. The
implications of the new design procedure for future generation of seismic design codes
are also discussed.
• In Chapter 4, a possible scheme to modify seismic behavior of existing
moment resisting frames to have a ductile yield mechanism is proposed. This upgrading
scheme consists of creating ductile rectangular openings reinforced with diagonal
members in the beam web near the middle of the span. These openings are designed such
that, under a severe ground motion, inelastic activity will be confined only to the yielding
and buckling of the diagonal members and the plastic hinging of the chord members of
the opening, while other members in the frame will remain elastic. This chapter presents
the experimental and analytical development of the ductile web opening system. Results
5
of reduced-scale experiments are presented. Based on the results of these experiments,
behavior of key members is discussed.
• Guided by the experimental results in Chapter 4, a design procedure for
seismic upgrading of steel moment frames is proposed in Chapter 5. The moment frame
structure in Chapter 2 was used again as an example structure. It was modified using the
proposed upgrading procedure. The response of the upgraded frame under severe ground
motions is presented and discussed. Finally, results from a full-scale test of a one-story
subassemblage are shown. These results were used to verify the proposed modification
procedure and to verify the results from computer analyses.
• Chapter 6, the final chapter, presents the summary and the concluding remarks
of this study. Suggestions for future studies are also presented.
• Appendix A describes the calibration of beam proportioning factor, which is
an important factor used in the drift and yield mechanism based design presented in
Chapter 3.
• Appendix B presents a design flowchart that summarizes the drift and yield
mechanism based design procedure. This appendix also provides a detailed design
example of a five-story moment frame using the proposed design procedure.
• Appendix C contains the abstract of this dissertation.
6
CHAPTER 2
A REVIEW OF SEISMIC DESIGN OF STEEL MOMENT FRAMES
2.1 INTRODUCTION
For the last three decades, extensive experimental research and post-earthquake
investigations have been carried out to better understand the response of multistory
buildings subjected to earthquake excitations. Many analytical and numerical procedures
as well as nonlinear finite element analysis codes have been developed to more
accurately estimate the response of structures. Despite all the advances in the field of
earthquake engineering, building codes and design provisions for earthquakes in the
United States and many other countries remain relatively unchanged. For example, the
Uniform Building Code [UBC 1994, UBC 1997], although it has gone through many
revisions, is still based on the 1959 recommendations of the Structural Engineers
Association of California [Seismology Committee 1959]. Similarly, the National
Earthquake Hazards Reduction Program or NEHRP provisions [NEHRP 1991] are based
on the 1978 ATC 3-06 [ATC 1978] provisions.
The primary design procedure for regular structures specified in most building
codes is still based on the Equivalent Lateral Static Force concept. Equivalent design
lateral forces are derived from expected maximum seismic forces assuming elastic
behavior, modified by suitable response reduction factors that depend mainly on the
ductility of the structural systems. The design work strives for providing adequate
strength and limiting lateral drifts to permissible values at the design (reduced or
working) level. The underlying philosophy is that the strength and drift criteria at the
7
design level assure that structures remain elastic and serviceable during small and
frequent earthquakes, and that, structural safety during a severe earthquake depends on
the capability of structures to dissipate the input energy in the inelastic range.
This chapter presents a review of underlying concepts behind current design
procedures for steel moment frames based on the Uniform Building Code. The
implications of current design philosophy for steel moment frames are discussed. An
existing moment frame structure designed by the conventional design method was taken
as a study case to investigate potential problems. This frame was subjected to an in-depth
study including nonlinear static and nonlinear dynamic analyses. The response of the
study building due to static forces as well as selected earthquakes is presented and
discussed.
2.2 EQUIVALENT LATERAL STATIC FORCE PROCEDURE (UBC-1994)
2.2.1 Design Base Shear
The minimum design base shear, V , for allowable stress design is given by
(UBC-94 Equation 28-1):
wR
ZICWV = (2.1)
where W is the seismic weight, Z is the seismic zone factor, I is the importance factor,
C is the elastic seismic coefficient, and wR is the response modification factor. The
seismic weight is the weight of the building mass that induces inertia forces which,
according to the UBC, includes the total dead weight. For some structures, the seismic
weight must include 25% of the live load and snow load if it is greater than 30 lb./sqft.
The factor Z is the seismic zone factor representing the peak ground acceleration (PGA)
of the design level earthquake at the building site. The peak ground acceleration depends
on the seismic zonation, originally adopted in ATC 3-06 [ATC 1978]. In high seismic
8
region, the zone factor, Z , has a value of 0.4. The factor I represents the relative
importance of the facility. I has a value of 1.0 for standard occupancy structures. For
essential or hazardous facilities, I is equal to 1.25. In essence, the UBC attempts to
increase the level of safety by increasing the magnitude of design forces, thereby
increasing strength and limiting the deformation of structure during earthquakes.
The factor C is the elastic seismic coefficient defined as:
3/2
25.1
T
SC = (2.2)
where S is the site coefficient and T is the fundamental period of vibration of the
structure. Factor C need not exceed 2.75 but the ratio of wR/C must be greater than
0.075. The site factor, S , accounts for the ground motion amplification due to local soil
conditions. The value of S ranges between 1.0 and 2.0 depending on the soil profile. The
fundamental period of the structure, T , can be estimated using an empirical formula. For
steel moment frames, the fundamental period in seconds is approximated by:
4/3h035.0T = (2.3)
where h is the total height of the structure in feet.
The response modification factor, wR , accounts for the ductility and energy
dissipation capacity of the structural system. The underlying basis of the response
modification factor is that ductile structures can dissipate a significant amount of energy
by means of inelastic material behavior. Hence, they can be designed to have a strength
smaller than required to remain elastic and to dissipate part of the input energy by using
inelastic material behavior. Ductile systems such as steel moment frames are assigned
larger values of wR than non-ductile system. In UBC, wR is taken as 12 for special steel
moment resisting frames and 6 for ordinary steel moment frames. The values of wR are
based on experience and performance of moment frames in past earthquakes. It should be
9
noted that many studies have questioned the wR values specified in the code and have
suggested lower values of wR [Bertero 1986, Riddell et al. 1989].
2.2.2 Distribution of Lateral Forces
The distribution of lateral forces over the height of the structure is given by:
∑=
+=n
1iit FFV (2.4)
where iF is the equivalent lateral force applied at level i , tF is an additional
concentrated force applied at the top floor of the structure, and n is the number of stories.
The force tF increases story shears in the upper stories to account for the contributions
from higher modes of vibration. tF is calculated as:
TV07.0Ft = if 7.0T > sec. (2.5)
0Ft = if 7.0T ≤ sec. (2.6)
The force applied at each level, iF , is given by:
∑=
−=n
1jjj
iiti
hw
hw)FV(F (2.7)
where iw is the weight of the structure at level i and ih is the height of level i . For a
structure with equal story mass and story height, lateral forces increase linearly from the
base to the top floor, corresponding to an assumed linear shape of the first mode of
vibration.
The effect of torsion must also be included in the design. The torsional design
moment at a given story can be found from the moment resulting form eccentricities
between applied lateral forces at levels above that story and the load-resisting elements in
that story plus additional moment due to accidental torsion. Accidental torsional moment
is calculated by assuming an additional eccentricity of 5% of the building dimension.
10
2.2.3 Drift Requirements
The UBC requires that structures must have sufficient lateral stiffness. The UBC
imposes drift limit in an attempt to keep the story drifts within an acceptable limit under
both small and frequent earthquakes as well as severe ones. Under design-level forces
(Equation 2.4), for a structure with a fundamental period less than 0.70 second, the story
drift is limited to the smaller of wR/04.0 or 0.005. For a structure with a fundamental
period greater than 0.7 second, the story drift is limited to the smaller of wR/03.0 or
0.004. By using this working level drift limit, the maximum inelastic story drift under a
design level earthquake expected by UBC should be in the order of 2-2.5% [Roeder et al.
1993]. It should be noted that the drift limit for special steel moment frames, with wR of
12, is very stringent. In most cases, this drift limit dictates the member sizes.
2.2.4 Beam and Column Strength Requirements for Controlling the Collapse Mode
A widely accepted design philosophy for moment frames is that columns should
be relatively stronger than beams. In other words, the inelastic activity should be
confined to beams only. This type of frame is generally known as a strong column - weak
beam frame (SCWB). The UBC imposes a condition that at any beam to column joint,
the following relationships be satisfied:
0.1FZ/)fF(Z ybbaycc >−∑ ∑ (2.8)
∑ ∑ >− 0.1M25.1/)fF(Z pzaycc (2.9)
where cZ is the plastic modulus of column, bZ is the plastic modulus of beam, af is the
axial compressive stress in the column, pzM is the beam moment when the connection
panel zone shear strength reaches the value specified in the code, ybF is the yield strength
of beam, and ycF is the yield strength of the column.
11
It has been shown that, although these strength requirements are necessary, they
are not sufficient to prevent flexural yielding in columns during a major earthquake
because these rules are not derived from a global limit state but rather from a localized
one [Lee 1996]. Very often, they do not prevent the occurrence of an undesirable collapse
mechanism. To date, no explicit checks of column yielding at ultimate load condition are
required by code.
The UBC code also provides exceptions when Equations 2.8 and 2.9 do not have
to be satisfied, which essentially means that a weak column-strong beam behavior is
permitted. This can be done if the axial force in the column is less than 40% of the
column yield force, if the shear resistance of the story is more than 50% greater than that
of the story above, and if the column is not part of the lateral load resisting system.
2.3 EQUIVALENT LATERAL STATIC FORCE PROCEDURE (UBC-1997)
Some significant changes have been introduced in 1997 version of the UBC. The
major changes include:
1) The change from a working stress-based design to a strength-based design.
2) The introduction of new design coefficients, notably the near source factors
and the reliability/redundancy factor.
In UBC-97, both working stress design and strength design are allowed. The
forces prescribed in UBC-97 are for strength design, and a factor of 1.4 is used to reduce
the magnitude of the forces if working stress design is to be used. The redundancy factor
accounts for the redundancy of the lateral load resisting system. The lower the degree of
redundancy, the higher the prescribed earthquake forces. The near source factors are a
result of recent findings that ground motions at sites close to a fault can be significantly
amplified. The near source factor is directly related to the distance of the structure to the
nearest fault. The closer to the fault, the higher the prescribed forces.
12
The deign base shear formula in UBC-97 is similar to that UBC-94 except for
several new coefficients and can be expressed as:
R
IWC5.2
RT
IWCV av ≤= (2.10)
where vC is the seismic coefficient (ranges between vN.320 and vN.960 in seismic zone
4) as per Table 16-R of UBC-97, aC is the seismic coefficient (ranges between aN.320
and aN.440 in seismic zone 4) as per Table 16-Q of UBC-97, R is the structural system
coefficient, aN is the near source acceleration factor (ranges between 1.0 and 1.5) as per
Table 16-S of UBC-97, and vN is the near source velocity factor (ranges between 1.0 and
2.0) as per Table 16-T of UBC-97. The base shear must not be less than:
IWC11.0V a= (2.11)
In addition, for seismic zone 4, the total base shear must not be less than:
R
IWZN8.0V v= (2.12)
The structural system coefficient, R , is similar to the response reduction factor,
wR , in the UBC-94 but the value has been reduced to account for the change to strength
based design. In UBC-97, the value of R for special moment resisting frames is 8.5,
while for ordinary moment resisting frames, R is 4.5. The soil profile factors have also
been revised. In UBC-97, there are six categories of soil profiles depending on the shear
wave velocity as opposed to four categories in UBC-94. The design base shear of the
UBC-97 depends considerably on the near source factors, the redundancy factor, and the
soil profile factor. Generally, the design base shear from the UBC-97 is larger than that
computed from UBC-94, when compared at the same strength-based design level or
working stress-based design level.
13
The drift limits have also been changed to reflect the strength-based design. The
expected maximum inelastic drift limit is computed from the design level drift by using
an empirical formula:
sm R7.0 ∆=∆ (2.13)
where m∆ is the expected maximum inelastic drift and s∆ is the elastic drift due design
level forces. The drift limit for m∆ is given as 0.025 for structures having a fundamental
period less than 0.7 second and 0.02 for structures with a fundamental period greater than
0.7 second. If the drift limits for a special steel moment resisting frame are back-
calculated to allowable stress design level as prescribed in the UBC-94, it becomes
apparent that drift limits in UBC-94 and UBC-97 are very similar. However, the drift
limits from UBC-97 are given in a more rational form and can be compared directly to
results of a time history analysis.
2.4 REVIEW OF RELATED RESEARCH
Many analytical and experimental studies have been carried out in the past to
study the implications of various design codes on the seismic behavior of steel moment
frames. Although most of the studies in the literature focus on the moment frames
designed by earlier versions of building codes, they can serve to evaluate moment frames
designed by newer codes since the underlying concepts of most building codes have not
been substantially changed. The focus of these studies ranges from cyclic tests of
building components and dynamic tests of full-scale and reduced-scale models to
analytical investigations of various aspects of seismic behavior of steel moment frames.
The major findings are summarized in the following sections.
2.4.1 Experimental Studies
Modern codes allow the use of both strong column-weak beam (SCWB) and weak
column-strong beam (WCSB) framing systems, as mentioned in Section 2.2.4, despite the
14
results of many cyclic tests of frame components that clearly show the superiority of
SCWB system. Tests of beam-column assemblages representing WCSB frames
[Schneider et al. 1993, Popov et al.1975] show that hysteretic behavior depends strongly
on the magnitude of axial loads in columns. During tests, columns with high axial loads
exhibited hysteretic behavior with rapid deterioration. Popov et al. [Popov et al. 1975]
suggested that the WCSB frames can be adequately used if axial forces are kept below
50% of the yield force. However, this suggestion is based on an assumption that the
ductility demands of SCWB and WCSB frames are similar, which is usually not the case
as will be discussed further. One shaking table test of a small-scale three-story WCSB
frame [Takanashi and Ohi 1984] has been reported and the frame collapsed during the
test. Although, this frame was designed according to Japanese standards and might not
directly reflect moment frames designed by U.S standards, the result strongly suggests
that WCSB frames should be avoided.
2.4.2 Analytical Studies
Many analytical studies on the difference in seismic behavior between weak
column-strong beam frames (WCSB) and strong column-weak beam frames (SCWB)
have been carried out in the past. Roeder et.al. [Roeder et al. 1993] studied the seismic
response of 3-, 8-, and 20-story moment frames designed with these two different
philosophies according to the UBC-88 standards, which are essentially identical to the
UBC-94 requirements. The results of inelastic time history analyses of these frames
under three earthquake records, the 1940 El Centro, the 1971 Pacoma Dam, and the 1979
Imperial Valley College, were reported. The results of these analyses indicated that the
SCWB frames are superior to WCSB in terms of both the global response and the local
damage capacity. The major findings were:
1) WCSB frames produced concentration of inelastic activity in a limited number
of elements, especially in columns, whereas SCWB frames distributed the inelastic
15
activity over many more elements. The local ductility demand and element damage
potential were much higher in WCSB frames.
2) The maximum interstory drifts of WCSB frames were very sensitive to the
increase in earthquake intensity. An increase in story drifts as much as 200% was
reported when the intensity of El Centro was increased by 50%, while only about 20%
increase was observed for the story drifts in SCWB frames.
3) Some plastic hinges formed in columns even when the frame was designed
according to SCWB requirements. The effect of plastic hinges in columns of SCWB on
the seismic behavior was not obvious in that study.
4) Both SCWB and WCSB frames experienced inelastic story drifts larger than
the 2% expected by the code, especially for frames with short periods. This suggests that
the design base forces of the UBC may not be large enough.
Osman et al. [Osman et al. 1995] studied the response of frames designed
according to Canadian standard and reported similar results about the seismic behavior of
WCSB and SCWB frames. The damage was found to be mostly concentrated in the first
story for WCSB frame, with highest plastic rotation at the base of the frame. The SCWB
frame had a better damage distribution over the height but the damage still mainly
localized in the first floor especially at the base of the frame.
It is important to note that, although some plastic hinges were observed in the
columns of the SCWB frames in both studies, no particular attention was paid to
investigate further. It was probably because the intensities of earthquakes used in those
studies were not so strong, therefore, the consequences of yielding in columns were not
obvious. More recent studies [Lee 1996, Park and Pauley 1975] have shown that the
requirements for SCWB in the UBC may not be adequate in preventing formation of a
soft story. The conditions set fourth by the building codes are very localized. They do not
recognize the actual distribution of plastic beam moments in columns. In some cases, the
elastic distribution may underestimate the demands by as much as 100%.
16
Lee [Lee 1996] studied the response of a six-story steel frame designed according
to ATC 3-06 under increasing static forces (pushover analysis) and concluded that the
ratio of sums of plastic moments implemented by the code can not prevent the occurrence
of plastic hinges in columns. It was observed that the ratio as high as 1.8 could not
prevent the formation of column hinges. The distribution of moments in columns
changed drastically from the elastic distribution after the formation of beam hinges. The
abrupt increase of moments in columns below joints and decrease of moments above
joints were observed and led Lee to propose a three-quarter rule for SCWB design.
Essentially, this rule means that three quarters of the sum of girder plastic moment should
be taken by the lower column.
Many similar findings have also been made by others [Park and Pauley 1975,
Goel and Itani 1994, Bondy 1996]. Goel and Itani [Goel and Itani 1994] observed that
moment frames designed by modern practice experienced unevenly distributed yielding
among the members of the frames. The reason for this uneven distribution can be
attributed to the difference in the distribution of internal forces at the ultimate level and at
the design level. This difference is due mainly to the redistribution of internal forces after
some significant yielding which typical elastic analysis can not capture. Park and Paulay
[Park and Pauley 1975] showed that the distribution of moments in columns under
dynamic excitations does not support a typical design assumption that the points of
contraflexure are located at mid-height of column. They suggested that the sum of girder
plastic moment should be resisted by only one column with an adjusting factor, which
takes into account the effect of higher modes and ranges from 0.8 to 1.3. Bondy [Bondy
1996] also arrived at the same conclusion and proposed a method to design a column
based on incremental displacement analysis using a pushover method. All the methods
recently proposed except that of Bondy, though based on extensive analyses, are still
based on localized joint behavior and do not recognize actual distribution of internal
forces.
17
In conclusion, an SCWB system provides much better seismic response than a
WCSB system. The WCSB should be avoided since it can result in serious damage
including the collapse of the building. In addition, the use of localized joint rule to ensure
SCWB in modern design codes is insufficient and a more rational method involving
global plastic distribution of moments should be used. Such method, based on plastic
analysis to determine the distribution of moments, will be discussed further in Chapter 3.
2.5 THE STUDY BUILDING
An existing six-story moment frame was selected to study the seismic response of
conventionally designed moment frames. This frame was a part of the lateral load
resisting system of a building located near the epicenter of the 1994 Northridge
earthquake. The frame suffered significant damage during the earthquake. More detailed
description of the damage has been reported elsewhere [Hart et al. 1995]. The damage
has raised serious questions regarding the performance of steel moment frames and has
clearly shown how ineffective the current design procedure can be.
The plan view of the study building is shown in Figure 2.1. The lateral stiffness in
the N-S direction of the frame is provided by four perimeter special moment-resisting
frames. Each of the moment frames is responsible for a quarter of the total mass. The
bottom story is below grade with extensive outside and interior basement walls. The floor
masses of the building are presented in Table 2.1. One of these three-bay frames, along
with its member sizes, is shown in Figure 2.2.
Table 2.1. Floor Masses of the Study Building.
Floor Floor Mass (kip•in/sec2)
Weight (kips)
Roof 6.26 2416.4 5 5.45 2103.6 4 5.45 2103.6 3 5.45 2103.6 2 6.93 2675.0
18
Figure 2.1. Plan View of the Study Building.
19
Figure 2.2. A Typical Three-Bay Moment Frame in the N-S Direction.
The UBC-94 lateral forces were used to represent the design forces for each
frame. The frame is a special moment resisting frame, thus wR =12. Other important
constants used to calculate the design forces were Z =0.4 (seismic zone 4), I =0.1
(standard occupance), and S =1.5 (soil type S3). The estimated period of the frame from
Equation 2.3 was 0.86 seconds. The total design base shear coefficient ( W/V ), including
the torsion effect prescribed in the code, was 0.09. The design lateral forces at each floor
level are summarized in Table 2.2. The computed story drifts under the UBC forces are
shown in Table 2.3. As can be seen, the frame satisfied the drift limits prescribed by
UBC-94.
20
Table 2.2.
UBC Design Lateral Forces for the Original frame. Floor
hi
(ft.)
wihi wihi/∑wjhj Ft (kips)
Fi (kips)
Fi/frame (kips)
Ftorsion+5% Ecc.
(kips)
Total Fi
(kips) Roof 71 171564.4 0.36 47.4 266.2 78.4 23.5 101.9
5 57 119905.2 0.25 - 184.9 46.2 13.8 60.0 4 43 90454.8 0.19 - 140.5 35.1 10.5 45.6 3 29 61004.4 0.13 - 96.1 24.0 7.2 31.2 2 15 40125.0 0.08 - 59.2 14.8 4.4 19.2
UBC Design Base Shear Coefficient (V/W) = 0.09
Table 2.3. Design Story Shears and Story Drifts.
Story Story Shear (kips)
Story Drift (%)
6 101.9 0.18 5 161.9 0.23 4 207.5 0.21 3 238.7 0.22 2 257.9 0.17
UBC Drift Limit = 0.03/12 = 0.0025 (0.25%)
2.6 NONLINEAR ANALYSES OF THE STUDY BUILDING
2.6.1 Methods of Analysis
Inelastic static as well as inelastic dynamic analyses were carried out to evaluate
the study frame. A nonlinear finite element code SNAP-2DX [Rai et al. 1996] developed
at the University of Michigan was used to perform the analyses. Inelastic static
(pushover) analysis was carried out by applying increasing lateral forces representing the
distribution of UBC design lateral forces. The purpose of the pushover analysis was to
determine the lateral load capacity, the failure mechanism, the sequence of inelastic
21
activity leading to collapse, and the progressive change in the internal force distribution.
For the inelastic dynamic analyses, the study frame was subjected to four selected
earthquake records. The 1940 El Centro, the 1994 Northridge (Sylmar Station), the 1994
Northridge (Newhall Station), and one synthetic ground motion were scaled and used as
base excitations. These records were chosen because of different characteristics of ground
shaking. The El Centro record is a classic base excitation and it contains a broad
frequency range. The 1994 Sylmar and 1994 Newhall records are recent records from the
Northridge earthquake. They were selected because of their near-source characteristics,
typically characterized by few large pulses concentrated over a relatively short duration.
The synthetic record was used to represent an ideal design level earthquake. This record
was generated in such a way that its response spectrum matches closely with that of the
UBC-94 [Gasparini 1976]. The other three actual earthquake records were scaled so that
their intensities are the same as the design earthquake.
The definition of the design earthquake is still somewhat vague. Many procedures
have been proposed for scaling earthquake records to represent a design level earthquake.
In this study, the scaling procedure was based on the definition of spectrum intensity by
Housner [Housner 1959]. The spectrum intensity of an earthquake is defined as the area
under damped elastic pseudo-velocity spectrum curve for periods between 0.1 to 2.5
seconds. The earthquake intensity can be defined mathematically as:
∫=52
10
.
.
ve dTSI (2.14)
where vS is the pseudo-velocity of a single degree of freedom system. For a particular
ground acceleration, the pseudo-velocity for a lightly damped system can be evaluated
from:
max0
)())(sin()(∫ −−−=t
tgv detaS ττωτ τζω (2.15)
22
where )(ag τ is the ground acceleration at time τ , ω is the natural circular frequency of
the system, ζ is damping as a fraction of critical damping, and t is the time at which the
integral is evaluated. The symbol )(τf denotes the absolute value of the mathematical
function.
The records used in this study were scaled to have the same earthquake intensity
as that computed from the UBC-94 design spectrum (with S =1.5 and I =1.0). The
pseudo-velocity spectra of the four scaled records (with 5% damping) and the one
corresponding to the UBC design acceleration spectrum are shown in Figure 2.3. The
scaled records are shown in Figure 2.4. Table 2.4 summarizes the characteristics and the
scaling factors of the four records.
Figure 2.3. Scaled Pseudo-Velocity Spectra of the Earthquakes Used in This
Study (5% Damping).
0
20
40
60
80
0 0.5 1 1.5 2 2.5
UBCSylmarNewhallEl CentroSynthetic
Sv (
in./s
ec)
Period (sec.)
23
-0.8
-0.4
0
0.4
0.8
0 5 10 15 20
El Centro
Acc
eler
atio
n (g
)
Time (sec.)
-0.8
-0.4
0
0.4
0.8
0 5 10 15 20
Sylmar
Acc
eler
atio
n (g
)
Time (sec.)
-0.6
-0.3
0
0.3
0.6
0 5 10 15 20
Newhall
Acc
eler
atio
n (g
)
Time (sec.)
-1
-0.5
0
0.5
1
0 5 10 15 20
Synthetic
Acc
eler
atio
n (g
)
Time (sec.)
Figure 2.4. Four Selected Earthquakes Used in this Study.
24
Table 2.4. Characteristics of Earthquake Records.
Earthquake Peak Acc. (g)
Intensity (g•sec2)
Scaled Peak Acc. (g)
Duration Used (sec.)
1940 El Centro 0.32 0.126 0.73 20 1994 Newhall 0.59 0.357 0.48 20 1994 Sylmar 0.84 0.395 0.61 20
Synthetic 1.00 0.292 1.00 20
UBC Spectrum Intensity (Soil Type S3) = 0.289 g•sec2
2.6.2 Analytical Modeling of the Study Building
An equivalent one-bay five-story frame of the original three-bay frame was used
in this study. The one-bay frame approach has been shown to represent the behavior of
the whole multi-bay frame well and has been used successfully in some past studies [Itani
and Goel 1991, Basha and Goel 1994]. The one-bay frame is a frame with average
properties of the original frame. The elastic properties (moment of inertia, area, and
modulus of elasticity) and the yield moment of beams in the one-bay frame are the same
as those of beams in the original frame. The elastic properties and the yield moment of
columns in the one-bay frame are equal to one-sixth of the sum of those in the original
frame. The frame was modeled as a five-story frame with fixed supports at the ground
level because its bottom story is below grade and consists of basement walls. The
original three-bay frame was assigned one quarter of the total mass of the building,
resulting in one-twelfth of the total mass in the one-bay frame model. The floor masses
were lumped at the beam-to-column connection nodes. The damping was taken as 2% of
the critical value and was taken proportionally to the mass matrix only as:
][][ 0 MaC = (2.16)
where ][C and ][M are the viscous damping and mass matrices of the system, and 0a is
the mass-proportional damping coefficient. With this damping model, the higher modes
of response were given very little damping. The mass-proportional damping coefficient
25
was calculated using the estimated period from UBC (Equation 2.3) and can be found
from:
na ζω20 = (2.17)
where ζ is the damping as fraction of the critical damping, 0.02 in this case, and nω is
the natural circular frequency. For the equivalent one-bay model, the period was
estimated as 0.86 second, resulting in 0a of 0.292.
