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Department of Civil and Environmental Engineering
Stanford University
MODELING OF ASSESSMENT OF SEISMIC PERFORMANCE
OF COMPOSITE FRAMES WITH REINFORCED
CONCRETE COLUMNS AND STEEL BEAMS
by
Sameh Samir Mehanny
and
Gregory G. Deierlein
Report No. 135
August 2000
Department of Civil and Environmental Engineering
Stanford University
MODELING OF ASSESSMENT OF SEISMIC PERFORMANCE
OF COMPOSITE FRAMES WITH REINFORCED
CONCRETE COLUMNS AND STEEL BEAMS
by
Sameh Samir Mehanny
and
Gregory G. Deierlein
Report No. 135
August 2000
The John A. Blume Earthquake Engineering Center was established to promote
research and education in earthquake engineering. Through its activities our
understanding of earthquakes and their effects on mankind’s facilities and structures
is improving. The Center conducts research, provides instruction, publishes reports
and articles, conducts seminar and conferences, and provides financial support for
students. The Center is named for Dr. John A. Blume, a well-known consulting
engineer and Stanford alumnus.
Address:
The John A. Blume Earthquake Engineering Center
Department of Civil and Environmental Engineering
Stanford University
Stanford CA 94305-4020
(650) 723-4150
(650) 725-9755 (fax)
earthquake @ce. stanford.edu
http://blume.stanford.edu
©2000 The John A. Blume Earthquake Engineering Center
MODELING OF ASSESSMENT OF SEISMIC
PERFORMANCE OF COMPOSITE FRAMES
WITH REINFORCED CONCRETE COLUMNS
AND STEEL BEAMS
by
Sameh Samir Fahmy Mehanny
and
Gregory G. Deierlein
Report No. 135
August 2000
ii
iii
Abstract
Composite moment frames consisting of steel beams and reinforced concrete columns (so
called RCS moment frames) are one of several types of hybrid systems gaining
acceptance as cost-effective alternatives to traditional steel or reinforced concrete frames
for seismic design. New design standards for composite moment frames have recently
been introduced in the United States, and composite RCS frames have been one focus
area investigated as part of Phase 5 (Composite and Hybrid Structures) of the US-Japan
Cooperative Earthquake Research Program. This research presents an extensive and
pioneering analytical study whose focus is on the seismic behavior of composite frames
with the objectives to (1) develop and improve existing analytical models and techniques
for the nonlinear inelastic static and time history analyses of composite RCS moment
frames, (2) propose damage indices and performance criteria to assess seismic
performance of such frames, (3) apply accurate nonlinear analysis methods to evaluate
building performance under varying seismic hazards, (4) develop and correlate stability
limit states to performance levels suggested by modern seismic codes, and (5) investigate
response dependency on ground motion parameters so as to reduce the uncertainty in
estimating median response. The ultimate goal is to achieve broader acceptance of RCS
frames in high seismic regions by demonstrating their reliability through a modern
performance-based methodology.
Our approach toward establishing a performance-based design basis for composite RCS
frames involves both evaluation of seismic damage indices with test data on member and
connection response and comparative behavioral studies between RCS and conventional
iv
structural steel moment frames. Trial designs of six- and twelve-story RCS and steel
framed buildings are developed to exercise the latest seismic design criteria and standards
in the United States including the recently approved International Building Code (IBC
2000) and the 1997 AISC Seismic Provisions. Nonlinear static and time-history analyses
are run under two sets of earthquake records (general versus near-fault records with
forward directivity) that were selected and scaled to different hazard levels representative
of performance levels ranging from immediate occupancy to near collapse. Peak and
cumulative performance (i.e., damage) indices are then developed, calculated and
compared with structural acceptance criteria established using data from tests and models
of structural components. A new methodology is proposed to quantify system stability
limit states by integrating the destabilizing effects represented by local damage indices
through modified second-order inelastic stability analyses. The proposed method avoids
the need for questionable ad-hoc averaging techniques to relate local to global damage
indices. Correlation parameters between ground motion intensity measures, such as
spectral acceleration, etc., and structural damage are presented, and statistical
performance measures of global response are reported.
Supported by test data on structural components, the analyses demonstrate excellent
seismic performance of composite framed structures when evaluated both on their own
merits and in comparison with steel frames. In particular, by permitting steel beams to
run continuous through the reinforced concrete columns, the composite frames avoid the
fracture critical details that have caused problems with welded steel moment frames. The
design studies do, however, suggest areas for improving current design criteria, in
particular, the minimum strength and stiffness requirements for proportioning beams and
columns to resist seismic loads. By improving understanding of the seismic response of
composite RCS frames this research should lead to their broader utilization for seismic
regions and will contribute towards the development of more transparent and reliable
performance-based design methodologies.
v
vi
Acknowledgements
This report is based on the PhD thesis of the first author under the supervision of the
second author. The research forms part of the US-Japan Cooperative Earthquake
Research Program Phase 5 - Composite and Hybrid Structures, supported in the United
States by the National Science Foundation under the leadership of Dr. S. C. Liu. The
authors gratefully acknowledge the National Science Foundation support (grant CMS9632502) and supplemental support from the Steel Structures Development Center of the
Nippon Steel Corporation. The authors conducted the research at Cornell (1996-98) and
Stanford Universities (1998-2000) and greatly appreciate the advice and support of
faculty, students, and staff of the John A. Blume Earthquake Engineering Center and the
departments of Civil and Environmental Engineering at Cornell and Stanford
Universities.
The authors would express their sincere gratitude to Dr. Hiroshi Kuramoto of the
Building Research Institute of Japan who spent a year in residence with the authors to
work on the project. Special thanks are also due to: Dr. Ryoichi Kanno of the Nippon
Steel Corporation and Dr. Sherif El Tawil of the University of Central Florida for their
participation, help and advice throughout the research; Professors C. Allin Cornell and
Helmut Krawinkler of Stanford University and Dr. Nilesh Shome of EQE, Inc. for their
advice regarding the seismic hazard analyses; Prof. Richard N. White of Cornell
University and Dr. Abdelkader K. Tayebi of Louisiana Tech for sharing their expertise on
modeling reinforced concrete structures; and Prof. Hiroshi Noguchi of Chiba University
and other participants of the US-Japan Cooperative Earthquake Research Program.
vii
Table of Contents
Abstract
iv
Acknowledgements
vii
List of Tables
xvii
List of Figures
xx
Chapter 1
Introduction
1
1.1 Evolution of Composite Construction ………………………….
3
1.1.1 Pros and Cons of Composite RCS Systems ………………
5
1.1.2 Background of Experimental and Analytical Work ………
6
1.1.3 Current Codes and Provisions for Composite Systems …..
9
1.2 Overview of Recent Developments in Performance-Based
Chapter 2
Engineering ……………………………………………………..
11
1.3 Objectives ………………………………………………………
13
1.4 Scope and Organization ………………………………………...
14
Analytical Models Using Spread-of-Plasticity Approaches
17
2.1 Overview of Inelastic Analysis Models ………………………...
18
2.2 Review of Bounding Surface Model …………………………...
19
2.2.1 Single-Surface Model …………………………………….
19
2.2.2 Two-Surface Bounding Model ……………………………
20
2.2.3 Motion of the Bounding Surface ………………………….
22
2.2.4 Plasticity Coefficients …………………………………….
23
x
2.3 General Bi-Symmetric Beam-Column Element in DYNAMIX ..
23
2.3.1 Element Formulation ……………………………………...
24
2.3.2 Modeling of Stiffness Degradation with Cycles ………….
29
2.3.3 Calculation of Plastic Rotation …………………………...
32
2.4 Composite Beam Model ………………………………………..
33
2.4.1 Limitations and Assumptions ……………………………..
34
2.4.2 Element Formulation, Moment-Curvature Skeleton and
Hysteresis Model ………………………………………….
35
2.4.3 Elastic Stiffnesses and Ultimate Strength Calculation for
Composite Beam ………………………………………….
41
2.4.4 Verification Study ………………………………………...
45
2.5 Composite Joint Panel Model …………………………………..
51
2.5.1 Joint Panel Kinematics ……………………………………
52
2.5.2 Joint Panel Moment-Distortion Hysteresis Models ………
53
2.6 Modeling of Geometric Nonlinearity …………………………..
55
2.6.1 Definitions, Assumptions and Limitations ………………..
56
2.6.2 Total Geometric Stiffness Matrix Based on Hermitian
Shape Functions …………………………………………..
57
2.6.3 Geometric Stiffness Matrix as a Function of Spread-ofPlasticity …………………………………………………..
58
2.6.4 General Comments ………………………………………..
59
2.7 Overview of the Scheme of the Numerical Integration of the
Chapter 3
Equation of Motion for Time History Analysis ………………...
62
2.8 Summary ………………………………………………………..
65
Stiffness Modeling of Reinforced Concrete Beam-Columns
67
3.1 Introduction ……………………………………………………..
68
3.2 Basic Behavior and Design Issues ……………………………...
70
3.2.1 Beam-Column Behavior ………………………………….
70
3.2.2 Frame Behavior and Design ………………………………
72
3.3 Inelastic Frame Analysis ………………………………………..
74
xi
Chapter 4
3.4 Review of Stiffness Guidelines …………………………………
75
3.4.1 ACI-318 Building Code (1995) …………………………..
77
3.4.2 FEMA 273 ………………………………………………...
78
3.4.3 New Zealand Standard (1995) ……………………………
78
3.4.4 CEB State-of-the-Art Report (CEB 1996) ………………..
79
3.4.5 Architectural Institute of Japan Standard (1991) …………
81
3.5 Proposed Stiffness Coefficients ………………………………...
82
3.6 Verification Study ………………………………………………
83
3.6.1 Description of Test Specimens …………………………...
84
3.6.2 Comparisons and Discussions …………………………….
84
3.6.3 Cyclic Behavior …………………………………………...
87
3.7 Effective Shear Stiffness (GAeff) ……………………………….
92
3.8 Summary and Concluding Remarks ……………………………
94
Seismic Damage Indices
96
4.1 Introduction ……………………………………………………..
97
4.2 When Do We Need Damage Indices? ………………………….
98
4.3 Definition of Damage Function and Damage Index ……………
99
4.4 Classification Schemes of Damage Indices and Categorization
of Damage ………………………………………………………
101
4.4.1 Local Versus Global Indices ……………………………...
102
4.4.2 Categorization of Damage ………………………………..
108
4.5 Proposed Damage Indices ………………………………………
109
4.5.1 Energy-Based Damage Index ……………………………..
110
4.5.1.1 Some details and advantages of the energy-based
damage model ………………………………………
114
4.5.2 Ductility-Based Damage Index …………………………...
116
4.5.2.1 Some details of the ductility-based damage index ….
117
4.6 Identification of Deformation and Energy Values
Corresponding to Failure ……………………………………….
118
4.6.1 Reinforced Concrete Columns ……………………………
118
xii
4.6.2 Steel and Composite Beams ………………………………
123
4.6.2.1 Case of steel beams and composite beams under
Chapter 5
hogging bending …………………………………….
125
4.6.2.2 Case of composite beams under sagging bending ….
128
4.6.3 Composite – Reinforced Concrete-Steel – Joint Panels …..
129
4.7 Calibration and Verification ……………………………………
132
4.7.1 Reinforced Concrete Columns ……………………………
133
4.7.2 Steel and Composite Beams ………………………………
138
4.7.3 Composite Reinforced Concrete-Steel Joints …………….
144
4.8 Useful Conclusions and Guidelines for Damage Categorization
150
4.9 Summary ………………………………………………………..
153
Case Study Buildings Design and Selection of Records
155
5.1 Overview of Different Seismic-Resistant Design Methods …….
155
5.1.1 Equivalent Lateral Force Static Procedure ………………..
156
5.1.1.1 Rationale of the R and Cd factors …………………...
161
5.1.2 Modal Response Spectrum Analysis ……………………...
165
5.1.3 Time History Analysis ……………………………………
166
5.1.4 Static Inelastic Pushover Analysis ………………………..
167
5.2 Case Study Building Designs …………………………………..
172
5.2.1 Overview of the ASCE Design Criteria for Composite
Chapter 6
Beam-Column Joints ……………………………………...
182
5.2.2 Summary of Design Values and Governing Criteria ……..
185
5.3 Selection of Ground Motion Records …………………………..
191
5.3.1 General Records …………………………………………..
194
5.3.2 Near-Fault Records and Directivity Effects ………………
195
5.4 Summary ………………………………………………………..
198
Detailed Performance Study of 6-Story RCS Frame
200
6.1 Modeling and Analysis Assumptions …………………………..
201
6.1.1 Frame Loading and Mass Characteristics ………………..
201
xiii
6.1.2 Numerical Models ………………………………………...
201
6.1.3 Modeling of Damping …………………………………….
203
6.2 Static Inelastic (Push-Over) Analysis …………………………..
205
6.2.1 Relating Global, IDR, and Local, θp, Responses for Static
Pushover Results ………………………………………….
210
6.3 Nonlinear Dynamic (Time History) Analyses ………………….
214
6.3.1 Incremental Dynamic Analysis (IDA) Concept …………..
214
6.3.2 Relationship between Spectral Acceleration and
Maximum Interstory Drift Ratio ………………………….
216
6.4 Identification of Collapse Limit State …………………………..
229
6.4.1 Methodology for the Determination of the State of Global
Collapse …………………………………………………...
229
6.4.1.1 New stiffness and strength values for updating the
damage state of the structure ………………………..
232
6.4.2 Relationship between Spectral Acceleration and Global
Failure Criterion, λu ………………………………………
233
6.4.2.1 Conditional regression of λu ………………………...
240
6.4.3 Relationship between Maximum Interstory Drift Ratio and
Global Failure Criterion, λu ……………………………….
241
6.4.4 Spatial Damage Distribution ……………………………...
246
6.5 Global versus Local Response ………………………………….
254
6.5.1 Relationship between ∆IDRmax and Peak θp,C …………….
254
6.5.2 Relationship between IDRp,max and Peak θp,B …………….
260
6.5.3 Estimates of Local Response Given Global Response and
Input Intensity Level – Benefits and Implications ………..
265
6.6 Global Response Dependency on Different Ground Motion
Input Parameters ………………………………………………..
267
6.7 Summary ………………………………………………………..
275
xiv
Chapter 7 Comparative Assessment of RCS and STEEL Moment Frames
282
PART I: 12-Story RCS Special Moment Frame
283
7.1 Modeling of the 12-Story RCS Frame ………………………….
283
7.2 Static Push-Over Analysis ……………………………………...
285
7.3 Incremental Dynamic Analyses ………………………………...
288
7.3.1 Story Incremental Dynamic Analysis Curves …………….
292
7.4 Global Failure Analysis of the 12-Story RCS Frame …………..
293
7.4.1 Relationship between Spectral Acceleration and Global
Failure Criterion, λu ………………………………………
296
7.4.2 Relationship between Maximum Interstory Drift Ratio and
Global Failure Criterion, λu ……………………………….
299
7.4.3 Spatial Distribution of Damage …………………………...
302
7.5 Global versus Local Response ………………………………….
308
7.5.1 Relationship between ∆IDRmax and Peak θp,C …………….
308
7.5.2 Relationship between IDRp,max and Peak θp,B …………….
309
7.5.3 Estimates of Local Response Given Global Response and
Input Intensity Level ……………………………………...
314
7.6 Global Response Dependency on Different Ground Motion
Input Parameters ………………………………………………..
322
PART II: 6-Story STEEL Special Moment Frame
328
7.7 Modeling of the 6-Story STEEL Frame ………………………..
328
7.8 Static Push-Over Analysis ……………………………………...
330
7.9 Incremental Dynamic Analyses ………………………………...
334
7.9.1 Story Incremental Dynamic Analysis Curves …………….
338
7.10 Global Failure Analysis of the 6-Story STEEL Frame ………..
338
7.10.1 Relationship between Spectral Acceleration and Global
Failure Criterion, λu ……………………………………...
340
7.10.2 Relationship between IDRmax and Global Failure
Criterion, λu ……………………………………………...
343
7.10.3 Spatial Distribution of Damage …………………………
346
xv
7.11 Global versus Local Response ………………………………...
349