The beams and columns in the frame were modeled by using the beam-column
element from the SNAP-2DX element library. This element is a concentrated plasticity
element with the ability to form plastic hinges only at its ends. The plastic hinge model
takes into account the interaction between the axial force and the plastic moment. Elastic-
plastic hysteretic behavior with 2% strain hardening was used to represent the inelastic
response of beam-column hinge. The panel zone deformations of the frame were not
considered in the analysis because the main purpose was to evaluate the global response.
The three-bay frame and the idealized one-bay frame are shown in Figure 2.5. The effect
of gravity loads was assumed to be small and was neglected in the analyses. This is
justified because the frame is at the perimeter of the building, therefore, the lateral loads
are much larger than the gravity loads.
Original Frame
One-Bay Idealized Model
Figure 2.5. The Original Frame and the Equivalent One-Bay Idealized Model.
26
2.6.3 Nonlinear Static Pushover Analysis
The plot of the base shear coefficient versus roof drift is shown in Figure 2.6.
Figure 2.7 shows the sequence of inelastic activity under increasing lateral forces. As can
be seen, the response of the frame was elastic up to a drift level of about 1% when the
first set of plastic hinges formed at the base of the frame. The inelastic activity then
quickly spread out into the beams resulting in significant reduction in lateral stiffness.
The first set of plastic hinges in the beams was at the fourth floor. It was almost instantly
followed by the formation of hinges in the second floor. The mechanism formed at the
roof drift level of about 1.5% when two plastic hinges formed at the top of the first story
columns creating a soft story type mechanism. Beyond this drift level, the resistance
came primarily from the strain hardening of the material at plastic hinges. The ultimate
strength of the frame was approximately 5 times the UBC design base shear.
The response of this study frame is typical of a conventional, elastically designed,
frame. Such response is generally characterized by early formation of plastic hinges at the
base, high degree of overstrength, and a soft story type collapse mechanism. Early
formation of plastic hinges at the column base can mean large ductility demands at a
rather critical location. The formation of a soft story mechanism can lead to more serious
consequences including collapse in some cases. The consequence of early formation of
base hinges was evident in the 1995 Kobe earthquake when numerous failure of column
base connections were observed. Both the early formation of base hinges and high
overstrength are the direct consequences of the inconsistency between the prescribed
strength and the drift limitation.
27
Figure 2.6. Base Shear - Roof Drift Response from Pushover Analysis.
Figure 2.7. Sequence of Inelastic Activity from Pushover Analysis.
3 3 4
1 1
4
2 2
0.40V
0.23V
0.18V
0.12V
0.07V
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0 0.5 1 1.5 2 2.5 3
Bas
e S
hear
Coe
ffici
ent (
V/W
)
Roof Drift (%)
1First Plastification
4 Mechanism
UBC DESIGN V = 0.09 W
28
In most cases, the member sizes of moment frames are governed by the drift
limits. Therefore, to increase the stiffness of the frame, designers generally increase the
sizes of beams while the sizes of columns remain relatively the same. The degree of
reserved strength in beams is, therefore, larger than that in columns. As the lateral forces
increase due to an earthquake, plastic hinge will form at the point where the degree of
reserved strength is lowest. For columns, the point where the reserved strength is lowest
is at the base because the applied moments are generally largest there. Consequently,
plastic hinges will form at the base at an early stage leading to large plastic rotational
demands. For a well designed frame, the sizes of beams and columns should be
proportioned to allow the plastic rotational demands to be more evenly distributed
throughout the structure.
The formation of a soft story type mechanism comes as a result of two major
factors. The first factor is that, as mentioned earlier, the code allows the use of WCSB
framing system in some cases when the axial load is not large. It should be emphasized
once more that WCSB frames, although have been found by experiments to have a stable
hysteretic response if the axial load is small, should not be used since the ductility
demands and the story drifts will generally be larger than those in SCWB systems.
Moreover, even though the joint requirements are satisfied, it does not necessarily mean
that the plastic hinging in columns will be prevented as mentioned earlier.
The second major factor is the drastic change in the internal distribution of forces
after the beam yielding has occurred. As Lee [Lee 1996] pointed out in his study, the
moment in the column below a joint may increase abruptly while the moment above the
joint decreases. To illustrate this, the distribution of beam moment in columns at the joint
of the second floor as the roof drift increases is shown in Figure 2.8.
29
Figure 2.8. Distribution of Beam Moment in Columns at the Second Floor Joint.
As can be seen, the moment above and below the joint started out at about 50% of
the beam moment. This corresponds to the assumption usually taken during elastic design
that the point of contraflexure is at the mid-height of the column. As yielding starts, the
distribution deviates more and more from the 50-50 distribution. Furthermore, the
moment above the joint decreases while the moment below the joint increases. This
eventually leads to the formation of a soft story mechanism. This analysis clearly shows
that the elastic analysis can not accurately represent the distribution of moments in the
inelastic state.
2.6.4 Nonlinear Dynamic Analyses
Several parameters were studied to investigate the performance of this
conventionally designed steel moment frame. These parameters, including the maximum
floor displacement, the maximum interstory drift, and the rotational ductility demand, are
0.40
0.45
0.50
0.55
0.60
0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4
Below JointAbove Joint
Dis
trib
utio
n of
Bea
m M
omen
t
Roof Drift (%)
Second Floor Joint
30
presented in Figures 2.9 through 2.11. The envelopes of maximum floor displacements
are shown in Figure 2.9. The envelopes of maximum story drifts are shown in Figure
2.10. The location of inelastic activity along with the rotational ductility demands at
plastic hinges are shown in Figure 2.11. The rotational ductility demand, in this study, is
defined as the ratio of the maximum end rotation of a member to the end rotation at the
elastic limit. The elastic limit rotation is the rotational angle developed when the member
is subjected to anti-symmetric yield moments at the ends. The rotational ductility, µ , can
then be calculated as:
y
p
y
py
θθ
θθθ
µ max,max, 1+=+
= (2.18)
where max,pθ is the maximum plastic rotation at plastic hinge and yθ is the yield rotation.
Figure 2.9. Maximum Floor Displacements due to the Four Selected Earthquakes.
0 5 10 15 201
2
3
4
5
6
El CentroSylmarNewhallSynthetic
Floor Displacement (in.)
Flo
or L
evel
31
Figure 2.10. Maximum Story Drifts due to the Four Selected Earthquakes.
Figure 2.11. Location of Inelastic Activity and Rotational Ductility Demands due to the
Four Selected Earthquakes.
(1.04) (1.04)
(2.11) (2.11)
(1.95) (1.95)
(1.20) (1.20)
(1.61) (1.61)
(2.23) (2.23)
(1.27) (1.27)
(2.70) (2.70)
(3.26) (3.26)
(1.82) (1.82)
(1.35) (1.35) (1.08) (1.08)
(1.03) (1.03)
(1.20) (1.20)
(1.24) (1.24)
(2.03) (2.03)
(1.15) (1.15)
(1.82) (1.82)
(1.38) (1.38)
(1.02) (1.02)
(1.24) (1.24)
(3.63) (3.63)
(2.09) (2.09) (1.87) (1.87)
(1.75) (1.75)
(1.37) (1.37)
(1.88) (1.88)
(1.08) (1.08)
El Centro Newhall Sylmar Synthetic
Note: Ductility Demands Shown in Parentheses
0 1 2 3 40
1
2
3
4
5
El CentroSylmarNewhallSynthetic
Story Drfit (%)
Sto
ry L
evel
32
As can be seen, the maximum floor displacements of the frame under the four
records were approximately in the same level. The story drifts under the four records
were kept at about 2% due to the large overstrength of the frame. The distribution of
story drifts over the height of the frame were similar in all cases, but more significantly,
the maximum story drift of the fourth story under the Newhall record and that of the first
story under the Sylmar record were almost twice of the others. This is because a story
mechanism formed during the excitation, as shown in the Figure 2.11. In fact, the soft
story mechanism was observed in three out of these four cases.
As mentioned earlier, the formation of a soft story mechanism can significantly
affect the response of a frame and it can lead to serious consequences such as collapse.
Another important effect of a story mechanism on the response of moment frames is the
permanent displacement after the excitation. Figure 2.12 shows the roof displacement
time histories of the study frame. It can be noticed that the frame had large permanent
displacements after the El Centro, Newhall and Sylmar records, but almost none after the
synthetic earthquake. This permanent displacement can seriously affect the function of
the building and impede or prevent repair work after major earthquakes.
The major reason for the formation of story mechanisms can be traced back, as
discussed earlier, to the unrealistic assumptions used during the design process. Column
design is usually carried out by sizing the columns based upon elastic distribution of
moment or simply assuming mid-height inflection points for plastic state and checking
the strength requirements at joints (Equations 2.8 and 2.9). Unfortunately, non-linear
inelastic time history analyses [Park and Pauley 1975, Bondy 1996] have shown that the
current design methods may underestimate the moment demands in columns especially
when beams have gone into the inelastic state.
33
-15-10
-505
1015
0 5 10 15 20
El Centro
Roo
f Dis
plac
emen
t (in
.)
Time (sec.)
-18-12
-606
1218
0 5 10 15 20
Newhall
Roo
f Dis
plac
emen
t (in
.)
Time (sec.)
-15-10
-505
1015
0 5 10 15 20
Sylmar
Roo
f Dis
plac
emen
t (in
.)
Time (sec.)
-15-10
-505
1015
0 5 10 15 20
Synthetic
Roo
f Dis
plac
emen
t (in
.)
Time (sec.)
Figure 2.12. Roof Displacement Time Histories under the Four Selected Earthquakes.
34
The distribution of beam plastic moments for column design as conventionally
assumed in practice (assuming mid-height inflection points) and the final, as provided,
column strength of the study frame are shown in Figure 2.13. The design moments are
compared with the maximum moments from the time history analyses in Figure 2.14. The
actual column strength is larger than the design moment due to several design provisions
that introduce reductions due to axial forces. Nevertheless, under strong earthquake
excitations, the strength eventually comes close to the design values due to the reduction
from axial force-moment interaction. In most cases, the design moments expected by the
designer were far lower than those computed during the time history analysis, as can be
seen in Figure 2.14. The columns, therefore, yielded under strong earthquakes. Figure
2.14 clearly substantiates the findings by Park and Pauley [Park and Pauley 1975] and
Bondy [Bondy 1996] that the conventional design procedure significantly underestimates
the moments in the columns. Although the structure was able to continue to dissipate the
input energy after the soft story mechanism had formed, its ability to do so under a more
severe earthquake is still questionable.
Figure 2.13. Distribution of Column Strength along the Height.
1 104 1.5 104 2 104 2.5 104 3 1040
10
20
30
40
50
60
70
80Design Moment
Provided Strength
Moment (kip-in)
Hei
ght A
bove
Gro
und
(ft.)
35
Figure 2.14. Maximum Column Moments Due to the Four Selected Earthquakes.
In addition to formation of soft-story mechanism, high ductility demands were
found in columns of the frame. As expected from the static pushover analysis results,
rotational ductility demands were largest at the column bases. This suggests that column
bases may fail under strong earthquakes, leading to a loss of ability to resist overturning
moment.
In conclusion, the results of the time history analyses showed that, although the
response of the frame was far from a collapsing state, the performance of the frame is not
satisfactory. Many subtle flaws exist in the current design practice and can lead to more
serious consequences if they are not treated properly.
2.7 SUMMARY AND CONCLUDING REMARKS
The current design procedures for steel moment resisting frames was discussed in
this chapter. Related experimental and analytical studies found in the literature were
briefly presented. An actual moment frame building located near the epicenter of the
1 104 1.5 104 2 104 2.5 104 3 1040
10
20
30
40
50
60
70
80Design MomentEl CentroSylmarNewhallSynthetic
Moment (kip-in)
Hei
ght A
bove
Gro
und
(ft.)
36
1994 Northridge earthquake was used as a study case to further investigate the
performance of conventionally designed moment frames. Nonlinear static and nonlinear
dynamic time-history analyses were carried out and the results were discussed. The major
findings are:
(1) SCWB frames are superior to WCSB frames. WCSB frames have been found
to produce concentration of inelastic activity in a limited number of elements, especially
in columns. SCWB frames have been found to distribute the inelastic activity over many
more elements. The ductility demands and damage potential are likely to be much higher
in WCSB frames than in SCWB frames.
(2) The maximum interstory drifts of WCSB frames have been found to be
sensitive to the increase in earthquake intensity. This is due to the formation of
undesirable mechanisms.
(3) Some plastic hinges can form in columns even when the frame is design
according to SCWB requirements. The use of localized joint strength requirements,
although important, is not sufficient to prevent the formation of plastic hinges in the
columns.
The distribution of moments in the columns after some beam yielding has
occurred was found to be drastically different from the elastic distribution. The
consequences of this redistribution are widespread inelastic activity and uncontrolled
mechanism.
(4) The response of a conventionally designed moment frame is typically
characterized by early formation of plastic hinges at the base, high degree of
overstrength, and a soft story type mechanism. Most of these characteristics are not
desirable and can lead to poor response under seismic excitation.
(5) Most of the problems associated with moment frames can be attributed to two
major factors. The first factor is the inconsistency between the strength and drift
(stiffness) criteria imposed by building codes. Most of the moment frames are designed to
37
conform to the drift requirements leaving the sizes of beams relatively large compared to
the sizes of columns. The inelastic activity, therefore, tends to occur in columns. The
second factor is the inability of the elastic design method to capture the distribution of
internal forces in the inelastic stage. Combination of these two factors leads to the
formation of undesirable yield mechanisms.
It is clear that new methods to design moment frames should be developed in such
a way that the level of force and drift requirements are compatible and the plastic
distribution of internal forces is explicitly recognized. One such method, based on plastic
analysis and the principle of energy conservation, will be presented in Chapter 3.
38
CHAPTER 3
DRIFT AND YIELD MECHANISM BASED DESIGN OF MOMENT FRAMES
3.1 INTRODUCTION
It was shown in Chapter 2 that building structures designed by modern code
procedures may undergo large cyclic deformations in the inelastic range when subjected
to a design level ground motion. Nevertheless, most seismic design work around the
world at present is carried out by elastic methods using equivalent static design forces.
Design codes in the United States, particularly the UBC [UBC 1994, UBC 1997], attempt
to provide sufficient strength and stiffness by imposing stringent drift limits at design
force level without any explicit checks pertaining to the ultimate state. By doing so, the
UBC offers an advantage that only elastic analysis needs to be performed. However, this
often, especially for steel moment frames, results in unpredictable and poor response
during severe ground motions with inelastic activity unevenly distributed among
structural members.
Typical seismic response of a structure designed by modern codes can be best
summarized using Figure 3.1. Point 0 in Figure 3.1 corresponds to the response of an
equivalent elastic system. Since modern structures are designed to undergo inelastic
deformation, the actual response will be as shown by the solid lines in the figure. Points 1
and 2 in Figure 3.1 correspond to the design points as specified in the UBC-94 and the
UBC-97 respectively. Point 1 is the allowable stress design level and Point 2 is the
strength design level. As was pointed out in Chapter 2, structural response at ultimate
state may vary significantly depending on the reserve strength and the failure mechanism
Generally, the ultimate response of a structure can be as follows:
39
1) The structure can develop a high degree of overstrength even though the
response may be poor due to the development of a non-ductile deformation mechanism,
such as a story type mechanism. This kind of behavior is depicted by Point 3 in Figure
3.1.
2) The structure is capable of developing a ductile mechanism but the degree of
overstrength is not sufficient. The result is excessive story drift as depicted by Point 4 in
Figure 3.1.
3) The structure is capable of developing a ductile mechanism and has adequate
strength. The result is a desirable response under both small, frequent, earthquakes as
well as severe ones (Point 5).
It is desirable to design structures so that they behave in a known predictable
manner during design level ground motions. This essentially means allowing for the
formation of a preselected desirable yield mechanism with adequate strength and
ductility. This chapter presents and discusses a new seismic design procedure in which
1
2
3
4
5
0
∆ max
Bas
e S
hear
Figure 3.1. Typical Response of Structures.
40
the structure is designed at the ultimate strength level (Point 5 in Figure 3.1). The
inelastic design base shear is derived corresponding to a target maximum drift using the
principle of energy conservation. Then, plastic (limit) design is used to design the
structure to achieve a selected mechanism without explicit checks of the drift criteria at
allowable stress level. Results of dynamic analyses of structures designed by the
proposed method will be shown and discussed. The implications of the new design
concept will also be presented.
3.2 PRINCIPLE OF ENERGY CONSERVATION
The principle of energy conservation is a well-known principle and has been
applied to solve many mechanics problems. In the field of earthquake engineering, the
use of energy as design criterion is not new. Most energy-based approaches are derived
from a concept first proposed by Housner [Housner 1956]. Further investigations were
carried out by many researchers [Akiyama 1985, Kato and Akiyama 1982, Uang 1988].
However, not many of these studies have found their way into design practice. A rare
example is the Japanese Seismic Design Code, which was developed by considering the
concept of energy balance [Kato 1995]. Most energy design methods are based on a
premise that the energy demand can be predicted, therefore, suitable member sizes can be
provided to dissipate the input energy within an acceptable limit state.
For a single degree of freedom system subjected to a horizontal ground motion,
the equation of motion at any given time can be written as:
gs mafxcxm −=++ ��� (3.1)
where m is the mass of the system, c is the viscous damping coefficient, sf is the
restoring force, and ga is the ground acceleration. Multiplying both side of Equaiton 3.1
by dx and integrating over the duration of the ground motion:
∫ ∫ ∫ ∫−=++ dxmadxfdxxcdxxm gs��� (3.2)
41
The first term on the left-hand side of this equation can be written as:
2
2xmxdxmdx
dt
xdmdxxm
���
��� ===∫ ∫∫ (3.3)
This is the kinetic energy, kE , of the system at the moment the ground motion
ceases. The second term on the left-hand side of Equation 3.2 is the damping energy, dE .
The third term on the left-hand side of Equation 3.2 is the absorbed strain energy which is
composed of elastic strain energy, esE , and cumulative hysteretic energy, pE :
∫ += pess EEdxf (3.4)
The fact that the integral is evaluated over the entire duration of the ground
motion and pE is irrecoverable implies that pE is the cumulative hysteretic energy
dissipated during the exicitation. The term on the right-hand side of Equation 3.2 is the
work done by the equivalent static force ( gma− ) or the total input energy from the
earthquake, E :
∫ =− Edxmag (3.5)
The principle of energy conservation can be written as:
EEEEE pesdk =+++ (3.6)
Housner [Housner 1956] defines the energy that contributes to damage of the
structure as the sum of the elastic vibrational energy, eske EEE += , and the cumulative
hysteretic energy only:
EEE pe ≤+ (3.7)
The right-hand side of this inequality is the energy demand and the left-hand side
is the energy supply. If the energy demand and supply can be determined, Equation 3.7
can be used to design a structure by conservatively rewriting it as:
EEE pe =+ (3.8)
42
3.3 INPUT ENERGY IN MUTI-DEGREE OF FREEDOM SYSTEMS
The characteristics of any earthquake can be measured by its effect on a single
degree of freedom system (SDOF). The maximum response of the SDOF system under a
particular earthquake is directly related to the input energy from that earthquake. The
response of an elastic, lightly damped, single degree of freedom system can be
characterized by a mathematical function:
max
t
0
)t(gv de))t(sin()(aS ∫ −−−= ττωτ τζω (3.9)
where vS is called the pseudo-velocity, )(ag τ is the ground acceleration at time τ , ω is
the natural circular frequency of the system, ζ is the damping as a fraction of the critical
value, and t is the time at which the integral is evaluated. The symbol max
)(f τ denotes
the maximum absolute value of the mathematical function. The maximum kinetic energy
attained by the elastic SDOF system during the ground motion can be found as:
2vk mS
2
1E = (3.10)
Housner [Housner 1956] showed that the plots of pseudo-velocity versus period
of the system, or the pseudo-velocity spectra, of typical earthquakes tend to remain
practically constant over a wide range of periods. This is particularly true for a spectrum
that is obtained from averaging several response spectra of earthquakes with similar
intensities.
Based on this assumption, if the pseudo-velocity spectra are almost constant over
a wide range of periods, then the maximum earthquake input energy for the system, on
the average, is:
43
2vmS
2
1E = (3.11)
Equation 3.11 is independent or only slightly dependent on the period of the
system. It can also be shown mathematically [Housner 1959] that the input energy for an
elastic multi-degree of freedom system is approximately:
2vMS
2
1E = (3.12)
where M is the total mass of the system. Equation 3.12 is independent of the size, shape
and stiffness of the system. It should be noted one more time that the derivation of this
input energy expression is based on a key assumption that the elastic velocity spectra for
several earthquakes tend to remain practically constant over a wide range of periods.
Although many actual response spectra are not strictly constant, they can be assumed to
be so for practical purposes.
The validity of Equation 3.12 for practical applications has been verified by
Akiyama [Akiyama 1985]. It should be mentioned here, at this point, that there is still a
controversy about the accuracy of Equation 3.12 in predicting the energy demand. Some
studies in the United States [Uang 1988, Akbas and Shen 1997] show that Equation 3.12
may sometimes significantly underestimate the energy demand. Nevertheless, this study
is based primarily on Akiyama’s study [Akiyama 1985] and the accuracy of Equation
3.12 is assumed.
3.4 ENERGY-BASED DESIGN BASE SHEAR
3.4.1. Design Energy Level
For energy-based design purposes, the design input energy level, as expressed by
Equations 3.11 and 3.12, can be found using the elastic design pseudo-acceleration
spectra given in many building codes. In this study, the design is based on the UBC
[UBC 1994] design spectrum which, for elastic systems, is specified as:
44
ZICgA = (3.13)
where A is the design pseudo-acceleration, I is the importance factor, Z is the zone
factor, g is the acceleration due to gravity, and C is the elastic seismic coefficient as
defined by Equation 2.2. The design pseudo-velocity can be found as:
ag2
TASv πω
== (3.14)
where ω is the natural circular frequency and:
ZICa = (3.15)
The design pseudo-acceleration spectrum and design pseudo-velocity spectrum
for 4.0=Z (seismic zone 4), 0.1=I (standard occupancy) and 5.1=S (soil type S3) are
shown in Figure 3.2. As can be seen, the design pseudo-velocity spectrum has a distinct
characteristic that its value tends to be relatively constant over a wide range of periods,
starting from period of approximately 0.5 second.
0
100
200
300
400
500
A (
in./s
ec2 )
0
20
40
60
80
100
0 0.5 1 1.5 2 2.5 3
Sv
(in./s
ec)
Period (sec.)
Figure 3.2 Design Pseudo-Acceleration and Pseudo-Velocity Spectra (UBC-94).
45
For design purposes, an average value of the design pseudo-velocity over a period
range can be used. An alternative for calculating the design value of pseudo-velocity
value, since it is rather constant over a wide range of periods, is to apply Equation 3.14
using an estimated fundamental period, T , for steel moment frames provided by the
UBC:
4/3h035.0T = (3.16)
where h is the total height of the structure in feet.
After the value of pseudo-velocity has been found, the energy demand can be
calculated using Equation 3.11 or Equation 3.12. Although these equations are only true
for elastic systems, it is postulated that the energy input for a structure remains the same
even when some parts of the structure are stressed beyond the elastic limit (Housner
1956). Therefore, the principle of energy conservation for a single degree of freedom can
be written as:
2vpe mS
2
1EE =+ (3.17)
where m is the mass of the system. Similarly, for multi-degree of freedom systems, the
principal of energy conservation can be written, in an approximate sense, as:
2vpe MS
2
1EE =+ (3.18)
where M is the total mass of the system.
3.4.2 Design Base Shear for Ultimate Response
It was shown in Chapter 2 that the deformation mechanism of a structure dictates
its behavior during an earthquake. Good or poor performance of the structure depends
significantly on whether it has a ductile mechanism, such as a strong column–weak beam
mechanism, or a non-ductile mechanism, such as a soft story mechanism. Considering the
46
fact that large uncertainty is involved in predicting ground motions, emphasis should be
placed on controlling the failure mechanism rather than on controlling the conventional
working level strength and drift values. The energy should be dissipated by means of a
controlled mechanism, which is capable of developing a stable hysteretic response within
an acceptable margin of drift.
An equivalent n-story one-bay moment frame subjected to equivalent inertia
forces in its maximum drift response state with a selected global mechanism is shown in
Figure 3.3. The plastic deformation of the frame takes place after the structure reaches its
yield point. After the formation of the yield mechanism, the deformation of the frame is
assumed to be uniform over the height of the structure and all the energy is dissipated
only in plastic hinges. The inelastic drift of the structure is related to the plastic rotation,
pθ , of the frame, i.e., the inelastic story drift is approximately equal to the plastic rotation
of the frame ( ppl θδ ≈ ). The principal idea is that, by limiting the amount of plastic
rotation, the global drift of the structure can be controlled.
Figure 3.3. Equivalent One-Bay Frame at Mechanism State.
pθ pθih
ipbMiF
47
The principle of energy conservation states that:
22 )2
(2
1
2
1ag
TMMSEE vpe π
==+ (3.19)
where eE is the elastic vibrational energy, and pE is the cumulative plastic work done by
the structure. Equation 3.19 is cast in terms of a because this value can be readily
obtained form the building codes as opposed to vS .
Akiyama [Akiyama 1985] showed that the elastic vibrational energy can be
calculated by assuming that the entire structure is reduced into a single degree of freedom
system, that is:
2e )g
W
V
2
T(M
2
1E ⋅⋅=
π (3.20)
where V is the yield base shear and W is the total seismic weight of the structure
( MgW = ). This simplification was justified by the results of several dynamic analyses
[Akiyama 1985]. Substituting Equation 3.20 into 3.19 and rearranging would gives:
))W
V(a(
8
gWTE 22
2
2
p −=π
(3.21)
Equation 3.21 gives the total cumulative plastic energy during the entire
excitation. During the peak response of the structure, only a portion of the total
cumulative plastic energy, ppE (where p <1), is dissipated by the structure. The exact
determination of the amount of energy dissipated during the peak response requires a full
dynamic analysis using the exact properties of both the structure and the ground motion
to which the structure will be subjected. In view of the uncertainty involved in predicting
the ground motions, the value of p is taken as unity for design purposes, implying that
all the plastic drift is assumed to be uni-directional. Although it is extremely unlikely
that the deformation will be uni-directional, the following two factors are taken into
consideration in setting p equal to unity.