7.11.1 Relationship between ∆IDRp,max and Peak θp,C ………….
350
7.11.2 Relationship between IDRp,max and Peak θp,B …………...
353
7.11.3 Explanation of Large Dispersion in Beams Plastic
Chapter 8
Rotation θp,B Values ……………………………………..
353
7.12 Response Dependency on Ground Motion Parameters ……….
357
7.13 Summary ………………………………………………………
360
Conclusions and Recommendations
365
8.1 Summary ………………………………………………………..
366
8.2 Main Findings and Conclusions ………………………………..
370
8.2.1 Large Static Lateral Overstrength ………………………...
371
8.2.2 Disaggregation of Response under Near-Fault Ground
Records ……………………………………………………
371
8.2.3 High Collapse Limit Hazard, Sa(λu=1.0) …………………
372
8.2.4 Relating λu=0.95λuo to λu=1.0 Performance Levels ………
373
8.2.5 Relating Performance to Hazard Levels ………………….
374
8.2.6 Consistency of Drift versus Stability criterion ……………
375
8.2.7 Spatial Distribution of Damage …………………………...
376
8.2.8 Local versus Global Response Relationships …………….
377
8.2.9 Reducing the Variability in the Response through a Dual
Earthquake Intensity Index ……………………………….
378
8.3 Suggestions for Future Work …………………………………...
379
Appendix A Selected Ground Records
383
Appendix B Story IDA Curves
416
Bibliography
429
xvi
List of Tables
2.1
Material properties for test specimens ………………………………..
46
3.1
Effective section properties per New Zealand Standard (NZS 1995) ..
79
3.2
Comparison of measured versus predicted stiffness ………………….
87
4.1
Summary of selected local damage indices …………………………..
106
4.2
Selected global damage indices ………………………………………
107
4.3
Useful values for calculation of RC columns damage indices ……….
133
4.4
Value of damage indices at failure state for RC columns …………….
134
4.5
Values for calculation of damage indices for steel and composite
beams …………………………………………………………………
138
4.6
Combined damage indices at failure for steel and composite beams ...
139
4.7
Values for calculation of damage indices for composite RCS joints ...
144
4.8
Combined damage indices at failure for composite RCS joints ……...
145
4.9
Structural performance levels and damage …………………………...
151
4.10
Correlation of damage index and damage state ………………………
152
5.1
Main design details and cross-sections dimensions of 6-story RCS
building ……………………………………………………………….
5.2
5.3
173
Main design details and cross-sections dimensions of 12-story RCS
building ……………………………………………………………….
174
Main design details and cross-sections of 6-story STEEL building ….
174
xvii
5.4
Seismic masses for case study frames ………………………………..
185
5.5
Summary of design parameters for case study buildings …………….
189
5.6
Comparisons of different Vdesign/W ratios for the case study frames …
189
5.7
Main characteristics of general records ………………………………
195
5.8
Main characteristics of near-fault records ……………………………
198
6.1
Stiffness and strength values of RC columns ………………………...
202
6.2
Stiffness and strength values of composite and steel beams …………
203
6.3
Properties of composite joint panels ………………………………….
203
6.4
Modal properties of the 6-story RCS frame …………………………..
205
6.5
Limiting values of rotation capacity for RC columns ………………...
209
6.6
Limiting values of rotation capacity for composite and steel beams …
210
6.7
Limiting values for composite joints distortion ………………………
210
6.8
Values of α and β for the regression fit of Equation 6.5 ……………..
218
6.9
Conditional dispersions and coefficient of determination for IDRmax ..
223
6.10
Values of a and ß for Equation 6.7 …………………………………
239
6.11
Indicative drift values at different performance levels (FEMA 273) ...
245
6.12
Regression equations for local response given global response and
input intensity level …………………………………………………...
266
6.13
R Sa values for different records ……………………………………...
272
6.14
Regression results for IDRmax conditioned on different input
parameters …………………………………………………………….
273
6.15
Regression results for λu conditioned on different input parameters …
274
7.1
Stiffness and strength values of RC columns ………………………...
284
7.2
Stiffness and strength values of composite and steel beams …………
284
7.3
Properties of composite joint panels ………………………………….
285
7.4
Values of α and β for the regression fit of Equation 7.1 ……………..
288
7.5
Values of a and ß for Equation 7.2 …………………………………
297
xviii
7.6
Regression equations for local response given global response and
input intensity level for the 12-story RCS frame ……………………..
7.7
319
Regression results for IDRmax conditioned on different input
parameters …………………………………………………………….
324
7.8
Regression results for λu conditioned on different input parameters …
325
7.9
Stiffness and strength values of steel columns ……………………….
329
7.10
Stiffness and strength values of composite and steel beams …………
329
7.11
Properties of joint panels ……………………………………………..
329
7.12
Regression parameters α and β for the 6-story steel frame …………..
334
7.13
Average regression parameters α and β for near-fault records ………
337
7.14
Values of a and ß for the 6-story steel frame ………………………
340
7.15
Average a and ß values for near-fault records ……………………...
343
7.16
Regression results for IDRmax conditioned on various input
parameters …………………………………………………………….
358
7.17
Regression results for λu conditioned on various input parameters …..
359
8.1
Summary of Sa statistical values at various performance levels ……...
373
xix
List of Figures
1.1
Schematic of typical composite RCS systems ……………………………...
2
2.1
Idealized elasto-plastic material behavior .…………………………………
20
2.2
Kinematics of the two-surface bounding model ……………………………
21
2.3
Beam-column element with distributed plasticity – DYNAMIX …………..
24
2.4
Schematic curvature distribution along a cantilever beam …………………
32
2.5
Constitutive model and moment curvature skeleton for composite beam
element ……………………………………………………………………...
40
2.6
Schematic diagram of nested bars movements ……………………………..
40
2.7
Cross-section main dimensions for a typical composite beam ……………..
43
2.8
Plastic stress distribution for a typical composite beam ……………………
44
2.9
Test setup and specimen for verification study problems …………………..
47
2.10
Experimental and analytical results – specimen Tagawa (1989) …………...
49
2.11
Experimental and analytical results – Bursi and Ballerini (1996) (Specimen
with full shear connection) …………………………………………………
50
2.12
Experimental and analytical results for specimen CG3 – Uang (1985) ……
50
2.13
Experimental and analytical results for specimen EJ-WC – Lee (1987) …...
51
2.14
Panel shear and bearing modes of failure …………………………………..
53
2.15
Composite joint panel model ……………………………………………….
54
2.16
Constitutive model for joint panel shear ……………………………………
54
2.17
Constitutive model for joint bearing ………………………………………..
55
xx
2.18
Comparison between FBSFs and Hermitian shape functions in the presence
of spread-of-plasticity (El-Tawil, 1996) ……………………………………
3.1
60
Behavior of reinforced concrete element in flexure (a) member subjected to
lateral load, (b) moment-curvature response, (c) load-deformation response
71
3.2
Load versus deflection behavior of a reinforced concrete frame …………...
73
3.3
Nonlinear beam-column element models for frame analysis (a)
concentrated-hinge type, (b) spread-of-plasticity type ……………………..
3.4
76
Stress-resultant yield surface model and idealized moment-curvature
response …………………………………………………………………….
76
3.5
Effective secant flexural stiffness per CEB (Filippou and Fardis, 1996) …..
80
3.6
Proposed EIeff model compared to test data and other models ……………..
85
3.7
Comparative of effective stiffness coefficients with test data ……………...
86
3.8
Test specimen WP9 by Watson and Park (a) variation in EIeff with axial
load, (b) moment-curvature response ………………………………………
3.9
Test specimen by Kuramoto (a) variation in EIeff with axial load, (b)
moment-curvature response ………………………………………………...
3.10
89
Comparison of cyclic load behavior for WP9 specimen (a) experimental,
(b) DYNAMIX analysis …………………………………………………….
3.11
88
90
Comparison of cyclic load behavior for Kuramoto specimen (a)
experimental, (b) DYNAMIX analysis ……………………………………..
91
3.12
Proposed shear stiffness model ……………………………………………..
92
4.1
Definition of PHCs and FHCs and load sequence effects ………………….
112
4.2
Different failure surfaces for different values of γ ………………………….
113
4.3
Stress-strain model for monotonic loading of confined and unconfined
concrete in compression (Paulay and Priestley, 1992) ……………………..
120
4.4
Moment-rotation relationship for steel beams ……………………………...
124
4.5
Idealized moment-rotation relationship for Ef calculation for steel beams ...
125
4.6
Values of cyclic joint panel distortion at failure by least square fit based on
results by Kanno (1993) …………………………………………………….
xxi
131
4.7
Idealized moment-distortion for composite joint panels, Sheikh et al.
(1989) ……………………………………………………………………….
132
4.8
Ductility-based damage index – Watson and Park (1994), Unit WP4 ……..
135
4.9
Energy-based damage index – Watson and Park (1994), Unit WP4 ……….
135
4.10a
Load-displacement relationship – Watson and Park (1994), Unit WP2 ……
136
4.10b
Results for combined ductility- and energy-based damage indices –
Watson and Park (1994), Unit WP2 ………………………………………..
136
4.11a
Load-displacement relationship – Watson and Park (1994), Unit WP4 ……
137
4.11b
Results for combined ductility- and energy-based damage indices –
Watson and Park (1994), Unit WP4 ………………………………………..
137
4.12
Ductility-based damage index – Kanno (1993), Unit OB1-1 ………………
140
4.13
Energy-based damage index – Kanno (1993), Unit OB1-1 ………………...
140
4.14
Ductility-based damage index – Uang (1985), Unit CG3 ………………….
141
4.15
Energy-based damage index – Uang (1985), Unit CG3 ……………………
141
4.16a
Beam-shear drift angle relationship – Kanno (1993), Unit OB1-1 …………
142
4.16b
Results for combined ductility- and energy-based damage indices – Kanno
(1993), Unit OB1-1 …………………………………………………………
142
4.17a
Load-displacement relationship – Uang (1985), Unit CG3 ………………...
143
4.17b
Results for combined ductility- and energy-based damage indices – Uang
(1985), Unit CG3 …………………………………………………………...
143
4.18
Ductility-based damage index – Kanno (1993), Unit OJS1-1 ……………...
146
4.19
Energy-based damage index – Kanno (1993), Unit OJS1-1 ………………..
146
4.20
Ductility-based damage index – Kanno (1993), Unit OJS4-1 ……………...
147
4.21
Energy-based damage index – Kanno (1993), Unit OJS4-1 ………………..
147
4.22a
Beam-shear drift angle relationship – Kanno (1993), Unit OJS1-1 ………..
148
4.22b
Results for combined ductility- and energy-based damage indices – Kanno
(1993), Unit OJS1-1 ………………………………………………………...
148
4.23a
Beam-shear drift angle relationship – Kanno (1993), Unit OJS4-1 ………..
149
4.23b
Results for combined ductility- and energy-based damage indices – Kanno
(1993), Unit OJS4-1 ………………………………………………………...
xxii
149
5.1
IBC 2000 Design response spectrum ……………………………………….
157
5.2
Elastic versus inelastic behavior as related by R and Cd factors …………...
159
5.3
Capacity spectrum superimposed over demand response spectra ………….
171
5.4
Architecture Plan of US-Japan Theme Structure …………………………...
172
5.5
Typical structural plan for 6-story RCS building …………………………..
175
5.6
Elevation of typical frames in both directions – 6-story RCS building …….
176
5.7
Cast-in-place RC column details …………………………………………...
177
5.8
Precast RC column details ………………………………………………….
178
5.9
Joint details for 6-story RCS building ……………………………………...
179
5.10
Gravity and design lateral loads for the 6-story RCS frame ………………..
186
5.11
Gravity and design lateral loads for the 12-story RCS frame ………………
187
5.12
Gravity and design lateral loads for the 6-story STEEL frame …………….
188
5.13
Comparison of acceleration response spectra of general records and the
2%in50years site response spectrum (IBC 2000) …………………………..
5.14
195
Comparison of acceleration response spectra of near-fault records and the
2%in50years site response spectrum (IBC 2000) …………………………..
198
6.1
Static pushover curve – IBC 2000 load pattern …………………………….
207
6.2
Distribution of interstory drift ratios up the height of the frame – pushover
results ……………………………………………………………………….
6.3
207
Distribution of damage indices and progression of damage – pushover
results ……………………………………………………………………….
208
6.4
Schematic of different deformed configurations …………………………...
211
6.5
Global, ∆IDR, versus local, θp,C, response – pushover results ……………..
213
6.6
Global, IDRp, versus local, θp,B, response – pushover results ……………...
213
6.7
Schematic of typical Incremental Dynamic Analysis Curves ……………...
215
6.8
Conditional regression relationship of IDRmax for general records ………...
219
6.9
Conditional regression relationship of IDRmax for near-fault records ……...
220
6.10
Spectral acceleration versus IDRmax for bin of general records …………….
221
6.11
Spectral acceleration versus IDRmax for bin of near-fault records ………….
221
6.12
Story IDACs for general records …………………………………………..
224
xxiii
6.13
Story IDACs for near-fault records ………………………………………...
226
6.14
Flow chart of the technique for global collapse determination …………….
231
6.15
Proposed stiffness reduction as a function of the damage index Dθ ………..
232
6.16
Spectral acceleration - λu relationship ……………………………………...
234
6.17
Schematic of the effect of residual displacements on λu …………………...
239
6.18
Conditional regression of λu given Sa ………………………………………
243
6.19
IDRmax - λu relationship …………………………………………………….
244
6.20
Distribution of Dθ at different λu values- Valparaiso (1985) record ………..
248
6.21
Distribution of Dθ at different λu values- Mendocino (1992) record ……….
249
6.22
Plastic rotation values at λu = 1.0 – Valparaiso (1985) record ……………..
250
6.23
Plastic rotation values at λu = 1.0 – Mendocino (1992) record …………….
251
6.24
Distribution of Dθ at different λu values – Erzincan (1992) record ………...
252
6.25
Plastic rotation values at λu = 1.0 – Erzincan (1992) record ……………….
253
6.26
Global versus local response (θp,C) for bin of general records at λu=1.0 …...
256
6.27
Global versus local response (θp,C) for bin of near-fault records at λu=1.0 ...
256
6.28
∆IDRmax-θp,C relationship for general and near-fault records at λu=1.0 ……
257
6.29
∆IDRmax-θp,C relationship at different levels of damage based on values of
λu ……………………………………………………………………………
258
6.30
Global versus local response (θp,B) for bin of general records at λu=1.0 …...
261
6.31
Global versus local response (θp,B) for bin of near-fault records at λu=1.0 ...
261
6.32
IDRp,max-θp,B relationship for general and near-fault records at λu=1.0 …….
262
6.33
IDRp,max-θp,B relationship at different levels of damage based on values of
λu ……………………………………………………………………………
6.34
Global versus local response at different hazard levels for bin of general
records ………………………………………………………………………
6.35
7.1
263
268
Global versus local response at different hazard levels for bin of near-fault
records ………………………………………………………………………
269
Static pushover curve – IBC 2000 lateral load pattern ……………………..
286
xxiv
7.2
Distribution of interstory drift ratios up the height of the frame – static
pushover results …………………………………………………………….
286
7.3
Spectral acceleration versus IDRmax relationship for bin of general records .
291
7.4
Spectral acceleration versus IDRmax relationship for bin of near-fault
records ………………………………………………………………………
7.5
Comparison of regression results of spectral acceleration versus IDRmax
relationship for general and near-fault records ……………………………..
7.6
292
Story IDACs for the 12-story RCS frame under the general record, Cape
Mendocino (1992) at Rio Del Overpass station …………………………….
7.7
291
294
Story IDACs for the 12-story RCS frame under the near-fault record,
Imperial Valley (1979) at Array 06 ………………………………………...
295
7.8
Spectral acceleration-λu relationship for bin of general records ……………
298
7.9
Spectral acceleration-λu relationship for bin of near-fault records …………
298
7.10
IDRmax-λu relationship for bin of general records …………………………..
301
7.11
IDRmax-λu relationship for bin of near-fault records ………………………..
301
7.12
Distribution of Dθ – Cape Mendocino (1992) record ………………………
304
7.13
Distribution of Dθ – Loma Prieta (1989) record at Lexington ……………...
306
7.14
Global versus local response (θp,C) for bin of general records at λu=1.0 …...
310
7.15
Global versus local response (θp,C) for bin of near-fault records at λu=1.0 ...
310
7.16
∆IDRmax-θp,C relationship for general and near-fault records at λu=1.0 ……
311
7.17
∆IDRmax-θp,C relationship at different levels of damage based on values of
λu ……………………………………………………………………………
312
7.18
Global versus local response (θp,B) for bin of general records at λu=1.0 …...
315
7.19
Global versus local response (θp,B) for bin of near-fault records at λu=1.0 ...
315
7.20
IDRp,max-θp,B relationship for general and near-fault records at λu=1.0 …….
316
7.21
IDRp,max-θp,B relationship at different levels of damage based on values of
λu ……………………………………………………………………………
7.22
317
Global versus local response at different hazard levels for bin of general
records ………………………………………………………………………
xxv
320
7.23
Global versus local response at different hazard levels for bin of near-fault
records ………………………………………………………………………
321
7.24
Static pushover curve – 6-story STEEL frame, IBC 2000 load pattern ……
332
7.25
Distribution of IDR up the height of the frame – static pushover results …..
332
7.26
Comparison of IDR values for 6-story RCS and STEEL frames – static
pushover results …………………………………………………………….
333
7.27
Sa-IDRmax relationship for bin of general records …………………………..
336
7.28
Sa-IDRmax relationship for bin of near-fault records ………………………..
336
7.29
Comparison of regression results of Sa-IDRmax relationship for 6-story RCS
and STEEL frames ………………………………………………………….
7.30
Story IDACs for the 6-story steel frame under the Cape Mendocino (1992)
record at Rio Del Overpass station – general record ……………………….
7.31
337
339
Story IDACs for the 6-story steel frame under the Erzincan (1992) record
in Turkey – near-fault record ……………………………………………….
339
7.32
Spectral acceleration-λu relationship for bin of general records ……………
341
7.33
Spectral acceleration-λu relationship for bin of near-fault records …………
341
7.34
IDRmax-λu relationship for bin of general records …………………………..
345
7.35
IDRmax-λu relationship for bin of near-fault records ………………………..
345
7.36
Distribution of Dθ at different λu values – Mendocino (1992) record ……...
347
7.37
Distribution of Dθ at different λu values – Erzincan (1992) record ………...
348
7.38
∆IDRp,max-θp,C relationship for general records at λu=1.0 …………………..
352
7.39
∆IDRp,max-θp,C relationship for near-fault records at λu=1.0 ………………..
352
7.40
IDRp,max-θp,B relationship for general records at λu=1.0 ……………………
354
7.41
IDRp,max-θp,B relationship for near-fault records at λu=1.0 …………………
354
7.42
Results from time history analysis under LP89-WAHO at λu=1.0 …………
356
A.1
Miyagi-oki 1978 ground record – Ofuna station …………………………...
384
A.2
Response Spectra (5% Damping) for Miyagi-oki (1978) record – Ofuna ….
385
A.3
Valparaiso 1985 ground record – Llol station ……………………………...
386
xxvi
A.4
Response Spectra (5% Damping) for Valparaiso (1985) record – Llol
station ……………………………………………………………………….
387
A.5
Loma Prieta 1989 ground record – Hollister City Hall …………………….
388
A.6
Response Spectra (5% Damping) for Loma Prieta (1989) record – Hollister
City Hall …………………………………………………………………….
389
A.7
Loma Prieta 1989 ground record – Hollister South & Pine ………………...
390
A.8
Response Spectra (5% Damping) for Loma Prieta (1989) record – Hollister
South & Pine ………………………………………………………………..
391
A.9
Loma Prieta 1989 ground record – WAHO ………………………………...
392
A.10
Response Spectra (5% Damping) for Loma Prieta (1989) record – WAHO .
393
A.11
Cape Mendocino 1992 ground record – Rio Del Overpass ………………...
394
A.12
Response Spectra (5% Damping) for Cape Mendocino (1992) record – Rio
Del Overpass ………………………………………………………………..
395
A.13
Landers 1992 ground record – Yermo Fire Station ………………………...
396
A.14
Response Spectra (5% Damping) for Landers (1992) record – Yermo Fire
Station ………………………………………………………………………
397
A.15
Mendocino 1992 ground record – Petrolia station ………………………….
398
A.16
Response Spectra (5% Damping) for Mendocino (1992) record – Petrolia
station ……………………………………………………………………….
399
A.17
Imperial Valley 1979 ground record – Array 06 …………………………...
400
A.18
Response Spectra (5% Damping) for Imperial Valley (1979) record –
Array 06 …………………………………………………………………….
401
A.19
Loma Prieta 1989 ground record – Los Gatos station ……………………...
402
A.20
Response Spectra (5% Damping) for Loma Prieta (1989) record – Los
Gatos station ………………………………………………………………..
403
A.21
Loma Prieta 1989 ground record – Lexington station ……………………...
404
A.22
Response Spectra (5% Damping) for Loma Prieta (1989) record –
Lexington station …………………………………………………………...
405
A.23
Erzincan 1992 ground record – Erzincan station …………………………...
406
A.24
Response Spectra (5% Damping) for Erzincan (1992) record – at Erzincan
station ……………………………………………………………………….
xxvii
407
A.25
Northridge 1994 ground record – Newhall station …………………………
A.26
Response Spectra (5% Damping) for Northridge (1994) record – Newhall
408
station ……………………………………………………………………….
409
A.27
Northridge 1994 ground record – Rinaldi station …………………………..
410
A.28
Response Spectra (5% Damping) for Northridge (1994) record – Rinaldi
station ……………………………………………………………………….
411
A.29
Northridge 1994 ground record – Sylmar station …………………………..
412
A.30
Response Spectra (5% Damping) for Northridge (1994) record – Sylmar
station ……………………………………………………………………….
413
A.31
Kobe 1995 ground record – JMA station …………………………………...
414
A.32
Response Spectra (5% Damping) for Kobe (1995) record – JMA station …
415
B.1
Story IDA curves for Miyagi-oki (1978) record – 12-story RCS frame …...
417
B.2
Story IDA curves for Valparaiso (1985) record – 12-story RCS frame ……
417
B.3
Story IDA curves for LP89-HCA record – 12-story RCS frame …………...
418
B.4
Story IDA curves for LP89-HSP record – 12-story RCS frame ……………
418
B.5
Story IDA curves for LP89-WAHO record – 12-story RCS frame ………...
419
B.6
Story IDA curves for CM92-RIO record – 12-story RCS frame …………...
419
B.7
Story IDA curves for LA92-YER record – 12-story RCS frame …………..
420
B.8
Story IDA curves for Mendocino (1992) record – 12-story RCS frame …...
420
B.9
Story IDA curves for IV79-A6 record – 12-story RCS frame ……………...
421
B.10
Story IDA curves for LP89-LG record – 12-story RCS frame ……………..
421
B.11
Story IDA curves for LP89-LX record – 12-story RCS frame ……………..
422
B.12
Story IDA curves for EZ92-EZ record – 12-story RCS frame ……………..
422
B.13
Story IDA curves for NR94-NH record – 12-story RCS frame ……………
423
B.14
Story IDA curves for NR94-RS record – 12-story RCS frame …………….
423
B.15
Story IDA curves for NR94-SY record – 12-story RCS frame …………….
424
B.16
Story IDA curves for KB95-JM record – 12-story RCS frame …………….
424
B.17
Story IDA curves for Miyagi (1978) record – 6-story STEEL frame ……...
425
B.18
Story IDA curves for Valparaiso (1985) record – 6-story STEEL frame …..
425
B.19
Story IDA curves for LP89-HCA record – 6-story STEEL frame …………
425
xxviii
B.20
Story IDA curves for LP89-HSP record – 6-story STEEL frame ………….
425
B.21
Story IDA curves for LP89-WAHO record – 6-story STEEL frame ………
426
B.22
Story IDA curves for CM92-RIO record – 6-story STEEL frame …………
426
B.23
Story IDA curves for LA92-YER record – 6-story STEEL frame …………
426
B.24
Story IDA curves for Mendocino (1992) record – 6-story STEEL frame ….
426
B.25
Story IDA curves for IV79-A6 record – 6-story STEEL frame ……………
427
B.26
Story IDA curves for LP89-LG record – 6-story STEEL frame …………...
427
B.27
Story IDA curves for LP89-LX record – 6-story STEEL frame …………...
427
B.28
Story IDA curves for EZ92-EZ record – 6-story STEEL frame ……………
427
B.29
Story IDA curves for NR94-NH record – 6-story STEEL frame …………..
428
B.30
Story IDA curves for NR94-RS record – 6-story STEEL frame …………...
428
B.31
Story IDA curves for NR94-SY record – 6-story STEEL frame …………...
428
B.32
Story IDA curves for KB95-JM record – 6-story STEEL frame …………...
428
xxix
Chapter 1
Introduction
Recent trends in the construction of moment-framed buildings show the increased use of
steel, reinforced concrete, and composite steel-concrete members functioning together in
what are termed composite, mixed and/or hybrid systems. Such systems make use of each
type of member in the most efficient manner to maximize the structural and economic
benefits. As shown in Figure 1.1, one example of a composite system consists of
reinforced concrete columns (with small steel erection columns for construction
purposes) and steel or composite beams. This system is also known as RCS system and it
is the focus of this research.
Over the past fifteen years, composite RCS moment frame systems have been used in the
US and Japan. Extensive research is currently underway to better understand the behavior
of such frames. Much of this research aims at experimentally investigating the
characteristics of joints between steel and reinforced concrete members and at
understanding the behavior of mixed sub-assemblies. System behavior on the other hand
has been much less researched and is not yet well understood. In most instances, system
1
design provisions are extrapolated from corresponding traditional steel or reinforced
concrete systems.
Erection Column
Steel Beam
Beam Splice
Composite Joint
Region with
Through Beams
RC Column
Figure 1.1 Schematic of typical composite RCS systems.
2
In view of the growing popularity and use of composite systems, there is the need for
rational nonlinear analysis tools suitable for better understanding the behavior of such
systems, especially when subjected to dynamic excitation, and for evaluating design
codes and procedures. Unfortunately, many of the available nonlinear analysis programs
are only suitable for modeling traditional steel or reinforced concrete systems and are not
directly applicable to composite frames. Part of the research presented herein is a
continuation of previous work at Cornell University (El-Tawil and Deierlein, 1996)
aimed at improving this situation by developing nonlinear analysis tools. Among the first
objectives of this research is to further the development of existing nonlinear inelastic
dynamic analytical models and techniques for composite systems. Using these analytical
tools, the next objective is to apply nonlinear static and dynamic analyses to evaluate the
performance of composite RCS frames under multi-level earthquake hazards. Efficient
“dual purpose” local damage indices detecting peak and cumulative type of damage of
various structural components are suggested. A newly proposed technique, which
integrates the local damage effects with system stability analysis, offers a reliable tool to
quantify “near collapse” performance. It further provides insight to relate the degradation
of global stability to performance and hazard levels suggested by seismic codes. This
investigation should lead to the improvement of current seismic codes requirements and
help the development of performance-based design methodologies for such composite
systems.
1.1 Evolution of Composite Construction
In the United States, composite RCS moment frames have been used in several high-rise
office buildings constructed during the 1980’s and 1990’s (Griffis, 1992, Heinge, 1992,
and Leon, 1990). These systems have evolved as a variation of traditional structural steel
framing systems where the floor framing is essentially the same as in a steel framed
structure, but where reinforced concrete columns have replaced steel columns. Among
the main reasons behind that evolution are economics and advances in concrete
technology that made it more cost effective for columns. The economics are simply the
3
relative price of concrete and steel, coupled with a construction industry that was willing
to try new schemes. Concurrent advances in concrete technology made higher strength
concrete commercially available and practical for use in tall buildings. There were also
some construction technologies that helped make concrete more viable in tall buildings
such as concrete pumping, flying forms, etc… Furthermore, as building heights increased
and framing systems became lighter in the last two decades, the required lateral stiffness
of the structural systems under service loads began to impose large penalties on the size
of columns in traditional steel moment frames (Leon and Deierlein, 1995). All of that
leads US designers to stiffening the steel columns by encasing them in concrete, while
the beams and braces are still steel. Further evolution of the mixed construction leads to
the replacement of composite columns by reinforced concrete columns into which the
steel beams frame (so-called RCS systems). Most applications of RCS frames have been
used almost exclusively in high rise construction (Sheikh 1995) in the central and eastern
US where wind forces control the lateral design and detailing of the frames. However,
there is now considerable interest in applying them to low- and mid-rise construction in
high-seismic zones.
In Japan, composite systems have also been used, however, they evolved differently
compared to the US because of differences in the construction practices in both countries.
Composite RCS moment frames have been applied in low-rise construction where they
are replacing traditional reinforced concrete (RC) and structural steel reinforced concrete
(SRC) construction (Kanno, 1993). This form of construction has then expanded because
of the perceived advantages it has in high seismic zones (Griffis, 1995).
Aside from construction sequence differences between the US and Japan (e.g. the
absence of steel erection columns in the Japanese practice), another difference is that in
Japan the composite RCS frames are usually space frames with beams framing into the
column in two directions, whereas in the US most systems have been built with planar
perimeter frames.
4
1.1.1 Pros and Cons of Composite RCS Systems
In general, since composite systems realize the most efficient use of steel, reinforced
concrete, and composite members in a structural system, this type of construction is often
more economical than traditional either all-steel or all-reinforced concrete construction.
Among main advantages of RCS frames are the efficiency of concrete (versus steel) in
carrying large column loads at much lower cost per unit strength and stiffness (Griffis,
1992), and the reduction in total construction time. Speed of construction may be
achieved through separation of trades. Accordingly, construction activity can be spread
vertically, with the help of the erection columns, thus allowing different trades to engage
simultaneously in the construction of the building.
Moreover, steel and composite beams in a floor system lead to reduced floor depth, and
lighter overall floor weights. This in turn leads to lower building mass and more
economical foundations. Furthermore, having steel beams running continuous through
the reinforced concrete columns offers stable hysteretic behavior of the joint region due
to the presence of the steel web. This construction detailing permits the elimination of
field welding at beam-column connections. This helps avoid fracture problems
experienced with welded steel connections that were observed after the Northridge
earthquake.
Among the drawbacks of the RCS construction is the congestion in the connections
regions with ties passing through steel beam webs or welded to them. In addition, more
on site activities are required, although prefabrication techniques may alleviate this
problem. Because of possible congestion, concrete mixes have to be highly workable. In
addition, differential creep and shortening effects and slip between concrete and
structural steel are other drawbacks of composite systems (Griffis, 1987). Yet, even with
these considerations, mixed construction remains a viable and efficient alternative to allsteel or all-reinforced concrete construction.
5
In spite of the economic and practical advantages of composite systems, their use has
been constrained by the lack of information on the behavior and design of composite