48
First, it is well known that higher modes of vibration can play an important role in
the seismic response of structures. The inter-story drift, which is a suitable damage index
for frame structures, is generally larger than the global drift assumed in the design
process. Qi [Qi and Moehle 1991] studied the response of reinforced concrete structures
due to several earthquakes and reported that the inter-story drift can be as much as 30%
larger than the roof drift, in some cases. For steel moment frames, the ratio between inter-
story drift and the global drift can even be larger. It may be as high as 1.4 [Collins 1995]
or even 2.0 for some cases [Krawinkler 1997]. This amplification is compensated by
assuming uni-directional plastic drift in the design process. Second, Qi [Qi and Moehle
1991] in the same study showed that the inelastic seismic response of a single degree of
freedom system in a certain period range, can be reasonably well captured by the
response due to largest earthquake acceleration impulse. This equivalent impulsive
loading produces mainly uni-directional plastic deformation, thereby, implying that the
assumed uni-directional plastic drift might be appropriate for design purposes. This is
particularly true for near field type earthquakes. Based on many trial and error analyses in
this study, it was found that the use of 1=p along with AISC-LRFD provisions produces
a satisfactory design. The results from dynamic analyses will be shown later in Section
3.6.
Following the assumptions stated above, the energy dissipated by plastic hinges in
the structure shown in Figure 3.3 must be equal to pE or:
p
n
ipcpbp MME
iθ)22(
1∑
=
+= (3.22)
where ipbM is the plastic moment of the beam at the level i and pcM is the plastic
moment of the columns at the base of the structure. Using the expression for pE from
Equation 3.21, Equation 3.22 can be rewritten as:
))W
V(a(
8
gWT)M2M2( 22
2
2
p
n
1ipcpbi
−=+∑= π
θ (3.23)
49
After yielding, the equivalent inertia forces must be in equilibrium with the
internal forces. Equating the internal work done in plastic hinges to the external work
done by the equivalent inertia forces gives:
∑∑==
=+n
1iii
n
1ipcpb hFM2M2
i (3.24)
where iF is the equivalent inertia force at level i and ih is the height of beam level i
from the ground. Assuming an inverted triangular force distribution along the height of
the structure, the inertia force at level i can be related to the base shear by:
Vhw
hwF
n
1jjj
iii ⋅=
∑=
(3.25)
where iw (or jw ) is the weight of the structure at level i (or j ). This assumed
distribution corresponds to the assumed linear shape of the first mode of vibration.
Substituting Equations 3.24 and 3.25 into Equation 3.23 gives:
))W
V(a(
8
gWT
hw
hwV 22
2
2
pn
1iii
n
1i
2ii
−=
∑
∑
=
=
πθ (3.26)
and consequently:
))((8 222
2
1
1
2
W
Va
gThw
hw
W
V pn
iii
n
iii
−=
∑
∑
=
= πθ (3.27)
Solving Equation 3.27 for WV / , the admissible solution of the quadratic equation
gives the required design based shear coefficient:
2
a4
W
V 22 ++−= αα (3.28)
50
where α is a dimensionless parameter, which depends on the stiffness of the structure,
the modal properties, and the intended drift level, and is given by:
gT
8
hw
hw
2
2p
n
1iii
n
1i
2ii πθ
α
=∑
∑
=
= (3.29)
Equation 3.29 gives the required design base shear corresponding to an intended
drift level, pθ . After the base shear has been determined the design force at each level
can be found from Equation 3.25. It is important to recognize two issues as follows:
First, this base shear produces the associated drift level only if the global
mechanism is maintained as assumed in Figure 3.3. Therefore, plastic design must be
used to ensure the formation of the intended global mechanism. Detailed plastic design
procedure for steel moment frames will be discussed later in Section 3.5.
Second, the drift level given in Equation 3.29 is the inelastic drift. The total drift
is the sum of the elastic and inelastic drift. Hence, it is important to estimate the yield
elastic drift of the structure so that the value of pθ can be prescribed according to an
intended ultimate drift level. For example, for a moment frame that has an estimated yield
drift of 1%, if the structure is to be designed for maximum drift of 2%, the value of pθ
can be taken as 1%(0.01). This approximately corresponds to 1% plastic drift.
Combining with 1% elastic drift, this gives approximately a total of 2% drift.
A better method to determine the yield drift is the pushover analysis. The yield
drift can be approximated as the drift corresponding to the inflection point of a bilinear
approximation of the pushover roof drift versus base shear response curve. Due to the
uncertainty involved in predicting earthquakes, only an estimated value of the yield drift
from past experience is sufficient for design purposes. Typically, this yield drift is rather
constant for steel moment frames. This is because structural steel members, especially for
51
wide flange sections, are manufactured in such a way that their ratios of the strength to
stiffness ( I/Z where Z is the plastic modulus and I is the moment of inertia) are quite
constant. Therefore, for two moment frames with similar mass and size, if the yield
mechanism is the same, then the yield drifts will also be similar. This will be shown later
in Section 3.6.
The calculated design base shear coefficients for one-bay frames with two, four,
six, eight, and ten stories, each with constant story mass and constant story height of 14
feet, are shown in Figure 3.4. For all frames, the yield drifts were assumed to be 1%
(0.01). The inelastic drifts ( pθ ) were selected as 0.005, 0.010, 0.015, and 0.020
corresponding to assumed total target drifts of 1.5%, 2.0%, 2.5%, and 3%, respectively.
Figure 3.4. Drift and Yield Mechanism Based Design Base Shear Coefficients.
As can be seen, when the target drift increases, the required design base shear
decreases. The underlying design philosophy for the proposed method is that lateral
forces are calculated corresponding to a selected displacement ductility. Choosing a
θ p
0.1
0.2
0.3
0.4
0.5
0.6
0 2 4 6 8 10 12
V/W
Number of Stories
0.005
0.010
0.015
0.020
52
target drift is conceptually equivalent to selecting a displacement ductility. This concept
has been discussed widely in the past for single-degree of freedom systems [Veletsos and
Newmark 1960, Veletsos and Newmark 1965]. However, the use of this concept for
multi-degree of freedom systems has not been fully appreciated, probably due to the fact
that the yield mechanism directly influences the ductility [Mahin and Bertero 1981].
Hence, the ductility demand for multi-degree of freedom systems varies significantly
from system to system, since they are not designed to have the same yield mechanism.
The proposed design method explicitly accounts for the effect of the failure mechanism
on the ductility demand by employing plastic (limit) analysis, which will be discussed in
Section 3.5.
3.4.3 Design Base Shear for Serviceability
Modern seismic design philosophy comprises of two levels of performance,
which are: serviceability and ultimate limit states. Structures should be capable of
resisting minor or moderate earthquakes without significant damage (serviceability limit
state) and resisting major earthquakes without collapse (ultimate limit state). In most of
the current building codes, the underlying philosophy is that structural safety during a
severe earthquake depends on the capability of structures to dissipate the input energy in
the inelastic range. The design forces for a structure are derived from the ultimate level
elastic design spectrum reduced by a response modification factor that accounts for
ductility of the structural system. The serviceability limit state is imposed implicitly by
limiting the story drifts under the above design forces. However, as was shown in
Chapter 2, attempting to combine both limit states into one design may result in an
inconsistency between strength and drift requirements. This leads to structures with ill-
proportioned member sizes and eventually leads to structures with undesirable yield
mechanisms.
53
A better way to satisfy this two-level seismic design is by considering the two
limit states independently, i.e., the required strength or stiffness of the structure is
calculated separately for each of the limit states. The governing case can then be selected
from the two cases.
The methodology presented thus far is intended to calculate the required yield
strength to satisfy the ultimate limit state (Equation 3.28). However, this design yield
strength does not ensure the serviceability criteria. The serviceability limit state can also
be considered in the design. This can be done by comparing the required yield strength of
the structure that satisfies the serviceability limit state with the one that satisfies the
ultimate limit state (Equation 3.28). The design can be carried out according to the larger
required yield strength of the two values computed from the two limit states.
The required yield strength of the structure that satisfies the serviceability limit
state can be found by the following procedure. First, the serviceability design level
spectrum is selected. This spectrum corresponds to an earthquake with moderate
intensity, which can be assumed to be in the order of one-sixth to one-eighth the intensity
of the UBC elastic design spectrum [Uang 1993]. Next, the serviceability drift limit is
established. This drift limit can be selected based on an allowable level of damage of the
structure. The objective of the serviceability limit state is that under the selected
serviceability design level spectrum, the structure should remain within the selected drift
limit.
Figure 3.5 shows the expected response of a structure designed to satisfy
serviceability limit state. Under the selected serviceability design level spectrum, since
the structure can be expected to behave elastically, the total force acting on the structure
can be estimated as:
g
Aa e
e = (3.30)
54
where ea is the base shear coefficient for serviceability (elastic) level, eA is the pseudo-
acceleration from serviceability level spectrum calculated using an estimated period
(Equation 3.16), and g is the acceleration due to gravity. From Figure 3.5, to satisfy the
serviceability design objective, the required yield strength can be related to the base shear
coefficient ea by using the following relation:
e
yea
W
V
δδ
= (3.31)
where eδ is the target serviceability level drift, yδ is the expected yield drift of the
structure. As was discussed earlier, the value of the yield drift can be estimated from past
experience. It will be shown later that the value of yield drift is rather constant regardless
of the strength of the structure and therefore can be predetermined. It should be noted that
the total force ea is only an estimate because the strength is not known at the design
stage, consequently, the period of the structure is also unknown. However, the estimated
period according to UBC (Equation 3.16) should be sufficient as a first approximation.
This required yield strength can be compared with the required yield strength for ultimate
limit state (Equation 3.28) to determine the governing value.
Figure 3.5. Expected Response of a Structure Designed to Satisfy Serviceability
δe δy
aeδy/δe
ae
Force-Displacement Response
Design Objective
Required Strength
55
3.5 PLASTIC DESIGN OF MOMENT FRAMES
As mentioned earlier, a desirable global deformation mechanism must be
maintained during the entire excitation in order to satisfy Equation 3.28. In Chapter 2, it
was shown that, during an earthquake, the distribution of internal forces at ultimate load
is drastically different than that predicted by elastic analysis. Therefore, the theory of
plastic design must be utilized since it provides internal force distribution at ultimate
level corresponding to a selected failure mechanism without considering the intervening
elastic-plastic range of deformation.
Theory of plasticity has long been utilized in design of framed structures [Beedle
1961]. Only recently has the concept been applied in design of structures for earthquakes.
For plastic design of steel structures for seismic loading, most of the studies found in the
literature focus on the design of braced frames such as eccentric or concentric braced
frame, with very little on the design of moment resisting frames [Hassan and Goel 1991,
Englehart and Popov 1989]. Recently, Mazzolani and Piluso [Mazzolani and Piluso
1997] proposed a plastic design method for steel moment frames based on kinematics
theorem of plastic collapse which includes the second order effects ( ∆−P ). Although
the method proposed is one of the most sophisticated and complete methods, it is based
on a premise that beam section properties are known ahead of time by designing the
beams to resist vertical loads. Only the sizes of columns are determined using the
procedure. Therefore, this method warrants the failure mode of the structure, but does not
assure the ultimate drift of the structure under dynamic excitations since the structure
designed by this method may not have enough lateral strength. From Equation 3.23, it is
clear that the sizes of beams are directly related to the amount of energy required to be
dissipated in order to maintain the target drift. Hence, in the method proposed herein,
both the sizes of beams and columns are treated as unknown.
56
Generally, moment frames are so placed in a structure that the influence of gravity
loads is much smaller than that of the earthquake loads. It will be shown later that this is
especially true when a structure is designed by using the proposed method which uses
significantly larger design lateral forces compared to typical elastic design forces
required by building codes. Therefore, in many cases, the gravity loads can be safely
ignored. Moreover, by controlling the ultimate drift within an acceptable limit, the second
order P-∆ effect can also be assumed to be small. Considering these two factors, the
plastic design of steel moment frames can be significantly simplified.
The primary aim of the proposed plastic design procedure is to eliminate the
possibility of formation of plastic hinges in columns. The well-known strong column-
weak beam mechanism is generally believed to be a desirable mechanism for seismic
design, and is selected for this design procedure. The equivalent n-story one-bay moment
frame subjected to design forces in its mechanism state is shown in Figure 3.6. It is
assumed that the design forces, iF , have already been computed from Equations 3.28 and
3.25. The plastic moment capacity of the beam at level i is denoted by a product rpbi Mβ ,
which will be defined later.
Figure 3.6. Frame with Global Mechanism.
pcM pcM
rpbiMβiF
ih
57
3.5.1 Design of Beams
Applying a virtual rotation θd at the base, the work done by the external force at
level i is θdhF ii , the internal work done at each plastic hinge in the beams is
θβ dMrpbi , and the internal work done at each plastic hinge at the base is θdM pc . By
selecting a suitable distribution of plastic moment capacity of beams and equating the
external work done to the internal work done in plastic hinges, the required beam strength
at each level can be determined, namely:
∑ ∑= =
+=n
1i
n
1ipcpbiii M2M2hF
rβ (3.32)
where iF is the design force at level i calculated from Equations 3.25 and 3.28, ih is the
height of beam level i from the ground, iβ is the beam proportioning factor for beam
strength at beam level i , rpbM is the reference plastic moment of beams, and pcM is the
required plastic moment of columns in the first story. In Equation 3.32, the beam
proportioning factor iβ represents the relative beam strength at level i with respect to
rpbM . The factor iβ can be predetermined as will be discussed later. The product
rpbi Mβ is the plastic moment capacity of beam at level i . Here, rpbM is common for
beams at all levels. If iF , ih , iβ , and pcM are all predetermined, the only unknown
variable in Equation 3.32 is rpbM .
For frames with fixed bases, the value of pcM must be appropriately chosen. A
desirable value of pcM should be such that the story mechanism in the first story is
prevented. As a first approximation, assuming plastic hinges form at the base and the top
of the first-story columns, the plastic moment of the first-story columns to prevent this
mechanism should be, from Figure 3.7:
4
Vh1.1M 1
pc = (3.33)
58
where V is the total base shear (from Equation 3.28), 1h is the height of the first story,
and the factor 1.1 is the overstrength factor to account for possible overloading due to
strain hardening as will be explained later.
Figure 3.7. Frame with Soft-Story Mechanism.
With a known value of pcM , the required nominal beam strength at each level,
ipbM , which is equal to ybb FZ where bZ is the plastic modulus and ybF is the nominal
yield stress of the beam, can be determined from the design inequality:
ri pbipb MM βφ ≥ (3.34)
where φ is the customary resistance factor and is equal to 0.9. It should be noted that the
resistance factor φ can be taken as 1 but it is recommended that it should be taken as 0.9
to comply with AISC-LRFD design specifications.
The beam proportioning factor, iβ , plays an important role in the seismic response
of a structure. It represents the variation of story lateral strength and stiffness along the
height of the structure. Indirectly, it reflects the variation of story drifts along the height.
pcMpcM pcM
pcM
V1.1
1h
59
Ideally, the optimum distribution of iβ values is achieved when the distribution of story
drifts under earthquakes over the height of the structure is uniform. In order to achieve
this, the beams should be proportioned according to the story shears. The distribution of
beam strength along the height should closely follow the distribution of story shear
induced by earthquakes. Based on the numerical analyses presented in Appendix A, it
was found that the relative distribution of maximum earthquake-induced story shears
along the height can be closely approximated by using the distribution of the static story
shears calculated from the design lateral forces given by Equation 3.25. It was concluded
that the beam at each level should be proportioned based on the square root of the ratio of
the static shear at that level to the shear at the roof level (level n ):
2/1
n
ii V
V
=β (3.35)
where iV and nV are the story shears at level i and at the roof level (level n ) due to the
design forces calculated from Equation 3.25. The value of the exponent ½ was
determined based on a least square minimization of the error between the actual story
shear distribution and the distribution given by static story shears. More details on this
process can be found in Appendix A. The actual shear distribution in some example
structures from nonlinear dynamic analyses will be shown later in Section 3.6.
3.5.2 Design of Columns
In order to ensure the selected strong column-weak beam plastic mechanism at the
ultimate drift level, it is important that columns are designed assuming that all plastic
hinges are fully strain-hardened when the drift is at the target ultimate level. The moment
generated by a fully strain-hardened beam is taken into account by multiplying its
nominal plastic moment by a factor called the overstrength factor, ξ . By assuming
appropriate overstrength factors, generally ranging from 1 to 1.1, the design moment for
each column can be calculated.
60
A free body diagram of a column of the frame in Figure 3.6 is shown in Figure
3.8. Since all the beams have gone into the strain-hardening range, the applied force at
each level, iF , must be updated to account for the increase in internal forces. The
magnitude of the updated forces, iuF , acting on one column can be found by equating the
overturning moment of the column produced by the updated forces iuF to the resisting
moment produced by the beams and column base. The distribution of the updated forces
can be assumed as inverted triangular along the height of the frame as used earlier.
Figure 3.8. Free Body Diagram of the Column in the Equivalent One-Bay Frame.
With the assumed inverted triangular distribution, the updated forces can be
related to the updated base shear for one column uV by:
un
jjj
iiiu V
hw
hwF ⋅=
∑=1
(3.36)
where iw and ih are as defined earlier. The equilibrium equation for one column can
then be written as:
∑∑==
+=n
ipbipc
n
iiiu i
MMhF11
ξ (3.37)
iuF
ih
)h(M c
pcM
ipbiMξ
61
where pcM is the plastic moment at the base of the frame from Equation 3.33, iξ is the
overstrength factor at level i , and ipbM is the nominal plastic moment of beam at level i
Substituting Equation 3.36 into 3.37 and solving for uV gives:
∑∑
∑=
=
= +=n
ipbipcn
iii
n
iii
u iMM
hw
hwV
1
1
2
1 ))(( ξ (3.38)
Upon substitution of Equation 3.38 back into 3.36, the updated force at each level
can be determined as:
))((1
1
2∑
∑ =
=
+=n
jpbjpcn
jjj
iiiu j
MMhw
hwF ξ (3.39)
In Equation 3.39, the subscript j is used to avoid the confusion between the
summation over the range and the symbol denoting level i .
After iuF has been determined, the distribution of moments in the column can be
found by treating the column as a cantilever, namely:
∑∑==
−−=n
1iiiui
n
1ipbiic )hh(FM)h(M
iδξδ (3.40)
where )h(M c is the moment in the column at a height h above the ground, and iξ ,
ipbM , ih , and iuF are as defined previously. iδ is a step function defined as:
1i =δ if ihh ≤ (3.41)
0i =δ if ihh > (3.42)
Similarly, the axial force in the column at a height h above the ground, )h(Pc ,
can be expressed as:
)()2
()(1
hPL
MhP cg
n
i
pbi
ici += ∑
=
ξδ (3.43)
62
where L is the span length of the beams, )(hPcg is the axial force due to gravity loads
acting at height h , and other symbols are as defined previously. After )(hM c and )(hPc
have been determined, the column can be designed as beam-column elements by any
suitable design provisions.
The purpose of introducing the overstrength strength factor, iξ , is to account for
the difference in the actual and the nominal yield strength as well as to account for the
increase in strength due to strain hardening. For example, typical modern A36 beams
have an average actual yield stress of 49 ksi [SAC 1995b]. This substantial difference
between the nominal and the actual yield stress could have detrimental effects on
columns. From Equation 3.40, it is apparent that the moment in a column could be
underestimated by as much as 40% if beam moments, ipbM , are calculated using the
nominal yield stress. For A572 GR.50 steel, the difference between the actual and the
nominal yield stress is much smaller than that of grade 36 steel. Typical A572 GR.50
steel has an average yield stress of about 55 ksi, or approximately 10% greater than the
nominal value. Conceptually, the effect of this overstrength due to the difference in yield
strengths can be neglected when the columns and the beams in the frame are of the same
steel grade. The effect of the difference is most pronounced when the beams in the frame
are A36 while the columns are A572 GR.50. Unfortunately, this is usually the case for
most structures.
The second source of overstrength is strain hardening. Since it is expected that
all beams will be deformed well beyond their elastic limit, strain hardening of material
will occur. The exact value of overstrength factor for strain hardening of beams in
bending is still not quite well known due to the fact that the overstrength for this purpose
is the one with respect to rotation as opposed to a more familiar, uniaxial, case. Due to
lack of sufficient information on the actual moment-rotation response of typical beams,
the overstrength due to strain hardening only may be taken between 1.0 to 1.10. The final
overstrength factor for column design is the product of the overstrength due to the
63
difference in nominal and actual yield strength and the overstrength due to strain
hardening:
sycac
ybabi FF
FFξξ
/
/= (3.44)
where ybF and ycF are the nominal yield strengths of the beams and columns,
respectively, abF and acF are their expected yield strengths, and sξ is the overstrength
factor due to strain-hardening. For a conservative design, the ratio ycac FF / in Equation
3.44 could be taken as 1.0.
In the proposed design method, all the beams are assumed to reach their
maximum overstrength at the same time. This may appear to be too conservative.
However, it should be realized that the proposed method is based on an assumed linear
distribution of inertia forces. In many cases, the actual distribution of inertia forces can be
quite different due to higher mode effects causing some members, especially in the upper
levels of the structure, to deform beyond the expected overstrength level. Therefore,
using an average overstrength factor for beams uniformly over the entire height of the
structure appears reasonable for calculating the design forces for the columns.
It should also be noted that the moment and axial force calculated from Equations
3.40 and 3.43 are the internal forces of the column in the equivalent frame. The results
can be projected back to the original multi-bay frame by considering that the internal
forces of the exterior columns in multi-bay frame are the same as those in the equivalent
frame. The internal forces of the interior columns in the multi-bay frame can be found by
assuming that the moment is twice as large as the moment in the column of the equivalent
frame, and that the axial force due to an earthquake load is practically zero. Hence, the
axial force at each level in the interior column of the multi-bay frame is the axial force
due to gravity loads only. This assumption is accurate enough for practical purposes
provided that the bay width of the multi-bay frame is constant or nearly constant. A
64
flowchart summarizes the design procedure and a frame design example are presented in
Appendix B.
3.6 PARAMETRIC STUDY OF THE PROPOSED DESIGN PROCEDURE
In order to validate the proposed design procedure, a parametric study was carried
out to investigate the effect of the number of stories and the effect of variation of the
design target drift. The effect of the variation in the number of stories was studied by
using 2-, 6-, and 10-story one-bay moment frames designed with the same target drift of
2%. The drift of 2% was chosen because it was consistent with the ultimate drift limit
expected by the UBC (Sections 2.2.3 and 2.3). The effect of the variation of target drift
was studied by using three 6-story moment frames designed with 1.5%, 2.5% and 3%
target drifts. All one-bay moment frames in this study consist of equal story mass, story
dimensions, and story gravity loads. A typical story height is 14 feet and the bay width is
25 feet. The gravity loads acting in each column at each story are assumed to be 25 kips.
The story weight is 190 kips. A typical story of the study frames is shown in Figure 3.9.
Figure 3.9. Typical Story of the Study Frames.
3.6.1 Variation in Number of Stories
The effect of the number of stories was studied first by designing the 2-, 6-, and
10-story frames with the proposed procedure. The target drifts for all three frames were
set at 2%. The design parameters were calculated as shown in Table 3.1. The design force
25 kips 25 kips
25 ft.
14 ft. W=190 kips
65
at each level was calculated using Equations 3.25 and 3.28 based on the UBC-94
spectrum with 4.0=Z and 0.1=I . The design forces for the three frames are shown in
Table 3.2. The internal forces were calculated from the plastic analysis as presented
earlier (Equations 3.32, 3.40, and 3.43).
All members in the study frames were designed by using AISC-LRFD
specifications (AISC 1994) assuming A572 GR.50 steel for all members. The
overstrength factor was taken as 1.05 for all beams except at the roof level where the
value of 1 was used. The reason for using the overstrength factor of one at the roof level
is because plastic hinges are allowed at that level without affecting the global behavior at
the mechanism state. Since same steel grade was used for the beams and columns, the
overstrength used in the design was the one associated with strain hardening only. The
final member sizes of the three frames are shown in Figure 3.10. It should be mentioned
that some members were selected such that their compactness ratios were below the
limits given by AISC-LRFD seismic provisions. This was done in order to keep their
strength as close to the required (calculated) values as possible. This exception was made
for the purpose of this parametric study only. In a real design situation, the compactness
limits must be strictly applied, such as the case of a study frame presented later in Section
3.7.1.
Table 3.1.
Design Parameters (2% Drift Limit). Number of
Stories Estimated
Period (sec.) a Estimated
Yield Drift pθ α Design
W/V 2 0.426 1.100 0.01 0.01 3.155 0.346 6 0.971 0.765 0.01 0.01 1.579 0.310 10 1.426 0.592 0.01 0.01 1.185 0.245
Note: Calculations based on 1415.3=π and 386=g in/sec2.
66
Table 3.2. Design Lateral Forces (in kips).
Floor Level 2-Story Frame 6-Story Frame 10-Story Frame 1 43.7 16.8 8.5 2 87.6 33.6 16.9 3 - 50.4 25.4 4 - 67.2 33.9 5 - 84.1 42.3 6 - 100.9 50.8 7 - - 59.2 8 - - 67.7 9 - - 76.2 10 - - 84.6
Figure 3.10. Member Sizes of the 2-, 6-, and 10-Story Frames with 2% Target Drift.
W24x55
W24x62
W14x90
W14x90
W36x135
W36x135 W14x398
W14x398
W33x130
W33x130
W14x398
W14x398
W27x94
W30x116
W14x311
W14x311
W36x182
W36x182
W14x550
W14x550
W36x170
W14x550
W36x160
W36x170 W14x550
W14x550
W36x160
W14x550
W36x135
W36x150 W14x500
W14x500
W30x116
W14x311
W27x94
W14x311
67
It should be emphasized that the sizes of beams and columns depend mainly on
the story weight and the number of bays used in the design process. The assumed one-
bay frames were used solely for the purpose of parametric study and they do not
necessarily represent moment frames in actual building structures, which are typically
multi-bay. The fundamental periods of the 2-, 6-, and 10-story frames from modal
analysis were 0.86, 1.02, and 1.6 seconds, respectively. It should be noted that the
estimated periods given by the UBC closely approximate the modal analysis periods of
the structures except for the 2-story frame, where the UBC estimate is significantly lower
than the actual value. The effect of this underestimation will be discussed later.
A series of non-linear analyses was carried out to study the response of the three
frames. The three study frames were subjected to inelastic static analysis and inelastic
time history analysis. Inelastic static analysis was carried out to investigate the yield
mechanisms of the study frames by applying increasing lateral forces representing the
inverted triangular distribution given by Equation 3.25. For the inelastic dynamic
analysis, the three frames were subjected to four different scaled ground motion records.
The records used in this study were the same as used in Chapter 2, which were the
Newhall record, the Sylmar record, the 1940 El Centro record, and a synthetic earthquake
whose response spectrum matched with that of the UBC.
All analyses were carried out by using a computer program SNAP-2DX [Rai et al.
1996] developed at the University of Michigan. Modeling assumptions were similar to
those used in Chapter 2. These assumptions included: floor masses were lumped at the
nodes, frame dimensions were taken at centerlines, gravity loads were neglected, yield
stress of 50 ksi with 2% strain hardening with respect to rotation was used for all
members, and 2% mass proportioning damping was used with the estimated period by the
UBC.