members and connections (Goel et al., 1992), and the lack of accurate and efficient
computational tools for the analysis of such systems. This is particularly crucial for
regions of moderate to high seismicity where there is concern about structural
performance in the inelastic range. This research is a contribution towards improving this
situation.
1.1.2 Background of Experimental and Analytical Work
As recently as ten years ago there was practically no information on the behavior and
design of connections between steel beams and reinforced concrete columns. Since then,
there has been extensive testing of composite beam-column connections which is now
resulting in the development of design guidelines in the US and Japan. In the US,
pioneering experimental work aimed at understanding composite joint behavior was
undertaken at the University of Texas at Austin (Deierlein et al., 1989, and Sheikh et al.,
1989) and at Cornell University (Kanno, 1993). Based on this research, proposed design
guidelines for composite RCS joints have been developed through ASCE (1994). More
extensive testing of various configurations, with the slab effect, is underway at the
University of Michigan (Wight, 1997,1998) and at Texas A&M University (Bugeja et al.,
1999). As discussed by Kanno (1993), research in this field has also been carried out in
Japan by universities, government research institutes, and private construction
companies.
Analytical work for modeling the behavior of either composite sub-assemblies or overall
composite systems is not abundant in the literature. For modeling the behavior of
composite joint panels, Sheikh et al. (1989) proposed a multi-linear relationship for
modeling the force-deformation of the joint. The model is only applicable to
monotonically increasing loading. Kanno (1993) proposed a more detailed model
differentiating between panel shear and bearing modes of deformation which are
characteristic of composite joints. However, as with Sheikh et al. (1989) model, Kanno’s
6
(1993) model is still only applicable to monotonically increasing loads. El-Tawil et al.
(1996) extended Kanno’s (1993) idea of separating joint deformations into shear panel
and joint bearing parts and proposed a joint panel model suitable for cyclic loading. The
model is implemented in DYNAMIX (the analysis software used in this research) and a
detailed explanation of the model is given in Chapter 2.
Several researchers have suggested various analytical models for composite beams
subassemblies (i.e., steel beam with a concrete slab and a metal deck). In general,
composite beams can show complex behavior due to slip between the reinforced concrete
slab and the steel beam, and the variation of longitudinal stress across the width of the
slab, which is dependent of the joint details and the loading pattern. In order to capture
this complex behavior, a three-dimensional finite element analysis may be needed.
However, some researchers (Lee 1987, Tagawa et al 1989, Engelhardt et al 1995)
developed two-dimensional discrete member models as a compromise between simplicity
and accuracy. In these models, it is assumed that the effect of slip and the variation of
longitudinal membrane stress on the behavior of composite beams can be implicitly
included in the constitutive moment-rotation relationships. Alternatively, a fiber beamcolumn model, with continuously distributed springs along the interface between the
concrete slab and the steel beam to represent shear connectors (studs), has been
developed by Salari et al (1996) to model the composite beam behavior in a more
accurate, but computationally much more expensive way.
Utilizing available information, a composite beam element is developed through this
research using a spread-of-plasticity flexibility formulation that tracks inelastic momentcurvature cross-section response along the member. This model aims to capture the
overall behavior of a composite beam, particularly differences in the member’s stiffness
and strength under positive versus negative bending, while maintaining computational
efficiency. The element does not explicitly model detailed behavior associated with
cracking in the slab, slip between the slab and beam, etc., but it accounts for these
behavioral characteristics empirically. Development of this model is explained in detail in
Chapter 2 of this thesis.
7
Throughout the literature, very few researchers have developed analytical models or
carried out inelastic analyses aiming at studying the overall system performance of
composite frames. Among these researchers are Elnashai and Elghazouli (1993) who
developed an advanced nonlinear model for the analysis of composite steel/concrete
frame structures subjected to cyclic and dynamic loading. Their formulation consists of
beam-column cubic finite elements accounting for geometric nonlinearities and material
inelasticity. The nonlinear cyclic concrete model considers confinement effects and the
constitutive relationship for steel includes the effect of local buckling and variable
amplitude cyclic degradation. Broderick and Elnashai (1996a,b) used this model to
evaluate the seismic response of moment-resisting composite frames with partially
encased columns sections through the application of nonlinear dynamic analysis
techniques.
El-Tawil and Deierlein (1996) developed a computer program, DYNAMIX – for the
DYNamic Analysis of MIXed (steel-concrete) structures, which is an extension of other
analysis programs from previous research at Cornell University dealing with inelastic
static and dynamic nonlinear analysis of steel structures. Employing a bounding surface
stress-resultant plasticity model, inelastic section behavior (i.e., moment-curvature
response captured through the bounding surface model) is integrated to simulate overall
member response through a flexibility element formulation. The resulting element
accounts for the interaction of axial loads and biaxial bending moments in steel, RC, and
composite beam-columns with bi-symmetric cross-sections, including the effects of
spread-of-plasticity, geometric nonlinearities (P-∆ and P-δ effects), and cyclic stiffness
degradation. A more detailed overview of the element formulation and capabilities is
presented in Chapter 2 of this thesis.
Building on El-Tawil and Deierlein (1996) work, the present research is a pioneering
analytical study aimed at improving available analytical models for composite structures,
and investigating the overall system behavior of composite RCS moment frames under
multi-level earthquake hazards using such reliable and efficient analytical models. It
8
further deals with cumulative damage modeling at the structural components level and
integrates such local damage effects through global collapse analysis techniques for
better seismic simulation and enhanced interpretation of response to random ground
motions. Such study is needed for the improvement of our understanding of the behavior
of such composite systems leading to their broader acceptance by demonstrating their
reliability through a modern performance-based methodology.
1.1.3 Current Codes and Provisions for Composite Systems
Given that composite RCS frames include both structural steel and reinforced concrete
members, many design provisions from the ACI-318 (1995) and AISC-LRFD (1993)
Specifications are directly applicable to composite frames. In certain instances, however,
there are differences in the treatment of fundamental issues in these specifications that
can lead to inconsistencies in design (Leon and Deierlein, 1995). For example, in the
AISC-LRFD Specification, frame stability and the design of beam-columns are handled
through the use of semi-empirical interaction equations which is different from the
approach taken in ACI-318. In large part, the differences are due to the ACI-318 and
AISC-LRFD Specifications treating the design of composite columns through extensions
to provisions for reinforced concrete and structural steel columns, respectively. Thus, for
composite frames with both steel and concrete members, it is not clear how to combine
the different approaches. Beyond this, there are shortcomings in each specification
related to the design and detailing of composite members and connections.
In much the same way that ACI-318 and AISC-LRFD treat composite members by
extension of reinforced concrete and steel provisions, the new IBC 2000 Standards and
the AISC Seismic Provisions (1997), although adopting new recommendations for
composite steel-concrete structures, treat these composite systems as extensions of
traditional steel or reinforced concrete systems. For instance, response modification and
displacement amplification factors (such as the R and Cd factors) are selected, based on
consensus opinion, from corresponding factors for comparable all-steel and/or allreinforced concrete systems. These extrapolations are necessitated by a lack of
9
information regarding the behavior of composite systems. Two reasons contribute to this:
(1) lack of relevant experimental research; and (2) most available inelastic analysis tools
handle only steel or only reinforced concrete members. It is generally recognized that
there is considerable room for improvement in current seismic design methods that are
based largely on such empirical factors (R and Cd) for determining seismic loads,
inelastic drifts, stability limits, etc. Not only do such methods greatly oversimplify the
underlying aspects of inelastic behavior under dynamic loads, but they do not provide the
means to accurately evaluate damage and structural limit states under various level
earthquakes.
Furthermore, while composite frames bear many similarities to traditional steel or
reinforced concrete structures, there are important differences that can change their
behavior but yet ignored by current seismic codes. For example, the relative proportions
of strength, stiffness, damping and mass of RCS composite frame buildings are different
than in pure steel or reinforced concrete construction. Thus, it is not known whether
member ductility demands are comparable to those for steel and concrete frames and
whether the same detailing rules should be applied.
The IBC and AISC provisions for composite construction are still new and largely
untried and will require further verification before being fully accepted by other model
codes and standards and the profession. By accurately modeling the inelastic dynamic
behavior of several prototype composite RCS structures under multi-level earthquake
hazards, the present work will help identify areas in seismic codes and earthquake
engineering practice that need improvement and will provide data and suggestions for
such improvements.
10
1.2 Overview of Recent Developments in Performance-Based Engineering
In recent years, a new design philosophy for building codes has been discussed among
the engineering community, namely performance-based design (Vision 2000, 1995). The
goal of any performance-based design procedure is to produce structures that have
predictable seismic performance. Additionally, performance-based design approaches
should be more transparent than current code provisions. 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 structure. In order to evaluate structural
performance, the following information is required (Bertero, 1996):
1. Sources of excitation during service life of structure
2. Definition of performance levels
3. Definition of excitation intensity
4. Types of failures (limit states) of components
5. Cost of losses and repairs.
One of the first requirements of performance evaluation is the selection of one or more
performance objectives, i.e.: select desired performance level and associated seismic
hazard level. Since the evaluation relies on analysis rather than experimentation, the
criteria should be stated in terms of a response that can be calculated. Depending on the
intensity of the ground motion, a different performance objective will be desired.
According to the expected intensity, the designer must analyze whether achieving the
desired objective will be economically feasible. For frequent events, the designer will
probably desire that the structure remains operational. For rare events, ensuring
prevention against collapse may be the only realistic goal. Ultimately, performance-based
design methods and codes will only be accepted if they improve the quality and costeffectiveness of constructed facilities. Significant work has been performed in the
development of performance-based design and evaluation, and good discussions on the
subject can be found in Bertero (1996), Cornell (1996), and Krawinkler (1996). Recent
guidelines, such as those in Vision 2000 (SEAOC 1995) and FEMA 273 (BSSC 1997),
11
provide a framework for the performance-based design and evaluation of structures under
seismic loads, including both qualitative and quantitative definitions for seismic hazard
and structural performance.
In the recently published FEMA 273 and ATC 40 guidelines, and similar to ideas
proposed in SEAOC’s Vision 2000, it is anticipated that three performance levels
(immediate occupancy, life safety, near collapse) would form the basis of seismic loading
and acceptance criteria for a performance-based design code. However, only two specific
levels of performance are adopted by the SAC Design Criteria, as mentioned by
Hamburger et al. (2000), which are subtly different from those adopted by FEMA 273.
The first, termed Collapse Prevention, is a state of incipient local or global collapse,
whereas the second, termed Incipient Damage, is that state in which structural damage
initiates. Structural acceptance criteria for each performance level are established through
FEMA 273 in terms of response quantities for individual components, assuming that the
demands on local elements are faithfully represented by the global structural analysis.
Structural analyses would be one of four types: linear static, linear dynamic, nonlinear
static (pushover), and nonlinear dynamic.
Acceptance criteria are generally distinguished between force and deformation controlled
based on the available ductility, and it is presumed that system design rules would be
applied to restrict inelastic action to deformation-controlled components. For linear
analyses, acceptance criteria for deformation-controlled components are expressed in
terms of limits on the calculated demand to capacity ratios. For nonlinear analyses,
criteria are described in terms of component deformations and/or generalized strains (e.g.,
curvature). Researchers should undertake a critical review of such acceptance criteria and
the source material upon which they are based, and further check their accuracy and
applicability to new structures. Furthermore, some shortcomings and challenges to
current proposals are yet to be addressed. For example, a key shortcoming of the
acceptance criteria is their reliance on a single peak deformation limit that does not
consider strong motion duration of ground records and other cumulative effects. More
12
importantly, current methods are totally lacking in providing techniques to reliably
address near collapse performance level from a system point of view.
Among other unresolved issues yet required to develop a performance-based design code
is the extent to which prescriptive system design requirements in current codes would
apply in performance based design. For example, to what extent should a performancebased design code attempt to categorize system types like “ordinary”, “intermediate”, and
“special”? Or, to what degree should capacity design principles be enforced? Much work
has yet to be done before finding accurate and convincing answers to these questions.
1.3 Objectives
This research is part of Phase 5 of the US-Japan Cooperative Earthquake Research
Program on Composite and Hybrid Structures. This thesis presents an extensive
analytical design and assessment study whose focus is on the seismic behavior of