Figure 3.11 shows the base shear versus roof displacement plots of the three
frames obtained form pushover analyses. Figure 3.12 shows the location of inelastic
68
activity in the three frames at 3% roof drift. As can be seen, all frames developed strong
column-weak beam mechanism as intended in the design. The yield drifts of the three
frames were within 1%, which were also consistent with the values assumed in the design
(Table 3.1). It should be noted that the frames could be redesigned with the refined values
of yield drifts from Figure 3.11. However, in this study, only one design iteration was
used for each frame to reduce computation time.
Figure 3.11. Base Shear versus Roof Drift Response of the Study Frames.
0
0.1
0.2
0.3
0.4
0.5
0 0.5 1 1.5 2 2.5 3
2-Story6-Story10-Story
Bas
e S
hear
Coe
ffici
ent (
V/W
)
Roof Drift (%)
69
Figure 3.12. Location of Inelastic Activity in the Three Frames at 3% Roof Drift.
Some selected results from the inelastic dynamic analysis of the three frames are
presented briefly in Figures 3.13 and 3.14. Figure 3.13 shows the envelopes of maximum
story drifts of the three frames due to the four selected ground motions. The envelopes of
maximum story drifts show that almost all story drifts were within the target design limit
of 2%. It can be noticed that the story drifts of the 2-story frame under El Centro and
Newhall records were larger than the 2% target drift. This is because the actual calculated
fundamental period of this 2-story frame, which is 0.86 second, falls in the range where
the pseudo-velocity spectra of the El Centro and Newhall records are significantly larger
than the design spectrum, as can be seen from Figure 2.3.
70
Figure 3.13. Maximum Story Drifts of the 2-, 6-, and 10-Story Frames.
It is also worth noting that the period estimated using the UBC formula was much
smaller than the computed period for the 2-story frame. Hence, the design input energy
was too small as can be seen from the pseudo-velocity spectra shown in Figure 2.3. This
trend has been observed in most short-period structures and has been reported in the
literature [Reddell 1989]. Design of short-period structures can be improved by using an
average (constant) value of pseudo-velocity based on the UBC to compensate for the low
value of input energy calculated from the pseudo-velocity spectrum.
Figure 3.14 presents the relative distribution of maximum story shears, enei VV / ,
where eiV and enV are the earthquake-induced story shear at level i and at the level n ,
respectively, along the height of the three frames under each earthquake. Figure 3.14 also
shows the distribution of beam strength and stiffness, iβ , used in the design process of
0 1 2 3 40
1
2
3
4
5
6
7
8
9
10
El Centro
Sylmar
Newhall
Synthetic
Tar
get
Dri
ft
Sto
ry L
evel
0 1 2 3 4
Tar
get
Dri
ft
Story Drift (%)
0 1 2 3 4
Tar
get
Dri
ft
71
the three frames. As can be seen, in most cases, the distribution given by Equation 3.35
agreed well with the distribution of story shears from time history analyses. The
difference is possibly due to the higher mode effects, and can be seen clearly in the 10-
story frame case. Nevertheless, the results demonstrate that the proposed design method
successfully controls the yield mechanism and maximum story drifts under design level
earthquakes.
Figure 3.14. Distribution of Maximum Story Shears from Dynamic Analyses.
0 1 2 3 40
1
2
3
4
5
6
7
8
9
10
El CentroSylmarNewhallSyntheticBeta
Sto
ry L
evel
0 1 2 3 4V
ei / V
en
0 1 2 3 4
72
3.6.2 Variation in Design Target Drift
The effect of the variation in design target drift was studied next by designing
three 6-story frames with target drifts of 1.5%, 2.5% and 3%. The target drift of 2% was
used earlier (Section 3.6.1) and was, therefore, excluded. The purpose was to study the
effectiveness of the proposed procedure to control maximum drift within the target drift
level. The design was carried out in similar fashion as for the frames presented in the
previous section. The design parameters are summarized in Table 3.3 and the design
lateral forces are summarized in Table 3.4.
From Table 3.3, it can be seen that the base shear becomes smaller when the
target drift becomes larger. This is consistent with the result for single-degree of freedom
systems studied by Chopra [Chopra 1995]. As the target ductility increases, the required
strength of the system decreases. The member sizes of the three frames are shown in
Figure 3.15.
Table 3.3.
Design Parameters. Number of
Stories Estimated
Period (sec.) a Estimated
Yield Drift pθ α Design
W/V 6 0.97 0.765 0.01 0.005 0.789 0.465 6 0.97 0.765 0.01 0.015 2.368 0.226 6 0.97 0.765 0.01 0.020 3.158 0.176
Note: Calculations based on 1415.3=π and 386=g in/sec2.
Table 3.4. Design Lateral Forces (in kips).
Floor Level 1.5% Target Drift 2.5% Target Drift 3.0% Target Drift 1 25.3 12.2 9.5 2 50.6 24.4 19.1 3 75.9 36.7 28.6 4 101.2 48.9 38.1 5 126.5 61.2 47.6 6 151.8 73.4 57.2
73
Figure 3.15. Three Six-Story Frames with 1.5%, 2.5%, and 3% Target Drifts.
The actual periods were calculated from modal analysis to be 0.91, 1.45, and 1.67
seconds for the frame with 1.5%, 2.5%, and 3% target drift, respectively. It can be
noticed that the periods estimated by using the UBC formula are significantly lower than
the modal analysis values when the frames are designed for a large value of target drift.
In the case of six-story frame the effect of this underestimation is not pronounced because
the design pseudo-velocity spectrum is quite constant at this range of periods. However,
care should taken when designing a low-rise structure with a large target drift, since the
error in the period may result in an unconservative design.
These three 6-story frames were subjected to the same series of inelastic analyses
as was done previously, using the same modeling assumptions. Figure 3.16 shows the
base shear versus roof displacement plots of the three frames from the results of pushover
analyses. As can be seen, the approximate yield drifts of the three frames were about the
same, regardless of their strength. This is due to the fact that steel members are usually
manufactured in such a way that their strength and stiffness are in proportion, as was
mentioned earlier. Although the yield drifts of the three frames were smaller than what
was assumed during the design, no further design iteration was carried out. All three
W30x116
W30x116
W14x283
W14x283
W30x99
W30x108
W14x283
W14x283
W24x76
W27x94
W14x257
W14x257
W27x102
W27x102
W14x233
W14x233
W27x84
W27x102
W14x233
W14x233
W24x62
W24x84
W14x193
W14x193
W40x183
W40x183
W14x550
W14x550
W36x170
W36x182
W14x550
W14x550
W33x118
W36x150
W14x455
W14x455
74
frames developed the same global (strong column-weak beam) mechanism as intended in
the design, as can be seen in Figure 3.16 where the location of inelastic activity in the
three frames at 3% roof drift is shown.
Figure 3.16. Base Shear versus Roof Drift Response of the Study Frames.
Figure 3.17. Location of Inelastic Activity in the Three Study Frames at 3% Roof Drift.
Target Drift 1.5%
Target Drift 2.5%
Target Drift 3.0%
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0 0.5 1 1.5 2 2.5 3
1.5%
2.5%
3.0%
Bas
e S
hear
Coe
ffici
ent (
V/W
)
Roof Drift (%)
75
Some selected results from the inelastic dynamic analyses are presented briefly in
Figures 3.18 through 3.20. Figure 3.18 shows the envelopes of maximum story drifts
along the height of the three frames due to the four selected ground motions. The overall
maximum story drift under each earthquake of all six-story frames, including the one
with a 2% target drift in Section 3.6.1, are presented again in Figure 3.19 where the target
drifts and maximum drifts are compared. The 45-degree line in Figure 3.19 represents a
1-to-1 relationship between the design target drift and the maximum drift.
Figure 3.18. Maximum Story Drifts under the Four Selected Earthquakes.
0 1 2 3 4
Tar
get
Dri
ft
Story Drift (%)0 1 2 3 4
El CentroSylmarNewhallSynthetic
0
1
2
3
4
5
6
7
8
9
10
Tar
get
Dri
ft
Sto
ry L
evel
0 1 2 3 4
Tar
get
Dri
ft
76
Figure 3.19. Comparison between Design Target and Attained Maximum Drifts.
Figures 3.18 and 3.19 show that almost all story drifts remained well within the
target design drift limits. The story drifts of the 6-story frame designed with a target drift
of 3% were slightly less than the target value. This result suggests that when the period of
the structure becomes large, the displacement tends to be bounded and depends mainly on
the characteristics of the ground motion and the elastic stiffness of the frame, regardless
of the strength of the system. This behavior is characteristic of structures in the
displacement sensitive period range. Nevertheless, in this case, the proposed design
procedure will give a conservative design. This fact is a direct result of the assumption
that all plastic energy is dissipated in one pulse ( p =1 in Equation 3.21) for all structures,
regardless of their periods.
1.0
1.5
2.0
2.5
3.0
3.5
1.0 1.5 2.0 2.5 3.0 3.5
El CentroNewhallSylmarSynthetic
Max
imum
Drif
t (%
)
Target Drift (%)
77
Figure 3.20. Distribution of Story Shears from Dynamic Analyses.
Figure 3.20 presents the distribution of maximum story shears along the height of
the three frames. Figure 3.20 also shows the distribution of beam strength and stiffness,
iβ , used in the design process of the three frames. As can be seen, the distribution given
by Equation 3.35 agreed well with the distribution of story shears from time history
analyses. The deviations under some records were probably due to the higher mode
effects. Nevertheless, these results agree well with the results from previous analyses.
0 1 2 3 4
El CentroSylmarNewhallSyntheticBeta
0
1
2
3
4
5
6
7
8
9
10
Sto
ry L
evel
0 1 2 3 4
Vei / V
en
0 1 2 3 4
Target Drift 1.5%
Target Drift 2.5%
Target Drift 3.0%
78
3.7. COMPARISON BETWEEN THE CURRENT AND THE PROPOSED
DESIGN PROCEDURES
3.7.1 Comparison of Seismic Response
In order to compare the difference between the current and the proposed design
procedures, the moment frame presented in Chapter 2 was used as a study case. This
moment frame was redesigned using the proposed design procedure. The objective was
to compare the seismic behavior of the original and the redesigned frame through
inelastic dynamic analyses.
The redesign process started by calculating the design base shear according to the
proposed design procedure. The period of the structure was estimated by Equation 2.3 to
be 0.86 second. By assuming 1% elastic drift, the value of pθ was taken as 1%(0.01) for
a total maximum drift of 2%. The parameters a (Equation 3.15) and α (Equation 3.29)
were found to be 0.83 and 1.73, respectively. With these parameters, the design base
shear coefficient W/V was calculated to be 0.333. Design steps and detailed
calculations are presented in Appendix B. The member sizes of the redesigned frame are
shown in Figure 3.21 along with the member sizes of the original frame for comparison.
Figure 3.21. Member Sizes of the Original Frame and the Redesigned Frame.
Redesigned Frame Original Frame
79
The base shear coefficient computed by the proposed design procedure is 0.333,
which is approximately three times the UBC-94 base shear coefficient for the same
building (Chapter 2). Thus, the design lateral forces in the proposed design procedures
are much larger making the relative effect of the gravity loads much smaller.
Consequently, the gravity loads can be safety neglected during the design process.
It is also noteworthy that, even though the proposed design method requires
significantly larger design base shear, the total weight of the original frame and the total
weight of the redesigned frame are almost equal (154.6 kips for the redesigned frame and
153.2 kips for the original frame). This is because member sizes in most of the frames
designed according to current code procedures are governed by drift requirements,
making the member sizes relatively large regardless of the base shear. The member sizes
of the frame designed by the proposed design method are smaller for beams but are larger
for columns when compared to the member sizes of the original structure. This is a result
of imposing the strong column-weak beam mechanism in the design process.
In order to compare the response of the frame designed by the proposed design
procedure to the response of the original frame, the series of nonlinear analyses carried
out in Chapter 2 was repeated. The redesigned frame was modeled using the same
assumptions as used in Chapter 2 for the original frame. A One-bay, five-story, model of
the redesigned frame was prepared for inelastic static analysis and inelastic time history
analysis. Inelastic static analysis was carried out by applying increasing lateral forces
representing the inverted triangular distribution of the UBC design lateral forces. For the
inelastic dynamic analysis, the frame model was subjected to the same ground motion
records as used in Chapter 2.
Figure 3.22 shows the base shear versus roof displacement pushover responses for
the two frames. The sequences of inelastic activity in the two frames are shown for
comparison in Figure 3.23. The horizontal drift at the UBC design lateral force level of
the original frame satisfied the UBC limit. The drift for the redesigned designed frame,
80
however, was above the UBC limit. Both frames possess significant overstrength above
the UBC design force level -- six times for the original frame and four times for the
redesigned frame.
The sequences of inelastic activity of the two frames under increasing static
lateral forces were significantly different. In the original frame, as mentioned in Chapter
2, the first set of plastic hinges to form was at the base and the yield mechanism was a
soft story in the first story. Both of these are not considered as good behavior. The
redesigned frame, on the other hand, behaved as expected, following selected strong
column-weak beam mechanism. All plastic hinges occurred only in the beams and in the
column bases with the later forming last, a major improvement over the original frame
where the hinges at column bases formed first.
Figure 3.22. Base Shear versus Roof Drift Response of the Original and the
Redesigned Frames.
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0 0.5 1 1.5 2 2.5 3
OriginalRedesigned
Bas
e S
hear
Coe
ffici
ent (
V/W
)
Roof Drift (%)
1First Plastification
4 Mechanism
UBC DESIGN V = 0.09 W
4
Mechanism
1 First Plastification
81
Figure 3.23. Sequences of Inelastic Activity under Increasing Lateral Forces.
Some selected results from the inelastic dynamic analyses of the redesigned frame
are presented in Figures 3.24 and 3.25. Figure 3.24 shows the maximum story drifts of
the redesigned frame as well as those of the original frame from Chapter 2. Figure 3.25
shows the locations of inelastic activity of the redesigned frame under selected ground
motions along with the ductility demands at plastic hinges. The maximum story drifts of
the redesigned frame, shown in Figure 3.24, are generally smaller than those of the
original frame. More significantly, the original structure developed a soft story
mechanism, as mentioned in Chapter 2, but the redesigned frame did not. The maximum
drifts of the redesigned frame agreed well with the target design limit of 2%, as expected.
Moreover, the inelastic activity in the redesigned frame was much better controlled and
limited to the locations as intended in the design. Another significant observation was
that the rotational demands at the base of the redesigned frame were significantly smaller
than those in the original structure. Hence, the chance of premature failure at the column
base in the redesigned frame was much smaller.
3
4 4
3 3
3 4
1 1
4
2 2
6 6
1 1
2 2
5 5
Original Redesigned
82
Figure 3.24. Maximum Story Drifts of the Original and the Redesigned Frames.
Figure 3.25. Location of Inelastic Activity under the Four Selected Earthquakes.
The results demonstrate that, even though the story drifts under static lateral
forces do not satisfy the drift criteria prescribed in the UBC, as seen in the frame
designed by the proposed method, the response under dynamic loading can be
significantly better. This is because the inelastic activity occurs in a control manner
following a desired yield mechanism.
(1.49) (1.49)
(2.47) (2.47)
(1.87) (1.87)
(2.05) (2.05)
(1.74) (1.74)
(1.83) (1.83)
(2.46) (2.46)
(2.48) (2.48)
(2.17) (2.17)
(2.67) (2.67)
(3.29) (3.29)
(1.04) (1.04)
(1.51) (1.51)
(2.17) (2.17)
(2.07) (2.07)
El Centro Newhall Sylmar Synthetic
(2.33) (2.33)
(2.17) (2.17)
(1.42) (1.42)
(2.98) (2.98)
(1.03) (1.03)
Note: Rotational Ductility Demands Shown in Parentheses
Original Frame Redesigned Frame
0 1 2 3 40
1
2
3
4
5
El CentroSylmarNewhallSynthetic
Story Drfit (%)
Sto
ry
0 1 2 3 4
El CentroSylmarNewhallSynthetic
0
1
2
3
4
5
Story Drfit (%)
83
3.7.2 Comparison of Design Forces
The calculated design base shear coefficients for one-bay frame models with two,
four, six, eight, and ten stories, based on the UBC-94 spectrum and a 2% target drift, are
shown in Figure 3.26. Also shown in the figure are the design base shear coefficients
from UBC-94 and UBC-97. The UBC-94 design base shear coefficients were calculated
based on 4.0=Z , 0.1=I , and 5.1=S . They were multiplied by a factor of 1.4 to
represent strength design levels. The UBC-97 design base shear coefficients were
computed assuming that both the redundancy/reliability factor and the near field factor
were equal to one.
Figure 3.26. Comparison of Design Base Shear Coefficients
As can be seen, the UBC-94 and UBC-97 design base shear coefficients are
approximately the same. On the other hand, the design base shear coefficients based on
the proposed method are about three to four times larger. This comparison shows that the
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
0.40
0 2 4 6 8 10 12
Proposed1.4 UBC-94UBC-97
Des
ign
Bas
e S
hear
Coe
ffici
ent
Number of Stories
84
wR factor in UBC-94 or R factor in UBC-97 are unreasonably high. Structures designed
using the UBC-94 and UBC-97 design base shear coefficients would likely experience
large displacements during a design level earthquake.
It should be noted that the final strength of structures designed using the UBC-94
and the UBC-97 is likely to be greater than the specified base shears. This is because of
the drift criteria and the larger values of redundancy/reliability and near filed factors.
Nevertheless, the comparison suggests that the UBC design procedure can be improved
significantly by simply using smaller values of the current R (or wR ) factor,
approximately one-third, and changing from an elastic design approach to a plastic design
approach.
As mentioned in Chapter 2, the response modification factor R (or wR ) is the most
controversial factor in the current design process. To date, no simple procedure to
evaluate these factors exists. However, it is clear that the use of a constant response factor
for all moment frames independent of their periods is not reasonable. The proposed
design method is a direct design method where the need for response modification factor
R is completely eliminated from the design process.
3.8 PERFORMANCE-BASED PLASTIC DESIGN
In recent years, a new design philosophy for building codes has been discussed
among the engineering community. New generation of design codes for earthquakes are
moving toward performance-based design framework [Vision 2000 1995]. The goal of
any performance-based design procedure is to produce structures that have predictable
seismic performance. Within the context of performance-based design, a structure is
designed such that, under a specified level of ground motion, the performance of the
structure is within prescribed bounds. These bounds depend mainly on the importance of
the structures.
85
Based on the Vision 2000 document [Vision 2000 1995], performance of
structures is categorized in four categories, which are 1) Fully Operational, 2)
Operational, 3) Life-Safe, and 4) Near Collapse. These performance levels are to be
selected corresponding to different design earthquake levels, which depend on the
frequency of occurrence. SEAOC recommends four levels of design earthquakes, which
are; 1) Frequent earthquakes that have 50% of probability of exceedance in 30 years, 2)
Occasional earthquakes that have 50% of probability of exceedance in 50 years, 3) Rare
earthquakes that have 10% of probability of exceedance in 50 years, and 4) Very rare
earthquakes that have 10% of probability of exceedance in 100 years. The performance
objectives can be best summarized in Figure 3.27. The drift and yield mechanism based
method developed in this study offers an opportunity to design a structure within a
performance-based framework.
Figure 3.27. Recommended Performance Objectives, Adapted from [SEAOC 1995].
The qualitative definition of performance objectives mentioned earlier is quite an
open question at present. Many response parameters can be used to measure performance.
Possible parameters include ductility demands, damage indices, and story drifts. Even
Near Collapse Basic
Life Safe Basic Essential
Operational Basic Essential Safety Critical
Fully Operational Basic Essential Safety Critical
Frequent Occasional Rare Very Rare
Ear
thq
uak
e P
erfo
rman
ce
Earthquake Design Level
Unacceptable Performance
86
though, SEAOC suggests possible values of transient and permanent drift levels for each
performance objective, these two parameters vary depending on the structural system.
The exact quantitative measurement of performance is not within the scope of this study.
In this study, drift levels are used as a measure of performance, based on the underlying
assumption in the UBC. Under a design level earthquake, the maximum story drift should
be in the order of 2-2.5%, according to the drift limits given in the UBC-97.
Based on this assumption, the drift levels corresponding to each performance
criterion are proposed in Table 3.5 and are used as examples in this study. It should be
noted that the suggested drift levels are not based on any theoretical basis and are used
only to illustrate the performance-based application of the proposed design method.
Similarly, earthquake design levels are used based on Housner’s intensity (Equation 2.14)
presented earlier. Table 3.6 shows a possible scenario between design earthquake levels
and Housner’s earthquake intensity. Based on Tables 3.5 and 3.6, the performance-based
design space can be quantified as shown in Figure 3.28.
Table 3.5. Performance Criteria.
Performance Maximum Story Drift Fully Operational 0%-1%
Fully Operational-Operational 1%-2% Operational-Life Safe 2%-3%
Life Safe-Near Collapse 3%-4% .
Table 3.6. Earthquake Design Levels.
Performance Intensity Frequent 0-0.5UBC
Frequent-Occasional 0.5-1.0UBC Occasional-Rare 1.0-1.5UBC Rare-Very Rare 1.5-2.0UBC
87
Figure 3.28. A Possible Quantification of the Performance-Based Design Space.
For a given structure, the design base shear can be computed using a combination
of interstory drift and earthquake intensity from Figure 3.28. To illustrate a performance-
based design procedure, the same 5-story moment frame used in Section 3.7.1 is taken as
an example. The design base shears were calculated, using the procedure proposed in this
study, based on the combination of the following parameters:
Target Drift = {0.5, 1, 1.5, 2.0, 2.5, 3, 3.5}%
Earthquake Intensity = {0.5, 1.0, 1.5, 2.0}UBC
After the values of design base shear for all combinations were calculated, a
contour plot of equal design base shears were obtained. The contour plot of the design
base shears for this particular frame is shown in Figure 3.29. The contour lines in the
figure can be used to select the design base shear level required to satisfy a given
performance objective. Note that the yield drift of 1% is assumed in the design process.
The base shear for the story drift of 0.5% is calculated using Equation 3.31.
0UBC 0.5UBC 1.0UBC 1.5UBC 2.0UBC Housner’s Intensity
4.0
3.0
2.0
1.0
0.0
Inte
rsto
ry D
rift
(%
)
Note: UBC Spectrum Intensity (Soil Type S3) = 0.289 g-sec2
Basic
Basic Essential
Basic Essential Safety Critical
Basic Essential Safety Critical
88
Figure 3.29. Design Base Shear for Different Performance Objectives.
It should be noted that Figure 3.29 is unique for a given moment frame. It readily
gives the minimum design value for a particular performance objective. For example, if
the structure is to be designed for the essential class of structures, the minimum design
base shear would be in a range of 0.6 to 0.8. On the other hand, the optimal design for the
basic performance objective corresponds to a design base shear in the range of 0.3-0.4.
As can be seen, the proposed design method presents a direct link between the design
objective and the design base shear. This direct link provides an obvious advantage over
conventional design methods.
3.9 SUMMARY AND CONCLUDING REMARKS
A new design procedure for steel moment frames was presented and discussed.
The new design concept is based on plastic (limit) design theory. The design forces are
4.0
3.0
2.0
1.0
0.0
Inte
rsto
ry D
rift
(%
)
Basic Essential Safety Critical
V/W=0.20
0.40
0.60
0.80
1.00
1.20 1.40
0.0UBC 0.5UBC 1.0UBC 1.5UBC 2.0UBC Earthquake Intensity
89
derived using the principle of energy conservation. In this proposed design method, the
story drift is directly specified as a design parameter and no explicit checks for drift
criteria under design forces are required. Also, the response modification factor ( R or
wR ) is also completely eliminated, making the design procedure consistent. Nonlinear
dynamic analysis was used to verify the proposed method. The results show that the
proposed method can produce structures that meet a preselected performance objective.
The implications of the proposed method were also presented. The major findings in this
chapter are:
(1) The use of plastic design principles in combination with the proposed design
forces leads to structures with better seismic response. The results of a parametric study
show that the proposed method produced structures with story drifts that comply well
with the target drift values. The results show that the proposed method works particularly
well for medium-rise structures. The proposed method, however, tends to produce over-
designed high-rise structures and under-designed low-rise structures. Another significant
observation was that the estimated period values given by the UBC were too small for
low-rise frames. Care should be taken when design such structures.
(2) Comparing with a structure designed using a conventional method, a structure
designed using the proposed method has relatively smaller beam sizes and larger column
sizes. The total weights of the structures design using both methods are similar. The
seismic responses of the two structures, on the other hand, are not.
The sequences of inelastic activity of the two frames under increasing static
lateral forces are drastically different. In the conventionally designed frame, the first set
of plastic hinges to form was at the column bases and the yield mechanism was a soft
story in the first story. The redesigned frame, on the other hand, behaved as expected,
with the selected strong column-weak beam mechanism. All plastic hinges were limited
to only the beams and at the column bases, the later forming last.
90
The results from dynamic analyses also showed a similar trend. The maximum
story drifts of the frame designed using the proposed method were consistently smaller
than those of the original frame. More significantly, the soft story mechanism was
eliminated. The maximum drifts of the redesigned frame agreed well with the target
design limit. In addition, the inelastic activity in the frame designed with the proposed
method was much better controlled and limited to the locations as intended in the design.
The plastic rotation demands at the base of the redesigned frame were much smaller
compared to those in the original structure.
(3) The use of elastic drift limit without considering the response at the ultimate
level is not quite meaningful for seismic design. It was shown that, even though the story
drifts under static lateral forces do not satisfy the drift criteria prescribed in the UBC, the
response under dynamic loading can be significantly better if the inelastic activity occurs
in a control manner, following a desired yield mechanism.
(4) By comparing the design base shear coefficients required by the proposed
method and those required by the UBC-94 and UBC-97, it was found that the required
design base shears from the UBC-94 and UBC-97 were far too small. This suggests that
the values of the response factors, R in the UBC-97 and wR in the UBC-94, are
unrealistically large. More appropriate values should be about 3 to 4 times smaller than
those currently used.
(5) The proposed method can be easily presented in a performance-based design
framework. The performance objectives can be defined based on the earthquake
intensities and interstory drift levels. An optimal design base shear corresponding to a
selected performance objective can be readily and directly obtained using the proposed
design approach.
The methodology presented thus far is a purely deterministic procedure. A
probabilistic approach can be incorporated into the proposed design process especially in
a performance-based design framework. Calibration of some important factors, especially
91
the p factor mentioned in Section 3.4.2, could improve the design significantly.
However, probabilistic study was not within the scope of this research.
92
CHAPTER 4
SEISMIC UPGRADING OF MOMENT FRAMES USING DUCTILE
WEB OPENINGS
4.1 INTRODUCTION
Moment-resisting steel frames have long been regarded as one of the best
structural systems to resist seismic forces. The performance of such frames under seismic
forces depends primarily on the strength and ductility of their beam-to-column
connections. Unfortunately, a large number of beam-to-column connection failures were
observed after the 1994 Northridge Earthquake and the 1995 Kobe Earthquake. These
failures clearly show that typical beam-to-column moment connections possess far less
ductility than expected.
Notwithstanding the question of ductile performance of connections, it was shown
in Chapter 2 that moment resisting frames designed by elastic method using equivalent
static forces may undergo inelastic deformations in a rather uncontrolled manner,
resulting in uneven and widespread formation of plastic hinges. Thus, combined lack of
ductility of the connections and the use of elastic design method could hold a major key
in explaining the recently observed poor performance of steel moment frames.
The drift and yield mechanism based design approach presented in Chapter 3 can
be utilized to design new moment frame structures so that they will behave in a
controlled and preferred manner. However, there is an urgent need for methods to retrofit
and upgrade the existing moment-resisting steel frames which were designed using the
pre-1994 practice. Two approaches are being employed in current design practice and
93
research studies to upgrade such frames: (1) A strengthening strategy, where the beam-to-
column connection is reinforced to meet the strength and ductility demand [Lee et
al.1997, Engelhardt and Sabol 1998]; (2) A “weakening” strategy, where the beam is
weakened (away from the connection) in order to create a “fuse” that limit the force
demand on the connection [Chen et al. 1997]. The strengthening strategy requires
checking the adequacy of columns and other critical regions of the frame for the
increased force demands. For this reason, weakening strategy (such as the “dog bone”
solution) is becoming increasingly popular.
As part of the weakening strategy, one possible scheme to modify the behavior of
moment resisting frames to have ductile yield mechanism is to create rectangular
Vierendeel openings in the girder web near the middle of the span. The shear capacity of
the openings can be increased, if needed, by adding diagonal and vertical members into
the openings to provide additional stiffness to the frame. The openings are designed so
that, under a severe ground motion, the inelastic activity will be confined only to
yielding, and buckling of the diagonal members, and the plastic hinging of the chords of
the opening while other members in the frame remain elastic.
This chapter presents the experimental and analytical development of the ductile
opening system. Results of reduced-scale experiments are presented. Based on the results
of these experiments, behavior of key members of the frame is discussed.
4.2 CONCEPT OF MOMENT FRAMES WITH WEB OPENINGS
The concept of using openings as ductile segments is derived from a structural
system known as Special Truss Moment Resisting Frame (STMF). This structural system
has been studied both analytically and experimentally by Goel et al. [Itani and Goel 1991,
Basha and Goel 1994] at the University of Michigan during the past ten years and has
been recently incorporated into the 1996 UBC Supplement [UBC 1996] and the AISC-
LRFD seismic provision [AISC 1997]. The system consists of truss frames with special
94
segments designed to behave inelastically under severe ground motions while other
structural members of the frame remain elastic. The special segments in STMF structures
can be either Vierendeel openings or Vierendeel openings with X-diagonal members,
depending on the desired level of shear strength of the special panels. When a STMF
structure is subjected to lateral forces induced by an earthquake, the shear forces in the
floor girders are resisted by the chord members and the X-diagonals in the openings.
After yielding and buckling of the diagonal members, plastic hinges will form at the ends
of the chord members. After the openings in all floor girders have yielded, complete
mechanism forms when additional plastic hinges occur at the column bases of the frame.
From the results of extensive analytical and experimental studies, STMFs have
been shown to have excellent hysteretic behavior under severe seismic forces and
perform well when compared to conventional open web and solid web framing systems.
Excellent energy dissipation, smaller story drifts, and smaller base shear were observed in
STMF system. Full-scale tests have also shown that STMFs possess excellent energy
dissipation and can sustain large cyclic displacements.
The idea of STMF structural system originated from a study of cyclic behavior of
the conventional open web frames. Itani and Goel [Itani and Goel 1991] carried out full-
scale tests of open web frames and observed that they exhibited hysteretic response with
rapid degradation due to buckling in major load carrying members. Guided by the
experimental results, they developed the STMF framing system. A design procedure that
explicitly accounts for the inelastic distribution of internal forces in the special segments
was also developed. In this design approach, the special segments are designed first.
Other structural elements are subsequently designed to remain elastic under the forces
generated by the fully yielded and strain hardened special segments. Using this design
approach, two full-scale special truss moment frames were designed and tested. These
test frames behaved as intended and demonstrated very ductile cyclic response.
Analytical study was also carried out to examine the seismic behavior of STMF framing
95
system. The results showed that STMF structures respond to seismic forces in an
excellent manner.
Based on the results of Itani and Goel’s STMF study, Basha and Goel [Basha and
Goel 1994] developed STMF with ductile Vierendeel opening segments. This type of
special segments eliminates the use of diagonal members, therefore, only the chord
members of the special opening segments are designed to resist the applied shear force. A
mathematical expression to calculate the maximum force generated by a fully yielded and
strain hardened opening was developed. Two full-scale test specimens, one with
simulated gravity loads and one without gravity loads, were designed and tested in that
study. The results were very satisfactory. The hysteretic response of the tested STMFs
was very stable and ductile. The “pinching” phenomenon, typically found in the
hysteretic response of truss-like structures, was completely eliminated. An analytical
study was carried out to verify the design procedure and the seismic response of STMF
structures. These results were also very satisfactory.
In moment frames with web openings, the openings serve the same function as the
special segments in STMF system. Under a severe ground motion, the inelastic activity
will be confined only to the openings. Mainly, the inelastic activity consists of yielding
and buckling of diagonal members and plastic hinging of the chord members of the
openings. In this proposed system, the chord, diagonal, and vertical members should be
designed such that, under their fully yielded and strain-hardened condition, the moment at
every beam-to-column connection generated by the shear force in the opening will be
smaller than a chosen critical value. This critical value is selected so as to reduce the risk
of premature failure of connections and confines all inelastic activity only to the
openings. Figure 4.1 shows a moment frame modified with girder web opening and a
STMF at the ultimate (mechanism) state.
96
a) Special Truss Moment Frame.
b) Moment Frame with Girder Web Opening.
Figure 4.1. Yield Mechanism of Special Truss Moment Frame and Moment
Frame with Girder Web Opening.
E
E
97
4.3 TESTING OF STEEL BEAMS WITH OPENINGS
In order to study the viability of the mentioned scheme, five, approximately half-
scale, specimens representing girders with an opening in moment frames were prepared
for cyclic tests. The main objective of the test program was to obtain the information
needed for the development of the design procedure and the analytical modeling of the
proposed structural systems. This information includes:
1) The behavior of the opening in both the elastic range and the inelastic ranges;
2) The yield mechanism and the overstrength of the key members;
3) The best detailing scheme to meet the ductility demand;
4) The reparability of the opening after being subjected to severe deformation.
Each specimen was fabricated differently to study various aspects of the proposed
upgrading system. The first and the second specimens were tested in order to obtain the
information about the overall behavior of the proposed upgrading scheme and also about
the stiffness of the support frame. The third test was done to verify a detailing scheme
around the opening region to meet the required ductility demand. The fourth test was
aimed at studying the behavior of an upgrading scheme using a Vierendeel type opening.
The fifth test was aimed at studying the reparability of the special opening after a severe
deformation history.
4.3.1 Test Set-Up
Each specimen was made to represent a girder with a special opening. Each test
specimen was approximately half-scale. Postulating the anti-symmetric behavior of
girders in moment frames, all specimens were made as half-span models. The test
specimens were mounted on a support frame, which consisted of 2C15x50 beams. The
specimens were braced against lateral–torsional buckling using two lateral supports at a
location below the openings.
98
The shear load was applied using a hydraulic actuator with 50 kips capacity and
5± inches stroke length. The actuator was connected between the tip of the test specimen
and a reaction wall. The force generated by the actuator represents the shear force
induced by an earthquake at the mid-span of a girder. A schematic diagram of a typical
test set up is shown in Figure 4.2. A photograph of a typical test set-up is shown in Figure
4.3.
Figure 4.2. Schematic Diagram of a Typical Test Set-Up.
99
Figure 4.3. Typical Test Set-Up.
4.3.2 Instrumentation and Test Procedure
The specimens were loaded following a cyclic displacement pattern, consisting of
cycles of increasing displacement magnitude. For each specimen, the magnitude and the
direction of the applied displacement were selected based on the size and shape of the
specimen. The tests were stopped when a considerable reduction in strength and stiffness
of the specimen was observed.
The hysteretic response of each specimen was obtained from the load cell and the
displacement transducer in the hydraulic actuator. Additional data on various key
members of the specimens were obtained using electrical strain gauges. Two
potentiometers were mounted to the support frame to measure the base rotation.
100
4.3.3 Material Properties
The specimens were made from dual grade W18x40 steel beams. In some
specimens A36 steel bars were used for diagonal members in the special openings. The
actual material properties were obtained from tensile tests of coupons from various parts
of the test specimens. For a given specimen, an average yield stress of coupons from the
flanges and coupons from the web was used to represent the yield stress of that specimen.
The average yield stress values of various key members are given in Table 4.1.
Table 4.1. Average Yield Stress of Key Members. Coupon Specimen Yield Stress
(ksi) W18x40 1,2 51.5 W18x40 3,4,5 54.5
1 x 3/8 PL. 1 48.3 1 x 1/4 PL. 2,3,5 50.4
1 5/8 x 1/4 PL. 4 48.5 L1 1/4 x 1 1/4 x 1/4 2,3,4,5 49.2 L1 1/2 x 1 1/2 x 3/16 1 48.0 L2 1/2 x 2 1/2 x 3/8 5 48.2
4.3.4. Specimen 1
Specimen 1 was designed to gather information about the basic behavior of the
proposed upgrading scheme as well as to gather the information about the test set-up
support frame. The most important objective was to verify the yield mechanism of the
opening. The opening was designed based on the information obtained from tests of
special truss moment frames. It was created by removing the material in the web using
flame cutting torch. Then, the cut surfaces were smoothed using a steel grinder. Specimen
1 is shown in Figure 4.4 along with the details in the vicinity of the opening. The size of
the opening was approximately 16 inches by 13.5 inches leaving about 2.25 inches for
each of the chord members. The diagonal members were made from 2, 1inch by 3/8 inch,
flat bars.
101
The test of the first specimen was done in two phases. In the first phase, the first
loading history was applied to investigate the combined yield mechanism of the diagonal
and the chord members. The maximum applied drift was 1.75%. Then, the diagonal
members were removed by flame cutting and the second loading history was applied in
the second phase of the test. The maximum applied drift was 2.75%. This was done in
order to investigate the behavior of the chord members alone without the diagonal
members. The two loading histories are shown in Figures 4.5 and 4.6. In Figures 4.5 and
4.6, the drift ratio is the ratio in percent of the applied displacement to the distance from
the tip of the beam to the centerline of the support frame. Figure 4.7 shows the specimen
before the diagonals were removed and Figure 4.8 shows the specimen after the diagonals
had been removed. Figures 4.9 and 4.10 show the hysteretic loops from the first and the
second phases of the test, respectively. In Figures 4.9 and 4.10, the applied displacements
have been corrected to obtain equivalent fixed-end displacements by subtracting the rigid
body displacements caused by the rotation of the support frame.
Figure 4.4. Test Specimen 1.
Note: Dimensions in inches
102
Figure 4.5. Loading History 1 of Specimen 1.
-4.0
-3.0
-2.0
-1.0
0.0
1.0
2.0
3.0
4.0
0 5 10 15 20 25 30Cycle
Drif
t (%
)
Figure 4.6. Loading History 2 of Specimen 1.
-4.0
-3.0
-2.0
-1.0
0.0
1.0
2.0
3.0
4.0
0 5 10 15 20 25 30Cycle
Drif
t (%
)
103
Figure 4.7. Specimen 1 before Removal of Diagonal Members.
Figure 4.8. Specimen 1 after Removal of Diagonal Members.
104
-40
-30
-20
-10
0
10
20
30
40
-3 -2 -1 0 1 2 3
For
ce (
kips
)
Corrected Displacement (in.)
Figure 4.9. Hysteretic Loops of Specimen 1 with Diagonal Members.
-40
-30
-20
-10
0
10
20
30
40
-3 -2 -1 0 1 2 3
For
ce (
kips
)
Corrected Displacement (in.)
Figure 4.10. Hysteretic Loops of Specimen 1 without Diagonal Members.
105
In the first phase of the test, Specimen 1 with diagonal members produced a very
stable hysteretic response. Some local yielding was observed prior to buckling of the
diagonal members at the corrected drift of approximately 0.6%. Yielding in the chord
members and in the horizontal member followed the buckling. Large degree of strain
hardening was observed, as can be seen by the steep rise of the force-displacement
response after yielding. Yielding and buckling in the first phase of the test are shown in
Figure 4.11. Some important observations from the first phase of the test were:
• The vertical member (horizontal member in the test set-up) suffered
significant yielding. Although it can be shown that the contribution of this member to the
overall shear resistance can be neglected, the result suggests that the compactness ratio of
this member should be large enough to prevent its premature fracture.
• The buckling load of the diagonal members was much larger than
expected. This suggested that the value of effective length factor was much lower than
initially assumed. From an analysis of the test data, it was found that the appropriate
effective length factor was in the order of 0.8 of the clear length of the braces.
The results of the first phase were very encouraging. The expected yield
mechanism formed, with inelastic activity confined in a few key locations only.
In the second phase of the test, the diagonal members were removed, as
mentioned earlier. The response was very stable at first, with yielding at the ends of the
chord members only. Yielding was mainly concentrated above the brace-to-chord
junctions. Plastic hinges were clearly seen before cracking occurred in a chord member
right above the location where the diagonal member used to be. Cracking resulted in a
reduction in the load carrying capacity, as can be seen in Figure 4.10. This early fracture
was probably due to high stress concentration at the corners of the opening. Yielding of
the chord members in the second phase of the test is shown in Figure 4.12. The cracking
in the chord member is shown in Figure 4.13.
106
Figure 4.11. Yielding and Buckling in Specimen 1 with Diagonal Members.
107
Figure 4.12. Yielding in Specimen 1 without Diagonal Members.
Figure 4.13.Cracking of the Chord Member.
108
4.3.5 Specimen 2
Specimen 2 was designed in a similar fashion as Specimen 1. Only a few changes
were introduced in this specimen. Major modifications included the use of a more
compact vertical member and the use of smaller diagonal members. The vertical member
was more compact as to prevent premature fracture that occurred during the previous test.
The size of the diagonal members was reduced because the results from the previous test
also suggested that the capacity of the actuator might not be sufficient to impose a drift of
3% if the member sizes remained the same as in Specimen 1. Specimen 2 is shown in
Figure 4.14. The loading history for Specimen 2 is shown in Figure 4.15. The response
of Specimen 2 is shown in Figure 4.16.
Figure 4.14. Test Specimen 2.
Note: Dimensions in inches
109
-4.0
-3.0
-2.0
-1.0
0.0
1.0
2.0
3.0
4.0
0 5 10 15 20 25 30Cycle
Drif
t (%
)
Figure 4.15. Loading History for Specimen 2.
-40
-30
-20
-10
0
10
20
30
40
-3 -2 -1 0 1 2 3
For
ce (
kips
)
Corrected displacement (in.)
Figure 4.16. Hysteretic Loops of Specimen 2.
110
The response of Specimen 2 was stable, similar to that of the Specimen 1. The
first inelastic activity occurred in the diagonal members when one of them buckled at the
corrected drift of approximately 0.3%. Following the buckling of both diagonal members,
yielding in the vertical member was observed. Finally, yielding of the chord members
completed the yield mechanism. As the test progressed, severe deformation of the
diagonal and vertical members was clearly seen, as shown in Figure 4.17. Nevertheless,
the load carrying capacity continued to rise. At the drift of 1.6%, small cracks developed
in both the chord and the vertical members. As the test continued, these cracks
propagated until they reached the flanges of the chords, as can be seen in Figure 4.18.
Although the crack in the vertical member continued to grow, it did not compromise its
ability to transfer the applied shear force.
Overall, the performance of Specimens 1 and 2 was not satisfactory. Although
both specimens could maintain their load carrying capacity, it was clear that the detailing
scheme used could not sustain the high ductility demands at the critical locations.
Figure 4.17. Yielding and Buckling in Specimen 2.
111
Figure 4.18. Cracking in the Chord Member of Specimen 2.
4.3.6 Specimen 3
The goal of Specimen 3 was to verify if an alternative detailing scheme could
sustain large ductility demands at the critical sections. The size of the opening was larger
than those in Specimens 1 and 2 to reduce the ductility demand. It has been shown by
Basha and Goel [Basha and Goel 1994] that the ductility demand is inversely
proportional to the square of the size of the opening. Results form tests of special truss
moment frames and tests of Specimens 1 and 2 suggest that the size of the opening in the
order of 20% of the span length is most desirable. Hence, the size of the opening in
Specimen 3 was set at about 20% of the span length, if it is taken as the distance from the
tip of the specimen to the center of the support frame. The detailing scheme near the
corners had also been improved. Schematic drawing of Specimen 3 is shown in Figure
4.19. The opening of Specimen 3 is shown in Figure 4.20. The loading history is shown
in Figure 4.21 and the response under this loading history is shown in Figure 4.22.
112
Figure 4.19. Test Specimen 3.
Figure 4.20. The Opening in Specimen 3.
Note: Dimensions in inches
113
-4.0
-3.0
-2.0
-1.0
0.0
1.0
2.0
3.0
4.0
0 5 10 15 20 25 30Cycle
Drif
t (%
)
Figure 4.21. Loading History for Specimen 3.
-40
-30
-20
-10
0
10
20
30
40
-3 -2 -1 0 1 2 3
For
ce (
kips
)
Corrected Displacement (in.)
Figure 4.22. Hysteretic Loops of Specimen 3.
114
The response of Specimen 3 was excellent. The hysteretic loops were very stable,
as can be seen from Figure 4.22. Similar to previous specimens, the inelastic activity
started with buckling of the diagonal members. Yielding of the chord and the vertical
members was observed in subsequent cycles. At the corrected drift of 2%, small cracks
developed at the ends of the vertical member, however, the load carrying capacity
remained unaffected. In later cycles, local buckling occurred at the end of the chord
members. This local buckling resulted in a small reduction in the load carrying capacity.
No fracture was observed in the chord members at the end of the loading history. Fracture
eventually developed after additional small displacements cycles, which were applied to
the specimen after the loading history shown in Figure 4.21 was complete. Additional
loading was applied in order to observe the failure mode only. No significant data were
recorded during these additional cycles. The deformation of the test specimen is shown in
Figures 4.23 and 4.24. Local buckling and the deformed shape of the diagonal members
are shown in Figure 4.25.
Figure 4.23. Deformation of the Test Specimen 3 (Positive Direction).
115
Figure 4.24. Deformation of the Test Specimen 3 (Negative Direction).
Figure 4.25. Local Buckling of Chord Members.
116
Specimen 3 clearly shows that it is possible to detail the opening to meet the high
ductility demand. Subsequent analysis shows that the ductile behavior of the opening
depends on both the geometry and the local detailing of the opening. An important factor
that influences local ductility is the geometry of the welds around the corners. It is
important to provide some free distance from the ends of welds to the edges of the chord
members, as can be seen in Figure 4.20, to allow plastic deformation of the material near
the edges. Such gaps should be maintained at all weld locations.
4.3.7 Specimen 4
Specimen 4 was designed to study if a Vierendeel opening can be used as an
alternative to the X-braced openings. This type of openings may be useful as it provides
space for ducts. The disadvantage in this case is that the chord members must provide all
the shear resistance. For this reason, additional plates were needed to reinforce the chord
members of Specimen 4. The additional plates were fully welded along the length of the
chord members. The details of Specimen 4 are shown in Figure 4.26 and 4.27. The
loading history for Specimen 4 is shown in Figure 4.28. The hysteretic response is shown
in Figure 4.29.
Specimen 4 did not perform as well as the other specimens. The specimen failed
prematurely at one of the corners of the opening. The additional plate suffered severe
local buckling at a very early stage of the test. The strength of the opening reduced
suddenly after cracking since this type of opening has very little redundancy in its load
carrying mechanism. This response is different from the ductile behavior observed in
special truss moment frames with Vierendeel opening. This is probably due to stress
concentration at the corners of the opening. Local buckling and fracture of Specimen 4
are shown in Figure 4.30.
117
Figure 4.26. Test Specimen 4.
Figure 4.27. A Close-Up View of Specimen 4.
Note: Dimensions in inches
118
0 5 10 15 20 25 30-4.0
-3.0
-2.0
-1.0
0.0
1.0
2.0
3.0
4.0
Cycle
Drif
t (%
)
Figure 4.28. Loading History of Specimen 4.
-40
-30
-20
-10
0
10
20
30
40
-3 -2 -1 0 1 2 3
For
ce (
kips
)
Corrected Displacement (in.)
Figure 4.29. Hysteretic Loops of Specimen 4.
119
Figure 4.30. Local Buckling and Fracture of Specimen 4.
4.3.8 Specimen 5
Specimen 5 was the last specimen in this series. The objective of Specimen 5 was
to investigate the reparability of the opening after severe deformation. Previously tested
Specimen 4 was moved from the test fixture and the chord members were removed by
flame cutting. The chord members of the opening were then replaced by 4 angles, two for
each chord member. New diagonal members were also added. This repair scheme offers
an advantage that the chord members are continuous from the opening into the web of the
girder making them behave similarly to the chord members in special truss moment
frames. The detailed drawing of the fifth specimen and a close-up photograph are shown
in Figures 4.31 and 4.32, respectively. The loading history for Specimen 5 is shown in
Figure 4.33. Its hysteretic response is shown in Figure 4.34. The deformation of the
specimen is shown in Figures 4.35 and 4.36.
120
Figure 4.31. Test Specimen 5.
121
Figure 4.32. Close-Up View of Specimen 5.
Figure 4.33. Loading History of Specimen 5.
Note: Unsymmetrical Loading Due to Out-of-Plane Twisting
-4.0
-3.0
-2.0
-1.0
0.0
1.0
2.0
3.0
4.0
0 5 10 15 20 25 30cycle
Drif
t (%
)
122
-40
-30
-20
-10
0
10
20
30
40
-3 -2 -1 0 1 2 3
For
ce (
kips
)
Corrected Displacement (in.)
Figure 4.34. Hysteretic Loops of Specimen 5.
Figure 4.35. Deformation of the Specimen 5 (Negative Displacement).
123
Figure 4.36. Deformation of Specimen 5 (Positive Displacement).
Specimen 5 produced a very stable response under cyclic loading, similar to the
response of Specimen 3. After buckling and yielding of the diagonal members, plastic
hinges started to form in the chord members. The chord members of this specimen were
very ductile, with plastic hinges well distributed along their length. In fact, the chord
members of Specimen 5 performed better than the chord members of Specimen 3 because
they were continuous far into the beam. The repaired specimen performed satisfactorily
up to the drift about 2.5% when out-of-plane twisting of the opening started. Twisting
intensified as the test continued. During the last few cycles, it was impossible to displace
the specimen to the intended level. Therefore, only one-sided loading was applied, as can
be seen in Figure 4.33. It was decided to stop the test to prevent damage to the actuator.
Specimen 5 shows that it is possible to replace the opening after severe
deformation and damage, provided that out-of-plane buckling of the chord members is
prevented. This can be done by providing lateral supports or keeping the size of the
124
opening sufficiently small. It can be noticed that the opening of Specimen 5 was
considerably larger than those in previous specimens. The length of this opening was
roughly 30% longer than that of the opening of Specimen 4. The results suggest that the
length of the opening should be maintained at about 20% of the span length. Doing so
provides sufficient stiffness and prevents excessive out-of-plane twisting.
4.4 ANALYSIS OF TEST DATA
The results form this series of tests provide invaluable information on the
behavior of the proposed upgrading scheme. As mentioned earlier, the goals of the
proposed system are to confine all the inelastic activity to the openings as well as to
reduce the strength demands at the beam-to-column connections. Therefore, the
overstrength of the opening plays a crucial role in the design of the upgrading scheme.
Any overstrength in the opening will result in the increase in strength demands in other
members. It is necessary to accurately estimate the overstrength developed in the opening
so that the ultimate strength of the opening can be predicted. Consequently, the demands
at critical members can be controlled.
The ultimate shear strength of an opening is the sum of the ultimate strengths
contributed by the chord members, the diagonal members, and the vertical member.
During the tests, it was observed that the reductions in the load-carrying capacity of the
specimens after the cracking of the vertical members were considerably small. Therefore,
by neglecting the contribution from the vertical member, the ultimate strength can be
expressed as:
xxcc VVV ξξ +=0 (4.1)
where oV is the ultimate shear strength of the opening, cV and xV are the nominal shear
strengths provided by the chord and diagonal members, respectively, and cξ and xξ are
the overstrength factors for the chord and diagonal members, respectively.
125
As mentioned in Chapter 3, the overstrength occurs primarily from two sources:
1) The difference in the actual and the nominal strength; 2) Strain hardening of materials.
In this proposed opening system, overstrength can be found in the diagonal members, the
chord members, and the vertical member. However, only the diagonal members and
chord members provide considerable resistance to the applied forces. Therefore, only
overstrength of the chord and the diagonal members is of concern in this study.
4.4.1 Overstrength of the Diagonal Members
In order to accurately estimate the shear strength of an opening, it is important to
account for both the maximum probable tensile strength and the probable post-buckling
strength of the diagonal members. During the tests, it was observed that the diagonal
members exhibited uniform yielding along their length. Therefore, the overstrength due
to strain hardening would not be significant. Strain hardening is typically more important
when the yielding is concentrated, such as in plastic hinge regions. For the diagonal
members, the difference in the actual yield stress and the nominal yield stress dominates
the overall overstrength of the members. Maximum tensile strength of the diagonals can,
therefore, be estimated as the product of the overstrength factor due to the difference
between the actual and nominal yield stresses and the nominal tensile strength.
It has been shown [Itani and Goel 1991] that flat bars exhibit large ductility
capacity. Therefore, it is recommended that this kind of structural members be used as the
diagonal members. Typically, flat bars are made of A36 steel. The expected yield stress
for A36 is on the order of 49 ksi. Hence, the overstrength factor xξ can be taken
approximately as:
40.136/49 ≈≈xξ (4.2)
The nominal post-buckling strength of the diagonal members is 0.3 of the
nominal buckling load as suggested in the AISC seismic provisions for STMFs [AISC
126
1997]. As mentioned earlier, test results show that the approximate buckling load can be
found by assuming the effective length factor k of 0.80. With these effective length and
the post-buckling strength factors, the shear contribution of diagonal members in the
opening can be estimated as:
xxcxyxx sin)P3.0P(V θξ += (4.3)
where xξ is the overstrength factor (1.40), xyP is the nominal yield force of the diagonal
member, xcP is the nominal buckling force of the diagonal member, and xθ is the angle
between the diagonal and the chord members.
4.4.2 Overstrength of the Chord Members
An expression for the overstrength factor for the chord members was first
proposed in the study of special truss moment frames by Basha and Goel [Basha and
Goel 1994]. This overstrength factor is a function of the length of the opening, the section
properties of the chord members, and the material properties. The overstrength factor can
be expressed as:
ch
cho
oc
c M
ML
LLEI )1()6(
2ηδη
ξ−+
−
= (4.4)
where δ is the story drift, η is the strain hardening factor, E is the young modulus, cI
is the moment of inertia of the chord member, L is the span length, oL is the length of
the special segment, and chM is the plastic moment of the chord member.