composite RCS moment frames. The main objectives of the present work can be
summarized in the following points:
1. Further develop and improve existing analytical models and techniques for the
nonlinear inelastic static and time history analyses of composite RCS momentframed buildings.
2. Synthesize and review existing knowledge on members and composite connections
design and behavior.
3. Exercise and evaluate current seismic design provisions for composite construction.
4. Develop accurate damage indices and performance criteria to assess seismic
performance of RCS moment frames.
13
5. Apply nonlinear analysis methods to evaluate building performance under varying
seismic hazards.
6. Develop and correlate stability limit states to performance levels suggested by
modern seismic codes.
7. Investigate correlation of structural response to various ground motion parameters so
as to reduce the uncertainty in estimating median response due to limited sample
size (i.e., limited number of ground records or limited number of time history
analyses).
8. Assess composite RCS moment frames through comparisons to well-established
steel moment-framed systems which will put into perspective all the issues that
should be addressed for improving the seismic performance of such new systems.
The ultimate goal is that by improving our understanding of the seismic response of
composite RCS frames under multi-level earthquake hazards, this investigation should
lead to their broader utilization for seismic regions and will contribute towards the
development of more transparent and reliable performance-based design methodologies.
1.4 Scope and Organization
This research is mainly divided into two parts. Part I deals with further development and
improvement of existing analytical tools and models for inelastic dynamic analysis of
composite RCS frames as well as development of performance acceptance criteria (i.e.,
seismic damage indices). Part II investigates the seismic performance of these composite
moment frames under multi-level earthquake hazards and compares their response to
traditional steel moment frames.
Chapter 2 describes analytical models implemented in the software DYNAMIX –
DYNamic Analysis of MIXed (steel-concrete) structures developed through this and
14
previous research (El-Tawil and Deierlein, 1996) with capabilities to perform inelastic
static and dynamic analyses of three-dimensional steel and RCS frames. Employing a
stress-resultant plasticity model, beam-column elements implemented in DYNAMIX
account for the interaction of axial loads and biaxial bending moments, including the
effects of spread-of-plasticity, geometric nonlinearities (P-∆ and P-δ), and cyclic stiffness
degradation. A new model for composite beams (i.e., composite floor decks on steel
beams) developed as part of this research is presented. The composite beam model is a
one-dimensional version of the 3-D bounding surface model used for general beamcolumns, including kinematic hardening for cyclic loading and stiffness degradation as a
function of the accumulated plastic energy in the member. Calibration and comparisons
to experimental results are provided. The chapter also summarizes a model for composite
connections between RC columns and steel beams which accounts for finite joint size and
inelastic panel shear and bearing deformations with cyclic stiffness/strength degradation.
Chapter 3 reviews various guidelines for flexural stiffness modeling of reinforced
concrete beam-columns for frame analysis. A formula is proposed to determine effective
initial flexural stiffness of reinforced concrete members, taking into account modest
degrees of cracking, amount of reinforcement, and stiffening effect of axial compression
load in the member. The flexural stiffness model has been verified by test results from
several beam-column specimens for a wide range of axial load ratios.
A brief literature review of seismic damage indices is presented in Chapter 4. Two new
local damage indices are proposed; a ductility-based index and an energy-based index.
The two damage indices are based on the idea of primary and follower half cycles in a
formulation that takes into consideration the ‘temporal’ effect of loading (i.e., loading
sequence) and cumulative damage. Results are compared to selected experimental data
including reinforced concrete columns, steel and composite beams, and composite RCS
joint sub-assemblages. Finally, data is reviewed to correlate the physical damage to the
value of the damage index.
15
Chapter 5 first provides an overview of various earthquake-resistant design methods
proposed by recent seismic codes and provisions. Full descriptions of the design of three
case study buildings (6-story RCS, 12-story RCS, and 6-story STEEL) are then
presented. All controlling design criteria are discussed in detail. The chapter also explains
the selection of earthquake records for the time history analyses of the case study
buildings. General characteristics and seismic properties of the records relevant to their
likely effect on the buildings are provided.
A detailed performance study of the 6-story RCS case study frame is described in
Chapter 6. Nonlinear static and time-history analyses results under two sets of earthquake
records (general versus near-fault records with forward directivity) are presented.
Incremental Dynamic Analyses are performed where the records are scaled to different
hazard levels representative of performance levels ranging from immediate occupancy to
near collapse. A new methodology is proposed to quantify system stability limit states by
integrating the destabilizing effects represented by local damage indices through
modified second-order inelastic stability analyses. These stability limit states are then
correlated to performance levels suggested by modern seismic codes. Relating local
(members plastic rotations) to global (interstory drift ratio) response has been also
investigated so as to estimate median local response at a given value of the global
parameter and compare it to acceptance criteria from ATC 40 or FEMA 273. Finally,
correlation parameters between ground motion intensity measures and structural damage
are presented, and statistical performance measures of global response are reported.
Chapter 7 presents a detailed comparative assessment study of RCS and STEEL moment
frames comparing the response of the 6-story RCS frame in Chapter 6 to that of the 12story RCS and the 6-story STEEL case study frames. All issues dealt with in Chapter 6
are revisited herein to confirm or modify the findings previously reported.
Finally, in Chapter 8, the main contributions and the general conclusions from this work
are discussed, and recommendations for future work are suggested.
16
Chapter 2
Analytical Models Using Spread-of-Plasticity
Approaches
One of the main objectives of this research is to develop efficient and accurate analytical
models for simulating the nonlinear behavior of composite RCS moment frames
subjected to static, cyclic or dynamic loading. This effort started by the development of
the frame analysis interactive program DYNAMIX for DYNamic Analysis of MIXed
systems (El-Tawil and Deierlein, 1996) which evolved from earlier versions used for
dynamic analysis of steel structures (CU-QUAND, Searer, 1994, and Zhao, 1993). As
part of this research further refinement of the analytical models implemented in
DYNAMIX with addition of a new element for composite beams has been accomplished.
In this chapter, analytical models for representing the inelastic beam-column elements
and composite joint panels under cyclic loading are described. These are all based on a
multi-dimensional force-space bounding surface model adopting a flexibility formulation,
which can model both the phenomena of gradual plastification and the interaction
between axial forces and moments. The models are also capable of capturing spread-ofplasticity effects along the member, geometric nonlinearities, and cyclic stiffness
degradation as will be discussed. Then, the formulation for a composite beam element
17
that tracks inelastic moment-curvature cross-section response along the member and its
implementation into DYNAMIX are described. A few examples are analyzed to verify
the accuracy of the composite beam element. Modeling strategies for geometric
nonlinearities are also presented. Finally, a brief overview of the numerical integration
procedure of the equation of motion for a time history analysis is provided.
2.1 Overview of Inelastic Analysis Models
Structural analysis models can be broadly categorized as either micro or macro models.
Micro models are generally considered to be more accurate since, as the name suggests,
pointwise stress-strain behavior is monitored throughout the structure. These models are
best suited to idealize individual members or very simple structural configurations
because of the computational effort involved. Micro models are usually based on either
the finite element method or the fiber element method. Background and examples of each
of these methods for the analysis of framed structures are presented by El-Tawil and
Deierlein (1996).
Macro models, on the other hand, form the basis for most practical large scale frame
analyses. In these models, the behavior is monitored at the member cross-sectional level
and emphasis is on cross-sectional force-strain or member end force-deformation
behavior. Macro models are typically categorized as either of the concentrated or the
distributed type. Concentrated models lump all inelasticity at the ends of the member, and
thus deal with inelastic material behavior in an approximate yet computationally efficient
manner. Although concentrated plasticity models imply behavior that is a physical
impossibility, i.e. infinite strains, they have the advantage of being conceptually simple in
addition to the computational convenience of having a stiffness matrix in a concise form.
The concentrated type (also known as plastic hinge model) can employ several strategies
to model bi-linear, multi-linear or nonlinear response. One is through a mathematical
assembly of multiple parallel elastic elements connected through rigid-plastic hinges that
capture abrupt change of stiffness at various load levels (i.e., parallel models).
18
Alternatively, linear or nonlinear elastic-plastic springs can be devised to achieve similar
behavior, i.e. series models (Powell and Chen, 1986).
Distributed macro models (also known as spread-of-plasticity models) are more accurate
and rational than concentrated plasticity models. There are many variations on spread-ofplasticity implementations, but most rely on modeling inelastic cross section behavior at
discrete sampling points along the member as opposed to only at the ends, either by
explicitly integrating stresses and strains or through a stress-resultant yield-surface
approach. Therefore, they are more computationally expensive. Note that the emphasis is
on force-sectional strain relationships, and not pointwise stress-strain relationship within
the cross section as is characteristic of micro models. This type of model therefore offers
an attractive compromise between the accuracy of micro models and the computational
efficiency of macro models.
2.2 Review of Bounding Surface Model
The model for representing material inelastic behavior in DYNAMIX is based on the
bounding surface model implemented in force space. This model was inspired by the
single-surface and bounding surface models developed in stress space by Dafalias and
Popov (1977), and the force space models previously implemented for frame analysis by
Orbison (1982), Hilmy (1984), and Zhao (1993).
2.2.1 Single-Surface Model
In classical plasticity theory for elastic-perfectly plastic materials, the stress-strain
relation under uniaxial loading is idealized as shown in Figure 2.1(a). When the stress
state reaches the yield point, plastic deformation occurs under a constant stress, σy. For
multi-axial states of stress, the elastic limit of material can be defined as a yield function
in terms of the various stress components, i.e., f(σij)=0. For two- or three-dimensional
stress space, the yield function can be interpreted geometrically as a closed, convex
19
surface such as the elliptical (von Mises) yield surface shown in Figure 2.1(b). When the
current stress state point is within the yield surface, the material behaves elastically.
When the current stress state point reaches the yield surface, plastic deformations occur.
Based on Drucker’s normality condition (Drucker 1951), plastic loading will cause the
current stress state point to move along the yield surface. The instantaneous detection of
plastic deformation is always parallel to the normal direction to the yield surface at the
current stress state point (Drucker’s postulate). Orbison (1982) implemented the single
surface model in force space to model the plastic response of beam-column members
with concentrated plastic hinges at the ends.
σ1
σ
Plastic Loading
(tangential)
σy
Unloading (direction
arbitrary but inside
surface)
Plastic Loading
σy
E
E
1
σy
1
Unloading
σ2
f(σ 1,σ 2)=0
ε
(a) Uniaxial Loading
(b) Biaxial Loading
Figure 2.1 Idealized elasto-plastic material behavior.
2.2.2 Two-Surface Bounding Model
The kinematics of the two-surface bounding model implemented in DYNAMIX is briefly
presented. Although the discussions herein are applicable to multi-dimensional force
space of any order, the loading and bounding surfaces are shown in Figure 2.2 in twodimensional, P-Mz, force space. The loading surface is assumed to be a scaled down
version of the bounding surface. Referring to Figure 2.2, the loading and bounding
surfaces are located by the vectors a and b respectively which are of zero length before
20
the application of loads. Usually the bounding surface does not move much even after
significant plastic loading because the assumed hardening parameter is typically small.
When the force point comes in contact with the loading surface this indicates initial yield
of the cross-section. Another point, known as the conjugate point, is then located on the
bounding surface. With continued plastic loading, the loading surface is pulled along in
force-space according to a kinematic hardening rule. The plastic modulii of the crosssection are functions of the proximity of the force point to the bounding surface, and a
memory parameter, defined as the ‘distance’ of the force point from the bounding surface
at initiation of yielding. The bounding surface as described above models pure kinematic
hardening.
Position of surfaces are
exaggerated for clarity.
g
A'
g
u
A
F
P
F'
a
b
Bounding Surface
Loading Surface
Mz
Figure 2.2 Kinematics of the two-surface bounding model.
At some stage unloading may occur, and eventually the force point may again come in
contact with the loading surface at another point A (Figure 2.2) where yielding is
assumed to re-initiate. The conjugate point, A’, is then located on the bounding surface
such that the normal, g, to the bounding surface at A’ is parallel to the normal to the
21
loading surface at A. As the force level is increased, the loading surface is pulled along.
The surface translates along the line, u, joining the force point to the conjugate point
(Mroz’s kinematic rule coupled with the consistency condition). The bounding surface is
assumed to translate in the same direction as the loading surface, but at a slower rate. The
ratio of the speed of the bounding surface to that of the loading surface is the ratio of the
residual plastic stiffness (strain hardening) at the bounding surface to the plastic stiffness
at the force point location (Zhao, 1993).
2.2.3 Motion of the Bounding Surface
The motion of the bounding surface is assumed proportional to the translation of the
loading point. Based on analogy to unidirectional plasticity, the motion of the bounding
surface is assumed to be
 k b
{db} = diag  p,P
 k p ,P
  k pb ,M z
, 
  k p ,M
z

  k bp ,M y
, 
  k p ,M
y


{da}


(2.1)
Where:
(k
b
p ,i
/ k p ,i
)
is the ratio between the residual plastic stiffness at the bounding
surface (strain hardening) and the current plastic stiffness for
principal direction i.
da = {da}
is the incremental shift of the loading surface
db = {db}
is the incremental shift of the bounding surface
In the limit, as the loading point reaches the bounding surface, the velocity of both the
loading surface (with the loading point laying on it) and the bounding surface match, and
the bounding surface is pulled along by the force point. Note however, that the movement
of the force point is in turn affected by the residual hardening modulii, k bp,i .
22
2.2.4 Plasticity Coefficients
Tangential plastic cross-section stiffnesses in the principal bending and axial directions
introduced in the previous section are represented by the following expression,
K p ,i
k3




d

= K e ,i  k 1 + k 2 


−
d
d
 in
  i

(2.2)
in which i is the principal direction under consideration (i.e., Mz, My, or P), Kp,i is the
plastic stiffness modulus of direction i, Ke,i is the elastic stiffness modulus of direction i,
d is the distance between the force point and the bounding surface, for direction i, din is
the distance, d, at initiation of the current plastic loading process, for direction i, and k1,
k2, k3 is a set of plasticity calibration parameters, for each direction i. Values for k1, k2,
and k3 differ from material to the other (i.e., steel versus RC versus composite beamcolumns).
The d / (din – d) term in Equation 2.2 represents the proximity of the force point to the
bounding surface. When d = din no plastification effects have occurred and the plastic
stiffness modulus is set to infinity, which implies elastic behavior. When d = 0, the force
point is at the bounding surface, implying that full plastification has occurred, and that
only a residual plastic stiffness (defined by the k1 parameter which models the element’s
strain hardening) is present. The plastic stiffness modulus changes smoothly between
these two limits as a function of the distance d.
2.3 General Bi-Symmetric Beam-Column Element in DYNAMIX
As shown in Figure 2.3, the bounding surface model in stress-resultant space, as
described in the previous section, is used to monitor the inelastic behavior at discrete
locations along a beam-column element. The tangent stiffness matrix derived according
to the bounding surface model for steel, composite, or reinforced concrete bi-symmetric
23
cross-sections reflects the nonlinear effects due to gradual concrete and steel
plastification and concrete cracking along the member length by integrating it along the
member length. Thus in this way the bounding surface model takes account of spread-ofplasticity through the cross-section while the numerical integration handles the spread-ofplasticity along the member length. This is specifically done by generating a sectional
flexibility matrix that is then integrated along the length to give the member flexibility
matrix. This matrix is inverted, and expanded to give the member stiffness matrix. In the
presence of the spread-of-plasticity, this method is more accurate than a displacement
based approach since it does not require any assumptions regarding the displaced shape
of the member. The flexibility approach does require assumptions regarding the member
force distribution along the member, but the distribution of forces is less sensitive to
spread-of-plasticity effects than is the displaced shape.
Behavior monitored at
Gauss locations along
member length
End forces and
corresponding
deformations
Gauss
point
Figure 2.3 Beam-column element with distributed plasticity - DYNAMIX.
2.3.1 Element Formulation
Following classical plasticity theory, the incremental strain vector, de, at a cross-section
can be separated into elastic and plastic components
24
de = dee + dep
(2.3)
where dee and dep are the vectors of incremental elastic and plastic strains, respectively.
In the following derivation, the torsional strains are neglected and they are assumed to
always remain elastic and uncoupled from the other deformations. Inclusion of the
torsional degrees of freedom is discussed later. Each part of the strain vector has three
components representing axial strain and bending curvatures in the two principal
directions.
dee = {dε e dφ z,e dφ y,e }
T
(2.4)
dep = {dε p dφ z,p dφ y,p }
T
(2.5)
where ε is the axial strain, φ is the curvature, and the subscripts e and p denote elastic and
plastic components respectively. The elastic strains, dee, are related to the cross-sectional
forces, dFsec by
dFsec = De dee
(2.6)
De is a diagonal matrix containing the sectional elastic stiffnesses,
[
De = diag EA EI z EI y
]
(2.7)
where EA is the elastic axial stiffness term, and EIz and EIy are elastic bending stiffness
terms.
Assuming that normality is enforced, the incremental plastic strain vector is proportional
to the normal, g, at the force point on the yield surface, hence:
25
dep = dλ g
(2.8)
where dλ is the plastic deformation parameter. For a force point F on the loading (or
bounding) surface defined by the function f, f(F)=0, g is a vector that contains the partial
derivatives of f with respect to the principal forces,
∂f
∂f (F )  ∂f
g=
=
∂F
 ∂P ∂M z
∂f 

∂M y 
T
(2.9)
The incremental force vector can be decomposed into two parts, one normal, dFn,sec, and
the other tangential, dFt,sec, to the yield surface such that
dFsec = dFn,sec + dFt,sec
(2.10)
The incremental plastic strain is assumed to be due to the normal component, dFn,sec, of
the incremental force vector. The relation between this normal force component and the
corresponding plastic strain is assumed to be uncoupled. In matrix form, the relationship
is written as
dFn,sec = Dp dep
(2.11)
where Dp is the matrix of plastic stiffnesses. Assuming Kp,i to be the plastic stiffness in
principal direction i (i = p,z,y), then
[
Dp = diag K p, p K p,z K p, y
]
(2.12)
Equations 2.3, 2.8, 2.10, and 2.11 represent the essence of the bounding surface model as
implemented herein. Each principal direction is calibrated independently with the
plasticity attributes of the model handling the necessary interaction by forcing the plastic
26
flow in a specific direction. For example, an applied bending moment will produce an
increase in the centroidal axial strain and vice versa.
Multiplying Equation 2.3 by De, and substituting from Equations 2.6 and 2.8,
dFsec = De de - dλ De g
(2.13)
Further, dFt,sec is normal to the gradient to the surface, g, hence:
dFt,secT g = (dFsecT – dFn,secT) g = 0
(2.14)
which together with Equations, 2.8, 2.10, 2.11 and 2.13 yields
[
dλ = g T (D p + D e ) g
]
−1
g T D e de
(2.15)
Substituting Equation 2.15 into Equation 2.13,

D g g T De 
dFsec = D e − T e
 de
g (D e + D p ) g 

(2.16)
This equation can be rewritten as,
dFsec = [De – Dr] de
(2.17)
where Dr, termed the plastic reduction matrix is given as,
Dr =
De g g T De
g T (D e + D p ) g
(2.18)
27
These matrix operations involved in these equations are relatively simple since they are at
most 3x3. Also the middle part that requires inversion, gT (De + Dp) g, is a scalar and
hence presents no difficulty. This term expands to
 ∂f 
  + D2,2
 ∂P 
2
T
g (De + Dp) g = D1,1
 ∂f