In Equation 4.4, all variables can be readily calculated except the strain hardening
factor η . Basha recommended that the value of the strain hardening factor, η , may be
taken as 0.05, if the plastic moment is calculated by using the actual yield strength, or it
may be taken as 0.10, if the plastic moment is calculated by using the nominal strength.
127
However, these values were based on built up chord angle members and might not be
applicable to wide-flange members.
Numerical simulations were carried out to calibrate the strain hardening factor for
wide-flange beam sections. Specimen 1, after the diagonal members had been removed,
was used as the model. Static pushover analyses were carried out to find the envelopes of
the force-displacement response corresponding to different values of the strain hardening
factor. The objective was to compare these envelopes with the test results.
The specimens were modeled in SNAP-2DX [Rai et al. 1994] using beam-column
elements. The simplified model consisted of four beam-column elements representing the
W18x40 beam, the chord members and the vertical member. The chord members were
connected to the element representing the W18x40 beam by means of rigid links, which
had the same dimension as the opening. All elements were modeled using the actual yield
strength obtained from coupon specimens. Two values of strain hardening, 5% and 10%,
were used. The hysteretic loops of Specimen 1 in the second phase of the test, before
cracking occurred, are compared with the results from computer analyses in Figure 4.37.
Figure 4.37. Comparison of Strain Hardening Values.
-15
-10
-5
0
5
10
15
-1.5 -1 -0.5 0 0.5 1 1.5
Experiment10%5%
For
ce (
kips
)
Corrected Displacement (in.)
128
As can be seen, the strain hardening value of 10% gives a better approximation of
the ultimate shear strength. It can also be noticed that the overstrength due to this strain
hardening alone is considerably large. Contrary to the diagonal members, the
overstrength due to the difference between the actual yield stress and the nominal yield
stresses in the chord members is not significant. The results from pushover analyses using
the actual yield stress and two assumed values of nominal yield stress are compared with
the response from the experiment in Figure 4.38. It can be seen that the difference in the
ultimate shear strength from the experiment and the ones computed using the nominal
values are insignificant. The result suggests that the nominal yield stress value can be
used in calculating the ultimate shear strength of the opening provided that a proper value
of strain hardening is used. Considering these two factors, it is recommended that
ultimate shear strength can be computed using the nominal yield stress and the strain
hardening value of 10%.
Figure 4.38. Comparison of Yield Stresses.
-20
-15
-10
-5
0
5
10
15
20
-1.5 -1 -0.5 0 0.5 1 1.5
Experiment
Fy = 51.5
Fyn = 50
Fyn = 36
For
ce (
kips
)
Corrected Displacement (in.)
129
It was also found form post-experiment investigations that the plastic hinges in
the chord members of most specimens were located at a small distance away from the
edge of the opening. Therefore, in calculating the ultimate shear strength of an opening, it
is recommended that the length of the opening be taken as 0.95 of the nominal length.
Using all this information, the overstrength of an opening can be computed as:
ch
cho
oc
c M
ML
LLEI )1.01(
)95.0(
95.0)6)(10.0(
2−+
−
=δ
ξ (4.5)
or ch
ch
o
oc
c M
ML
LLEI 90.0
95.0665.0
2+
−
=δ
ξ (4.6)
4.4.3 Ultimate Shear Strength of the Openings
Figure 4.39 shows the equilibrium of the internal forces at the ultimate state of a
frame modified with a web opening. The shear strength of the opening consists of the
shear resistance contributed by the chords, the diagonals, and the vertical member. By
assuming that the points of inflection of the chord members are at the middle of the
opening and by neglecting the shear contribution from the vertical member, the vertical
resultant force oV in the opening can be expressed as:
)sin()3.0(95.0
)(4xxcxyx
o
chco PP
L
MV θξξ
++= (4.7)
where cξ and xξ are the overstrength factors for the chord and the diagonal members,
respectively, chM is the plastic moment of T-section chord members, oL is the length of
the opening, xyP is the yield force of the diagonal members, xcP is the buckling force of
the diagonal members, and xθ is the angle between the diagonal and the chord members.
130
Figure 4.39. Equilibrium of Internal Forces in the Opening.
Using Equations 4.2, 4.6, and 4.7, the expected ultimate shear strength of
Specimens 1 to 5 can be calculated. The shear forces contributed by the chord members
(the first term on the right-hand side of Equation 4.7) in Specimens 1 to 5 are calculated
and shown in Table 4.2. The shear forces contributed by the diagonal members (the
second term on the right-hand side of Equation 4.7) in Specimens 1 to 5 are calculated
and shown in Table 4.3. The calculated values of the ultimate shear strength, the sum of
the shear contributed by the chord and the diagonal members, of Specimens 1 to 5 are
compared with the attended loads from experiments in Table 4.4. In all calculations, it
was assumed that the story drift for each specimen was approximately equal to the fixed-
ξcMch
ξcMch
2ξcMch/0.95Lo
2ξcMch/0.95Lo
θx
0.3ξxPxc
ξxPxy
Resultant Force Vo
BMD
SFD
b) Equilibrium of Internal Forces
a) Moment and Shear Force Diagrams
131
end drift from the experiment. The fixed-end drift was the ratio of the applied
displacement to the distance from the tip of the beam to the center of the support frame.
The nominal yield stresses were taken as 36 ksi for the diagonal members and as 50 ksi
for the chord elements. The modulus of elasticity was assumed to be 29000 ksi. Lengths
L and oL were taken as twice the specimen length and twice the opening length,
respectively. This is because the test specimens were all half-span models.
Table 4.2. Shear Force Contributed by Chord Members.
Specimen δ (%)
cI
(in4)
L
(in.)
oL
(in.) chM
(kip•in) cξ cV
(kips) 1 1.44 0.79 176 32 52.20 1.50 10.31 2 2.24 * 0.79 123 32 52.20 1.49 10.26 3 2.88 0.79 176 36 55.23 1.77 11.46 4 1.98 * 2.50 123 32 94.13 1.53 18.96 5 3.9 1.97 123 41 102.00 1.62 17.03
Note: ( * ) Drifts at first fracture.
Table 4.3.
Shear Force Contributed by Diagonal Members. Specimen
xL
(in.) xA
(in2) xθ
(rad.) xyP
(kips) xcP
(kips) xξ xV
(kips) 1 8 0.375 0.818 13.5 11.23 1.4 17.24 2 8 0.250 0.818 9 5.95 1.4 11.02 3 8.5 0.250 0.759 9 5.64 1.4 10.30 5 10 0.250 0.626 9 4.71 1.4 8.54
Table 4.4. Comparison between Expected and Experimental Ultimate Shear Strengths. Specimen
xV
(kips) cV
(kips)
Expected oV
(kips)
Experiment * (kips)
Exp./ oV
1 17.24 10.31 27.55 30.85 1.12 2 11.02 10.26 21.28 26.10 1.23 3 10.30 11.46 21.76 20.50 0.94 4 - 18.96 18.96 19.22 1.01 5 8.54 17.03 25.57 25.31 0.99
Note: ( * ) Average maximum positive and negative loads.
132
Table 4.4 shows that the expected values compare well with the results from
experiments. For Specimens 1 and 2, the expected values are somewhat smaller than the
actual values. This is probably because, in these specimens, the diagonal members were
connected to the chord members inside the opening. The consequence was that the plastic
hinges were pushed further into the openings making the length of the opening smaller
and the strain-hardening rate larger. This finally resulted in additional shear forces.
Although Equation 4.6 recognizes the rigid zones at the ends of the chord members by
introducing a factor of 0.95, this value was calibrated primarily from Specimen 3 test
results. Thus, the detailing scheme used in Specimen 3 is recommended for future use.
4.4.4 Modeling the Openings under Cyclic Loading
Cyclic behavior of a beam with an opening can be captured by using a finite-
element based computer code that has the capability to model beam-column elements as
well as axial compression elements. One of the issues involved in the modeling of any
truss-like structure under cyclic loading is the modeling of the axial compression
elements. Axial compression elements exhibit a complex post-buckling strength
degradation pattern, which affects the overall hysteretic response of the structure.
Pinching observed in the hysteretic loops of the test specimens is due primarily to the
strength degradation after buckling and the increase in the member length after
significant yielding [Jain et al. 1980].
In this study, test specimens were modeled using SNAP-2DX [Rai et al. 1997].
The axial compression elements in SNAP-2DX uses the Jain’s hysteretic model [Jain et
al. 1978], which is capable of modeling the post-buckling strength and stiffness as well as
the elongation in the element. The model uses several straight-line segments depending
on several control parameters in compression, but it is bilinear in tension. In compression,
the compressive strength reaches the buckling strength in the first cycle. In subsequent
cycles, the compressive strength reduces to the post–buckling strength specified by a
133
strength reduction factor. The strength reduction factor of 0.3 has been found to correlate
well with the results from experiments. In tension, the element yields at the yield strength
of the element. After the yield point, the load carrying capacity remains at the same level
without strain hardening. Jain’s hysteretic model is shown in Figure 4.40.
An analytical model using the same modeling assumptions mentioned in Section
4.4.2 was created to represent Specimen 3. Axial compression elements were used to
represent the diagonal members and beam-column elements were used to represent the
chord members and the beam. The model was subjected to the corrected displacement
from the test. The results from the analysis are shown in Figures 4.41. The figure shows
that the behavior of a beam with an opening can be modeled very well by using the
modeling assumptions presented earlier. Such modeling is essential for the seismic
evaluation of the proposed upgrading scheme, which will be presented in Chapter 5.
Figure 4.40. Axial Hysteretic Model for Diagonal Members [Jain et al. 1978].
5∆y
Pxy
Pxc
αPxc
Compression
Tension
Displacement
∆y = Yield Displacement in Tension α = Strength Reduction Factor
134
-30
-20
-10
0
10
20
30
-3 -2 -1 0 1 2 3
ExperimentalAnalytical
For
ce (
kips
)
Corrected Displacement (in.)
Figure 4.41. Analytical Modeling of Specimen 3.
4.5. SUMMARY AND CONCLUDING REMARKS
An upgrading scheme for steel moment resisting frames was proposed in this
chapter. This upgrading scheme consists of creating ductile rectangular openings in the
middle of the beam web to control the yield mechanism of the frame. Five half-scale half-
span specimens were tested to study the behavior of each key member of the opening.
The major findings in this chapter are:
(1) In moment frames with beam web openings, the openings serve the same
function as the special segments in STMF structures. Under a severe ground motion, the
inelastic activity will be confined only in the opening region. It consists mainly of
yielding and buckling of diagonal members and the plastic hinging of the chord members
of the openings. In this proposed system, the chord, diagonal, and vertical members
should be designed such that, under their fully yielded and strain-hardened condition, the
moment at the every beam-column connection generated by the shear force in the
135
opening will be smaller than a critical value to reduce the risk of premature failure of
connections.
(2) Results of tests of beams with openings show that the proposed upgrading
scheme is feasible. Specimens with proper detailing provided a stable hysteretic response.
The inelastic activity was confined only in the designated locations as intended in the
design.
(3) Out of five specimens, Specimen 3 was the best. Its stable response was due to
the proper detailing of critical locations. Special detailing should be provided to minimize
stress concentration and also to move the locations of plastic hinges away from the
critical corners. The detailing scheme used in Specimen 3 provides the needed ductility
and it is also practical. The opening can be flame-cut but the surface of the cut close to
the corners should be smoothed out by grinding. It is desirable to have a radius in all
corners, although test results suggest that it may not be necessary. The vertical members
that reinforce the ends of the opening should be placed at an offset of about 0.5 in. from
the ends of the opening. This is done so that the plastic hinges are pushed away from the
critical areas. The diagonal bars also help in reinforcing the critical areas. In addition, all
welds should have at least 0.25 in. clearance from the edges to allow for plastic flow, thus
increasing local ductility.
(4) Shear in the openings is primarily resisted by the chord members and the
diagonal members. The ultimate shear strength of openings can be predicted by
multiplying the nominal shear strength of the chords and diagonals by the corresponding
overstrength factors.
The overstrength factor of the diagonal members can be primarily attributed to the
difference in the nominal and the actual yield strengths. The overstrength factor due to
strain hardening is not significant for diagonals. A value of 1.4 has been found to be a
reasonably accurate value of the overstrength factor for the chord members.
136
Unlike the diagonal members, the overstrength of the chord members is
dominated by strain hardening. The overstrength factor for the chord members is a
function of the length of the opening, the section properties of the chord members, and
the material properties. A strain hardening of about 10% has been found to correlate well
with the experimental data. It is recommended that ultimate shear strength be computed
using the nominal yield stress value with 10% strain hardening.
(5) Openings can be easily replaced after severe deformation as shown by
Specimen 5. The chord members of the opening can be replaced by angles, provided that
the length of the opening is not too large. Lateral instability may occur if the chord
members are not properly braced against lateral movement. This is also true for the
original wide-flange beam openings before retrofitted.
(6) Cyclic behavior of beams with openings can be modeled using a finite-
element based software, provided that proper hysteretic behavior of the axial compression
elements is used. Axial compression elements exhibit a complex post-buckling strength
degradation pattern, which affects the overall hysteretic response of the structures. An
analytical model in Section 4.4.4 was shown to capture the hysteretic behavior a beam
with an opening very well.
137
CHAPTER 5
SEISMIC DESIGN AND BEHAVIOR OF MOMENT FRAMES WITH
DUCTILE WEB OPENINGS
5.1 INTRODUCTION
The results of small-scale experiments described in Chapter 4 show that the
proposed upgrading scheme can be utilized to control the strength and deformation of
existing moment frames. To verify this, it is necessary to conduct an analytical
investigation on the seismic response of full-scale structures as well as to experimentally
verify the analytical models used to predict their response. Guided by the experimental
results presented in Chapter 4, a design procedure for seismic upgrading of steel moment
frames is presented in this chapter. The moment frame structure discussed in Chapter 2
was used as an example structure. It was modified using the proposed upgrading
procedure. The response of the upgraded frame under severe ground motions was studied
using the same series of nonlinear analyses as in Chapter 2. Finally, a full scale testing of
a one-story subassemblage was carried out to verify the proposed modification procedure
and to verify the results of computer analyses.
5.2 PROPOSED DESIGN APPROACH
As mentioned in Chapter 4, conventional beam-to-column connections may
possess far less ductility than expected [Englehardt and Husain 1993, SAC 1996]. In the
system proposed herein, beam openings can be conservatively designed so that, under a
severe ground motion, the connection moments generated by the shear force in the
138
opening will be smaller than a chosen critical value during the entire earthquake
excitation. This critical value can be selected based on the type of connections used in the
frame. For conventional beam-to-column connections (welded-flange-bolted-web
connections), one possible design approach is that the openings, under their fully yielded
and strain-hardened condition at about 3% of story drift, would generate moments at the
beam-to-column connections smaller than their flexural yield capacity. This reduces the
risk of having premature failure of those connections. The story drift of 3% is selected
based on an observation that strong column moment frames would generally experience
story drift less than 3% for design level earthquakes. The maximum shear strength of an
opening can be estimated by using the equations given in Chapter 4.
The design of an opening begins by calculating the maximum allowable shear
force in the opening from the design requirement that the connection moment created by
the opening shear force is approximately equal or smaller than the yield moment of the
connection. Generally, moment frames are exterior frames, therefore, the effect of gravity
loads is small when compared to that of the lateral loads. Therefore, it is neglected in the
following design procedure. Figure 4.39 shows the internal forces in a frame with an
opening due to lateral loads only. From the simplified moment and shear force diagrams
shown in Figure 4.39a, assuming that the opening is placed at the mid span and
neglecting the moment due to the vertical member and the axial forces in the chord
members, the opening shear force, oV , should be (using center line dimensions):
L
MV y
o
φ2≤ (5.1)
where yM is the yield moment of the connection,φ is the resistance factor , 0.90, and L
is the span length.
139
5.2.1 Design of Chord Members
As shown in Chapter 4, the overstrength factor for the chord members is a
function of both the length of the opening and the section properties of the chord
members. The overstrength factor for the chords of a wide-flange beam with an opening,
cξ , can be expressed as:
ch
cho
oc
c M
ML
LLEI 90.0
95.0665.0
2+
−
=δ
ξ (5.3)
where δ is the story drift, E is the young modulus, cI is the moment of inertia of the
chord member, L is the span length, oL is the length of the special opening, and chM is
the nominal plastic moment of the chord members. By substituting δ of 0.03 (3% drift),
the overstrength factor for the chord members can be evaluated as:
ch
cho
oc
c M
ML
LLEI 90.0
95.002.0
2+
−
=ξ (5.4)
The overstrength factor is directly related to the ductility demand at the plastic
hinges in the chord members–the larger the ductility demand, the larger the overstrength
factor. Therefore, in order to prevent severe damage in the chords during a major
earthquake, the overstrength factor should be maintained in the range of 1.8-2.0.
Overstrength values in this range have been found by experiments to be practical.
From Equation 5.4, the overstrength factor of the chord members is a function of
both the length of the opening and the section properties of the chord members. The
design process for a chord member is based primarily on a trial and error approach to
converge on a reasonable value of the overstrength factor.
The design begins by determining the length of the opening. From the tests of
STMF frames and the tests of small-scale specimens, the length of the opening on the
140
order of about 0.20 to 0.25 of the span length has been found to perform well and provide
a good combination of frame stiffness and strength. After the length of the opening has
been selected, since cI and chM can be expressed in terms of the depth of the chord, the
chord of the opening can be designed by varying the depth of the chord until the
overstrength factor calculated by Equation 5.4 converges to the target range of 1.8-2.0.
5.2.2 Design of Diagonal Members
With a known depth of the chord members, the shear contribution of the chords
can be determined, and consequently, the size of the diagonal bars can be computed.
From Equation 4.7, it follows that:
o
chcoxxcxyx L95.0
)M(4Vsin)P3.0P(
ξθξ −=+ (5.5)
where oV is the required shear force in the opening calculated from Equation 5.1.
Forces xyP and xcP can be calculated by using the formulas given in the AISC-
LRFD specifications [AISC 1994] by using the clear length of the diagonal members and
the effective length factor, k , of 0.80. The design process for the diagonal members is
also based on trial and error approach to satisfy Equation 5.5.
5.2.3 Design of the Vertical Member
With the designed chord members and diagonal bars, the force in the vertical
member, vP , can be found from equilibrium. From Figure 5.1, where equilibrium of
forces at the vertical-to-chord junction is shown, this axial force can be calculated as:
xxcxyv sin)P3.0P(4.1P θ−= (5.6)
Conservatively, the compression force in vertical member can be taken as:
xxyv sinP4.1P θ= (5.7)
141
Using this force, the vertical member can be conservatively designed by using the
AISC-LRFD specifications, assuming a clear length and the effective length factor, k , of
1.0.
Figure 5.1. Equilibrium of Forces at the Middle Joint.
5.2.4 Design of the Welds
The welds for the diagonal bars should be designed to take the fully strain
hardened forces created by the diagonal members, i.e., xyP4.1 . The welds for the vertical
member should be designed such that the full plastic moment of the vertical member can
be developed.
5.2.5 Required Strength of the Opening under Gravity Loads
The previously mentioned design procedure for the girder web opening is a limit
state design procedure, which considers the force distribution at the ultimate lateral load
condition. It is based on a premise that the dead load is small. However, the opening
should also be checked against the gravity load combination, 1.4DL+1.6LL, even though
its effect may be small. Under this condition, all the members in the opening should
ξcMch
ξcMch
2ξcMch/0.95Lo
2ξcMch/0.95Lo
0.3ξxPxc
ξxPxy
Resultant Force Vo
θx
ξxPxy 0.3ξxPxc
Pv
2ξcMch/0.95Lo 2ξcMch/0.95Lo
142
remain in the elastic range. The design philosophy used herein is based on permitting
inelastic activity in the openings only in the event of extreme earthquake lateral loads.
5.2.6 Detailing of the Openings
Stress concentration is a major cause of damage and cracking in steel structures.
In case of the proposed upgrading scheme, stress concentration is highest at the corners
of the opening. Therefore special detailing should be provided to minimize the stress
concentration and to move the location of plastic hinges away from the critical corners.
The detailing scheme used in Specimen 3, presented in Chapter 4, provides the needed
ductility and is practical too. Therefore, it is recommended that this kind of detailing be
used. The opening can be flame-cut, but the surface of the cut in the vicinity of the
corners should be smoothed out by grinding. It is desirable to have a radius in all corners,
although test results suggest that it might not be necessary. The vertical members at the
ends of the opening should be placed at an offset of about 0.5 in. from the ends of the
opening. This is done so that the plastic hinges are pushed away from the critical areas.
The diagonal bars, as used in Specimen 3 described in chapter 4, also help in reinforcing
the corners. All welds should have at least 0.25 in. clearance from the edges to allow for
plastic flow and increase local ductility.
5.3 THE STUDY BUILDING
The building selected for this study is the six-story moment frame structure used
earlier in Chapters 2 and 3. In order to study the response of the proposed structural
system, the moment frame structure of the example building was modified according to
the proposed procedure.
Taking the fourth floor girder (W36x150) as an example, and assuming grade 50
steel, the maximum allowable shear at mid span is:
143
2.151300
)50)(504)(9.0(22==
L
M yφ kips (5.8)
Choosing the length of the opening as 0.20 times the span length and a target
overstrength factor of 2.0 for the chord members, the appropriate depth of the chord
members was found by trial and error to be 4.25 inches, with an exact overstrength value
of 2.09 and a plastic moment, chM , of 347.6 k-in. Therefore, the shear provided by the
chords is:
02.51)300)(2.0(95.0
)6.347)(09.2(4
95.0
4==
o
chc
L
Mξ kips (5.9)
By taking 40.1=xξ , the shear contribution from the diagonal members should
be, from Equation 5.5:
18.10002.512.151sin)P3.0P(4.1 xxcxy =−=+ θ kips (5.10)
Using 17/8 x 11/8 bars interconnected at the mid-length, the yield force and the
buckling force (with k = 0.80 and xl = 23 in.) were found to be 76.0 kips and 64.3 kips,
respectively. Taking xθ to be approximately 49 degrees (Figure 5.2), the shear
contribution of the diagonal members is:
2.100sin)P3.0P(4.1 xxcxy =+ θ kips (5.11)
The total shear force provided by the opening is:
2.1512.512.100 =+=oV kips (5.12)
which is adequate. With the selected bar size, the compression force in the vertical
member is:
9.79sinP4.1P xxyv == θ kips (5.13)
Therefore, double angles 2L21/2 x 21/2 x 3/8 with a calculated critical load of 95
kips were used for the vertical member. The modified frame is shown in Figure 5.2.
Calculations for the other floor girders are summarized in Tables 5.1 and 5.2.
144
Figure 5.2. The Modified Frame with Beam Web Openings.
145
Table 5.1. Design of Web Openings.
Beam Size
ØMy (kip-in)
V-allowable (kips)
Chord Depth (in)
cξ Vc
(kips) Diagonal Members
Vx
(kips)
W27x94 10935 72.9 3.75 2.00 28.7 1 1/2 x3/4 42.5
W36x135 19775 131.7 3.75 1.97 35.5 1x17/8 88.0
W36x150 22680 151.2 4.25 2.09 51.0 17/8x11/8 100.2 W36x210 32355 215.7 4.50 2.01 80.0 2x13/8 133.0 Note: Calculations based on Fy = 50 ksi for chord members and Fy=36 ksi for diagonal
members.
Table 5.2. Member Sizes of the Modified Frame with Web Openings.
Floor Beam Size
Opening Length
(in)
Chord Depth (in)
Diagonal Members
(inxin)
Vertical Members
Roof W27x94 60 3.75 11/2 x 3/4 2L 2 x 2 x 3/16
5 W36x135 60 3.75 1 x 17/8 2L 2 x 2 x 3/8 4 W36x150 60 4.25 17/8 x 11/8 2L 21/2 x 21/2 x 5/6
3 W36x210 60 4.50 2 x 13/8 2L21/2 x 21/2 x 1/2
2 W36x210 60 4.50 2 x 13/8 2L21/2 x 21/2 x 1/2
5.4 NONLINEAR ANALYSES OF THE STUDY BUILDING
In order to study the behavior of the structure modified by the proposed scheme,
nonlinear analyses were performed to compare its behavior before and after the
modification. One-bay, five story, models of the original three-bay moment frame and
the modified frame with web openings were prepared for inelastic static ("pushover") and
inelastic dynamic analysis. SNAP-2DX [Rai et al. 1994] computer program was used for
the inelastic analyses. Modeling assumptions were similar to the ones used in previous
studies. These assumptions include: 1) Gravity loads were neglected; 2) Floor masses of
the frame were lumped at the beam-to-column connection nodes; 3) 2% mass
proportional damping was used in dynamic analyses with the estimated frame period
calculated from the 1994 UBC. In this study, girders and columns were modeled using
146
beam-column elements with 2% strain hardening in the end moment-rotation models. The
chord members were also modeled using the beam-column elements, but with 10% strain
hardening. Diagonal members of the openings were modeled using axial buckling
elements with Jain’s hysteretic model as explained in Chapter 4.
In order to accurately simulate the response of the modified frame, it is important
to correctly model the overstrengths of the structural members. This is because
controlling the overstrength is one of the main objectives of this upgrading system.
Therefore, the yield stress for each member was taken as the expected yield stress. The
yield stresses for the girders and columns were taken as 55 ksi (expected for A572 GR.50
steel). For diagonal members, the yield stress was taken as 49 ksi (expected for A36
steel). For comparison purposes, the original frame was also modeled using an expected
yield stress of 55 ksi for all members. The panel zone deformations of the original and
the modified frames were not considered since the main purpose was to compare the
overall behavior of the modified frame to that of the original frame. The analytical
models of the modified frame with openings and the original frame are shown in Figures
5.3 and 2.5, respectively.
Figure 5.3. The Modified Frame and its Analytical Model.
147
Similar to the analyses performed in Chapter 2, the static pushover analysis was
carried out by applying lateral forces representing the UBC distribution of design lateral
forces. For the inelastic dynamic analysis, these two models were subjected to the four
scaled earthquake records used in previous chapters. These records were: the 1940 El
Centro record, the 1994 Northridge (Sylmar Station) record, the 1994 Northridge
(Newhall Station) record, and one synthetic record. The results from the analyses are
presented and discussed in the following sections.
5.4.1 Inelastic Static Pushover Analyses
The results from static pushover analyses are summarized in Figures 5.4 and 5.5.
Figure 5.4 shows the base shear versus roof displacement plots for the original and the
modified frames. In the modified frame, the first inelastic activity was the buckling of the
diagonal members in the fourth floor girder when the roof drift was approximately 0.8%.