 ∂M z
2

 + D3,3

 ∂f

 ∂M
y





2
(2.19)
and D1,1 = EA + Kp,p, D2,2 = EIz + Kp,z, and D3,3 = EIy + Kp,y.
For the implementation in DYNAMIX, the axial and bending strains are uncoupled when
the cross-section is partially plastified. In other words, the axial effects are assumed to
follow one dimensional plasticity theory, while the biaxial bending terms are coupled and
follow the plasticity rules described above. This separation is easily taken care of by
adjusting the De and Dp matrices. Interaction between axial and bending strains is
reinstated when the force point reaches the bounding surface.
The resulting stiffness relationship between dFsec and de relates behavior at the crosssection. The member flexibility matrix is obtained by assuming member force
distribution functions, B (representing an equilibrium matrix), and then integrating the
following relationship along the length:
L
fM =
∫
BT (De – Dr)-1 B dx
(2.20)
0
The integration is performed numerically using a Gauss-Lobatto scheme. The GaussLobatto is chosen over other methods since it allows monitoring points at the beginning
and at the end of the member where plastic effects are maximum. The resulting member
flexibility matrix is a 5x5 matrix. Elastic shear deformation terms (1/GAz,eL and
1/GAy,eL) are added appropriately to the relevant matrix cells, where GAz,e and GAy,e
represent the elastic shear stiffness in the major and minor principal direction,
respectively, and L is the member length. fM is then inverted to get the 5x5 member
28
stiffness matrix, SM, without rigid body modes. To include rigid body modes, the
stiffness matrix is pre- and post-multiplied by a transformation matrix, T, resulting in the
10x10 element stiffness matrix, KM. Torsional stiffness components are then added,
resulting in a 12x12 local member stiffness matrix.
2.3.2 Modeling of Stiffness Degradation with Cycles
Stiffness degradation is an important phenomenon that may affect the analysis and the
behavior of reinforced concrete and composite structures. Based on a concentrated
plasticity approach, Gourley and Hajjar (1994) developed a model for stiffness
degradation as a function of the accumulated plastic energy at a member end. DYNAMIX
adopts a similar formulation based on a normalized accumulated plastic energy per unit
length at a point along the element; a treatment which is suitable for a distributed
plasticity approach that involves integration along the member length. The normalized
plastic energy terms for the axial, major bending, and minor bending effects are handled
separately.
The accumulated plastic strain energy per unit length, Wp, accumulated over n load
increments is,
Wp =
n
∑F
T
sec
(2.21)
de p
1
T
where Fsec
are the total member forces at a cross-section. It can be explicitly separated
into its components as follows
n
Wp =
∑
(P dεp + Mz dφz,p + My dφy,p)
(2.22)
1
where dεp, dφz,p and dφy,p are the incremental generalized plastic cross-sectional strains,
and P, Mz and My are the total member forces at a specific cross-section.
29
This plastic energy density is then normalized to allow a calibration that is independent of
the section properties. It is thus divided on a term by term basis (i.e. normalizing each of
the axial, major bending, and minor bending terms separately) by the elastic strain energy
Mz
P
y
, Wnorm
, Wnorm
. Those normalizing terms consist of the elastic strain
densities: Wnorm
M
energy densities associated with the axial, major axis bending, and minor axis bending
capacities of the cross-section, respectively. These are calculated as follows
P
norm
W
Pcn2
=
2.E.A
Mz
Wnorm
=
My
norm
W
=
(2.23)
M 2znb
2.E.I z
(2.24)
M 2ynb
(2.25)
2.E.I y
where Pcn is the sectional squash load, Mznb is the major axis bending capacity at the
balanced load, and Mynb is the minor axis bending capacity at the balanced load.
Accordingly, the normalized accumulated plastic energy index at the cross-section level
is defined as follows
Ωp =
n
∑
1
 P dε p M z dφ z ,p M y dφ y,p 
+

 P +
My
Mz
Wnorm
Wnorm
 Wnorm

(2.26)
Note that Ωp is a cumulative measure of the plastic work (or plastic dissipated energy) for
each cross-section throughout the entire load history.
30
Stiffness degradation is then simulated through: (a) a degradation of the unloading elastic
stiffness; and (b) a degradation of the plastic loading parameter k2, as a function of Ωp.
The following expressions are introduced for the unloading stiffness degradation ratio,
−ξi Ω p
rKi = 0.1 + 0.9 x 11
.
(2.27)
and for the k2 degradation ratio,
-ς i Ω p
rki 2 = 1.01
(2.28)
where ξi and ζi are calibration parameters for the unloading stiffness degradation ratio
and the plastic loading parameter, k2, degradation ratio, respectively for axial, major
bending, and minor bending principal directions. Accordingly, at any time step, the
instantaneous unloading elastic stiffnesses, Ki, at a specific cross-section along a member
can be given as
Ki = rKi Kiinitial
(2.29)
and the instantaneous k2,i parameters can be written as
k2,i = rki 2 k2,iinitial
(2.30)
where Kiinitial and k2,iinitial are the initial elastic stiffness and initial k2 plastic loading
parameter of the cross-section in principal direction i. Accordingly, the 3x3 cross-section
stiffness matrix is updated at each time step considering the suitable amount of stiffness
degradation according to the above model. Among other advantages of considering this
treatment of stiffness degradation is that it approximately captures some of the pinching
effect usually observed in the behavior of reinforced concrete and composite elements
under cyclic loading.
31
2.3.3 Calculation of Plastic Rotation
The inelastic beam-column model in DYNAMIX follows a spread-of-plasticity approach
where generalized strains (e.g., curvatures) are monitored along the member at a
predetermined number of integration sampling points selected by the user. Curvatures are
thus monitored as the basic element/material deformation measure in the analysis.
Although this scheme provides an effective way of modeling spread-of-plasticity, the
monitoring of very localized curvatures presents some disadvantages for practical
interpretation of the analysis results. One disadvantage is that plastic rotations, as
opposed to curvatures, are more commonly cited as a basic behavioral index in
experimental tests and in seismic design/evaluation standards (e.g., FEMA 273).
Therefore, by monitoring curvatures, one is faced with the question of relating curvatures
to hinge rotations. Another concern is that curvature only describes the behavior of a
specific point along a member, usually the peak value at the member end, and thus does
not reflect the cumulative distribution of damage along the member. Moreover, the peak
value of curvature at the highly strained end section of a member can be very sensitive to
numerical analysis parameters such as the number of Gauss integration points,
convergence criteria, etc. Accordingly, a routine is implemented to permit the monitoring
of both curvatures and plastic rotations for beam-column members.
Plastic
Curvature
φ p,i
Elastic
Curvature
Gauss
point
φ e,i
L
Figure 2.4 Schematic curvature distribution along a cantilever beam.
32
Using basic principles of structural mechanics and beam theory, it is straightforward to
determine plastic rotation at the end of a member by integrating the plastic curvature
distribution along a member. This is sometimes done approximately (e.g., Paulay and
Priestly 1992), by multiplying the maximum plastic curvature at the most stressed section
along the member by an equivalent effective plastic hinge length obtained through
empirical formulae. In DYNAMIX a more rigorous method of calculating the plastic
rotation is used by first sampling the plastic curvature at several integration points along
each element. Next, referring to Figure 2.4, and by using the Gauss-Lobatto quadrature
scheme, the plastic rotation at the end of the beam is computed by integrating the plastic
curvatures according to the following formula
θp =
∫
L
0
φ p dx =
GP
∑φ
p,i
(2.31)
Wi L
i=1
where θp is the plastic rotation at the member end, φp,i is the plastic curvature at Gauss
point i along the member, Wi is the weight of the Gauss-Lobatto integration scheme at
Gauss point i, and L is the member length.
2.4 Composite Beam Model
In design of composite moment frames using elastic analysis, it is generally acceptable to
use approximate techniques for modeling the behavior of the composite beam, i.e., the
reinforced concrete floor slab with steel deck and steel beam. For instance, an average
stiffness of the positive (composite) section and the negative (steel) section can be used
as a good approximation to model the overall effective stiffness of the composite beam.
However, when inelastic analysis is used, either for design practice or in research, it is
important to accurately represent the composite beam.
A composite beam shows complex behavior due to slip between the reinforced concrete
slab and the steel beam, and the variation of longitudinal stress across the width of the
33
slab, which is dependent of the joint details and the loading pattern. In order to capture
this complex behavior, a three-dimensional finite element analysis may be needed.
However, some researchers (Lee 1987, Tagawa et al 1989, Engelhardt et al 1995)
developed two-dimensional discrete member models as a compromise between simplicity
and accuracy. In these models, it is assumed that the effect of slip and the variation of
longitudinal membrane stress on the behavior of composite beams can be implicitly
included in the constitutive moment-rotation relationships. It is also worthy to mention
that these models adopt a concentrated plasticity approach. On the other hand, a fiber
beam-column model, with continuously distributed springs along the interface between
the concrete slab and the steel beam to represent shear connectors (studs), is developed
by Salari et al (1996) to model the composite beam behavior in a more accurate, but
computationally much more expensive way.
In the present work, a two-dimensional beam-column element is developed to model the
behavior of composite beams using a spread-of-plasticity approach. The constitutive
model is a moment-curvature relationship based on an adaptation of the bounding surface
model described in the previous sections which employs a kinematic hardening rule for
cyclic loading. The model also accounts for stiffness degradation as a function of the
accumulated plastic energy in the member as done for the general beam-column element
with bi-symmetric cross-section.
2.4.1 Limitations and Assumptions
The analytical method employed in this work assumes plane sections to remain plane
after bending. This implies perfect bonding between the steel beam and the concrete slab
or in other words full shear connection and suitable number of studs to ensure full
capacity. As mentioned by El-Tawil and Deierlein (1996), the “plane sections remain
plane” assumption is reasonably good even well into the inelastic range.
In concept, the bounding surface model consists of two nested surfaces described in
multi-dimensional force (or stress) space. However, for the suggested composite beam
34
model, as the expected level of axial force in the element will be generally negligible
compared to the axial capacity, it can be assumed that the axial behavior will remain
elastic. Moreover, it is assumed that the out-of-plane (minor axis) bending will also
remain elastic. Accordingly, the only remaining parameter in the force space is the major
axis bending moment and thus the bounding “surfaces” reduce to a set of nested “bars”.
The model’s details will be discussed in the sequel.
Moreover, member slenderness effects, steel bar buckling and local buckling in the
structural steel section are not considered. Elastic shear deformations are included in the
model but shear and torsion interaction is not considered. Warping, creep and shrinkage
effects are also not accounted for in the present model.
2.4.2 Element Formulation, Moment-Curvature Skeleton and Hysteresis Model
While the incremental elastic strain vector is as given by Equation 2.4, the incremental
plastic strain vector, according to the assumption stated in the previous section, has only
one non-zero term corresponding to the major axis bending curvature.
dep = {0 dφz,p 0}T
(2.32)
Assuming that the normality rule is still enforced, Equation 2.8 is also valid with the
normal, g, to the yield surface, in this special case, having a value of {0 ±1 0} at all
force points in the force space. The sign + or - depends on the direction of loading.
Equation 2.12 describing Dp, the matrix of plastic stiffnesses, is rewritten here for the
composite beam element with only one non-zero term corresponding to the plastic
stiffness, Kp,z, in the principal major axis bending direction. Then,
Dp = diag[0
Kp,z
0]
(2.33)
35
Specific equation for the plastic stiffness, Kp,z, is given later according to a suggested
hysteresis model. All other Equations 2.3 to 2.20 hold. Equation 2.17 relating the
incremental strain vector to the incremental force vector at the cross-section level through
the total cross-section stiffness, Dsec, is written again herein for completeness

D g g T De 
dFsec = D e − T e
 de = Dsec de
g (D e + D p ) g 

(2.34)
with Dsec, based on the assumptions given before, can be given in a condensed form as

Dsec = diag  EA

EIz Kp, z
EIz + Kp, z

EIy 

(2.35)
The constitutive relations for the composite beam element - as mentioned before - are
given in the form of moment-curvature relationship at the cross-section level to be
suitable for a formulation based on a distributed plasticity approach. According to the
assumptions stated in the previous section, the bounding surface model reduces to a onedimensional model composed of two nested bars: loading or onset of yielding bar, and
bounding or full plastification bar. The basic concept of this model is described through
Figures 2.5 and 2.6 which show the response through cycles of loading and unloading
following a standard kinematic hardening rule. The letters A through F indicate
corresponding load points in the two figures. The inner or “loading” bar shown in Fig. 2.6
describes a region inside which the response is elastic. For force point movement inside
this bar (e.g. from point A to B, C to D, or E to F) the response is elastic, whereas when
the force point contacts the bar, the structure starts to load inelastically. During plastic
loading the inner bar is pulled along with the force point, i.e., dML = dM where dML is
the incremental movement of loading bar and dM is the incremental composite beam
cross-section moment. On the other hand, movement of the outer bounding bar, dMB, is
governed by dMB = η dML where η is given below. The loading bar is confined within
the outer or “bounding” bar, and during plastic loading the relative proximity of the two
defines the inelastic stiffness. As shown in Figs 2.5 and 2.6, the proximity between the
36
bars is measured by the distances d, and din which refers to the value of d measured at the
initiation of each plastic loading excursion.
Another main issue in the constitutive relations given in Figure 2.5 is the different
moment capacities, Mzp+ or Mzp-, depending on whether it is a positive or negative crosssection along the composite beam length, respectively. By positive section, it is meant
that the top fibers of the concrete slab are in compression while the bottom fibers of the
steel beam are under tension and the composite action is thus mobilized. On the other
hand, a negative cross section takes place when the bending moment gives rise to
compressive stress at the bottom fibers of the steel beam while the concrete slab is
cracked under tensile stresses and the rebars carry part of the tension. Moreover, the
elastic stiffness may take two values, Ke+ or Ke-, for the positive and negative sections
along the composite beam, respectively. Recommendations for calculation of positive and
negative stiffness and moment capacity are given later.
It is worthy to mention herein that according to the constitutive model given in Figure
2.5, once the cross-section has yielded in the negative moment direction, the cracks in the
concrete slab is considered to take place causing the loss of the slab’s share in the
stiffness. Then, the element assumes the negative stiffness, Ke-, whether continuing
loading or unloading. However, once the positive moment applied surpasses again the
magnitude of the negative moment capacity, Mzp-, but while loading in the positive
direction, the cracks are considered closed again and the element assumes the positive
stiffness, Ke+, whether loading or unloading. Another assumption in the present model is
that the composite beam element assumes a negative stiffness values for any applied
negative moment. In this model, we are looking for capturing the overall behavior of a
composite beam concerning different stiffness and strength capacities depending on the
state of loading (i.e., positive or negative direction). To maintain computational
efficiency, the element does not explicitly model detailed behavior associated with
cracking in the slab, slip between the slab and beam, etc.
37
Referring to Figure 2.5, the tangent stiffness in major axis bending direction of a crosssection along the composite beam element, Kt, is defined as follows:
Regions AB, CD, and EF (elastic response)
Kt = Ke+ or Ke-
(i.e., EIz+ or EIz-)
(2.36)
depending on whether the cracks are closed or open and on other assumptions and
conditions governing the constitutive relations already mentioned above.
Region BC (positive plastic loading - closed cracks)
Kt =
K p,z
Ke
+
K p,z
(2.37)
K e + K p,z
+
k3