After all the diagonal members in other floor openings had buckled, yielding of the
diagonal members started. Then, it was followed by yielding in the chord members. At
the roof drift of about 2.3%, last set of plastic hinges formed in the chord members of the
roof girder opening. The sequence of inelastic activity up to the onset of mechanism is
shown in Figure 5.5.
The strength corresponding to the first significant non-linearity in the force-
displacement plot of the modified frame was approximately half that of the original
frame. Both frames showed significant overstrength above the UBC-94 design force
level, approximately 6 times for the original frame and 5 times for the modified frame.
The stiffness of the modified frame was somewhat smaller than that of the original frame
because of the openings. However, in the original frame, the first set of plastic hinges to
form was at the column bases and the yield mechanism was a soft story in the first story,
as shown in previous chapters. The modified frame, on the other hand, behaved in a truly
strong-column weak-beam fashion with inelastic activity essentially limited to the
148
openings in the girders and minor flexural yielding at the column base forming almost
last in the plastic hinging sequence.
The behavior of the modified frame was very similar to the plastic designed frame
presented in Chapter 3. It is also noteworthy that no yielding was observed in the beam-
to-column connections. Therefore, in this case, the chance of having premature failure at
the connections was significantly smaller than in the original case.
Another objective of the pushover analysis of the modified frame was to
determine the maximum overstrength values in the chord members at 3% roof drift. As
mentioned earlier, one of the objectives of the proposed design procedure is to control the
value of the overstrength factor within an acceptable range,1.8 to 2.0. Table 5.3 shows
the overstrength values in the chord members at 3% roof drift and the design values.
Maximum overstrength ratio at a plastic hinge was calculated by dividing the magnitude
of the plastic moment occurred at 3% roof drift by the yield moment of the corresponding
chord member.
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0 0.5 1 1.5 2 2.5 3
ModifiedOriginalB
ase
She
ar C
oeffi
cien
t (V
/W)
Roof Drift (%)
1First Plastification
4 Mechanism
UBC DESIGN V = 0.09 W
4 Mechanism
1 First Buckling
Figure 5.4. Base Shear – Roof Drift Response of the Original and the Modified Frames
(Based on Expected Yield Strength).
149
Figure 5.5. Sequence of Inelastic Activity in the Modified Frames.
Table 5.3. Comparison Between Design and Attained Overstrength Values. Floor Level Beam
cξ (Analysis) Design Value
Roof W27x94 1.47 2.00 5 W36x135 1.80 1.97 4 W36x150 2.06 2.09 3 W36x310 1.87 2.01 2 W36x210 1.74 2.01
All of the overstrength values were smaller the design values. The maximum
overstrength occurred in the fourth floor girder (2.06) where the inelastic activity first
started. This value agreed well with the design value (2.09).
5.4.2 Inelastic Dynamic Analyses
Selected results from the inelastic dynamic analyses of the two frames are
presented in Figures 5.6 through 5.9. The envelopes of maximum story drifts and
maximum floor displacements of the two frames are compared in Figures 5.6 and 5.7,
4
0.40V
0.23V
0.18V
0.12V
0.07V
1
2
3
6
5 5
Note: Inelastic Activity in Openings Includes Yielding and Buckling of Diagonal Members and Plastic hinging of Chord Members.
150
respectively. Note that the response of the original frame was slightly different than the
ones shown in Chapters 2 and 3. This is because the modeling assumptions were slightly
different. The floor displacements and the maximum story drifts of the modified frame
with web openings were similar to those of the original frame. However, the patterns of
inelastic activity in the two frames were different. A story mechanism in the first story
formed in the original frame, as discussed in Chapters 2 and 3. It can be noticed from
Figure 5.8, where the ductility demands and location of inelastic activity are shown, that
the inelastic activity in the modified frame was limited only to the openings, as was
intended in the design. No story mechanism was observed in the modified frame. In
addition, no yielding in the beam-to-column connections was observed, thereby, reducing
the risk of having premature failure. As emphasized earlier, design for controlled inelastic
activity results in better response. Damage inspection and repair work after an earthquake
would also be relatively easier and less costly.
Modified Frame Original Frame
Figure 5.6. Maximum Floor Displacements of the Modified and the Original Frames.
0 5 10 15 201
2
3
4
5
6
El CentroSylmarNewhallSynthetic
Floor Displacement (in.)
Flo
or L
evel
0 5 10 15 201
2
3
4
5
6
El CentroSylmarNewhallSynthetic
Floor Displacement (in.)
Flo
or L
evel
151
Modified Frame Original Frame
Figure 5.7. Maximum Interstory Drifts of the Modified and the Original Frames.
El Centro Newhall Sylmar Synthetic
Figure 5.8. Location of Inelastic Activity in the Modified Frame under the Four Selected
Records.
(5.38)
(3.76)
(1.30)
(4.35)
(5.10)
(6.24)
(3.63)
(2.04)
(3.42)
(2.91)
(2.50)
(1.10)
(2.04)
(2.47) (2.47)
(2.77)
(4.93)
(1.44)
(2.59)
Note: Rotational Ductility Demands of the Chord Members Shown in Parentheses.
0 1 2 3 40
1
2
3
4
5
El CentroSylmarNewhallSynthetic
Story Drfit (%)
Sto
ry L
evel
0 1 2 3 40
1
2
3
4
5
El CentroSylmarNewhallSynthetic
Story Drfit (%)
Sto
ry L
evel
152
Figure 5.8 shows that the ductility demands were quite large in the chord
members of the openings. These large ductility demands were expected. They are
characteristic of structural systems that have “fuse” elements, such as eccentrically
braced frames and special truss moment frames. These high ductility demand values
imply that the chord members must have large ductility capacity.
Experiments have shown that the fuse elements in this upgrading system can be
very ductile. It was found that the ductility demand corresponding to the overstrength on
the order of 2.0 is acceptable. Figure 5.9 shows the maximum overstrength values in the
chord members due to the four selected records. As can be seen, the maximum
overstrength values in the chord members were well within the acceptable limit. The
overstrength values were lower than the design values (Table 5.1) because the attained
drifts were smaller than the value assumed during the design (3%).
El Centro Newhall Sylmar Synthetic
Figure 5.9. Maximum Overstrength Values Under the Four Selected Records.
(1.54)
(1.34)
(1.04)
(1.41)
(1.51)
(1.64)
(1.32)
(1.12)
(1.30)
(1.24)
(1.18)
(1.01)
(1.12)
(1.21)
(1.48)
(1.02)
(1.19)
153
The results of the analyses show that it is possible to improve the performance of
a moment frame using the proposed upgrading scheme. One critical issue is the ductility
capacity of the chord members. Although results form the test program presented in
Chapter 4 suggest that the ductility demand corresponding to an overstrength factor of
about 2 can be satisfied with the ductility capacity created by the detailing scheme similar
to the one used in Specimen 3, these results were based on small-scale tests with
simplified boundary conditions. A full-scale experiment is necessary to fully verify these
assumptions. The results of a full-scale test conducted as part of this study will be
presented and discussed in the following sections.
5.5 EXPERIMENTAL PROGRAM
In order to study the behavior of the proposed upgrading scheme experimentally,
a full-scale specimen representing a one-story sub-assemblage of a moment frame with
openings was designed and fabricated for a cyclic test. The objective of this test program
was to verify the design and the analytical modeling procedures, which were developed
based on tests of small-scale specimens. In addition, the full-scale test provided an
opportunity to verify the detailing scheme used in the critical locations. The following
sections describe the test procedure and the test results.
5.5.1 Test Set-Up
A one-story subassemblage, consisting of a full-scale 28 feet long W24x62 beam
with a web opening and two 13 feet long W14x82 columns at the ends of the beam, was
designed and fabricated for this experimental study. The columns were half-story high
above and below the beam with pinned ends at both the top and bottom of the columns.
The specimen represented a story in a one bay frame assuming that inflection points were
at the mid-height of the story. Even though this assumption is not entirely accurate as
shown in Chapter 2, it provides a convenient way to simulate the cyclic behavior of the
154
proposed structural system in the laboratory. The load was applied at the top of one of the
columns via a 100-kip actuator with a maximum stroke length of 5± inches. In order to
simulate earthquake loading, the actuator force must be transmitted through both
columns. This was accomplished by using one W12x50 link beam, which was pin-
connected between the column ends. The applied load represented the story shear
induced by an earthquake. Lateral braces were provided at one-third of the span to
simulate the presence of cross beams in the real structure. An overview of the test set-up
is shown in Figures 5.10 to 5.13. The dimensions of the test frame are presented in Figure
5.14.
Figure 5.10. Overall View of the Test Set-Up.
155
Figure 5.11. Close-Up View of the Test Specimen.
Figure 5.12. Lateral Bracing of the Test Specimen.
156
Figure 5.13. Beam-to-Column Connection of the Test Specimen.
157
Figure 5.14. Dimensions of the Test Specimen.
Note: Dimensions in inches
158
5.5.2 Design of the Girder and the Web Opening.
In this test, the size of the girder was selected based on the capacity of the
columns. These columns were not specifically designed for this experiment. Instead, they
were previously designed and used for other purposes [Basha and Goel 1994, Itani and
Goel 1991]. It was decided that, in order to prevent any significant yielding in the
columns, the maximum actuator force should not exceed 65 kips. The size of the girder
was selected such that the moment generated by this actuator force would create
moments at the beam-to-column connections of about 85% of the yield moment of the
beam. With the selected girder size, the opening was then designed according to the
previously described procedure such that the maximum applied force at 3% story drift
was at the target value (65 kips). The chords of the opening in the specimen were
designed to have an expected overstrength factor of about 1.80 at 3% story drift. The
dimensions of the opening are shown in Figures 5.15 and the close-up photographs of the
opening are shown in Figures 5.16 to 5.18.
Figure 5.15. Dimensions of the Web Opening in the Test Specimen.
159
Figure 5.16. Close-Up View of the Special Opening.
Figure 5.17. Diagonal-to-Chord Junction.
160
Figure 5.18. Vertical-to-Chord Junction.
5.5.3 Instrumentation and Test Procedure
Specimen displacement was applied at the top of the test frame in a quasi-static
manner using a predetermined cyclic displacement pattern. Two loading histories were
used in this experiment. Initially, only one loading history was intended to be used.
However, during the test, one of the bolts that connected the actuator to the reaction wall
unexpectedly became loose, causing the actuator to twist. It was necessary to stop the
experiment to replace the bolt. The test was resumed with the second loading history.
The first loading history consisted of cycles of increasing displacements up to
about 0.9% story drift where buckling and yielding first initiated. The second loading
history consisted of cycles of large displacement amplitudes up to 3% story drift. The
3% story drift limit was used because the value assumed in the design corresponding to
3% story drift, and also because the maximum stroke of the actuator was on the order of
161
3% story drift of the frame. The first and the second loading histories are shown in
Figures 5.19 and 5.20, respectively.
-4.0
-3.0
-2.0
-1.0
0.0
1.0
2.0
3.0
4.0
0 2 4 6 8 10 12
Sto
ry D
rift (
%)
Cycle
Figure 5.19. First Loading History.
-4.0
-3.0
-2.0
-1.0
0.0
1.0
2.0
3.0
4.0
0 5 10 15 20
Sto
ry D
rift (
%)
Cycle
Figure 5.20. Second Loading History.
162
The hysteretic response of the test frame was obtained from the load cell and the
displacement transducer in the hydraulic actuator. Additional data on various key
members were obtained using electrical strain gauges at selected points. Photographic
records were also made throughout the experiment.
5.5.4 Material Properties
The girder was made of a W24x62 dual grade steel. All diagonal web members
were made of A36 steel flat bars. The material properties were obtained by means of
tensile tests of coupons from various parts of the specimen. An average yield stress from
coupons was used to represent the yield stress of the girder. The average yield stresses of
various key members are given in Table 4.1.
Table 5.4. Average Yield Stresses of Key Members.
Coupon Yield Stress (ksi)
W24x62 52.8 L21/2 x 21/2 x 3/8 48.9 1 7/8 x 1 1/8 PL. 50.0
5.5.5 Test Results
The hysteretic loops from the first phase and the second phase of the test are
shown in Figures 5.21 and 5.22, respectively. In the first phase of the test, the response
started to deviate from elastic behavior at a drift of about 0.75%. Two of the diagonal
members buckled at this displacement. At the story drift of about 0.9%, some yielding in
the diagonal members was observed. An actuator bolt became loose at this point. The test
was resumed using the second loading history after the bolt was replaced.
In the second phase of the test, after the diagonals had completely buckled and
yielded, the chords of the opening started to plastify. Plastic hinges clearly formed at the
end of the 1.8% story drift displacement cycles. Progressing further into larger
163
displacement cycles showed a mechanism pattern as intended by the design, i.e., yielding
and buckling of the diagonal members followed by plastic hinging at the ends of the
chord members. The deformation and the complete mechanism of the test frame are
shown in Figures 5.23 to 5.26.
The test specimen was able to sustain many cycles of large displacements without
fracture. Only local buckling in the chords and local necking in the diagonal bars due to
very high local strain were observed. These local instabilities resulted in a small
reduction in the load carrying capacity of the frame during the 3% story drift cycles. The
chord members fractured much later after the frame was subjected to additional
decreasing displacement cycles, which are not shown here. These additional displacement
cycles were used to observe the failure mode only, thus no significant data were recorded.
The chord member cracks stopped propagating when they reached the flange of the chord
members. This fracture can be seen in Figure 5.27.
-80
-60
-40
-20
0
20
40
60
80
-5 -4 -3 -2 -1 0 1 2 3 4 5
-3 -2 -1 0 1 2 3
Load
(ki
ps)
Displacement (in.)
Story Drift (%)
Figure 5.21. Hysteretic Loops from the First Loading History.
164
-80
-60
-40
-20
0
20
40
60
80
-5 -4 -3 -2 -1 0 1 2 3 4 5
-3 -2 -1 0 1 2 3
Load
(ki
ps)
Displacement (in.)
Story Drift (%)
Figure 5.22. Hysteretic Loops from the Second Loading History.
Figure 5.23. Deformation of the Test Frame (Positive Displacement).
165
Figure 5.24. Deformation of the Test Frame (Negative Displacement).
Figure 5.25. Inelastic Activity in the Opening.
166
Figure 5.26. Yielding of the Chord and the Diagonal Members.
Figure 5.27. Crack in the Chord Member.
167
Overall, the test specimen performed as intended. The results show that the
proposed upgrading system is very ductile. All inelastic behavior was confined to the
designated elements of the web opening only. Even though pinching was observed in the
hysteretic loops, they were very stable. More importantly, the results show that, with a
proper detailing scheme, the chord members can sustain large cyclic deformations.
5.6 EVALUATION OF THE PROPOSED DESIGN PROCEDURE AND THE
ANALYTICAL MODELING
As can be seen from Figure 5.22, the maximum load obtained from the test was
65.8 kips in the first 3% story drift cycle. This maximum load agreed very well with the
design value of 65 kips. This clearly shows that the design procedure can accurately
estimate the ultimate strength of the opening. In addition, it is also important to
demonstrate that the analytical model can accurately capture the entire behavior of the
proposed upgrading system.
The modeling techniques used in the nonlinear analyses presented earlier were
used to create an analytical model of the test frame. The modeling assumptions were: The
expected yield stress value for diagonal members was taken as 49 ksi, and 55 ksi for
other members; centerline dimensions were used and the chord-to-beam junctions were
modeled as rigid. The analytical model of the test frame consisted of 12 beam-column
elements and 4 axial buckling elements. The beam-column elements represented the
columns, the girder, the link beam, and the chord members of the opening. The axial
buckling elements represented the diagonal members. The only change in modeling
assumptions was in modeling the panel zone deformation of the columns: the finite joint
dimensions were modeled explicitly using rigid link elements but the elastic deformation
in the panel zone was included as suggested by Krawinkler [Krawinkler 1978].
If the panel zone remains elastic at all times, the approximate story displacement
due to this panel zone deformation, pδ , can be calculated from:
168
pwcc
bsp V
Gtd
dh −=δ (5.14)
where sh is the story height, bd is the depth of the beam, cd is the depth of the column,
wct is the column web thickness, G is the shear modulus, and pV is the shear force in the
panel zone.
The analytical model of the test frame is shown in Figure 5.28. The results from
the simulations of the test are shown in Figures 5.29 and 5.30. Figure 5.29 shows the
results from the simulation with the first loading history. Figure 5.30 shows the results of
the simulation with the second loading history.
Figure 5.28. Analytical Model of the Test Specimen.
δ Applied Displacement
169
-80
-60
-40
-20
0
20
40
60
80
-5 -4 -3 -2 -1 0 1 2 3 4 5
-3 -2 -1 0 1 2 3
ExperimentalAnalytical
Load
(ki
ps)
Displacement (in.)
Story Drift (%)
Figure 5.29. Analytical Simulation of the Experiment with the First Loading History.
-80
-60
-40
-20
0
20
40
60
80
-5 -4 -3 -2 -1 0 1 2 3 4 5
-3 -2 -1 0 1 2 3
ExperimentalAnalytical
Load
(ki
ps)
Displacement (in.)
Story Drift (%)
Figure 5.30. Analytical Simulation of the Experiment with the Second Loading History.
170
As can be seen, the analytical model can accurately capture the response of the
test frame. The actual stiffness of the test frame was smaller than that predicted by the
analysis. This suggests that the dynamic responses presented in Section 5.4 might be
somewhat underestimated. However, the important aspect of this comparison is that the
inelastic activity, the yield mechanism, and the strength of the test frame can be
accurately simulated.
5.7. SUMMARY AND CONCLUDING REMARKS
A design procedure for the proposed steel moment frame upgrading scheme was
proposed in this chapter. The results from the experimental program presented in Chapter
4 were used as a basis for development of the design and the analytical modeling
procedures. Based on the design and modeling procedures presented herein, an analytical
study was conducted to investigate the dynamic behavior of the proposed system under
seismic excitations. The example frame used earlier in Chapter 2 was upgraded according
to the design procedure presented in this chapter. The results of both nonlinear static and
nonlinear dynamic analyses of the modified frame were presented and discussed. A full-
scale test was carried out to validate the key assumptions used in analyses and design.
The major findings in this chapter are:
(1) In the proposed upgrading system, beam openings can be designed so that,
under a severe ground motion, the moments at beam-column connections generated by
the shear forces in the openings will be smaller than a chosen critical value during the
entire excitation. One possible design approach is that the openings, under their fully
yielded and strain-hardened condition at about 3% story drift, would generate moments at
the beam-column connections smaller than their flexural yield moments. This reduces the
risk of premature failure of those connections and confines all inelastic activity only to
the openings. The maximum shear strength of an opening can be estimated by the
procedure as discussed in Chapter 4.
171
(2) The overstrength in the chord members is one of the most critical parameters
in the proposed upgrading system. It is directly related to the ductility demand in the
chord members. Therefore, it is necessary to control the ductility demand within an
acceptable limit. The maximum overstrength value depends on the maximum interstory
drift, the length of the opening, and the section properties of the T-shaped chord
members. The use of expected interstory drift of 3% in the design provides a realistic
upper-bound estimate of the maximum response. The target overstrength value of 2 is
recommended for chord members. The length of the opening in the order of 20% of the
span length provides a balance between strength and stiffness, and is recommended.
(3) The results of both nonlinear static and dynamic analyses of the upgraded
frame showed that the upgraded frame responded as expected and behaved very well.
From the static pushover analysis, the modified frame showed rather insignificant
decrease in stiffness because of the presence of the openings, when compared to the
original frame. However, from the time history analyses, the upgraded frame responded
to a severe ground motion in a desirable manner with controlled inelastic activity at
designated locations. Moreover, the risk of premature failure of beam-to-column
connections was essentially eliminated since no plastic hinges formed at the connections
of the upgraded frame.
(4) A full-scale one-story subassemblage, consisting of a 28 feet long W24x62
beam with a web opening and two 13 feet long W14x82 columns, was designed and
fabricated for a cyclic test. The subassemblage specimen responded as expected. The
inelastic activity was confined to only the designated members of the opening. The
maximum load was accurately predetermined. The test specimen also confirmed that the
detailing scheme devised earlier using small-scale specimens can be successfully used to
provide the required ductility. Detailing is one of the most important aspects of the
proposed upgrading system. If properly detailed, the opening can sustain large cyclic
deformations without fracture.
172
5) Analysis of the test data showed that the analytical modeling procedure
presented in this chapter slightly overestimated the stiffness of the test frame.
Nevertheless, the strength and the yield mechanism, which are more important, can be
accurately predicted.
In conclusion, it is possible to upgrade an existing moment frame by using the
special beam web openings, so that the strength and plastic mechanism of the upgraded
frame is controlled in a desirable manner. The proposed upgrading scheme provides an
effective alternative to the strengthening schemes that are currently being employed in
practice.
173
CHAPTER 6
SUMMARY AND CONCLUSIONS
6.1 SUMMARY
6.1.1 Introduction
The behavior of an existing moment-resisting frame designed by conventional
method was studied using extensive nonlinear static and nonlinear dynamic finite element
analyses. The results show that moment-resisting frames designed by the conventional
elastic method, using equivalent static forces, may undergo inelastic deformations in a
rather uncontrolled manner resulting in uneven formation of plastic hinges.
Guided by the performance of this conventionally designed frame, a new design
concept was proposed based on the principle of energy conservation and theory of
plasticity. Parametric studies were carried out to verify the validity of the proposed
design procedure. The results show that the proposed method can produce structures that
meet a pre-selected performance objective in terms of both the maximum drift and the
yield mechanism.
The study was then extended to include seismic upgrading of existing steel
moment frames for future earthquakes. A possible scheme to modify the behavior of
existing moment-resisting frames to have a ductile yield mechanism is proposed. This
upgrading scheme uses rectangular openings in the girder webs reinforced with diagonal
members as ductile “fuse” elements. A series of small-scale experiments were carried out
to study the feasibility of the proposed upgrading system.
174
Based on the results of these experiments, a detailed design procedure for seismic
upgrading of steel moment frames was presented. The results of nonlinear static and
dynamic analyses show that it is possible to upgrade an existing moment frame by using
special openings, so that its strength and plastic mechanism can be controlled in a
desirable manner. Finally, a full-scale test of a one-story subassemblage was carried out
to verify the proposed modification concept experimentally. The test results were very
satisfactory.
6.1.2 Conventional Moment Frame Behavior
The current design procedures for steel moment resisting frames were discussed
in Chapter 2. Related experimental and analytical studies found in the literature were
briefly presented. Nonlinear static and nonlinear dynamic time-history analyses were
carried out on an example structure. The results can be summarized as follows:
(1) Strong column – weak beam (SCWB) frames are superior to weak column –
strong beam (WCSB) frames. WCSB frames were found to produce concentration of
inelastic activity in a limited number of elements, especially in columns. SCWB frames
were found to distribute the inelastic activity over many more elements. The ductility
demands and damage potential are likely to be much higher in WCSB frames than in
SCWB frames. For example, the maximum interstory drifts of WCSB frames were found
to be quite sensitive to the increase in earthquake intensity. This is due to the formation
of undesirable yield mechanisms.
(2) Some plastic hinges can form in columns even when the frame is designed
according to SCWB requirements. The use of localized joint strength requirements,
although important, is not sufficient to prevent the formation of plastic hinges in the
columns. The distribution of moments in the columns after some beam yielding has
occurred is drastically different from the elastic distribution. The consequences of this
175
redistribution are uneven and unpredictable inelastic activity and uncontrolled
deformation mechanisms.
(3) The response of a conventionally designed moment frame is typically
characterized by early formation of plastic hinges at the column bases, high degree of
overstrength, and a soft story type mechanism. These problems are generally attributed to
two major factors. The first factor is the inconsistency between the strength and the drift
(stiffness) criteria imposed by building codes. Most of the moment frame designs are
governed by drift requirements leaving the sizes of beams relatively large compared to
the sizes of columns. The inelastic activity, therefore, tends to occur in the columns. The
second factor is the inability of the elastic design method to capture the distribution of
internal forces in the inelastic response stages. Combination of these two factors leads to
the formation of undesirable yield mechanisms.
6.1.3 Drift and Yield Mechanism Based Design (DYMB)
A new design procedure for steel moment frames was presented and discussed in
Chapter 3. The new design concept is based on plastic (limit) design theory. The ultimate
design base shear for plastic analysis is derived by using the input energy from the design
pseudo-velocity spectrum, a pre-selected yield mechanism, and a target drift. The
procedure also includes a step to determine the design forces in order to meet specified
target drifts in the elastic stage under moderate ground motions. Thus, the proposed
design procedure eliminates the need for drift check after the structure is designed for
strength as is done in the current design practice. Also, the need for response
modification factors is completely eliminated since the load deformation characteristics
of the structure, including ductility and post-yield behavior, are explicitly used in
calculating the design forces. The implications of the proposed method were also
presented. The major findings in Chapter 3 are:
176
(1) The use of plastic design principles in combination with the proposed design
forces derived by using the principle of energy conservation leads to structures with
better seismic response. The results of a parametric study showed that the proposed
method produced structures with story drifts that complied well with the target drift
values.
(2) Comparing with a structure designed by conventional method, a structure
designed by the proposed energy-based method had relatively smaller beam sizes and
larger column sizes. The total weights of the structures design using both methods were
similar. The seismic responses of the two structures, on the other hand, were not. The
sequences of inelastic activity of the two frames under increasing static lateral forces
(pushover analysis) were also drastically different. In the conventionally designed frame,
the first set of plastic hinges to form was at the column base and the yield mechanism
was a soft story in the first story. The redesigned frame, on the other hand, behaved as
expected, with a desirable strong column-weak beam mechanism. Plastic hinges
occurred only in the beams and at the column bases, the later forming last. The results
from dynamic analyses also showed a similar trend. The maximum drifts of the
redesigned frame agreed well with the target design limits.
(3) The use of elastic drift limit without considering the response at the ultimate
level is not quite meaningful for seismic design. It was shown that, even though the story
drifts under static lateral forces do not satisfy the drift criteria prescribed in the UBC, the
response under dynamic loading can be significantly better if the inelastic activity occurs
in a controlled manner, following a desired yield mechanism.
(4) By comparing the design base shear coefficients required by the proposed
method and those required by the UBC-94 and UBC-97, it was found that the design base
shears from the UBC-94 and UBC-97 were far too small. This suggests that the values of
the response modification factors, R in the UBC-97 and wR in the UBC-94, are
177
unrealistically large. More appropriate values should be about 3 to 4 times smaller than
those currently used.
(5) The proposed method can be easily presented in a performance-based design
framework. The performance objectives can be defined based on the earthquake
intensities and interstory drift levels. An optimal design base shear corresponding to a
chosen performance objective can be readily and directly obtained.
6.1.4 Seismic Upgrading with Beam Web Openings
An upgrading scheme for steel moment resisting frames was proposed in Chapter
4. The upgrading scheme consists of creating ductile rectangular openings in the middle
of the beam web to control the yield mechanism of the frame. Five half-scale half-span
specimens were tested to study the behavior of the key members in the opening. The
following is the summary of Chapter 4:
(1) In moment frames with web opening, the openings serve the same function as
the special segments in special truss moment frame system. Under a severe ground
motion, the inelastic activity will be confined only in the openings, which mainly consists
of yielding and buckling of diagonal members and plastic hinging of the chord members
of the openings. In this proposed system, the chord, diagonal, and vertical members
should be designed so that, under their fully yielded and strain-hardened condition, the
moment at every beam-column connection, generated by the shear force in the opening,
is smaller than a critical value. This critical value is selected so as to reduce the risk of
premature failure of connections.
(2) Tests of beams with opening shows that the proposed upgrading scheme is
feasible. A properly detailed specimen can provide a stable hysteretic response. In all
tests, the inelastic activity was confined only in the designated locations, as intended by
design.
178
(3) Stable responses of test specimens were due to the proper detailing at critical
locations. Stress concentrations are highest at the corners of the opening, therefore,
special detailing should be provided there to minimize stress concentrations and also to
move the locations of plastic hinges away from the critical corners. The detailing scheme
used in Specimen 3 provides the needed ductility and is also practical. The opening can
be flame-cut but the surface of the cuts near the corners should be smoothed out by
grinding. It is desirable to have a radius in all corners, although test results suggested that
this might not be necessary. The vertical members at the ends of the opening should be
placed at an offset of about 0.5” from the ends of the opening. This is done so that the
plastic hinges are pushed away from the critical areas. The diagonal bars also help in
reinforcing the critical areas. All welds should have at least 0.25” clearance from the
edges of the opening to allow for plastic flow, thus increasing local ductility.
(4) Shear in the opening is primarily resisted by the chord members and the
diagonal members. The ultimate shear strength of the opening can be accurately predicted
by multiplying the nominal shear strength of the chords and diagonals by the
corresponding overstrength factors. The overstrength factor of the diagonal members is
primarily attributed to the difference in the nominal and the actual yield strength. The
overstrength factor due to strain hardening is not significant for the diagonal members. A
reasonable value for the overstrength factor for the diagonal members was found to be
1.4.
Unlike the diagonal members, the overstrength in the chord members is
dominated by strain hardening. The overstrength factor for the chord members is a
function of the length of the opening, the section properties of the chord members, and
the material properties. It is recommended that the ultimate shear contributed by the
chord members can be computed using the nominal yield stress values and a 10% strain
hardening factor.
179
(5) Openings can be easily repaired after severe deformations. The chord
members of the opening can be replaced by double angles, provided that the length of the
opening is not too large. Lateral instability may occur if the chord members are not
properly braced against lateral movement when the length of the opening is large. This is
also true for the regular openings before repair.
(6) Cyclic behavior of a beam with an opening can be modeled using a finite-
element based code, provided that a proper hyteretic model for the axial compression
elements is used. Axial compression elements exhibit a complex post-buckling strength
degradation pattern which affects overall hyteretic response of the structure. An
analytical model presented in Section 4.4.4 was able to capture the hysteretic behavior of
the beam with opening very well.
6.1.5 Seismic Behavior of Upgraded Frames
A design procedure for the proposed upgrading scheme for steel moment resisting
frames was proposed in Chapter 5. The results from the experimental program presented
in Chapter 4 were used as a basis for development of the design and analytical modeling
procedures. Based on the design and modeling procedures developed in Chapter 5, an
analytical study was conducted to investigate the seismic behavior of an upgraded frame.
The results of nonlinear static and nonlinear dynamic analyses of the modified frame
were presented and discussed. Results of a full-scale test were also presented and
discussed. The following is the summary of Chapter 5:
(1) In the proposed upgrading scheme, openings can be designed so that, under a
severe ground motion, the moments at beam-to-column connections generated by the
shear forces in the openings are less than a selected critical value during the entire
excitation. One possible design approach is that the openings, in their fully yielded and
strain-hardened condition at about 3% story drift, generate moments at the beam-column
connections smaller than their flexural yield moments. This reduces the risk of premature
180
failure of those connections and confines all inelastic activity to the openings. The
maximum shear strength of an opening can be estimated by the procedure presented in
Chapter 4.
(2) The overstrength in the chord members is one of the most critical parameters
in the proposed upgrading system. It is directly related to the ductility demand in the
chord members. Therefore, it is necessary to control the ductility demand within an
acceptable range. The maximum overstrength value depends on the maximum interstory
drift, the length of the opening, and the section properties of the T-shaped chord
members. The use of expected interstory drift of 3% in the design provides a realistic
upperbound estimate of the maximum response. The target overstrength value of 2 for the
chord members is recommended. The length of the opening on the order of 20% of the
span length provides a good balance between strength and stiffness, and is recommended.
(3) The results of nonlinear static and nonlinear dynamic analyses of an upgraded
frame showed that the frame behaved well as expected. From static analysis, the
upgraded frame showed an insignificant decrease in stiffness because of the presence of
the openings, when compared to the original frame. More importantly, the frame
responded to a severe ground motion in a desirable manner with controlled inelastic
activity at designated locations. Moreover, the risk of premature failure of beam-to-
column connections was substantially reduced since no plastic hinge formed at the
connections.
(4) A full-scale one-story subassemblage, consisting of a 28 feet long W24x62
beam with a web opening and two 13 feet W14x82 columns, was designed and fabricated
for the experimental study. The subassemblage test specimen responded as expected. The
inelastic activity was confined to only the designated members in the opening. The
maximum load could be accurately estimated. The test specimen confirmed that the
detailing scheme used earlier in a small-scale specimen could be successfully used to
provide the required ductility in full-scale structure. Detailing is one of the most
181
important aspects of the proposed system. If properly detailed, the opening can sustain
large cyclic deformations without fracture. Test of the full-scale one-story
subassemblage also showed that the analytical modeling procedure presented in Chapter
5 could accurately estimate the response of a frame upgraded by the proposed system.
6.2 CONCLUDING REMARKS AND SUGGESTED FUTURE STUDIES
The research work presented herein is a pilot study on a new seismic design
philosophy. In this regard, some of the underlying assumptions may be debatable. Some
aspects of the research work in the design and upgrading of steel moment frames need to
be further studied. Some future study topics are suggested in the following sections.
6.2.1 Drift and Yield Mechanism Based Design
It was shown that the proposed drift and yield mechanism based procedure offers
an opportunity to design a structure within the performance-based framework. Two major
issues must be resolved before this new design philosophy can be successfully used in
practice. These issues are: the quantification of design earthquake levels and the
quantification of damage levels for different performance objectives. The performance-
based methodology presented in Chapter 3 is a purely deterministic procedure based on
assumed values of earthquake intensities and story drifts. The design procedure can be
improved by incorporating a probabilistic approach into the design process, particularly
for quantifying the design earthquake levels and the damage levels.
With respect to the quantification of design earthquakes, many studies in the past
have focused on combining a probabilistic approach with the response spectrum method.
A probability-based design spectrum, generally known as an equal hazard design
spectrum, is one example of a design spectrum where the intensity is directly specified in
terms of a probability of exceedance. This type of spectrum can be readily incorporated
into the proposed design procedure. However, with respect to the quantification of
182
damage levels, the quantitative definitions are still somewhat arbitrary. Extensive field
studies and response monitoring systems are needed in the future to produce reliable
quantitative definitions of damage levels. Until such data are collected, the damage
levels based on experience and performance of moment frames in past earthquakes can
be used.
The drift and yield mechanism based design approach can also be applied in the
design of other structural systems, such as concentrically braced frames, eccentrically
braced frames, or special truss moment frames. Although the essence of the procedure
remains the same, calibrations of some design parameters may be necessary. This is
because the fundamental behavior of braced frames (or truss frames) and moment frames
are different. Major differences that might affect the design procedure include the
differences in hysteretic behavior and the differences in the distribution of story drifts
along the building height.
6.2.2 Moment Frames with Ductile Web Openings
In this study, the analysis and design of the proposed upgrading scheme were
based on the moment frames subjected to lateral loads only. The presence of gravity
loads could be important especially when the governing lateral loads are small, such as in
moderate seismic zones. The relative effects of gravity loads in that case deserve a closer
examination. In addition, the presence of floor slabs might influence the yield mechanism
and therefore should also be investigated.
Another important issue is the selection of the critical moment for beam-column
connections. In this study, this value is taken as the yield moment. However, tests of
beam-to-column connections suggested that this value may not be safe, especially for
pre-Northridge connections. Tests are needed to find more appropriate values.
183
APPENDICES
184
APPENDIX A
CALIBRATION OF BEAM PROPORTIONING FACTOR
As mentioned in Chapter 3, the beam proportioning factor, iβ , plays an important
role in the seismic response of a structure. It represents the variation of story strength and
stiffness over the height of the structure. In order to have uniform story drift along the
height, the stories with relatively high input story shears should have relatively large
beam strength and stiffness. Similarly, stories with relatively low input story shears
should have relatively small strength and stiffness. Thus, the distribution of beam
strength at each level along the height should follow closely the distribution of story
shears induced by earthquakes. The strength of the columns is subsequently determined
to ensure formation of a strong column-weak beam plastic mechanism.
In order to find an optimal distribution of beam strength, a numerical simulation
was carried out. The goal was to find a function to represent the earthquake-induced story
shears from a variety of earthquakes and use it to represent the beam proportioning
factor. The problem is an iterative process, meaning that the distribution of beam
strengths should follow the distribution of earthquake-induced story shears, which in turn
depends on the distribution of beam strengths. The earthquake-induced story shears along
the height of the frame are unknown at the time of design. As a first approximation, the
relative distribution of earthquake-induced story shears can be represented by the relative
distribution of static story shears computed from the design forces presented in Chapter 3.
The ratio of the earthquake-induced story shear at level i to that in the top level, n , is
assumed to be in the form of:
185
b
n
ii V
V
=β (A1)
where iV and nV respectively are the static story shears at level i and at the top story
computed from design forces given by Equation 3.25, and b is a numerical factor to be
determined. Ideally, the beam proportioning factor should follow the same relative
distribution of the earthquake-induced shears. Factor b can be found by solving a least
square minimization problem, which will be presented next.
A six-story one-bay moment frame was used to calibrate the b factor. The
properties of the frame were a constant story height of 14 feet, a bay width of 25 feet, and
a constant story mass of 190 kips. The gravity load on the frame was assumed to be 25
kips per floor per column. A typical story of the six-story frame is shown in Figure A1.
Figure A1. Typical Story of the Six-Story Frame Used to Calibrate iβ .
The frame was designed four times using the proposed method in Chapter 3, using
four possible functions for the beam proportioning factor. These functions were
25.0)/( nii VV=β , 50.0)/( nii VV=β , 75.0)/( nii VV=β , and 0.1)/( nii VV=β . A total of
sixteen cases were used to calibrate the b factor by subjecting each of the frames to four
earthquake records used previously. Those records were the El Centro, the Newhall, the
25 kips 25 kips
25 ft.
14 ft. W=190 kips
186
Sylmar, and a synthetic ground motion. The distribution of maximum story shears in each
case was computed from nonlinear dynamic analyses. The optimal value of b is obtained
when, at each story level of all frames, the difference between the prescribed iβ function
and the relative distribution of maximum story shears is minimized. If the error is defined
as the sum of the squares of the difference between the relative distribution of story
shears and the prescribed iβ function from each case, then the overall error can be
written as:
∑∑∑∑= == =
−=−=n
i j
bnienjeij
n
i jienjeij VVVVVVX
1
16
1
2
1
16
1
2 ])/()/[(])/[( β (A2)
where eijV and enjV are the attended maximum earthquake-induced story shears at story
level i and at the top story n in case j of the sixteen cases, and iV and nV are the static
story shears computed from design forces (Equation 3.25) at level i and at the top story
n . The subscript j denotes each of the sixteen cases considered, the subscript i denotes
story level, and n denotes the number of stories (6 in this case).
The optimal value of b corresponds to the minimum value of the error X , that is
the solution of a minimization problem:
(A3)
The study began by designing the six-story frames. After the design forces were
calculated, the required beam strength at each level was found using four beam
proportioning functions, iβ , as mentioned earlier. Table A1 shows the distribution of
beam strengths at each level for each of the four cases. The sizes of beams and columns
were selected using AISC-LRFD specifications. The resulting four frames are shown in
Figure A1. Nonlinear analyses were carried out using same modeling assumption as in
Chapter 2. The distribution of maximum earthquake-induced story shears for each case is
shown in Figure A3.
Min b
Min b ∑∑
= =
−n
i jj
bnienei VVVV
1
16
1
2])/()/[(=X
187
Table A1. Distribution of Beam Strength.
Case 1 Case 2 Case 3 Case 4 Story ni VV /
( ni VV / )0.25 ( ni VV / )0.50 ( ni VV / )0.75 ( ni VV / )1.0
6 1.00 1.00 1.00 1.00 1.00
5 1.83 1.16 1.35 1.57 1.83
4 2.50 1.26 1.58 1.99 2.50
3 3.00 1.32 1.73 2.28 3.00
2 3.33 1.35 1.83 2.47 3.33
1 3.50 1.37 1.87 2.56 3.50
Figure A2. Four Six-Story Frames Used to Calibrate iβ .
W33x135
W30x135
W14x398
W14x398
W33x130
W30x130
W14x398
W14x398
W24x94
W30x116 W14x311
W14x311
W36x150
W36x150 W14x342
W14x342
W33x130
W36x135
W14x342
W14x342
W24x84
W30x108
W14x233
W14x233
W33x130
W33x130 W14x426
W14x426
W33x130
W33x130
W14x426
W14x426
W30x108
W33x118
W14x370
W14x370
W36x150
W36x150 W14x311
W14x311
W30x132
W33x141
W14x283
W14x283
W24x68
W27x102
W14x211
W14x211
25.0)/( nii VV=β 50.0)/( nii VV=β 75.0)/( nii VV=β 0.1)/( nii VV=β
188
Figure A3. Distribution of Maximum Story Shears under the Four Selected Records.
After the distribution of story shears under each case had been obtained, Equation
A3 was solved numerically to find the best value of b . The result showed that the error
function (Equation A2) was minimized when b was equal to 0.527. The ratios of the
error function when b equals 0.1,0.2,…, 1.0 to the error when b equals to 0.527 are
shown in Figure B4.
For practical purposes, the rounded value of b of 0.50 is recommended and was
used in this study. It should be noted that this value may not apply to all cases due the
uncertain nature of earthquakes. This value should be site specific depending on the
characteristics of earthquakes at a given site. In this study, it is assumed that the four
earthquakes are representatives of the earthquakes that could occur at a site. Finally, the
distributions of story shears for all cases are shown in Figure A5 once again to compare
them with the distribution given by 50.0)/( nii VV=β . This figure shows that the proposed
function is reasonable.
25.0)/( nii VV=β 50.0)/( nii VV=β 75.0)/( nii VV=β 0.1)/( nii VV=β
0 1 2 3 4
El Centro
Sylmar
Newhall
Synthetic
0
1
2
3
4
5
6S
tory
Lev
el
0 1 2 3 4 0 1 2 3 4 0 1 2 3 4Vei/Ven
189
Figure A4. Variation of Error Function X .
Figure A5. Comparison between 50.0)/( nii VV=β and Relative Shear Distributions
from Dynamic Analyses.
0.5 1 1.5 2 2.5 30
1
2
3
4
5
6
7
Vei
/ Ven
Sto
ry L
evel
Beta = (Vi / V
n )0.5
0
5
10
15
20
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1b
X(b
) / X
(0.5
27)
190
APPENDIX B
DESIGN EXAMPLE
The example frame in Chapter 2 is redesigned using the proposed drift and yield
mechanism based design procedure in this appendix. The design procedure is
summarized in the flowchart presented in Figure B1.
The estimated design loads for the frame are as follows:
Live Load: Average design live load for all floors 50 psf
Dead Load: Floor (3” Steel Composite Deck) 45 psf
Ceiling 8 psf
Beams and Columns 12 psf
Earthquake: Seismic Zone 4, Soil Type S3, and Standard Occupancy
Wind: Not Governing (Assumed)
Factored Design Gravity Load: 1.2DL+0.5LL = 103 psf
The gravity loads on the beams of the moment frame are the reaction forces
transferred from floor beams at every one-third point of the span length:
40.72
1)
12
317()
3
25(103.0 =×+××=bR kips (B1)
The gravity loads on the columns of the moment frame at each floor level is
calculated based on the tributary area:
4.44103.0)12/317(25 =×+×=cgP kips (B2)
191
Formulate Equivalent One-Bay Frame
1) Estimate Fundamental Period 2) Estimate Yield Drift 3) Select Target Drift
Calculate Design Base Shear (Eq. 3.28) and Design Forces (Eq. 3.25)
Beam Design 1) Select Mpc (Eq. 3.33) 2) Calculate βi (Eq. 3.35) 3) Calculate Required Beam Strengths (Eq. 3.32) 4) Determine Beam Sizes (AISC-LRFD)
Column Design 1) Select Overstrength Factors ξi (Eq. 3.44) 2) Calculated Updated Forces Fiu (Eq. 3.39) 3) Calculate Design Moment and Axial
Forces (Eqs. 3.40 and 3.43) 4) Determine Column Sizes (AISC-LRFD)
Stop
Figure B1. Drift and Yield Mechanism Based Design Procedure Flowchart.
Verify with Nonliner Static and Nonlinear Dynamic Analyses
192
The three-bay moment frame is transformed into an equivalent one-bay moment
frame for analysis. The weight of the equivalent one-bay frame is calculated in Table B1.
Assuming fixed supports at the base, the frame is treated as a five-story frame.
Table B1.
Weight of the Equivalent One-Bay Frame. Floor Weight
(kips) Weight/frame
(kips) Weight One-Bay
(kips) 5 2416.4 604.1 201.4 4 2103.6 525.9 175.3 3 2103.6 525.9 175.3 2 2103.6 525.9 175.3 1 2675.0 668.8 222.9
Using the proposed design method as presented in Chapter 3, the frame is to be
designed for a maximum target drift of 2%. The estimated period of the frame is:
86.0)71(035.0035.0 4/34/3 =×=×= hT second. (B3)
The design base shear is calculated as follows:
07.286.0
5.125.125.13/23/2
=×==T
SC (B4)
828.007.20.14.0 =××== ZICa (B5)
6.40256hw5
1iii =∑
=
kip•ft (From Table B2) (B6)
3.21065165
1
2 =∑=i
ii hw kip•ft2 (From Table B2) (B7)
01.0p =θ (Assuming 1% Elastic Drift) (B8)
735.12.3286.06.40256
801.03.2106516
gT
8
hw
hw
2
2
2
2p
5
1iii
5
1i
2ii
=××
×××=
=∑
∑
=
= ππθα (B9)
193
332.02
)828.0(4735.1735.1
2
4 2222
=++−
=++−= a
W
V αα (B10)
5.3152.950332.0 =×=V kips (B11)
Lateral force at each level is calculated in Table B2 and beam proportioning
factors are calculated in Table B3 using Equation 3.35.
Table B2.
Design Lateral Forces. Floor
ih
(ft.) iihw
(kip•ft) 2iihw
(kip•ft2) iF
(kips) 5 71 14299.4 1015257.4 112.0 4 57 9992.1 569549.4 78.2 3 43 7537.9 324129.7 59.1 2 29 5083.7 147427.3 39.8 1 15 3343.5 50152.5 26.2
Table B3. Calculation of Beam Proportioning Factors.
Floor iF
(kips) iV
(kips) iβ
5 112.0 112.0 1.00 4 78.2 190.2 1.30 3 59.1 249.3 1.49 2 39.8 289.0 1.61 1 26.2 315.2 1.67
B1. Design of Beams
From Figure 3.6, the first approximation of pcM is:
2.13014
155.3151.1
4
1.1 1 =××==Vh
M pc kip•ft (B12)
After pcM has been determined, the required beam strength at each level can be
calculated from Equation:
194
∑ ∑= =
+=5
1
5
1
22i i
pcpbiii MMhFr
β (B13)
)2.1301(2)07.7(29.16497 +=rpbM (B14)
7.982=rpbM kip•ft (B15)
Based on the value of rpbM , beam section at each level can be found based on
minimum weight and compactness criteria. Selected beam sections are given in Table B4.
Note that all beams are assumed to be fully laterally supported by the crossbeams and
floor slab ( ybn ZFM φφ = ). Also, the compactness requirements of AISC-LRFD seismic
provisions must be satisfied ( ybff Ftb /522/ ≤ and ybw Fth /520/ ≤ ).
Table B4.
Minimum Weight Beam Sections. Floor
rpbi Mβ
(kip•ft)
Section (Min. Wt.) nMφ
(kip•ft) 5 982.7 W27x94 1042.5 4 1277.5 W30x118 1297.5 3 1464.2 W30x124 1530.0 2 1582.1 W33x130 1751.3 1 1641.1 W33x130 1751.3
After member sizes have been determined, the beams are checked for gravity
loads to find the correct mechanism. Using the roof beam as an example, the moment
diagram of the beam assuming plastic hinges at both ends is shown in the Figure B2.
Since the moment does not exceed the plastic moment anywhere in the beam span, the
assumed mechanism of the beam is the correct mechanism (with plastic hinges at the
ends).
195
B2. Design of Columns
From Chapter 3, the distribution of the internal forces in the column of the
equivalent one-bay frame can be calculated by Equations 3.40 and 3.43. The fully strain-
hardened beam moments can be computed by assuming the value of overstrength factor
at each level. In this design, the overstrength factor will be taken as 1.0 for the roof beam
and 1.05 for other floor beams. The overstrength factor of 1.0 is used at the roof level
since plastic hinge is allowed to form in columns at that level. Plastic hinges at the roof
level will not affect the overall mechanism.
Lateral forces at ultimate drift level, iuF , can be found using Equation 3.39. The
values of ipbi Mξ and iuF are shown in Table B5.
Table B5. Lateral Forces at Ultimate Drift Level.
Floor Level ipbi Mξ
(kip•ft) iuF
(kips) 5 1158.3 66.8 4 1513.8 46.7 3 1785.0 35.2 2 2043.2 23.8 1 2043.2 15.6
7.4 k 7.4 k
1042.5 k-ft -1042.5 k-ft
76 k 90.8 k
1042.5 k-ft
409.2 k-ft
285.7 k-ft
1042.5 k-ft
Figure B2. Internal Forces in the Roof Beam.
196
The distribution of moments in an exterior column of the frame can be found from
Equation 3.40. The moment at each level is shown in Figure B3. The design axial forces
for an exterior column are calculated using Equation 3.43 and are shown in Table B6.
Table B6.
Axial Forces in an Exterior Column (kips). Column
bpbii LMi/2ξδ∑
(kip•ft) cgP
(kips) cP
(kips) 5 92.7 44.4 137.1 4 213.8 88.8 302.6 3 356.6 133.2 489.8 2 520.0 177.6 697.6 1 683.5 222.0 905.5
Figure B3. Distribution of Moment in an Exterior Column (Units in kips and ft.).
For interior columns, the design moments are assumed to be twice of those in
exterior columns while the design axial forces are taken from the gravity loads only.
Using the AISC-LRFD specifications for beam-column members, the member sizes of
columns can be determined.
1158.3
223.1 1736.9
147.9 1932.9
-148.9
1522.5 -520.7
1894.3
1301.2
35.2
23.8
1158.3
1513.8
1785.0
2043.2
2043.2
1301.2
197
The exterior column at the third level will be used as an example. From Figure
B2, the design moments of the column are 1932.9 kip-ft and –148.9 kip-ft. The axial
force is calculated as 492.2 kips. The beams above and below this column are W30x124
and W33x130, respectively. Assuming W14x283 for column between roof to the third
floor and W14x311 from the third floor to the ground, the design check can be carried out
according to AISC-LRFD [AISC 1994] as follows:
72.225/5360
14/433014/3840 =+=aG (B16)
30.225/6710
14/433014/4330 =+=bG (B17)
71.1≈xk (B18)
Assuming the column is laterally braced in y - direction:
0.1=yk (B19)
xx rlk / > yy rlk / , the major axis controls crc Fφ . From AISC-LRFD E2, the
nominal compressive strength of the column is:
1.3418=nc Pφ kips (B20)
The nominal flexural strength is determined to be:
8.2260=== ypn ZFMM φφφ kip•ft (B21)
For 2.0/ ≤ncu PP φ , the member strength must satisfy:
0.12
≤+nx
ux
nc
u
M
M
P
P
φφ (B22)
where ltntux MBMBM 21 += (B23)
For plastic design, It is conservative to determine uxM based on ×2B (Total
Moment) [Salmon and Johnson 1990], therefore:
198
)(1
12
LH
PB
u ∆−=
∑∑
(B24)
Substituting cgP for uP , story shear at the third story for H , and a target drift of
2% for L/∆ , 2B can be taken as:
02.1)02.0(
3.249
2.13321
12 =
×−=B (B25)
Finally, Equation B22 can be evaluated as:
0.194.08.2260
9.193202.1
1.34182
8.489 ≤=×+×
(B26)
Since no plastic hinge is expected in the column, only compactness for elastic
design is required:
7.6045709.0
2.49275.21
50
64075.21
6401.8 =
××−=
−≤
=
y
u
yw P
P
Ft
h
φ (B27)
Similar calculations can be carried out for other columns in the frame. The final
member sizes of the redesigned frame are shown in Figure B4.
Figure B4. Member Sizes of the Redesigned Frame.
199
APPENDIX C
ABSTRACT
DRIFT AND YIELD MECHANISM BASED SEISMIC DESIGN AND UPGRADING
OF STEEL MOMENT FRAMES
The behavior of an existing moment-resisting frame designed by conventional
method was studied using nonlinear static and nonlinear dynamic finite element analyses.
The results show that moment-resisting frames designed by conventional, elastic, method
using equivalent static forces may undergo inelastic deformations in a rather uncontrolled
manner resulting in uneven distribution of plastic hinges.
Guided by the performance of this conventionally designed frame, a new design
concept is proposed. The new design concept is based on plastic (limit) design theory and
principle of energy conservation. The ultimate design base shear for plastic analysis is
derived using the input energy from the design pseudo-velocity spectrum, a pre-selected
yield mechanism, and a target drift. Parametric studies were carried out to verify the
validity of the proposed design procedure. The results show that the proposed method can
produce structures that meet a pre-selected performance objective in terms of both the
maximum drift and the yield mechanism.
The study was then extended to include seismic upgrading of existing steel
moment frames. A possible scheme to modify the behavior of existing moment-resisting
frames to have a ductile yield mechanism is proposed. This upgrading scheme uses
rectangular openings in the girder webs reinforced with diagonal members as ductile
“fuse” elements. A series of small-scale experiments were carried out to study the
200
feasibility of the proposed upgrading system. A detailed design procedure for seismic
upgrading of steel moment frames was presented. The results of nonlinear static and
dynamic analyses show that it is possible to upgrade an existing moment frame using the
special openings. Finally, a full-scale test of a one-story subassemblage was carried out to
verify the proposed modification concept experimentally. The test results were very
satisfactory. All inelastic behavior was confined to the designated elements of the web
opening only. The results also confirm that the proposed upgrading system is very
ductile.
201
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