 d +  
+


Ke+
= k 1 + k 2  +
+


 d in − d  

(2.38)
where Kp,z is the instantaneous positive plastic stiffness.
Region DE (negative plastic loading - open cracks)
Kt =
K p,z
K e - K p,z
(2.39)
K e - + K p ,z
k3

 d−  
=  k1 + k 2  −
  Ke−

 d in − d  

(2.40)
where Kp,z here is the instantaneous negative plastic stiffness.
d+ or d -
current distance between the force point and the bounding
surface for positive and negative directions, respectively.
(Figs. 2.5 and 2.6).
din+ or din-
distance between the force point at initiation of yielding and
the bounding surface for positive and negative directions,
respectively (Figs. 2.5 and 2.6).
38
k1+ and k1-
plasticity parameter defining strain hardening ratio for positive
and negative directions, respectively (Figs. 2.5 and 2.6).
k2 and k3
plasticity parameters that are calibrated to experimental response.
Note that when d = din, Kp,z = ∞ and the response is elastic, whereas d = 0, Kp,z = k1 Ke,
and the slope of the bounding line in Fig. 2.5 approaches the value Kt = Ke/(1+1/ k1).
During plastic loading, the movement of the outer bounding bar is based on the ratio of
the bounding stiffness to the current plastic stiffness, as follows
dMB = η dML
(2.41)
η = (k1 Ke) / Kp,z
(2.42)
In this way, the two surfaces (i.e., the two bars) move together when they are in contact,
i.e., when d = 0.
Finally, based on calibration to test results that will be discussed later, the following
values are assumed for parameters in the composite beam element constitutive model:
k1+ = 0.01, k1- = 0.02, k2 = 1.0 and k3 = 1.2
(2.43)
α=
Size of loading bar (-ve side)
= 0.55
Size of bounding bar (-ve side)
(2.44)
β=
Size of loading bar (+ve side)
= 0.44
Size of bounding bar (+ve side)
(2.45)
In the present model of the composite beam, an adaptation of the stiffness degradation
computation strategy given in Section 2.3.2 is implemented taking into consideration the
fact that plastic deformation takes place only for the major bending direction. The method
39
also accounts for different stiffness and strength capacities for the positive and negative
loading sides associated with the composite beam behavior.
Moment
Pos itive bounding line
k1+
C
M zp +
d+
Ke+
Ke+
din+
M zp Initiation
of yielding
F
β M zp +
KeKe-
B
K e+
KeK e+
A
α M zp -
D
K e-
dind-
M zp -
Curvature
Reinitiation
of yielding
k1Negative bounding line
E
Figure 2.5 Constitutive model and moment curvature skeleton for composite beam
element.
Moment
Reinitiation of
yielding
C
Initiation
of yielding
M zp +
din+
β M zp +
d+
B
D
F
Loading bar
din-
A
α M zp -
d-
Bounding bar
M zp ELASTIC
E
INELASTIC
INELASTIC
ELASTIC
Figure 2.6 Schematic diagram of nested bars movements.
40
ELASTIC
2.4.3 Elastic Stiffnesses and Ultimate Strength Calculation for Composite Beam
To define the positive elastic stiffness, Ke+, of the composite beam, the effective width of
the concrete slab is required. Assuming fully composite action, Lee (1987) conducted
three-dimensional elastic finite element analyses to investigate the effects of several
parameters influencing the effective width of composite beams. He then came up with a
rather complicated equation for the effective width that can be simplified, for practical
purposes, as follows:
beff = 0.19 L + bcf
(2.46)
where beff is the effective width of the slab, L is the beam length from the column face to
the end of the beam (inflection point), and bcf is the column flange width.
Moreover, as stated by Engelhardt et al (1995), using the partial interaction theory
(Newmark 1951; Robinson 1969) and experimental results, Uang (1985) and Lee (1987)
investigated the influence of a slip on the effective width of composite beams. They
found that when the partial interaction theory employs the stiffness of a shear stud
calibrated to the average value of experimental slip data over the beam length, the theory
can properly reflect the effect of slip on the positive elastic stiffness. However, for the
partial interaction theory to be generally applied to obtain the positive elastic stiffness,
more experimental data are required for the flexible behavior of a shear stud along the
composite beam length. The experimental positive elastic stiffness obtained at design
load is smaller by about 15% than that computed under the full interaction assumption.
Later, Lee et al. (1989) used one quarter of the beam length (column face to the inflection
point) as the effective width to account for the influence of a slip on the positive elastic
stiffness of composite beams.
In their work, Engelhardt et al (1995) used the minimum of the following three criteria
(AISC-LRFD Specification 1993) as the effective width of the concrete slab on each side
of the beam center-line for computing positive elastic stiffness:
41
L/8

beff ≤ b o / 2
 b
 es
(2.47)
where L is the beam span center to center of supports, bo is the distance from the beam
center-line to the center-line of the adjacent beam, and bes is the distance from the beam
center-line to the edge of the slab. To account for the influence of slip between the
concrete slab and the steel beam on the positive elastic stiffness, and based on calibration
with experimental results, Engelhardt et al (1995) assumed a value, for the moment of
inertia I + applied to the positive elastic stiffness, of 0.85 Itr, where Itr is the transformed
moment of inertia of the composite cross-section. For the negative elastic stiffness, I -,
they considered the steel beam section and reinforcing steel bars within the effective slab.
Tagawa et al. (1989) used a value of the effective width of concrete slab as given by the
specifications for the design and fabrication of composite structures (Japan, 1985):
beff = b + 2 ba
(2.48)
where b is the flange width of steel beam, and ba is the smaller of {[0.5-(0.6a/L)]L ,
0.1L}, in which L is the span length of the composite beam, and a is the clear spacing of
adjacent beams.
In the present work, the effective width of slab required for the elastic stiffness
calculation follows the recommendations by Lee (1987) and given in Equation 2.46 but
provided that it is less than the value proposed by AISC-LRFD Specification (1993) and
given in Equation 2.47.
As shown in Figure 2.7, the actual cross section is transformed to an idealized section
with exactly same dimensions except the effective width of concrete slab, beff. The
42
concrete slab is converted to an equivalent steel slab with the same thickness, tc, but with
a width given by
bs = beff (Ec / Es)
(2.49)
where Ec and Es are Young’s modulus for concrete and steel, respectively.
Given the value of yt, the moment of inertia, I +, can be easily calculated as follows:
or
I + = bs yt3/3 + Is + As (ys - yt)2
,
yt < tc
(2.50)
I + = bs tc3/12 + Is + bs tc (yt - 0.5tc)2 + As (ys - yt)2
,
yt > tc
(2.51)
where Is is the moment of inertia of the steel beam about its centroid.
In calculating the negative elastic moment of inertia, I -, the reinforcing steel bars within
the effective width, beff, are considered while the concrete slab itself is neglected since it
is assumed to be cracked.
bs
beff
h
tc
ys
C.G. Steel
Section
As
yt
As
a) Actual Section
b) Idealized Section
Figure 2.7 Cross-section main dimensions for a typical composite beam.
As mentioned by Tagawa et al. (1989), the effective width of concrete slab of a
composite beam was originally specified for the evaluation of the elastic stiffness of the
beam; so, it is uncertain whether it could also be applied to the evaluation of the moment
43
capacity of the beam. Moreover, it has been reported that the ultimate strength of
composite beams is dependent on the slab area which is in contact with the column flange
and, as a lower bound, the concrete compressive strength can be increased to 1.3f’c due to
the confinement of concrete near the column (duPlessis et al. 1972). Lee (1987)
considered the contribution of the concrete slab to the ultimate strength by using the
column width and a concrete compressive strength of 1.4f’c. Tagawa (1989) also assumed
that the contribution of the concrete slab takes place over the whole width of the column
but used a concrete compressive strength of 1.8f’c. Engelhardt et al. (1995) considered the
recommendations by duPlessis et al. (1972). In the present work, the same
recommendations are also adopted.
bcf
1.3f’c
tc
yn
Cc
Cr
Fyr
yr
Cs
y sc
P.N.A
Fy
ys
T
Fy
a) Cros s section
b) Plastic stress distribution
Figure 2.8 Plastic stress distribution for a typical composite beam.
The negative moment capacity, Mzp-, of the composite beam can be computed as the
plastic moment of the steel beam section and the reinforcing steel bars within the
effective beam width. The positive moment capacity, Mzp+, can be determined according
to Figure 2.8, and based on the assumptions stated in the previous paragraph.
Accordingly, the plastic neutral axis is determined by solving the following equation for
the compressive steel area, Asc: (Equations 2.52 and 2.53 are given by Engelhardt et al.,
1995)
2 Asc Fy = As Fy - 1.3 f’c bcf tc - Ar Fyr
44
(2.52)
Then, the positive moment capacity, Mzp+, can be calculated by

t 
Mzp+ = 1.3 f c' b cf t c y n  1- c  + A sc Fy y sc + (A s − A sc ) Fy y s + A r Fyr y r
 2 yn 
(2.53)
where yn
: distance from the plastic neutral axis (P.N.A.) to the top surface of the slab,
ysc : distance from P.N.A. to the compression resultant of steel,
ys
: distance from P.N.A. to the tension resultant of steel,
yr
: distance from P.N.A. to the compression resultant of reinforcing bars,
bcf : column flange width,
tc
: concrete slab thickness from the top surface to the top of metal deck,
As : total area of steel beam cross-section,
Ar : area of reinforcing bars within the effective width bcf,
Fy : yield stress of steel beam,
Fyr : yield stress of reinforcing bars.
2.4.4 Verification Study
In this section, the accuracy of the proposed composite beam model, as implemented in
the computer program DYNAMIX, is tested. Comparisons of analytical and experimental
results for available composite beam test specimens are presented. It is important to
mention that it is not the objective of this verification study to fit the analytical data to
test results by ‘tweaking’ calibration parameters for each specific specimen. Rather, the
main goal is to obtain a set of calibration parameters that works quite reasonably for all
specimens and that is capable of capturing to a good extent the overall behavior of the
composite beam and the whole specimen.
The test setup and the specimens details for the four tests considered in this study are
given in Figure 2.9. One of the tests, specimen CG3 - Uang (1985), is a small-scale test.
45
The others are full-scale tests: Tagawa et al. (1989), Bursi and Ballerini (1996) specimen with full shear connection, and specimen EJ-WC by Lee (1987). Table 2.1
gives the material properties of the different test specimens.
Test
Specimen
Tagawa
1989
Bursi
1996
Uang
1985
(CG3)
Lee 1987
(EJ-WC)
Table 2.1 Material properties for test specimens.
Reinforc.
Structural Steel
Steel
Yield
Yield
Stress
Stress
(ksi)
(ksi)
Beam
Column
Web
Flange
Web
Flange
47.86
41.19
54.68
41.48
51.63
Concrete
Strength
(ksi)
3.55
43.50
43.50
43.50
43.50
69.90
5.66
41.50
37.00
-----
-----
79.00
4.26
37.80
36.65
39.20
36.40
60.00
5.10
Figure 2.10 shows the comparison of experimental and analytical results for the specimen
by Tagawa et al. (1989). Figure 2.10a gives the horizontal load versus the horizontal
displacement of the specimen monitored at point A shown in Figure 2.9a, while Figure
2.10b shows beam moment versus beam rotation at the same point. The agreement
between the experimental and the analytical results is quite reasonable within a range of
predicting strength of about 6%. Furthermore, it can be observed that the overall behavior
is reasonably captured by the analytical model.
46
147.6”
P, ∆
41”
Loading Beam
1.18”
Concrete slab
134”
W 14x30
Point A
W 16x57
Reaction
Fra me
3.54”
2.95”
Wire mesh
φ0.236” @ 3.94”
W 14x30
118”
295”
(a) Test setup and specimen - Tagawa et al. 1989
47.24”
P, ∆
Concrete slab
0.79”
55.12”
IPE 330
1.97”
2.75”
8 bars - φ0.47”
HE 360B
IP E 330
157.48”
(b) Test setup and specimen - Bursi and Ballerini 1996
30”
0.56”
P, ∆
Concret e slab
1”
1”
Wire mesh
φ0.0625” @ 1”
M 6x4.4
M 6x4.4
45”
(c) Test setup and specimen (CG3) - Uang 1985
66.93”
66.93”
47.24”
W 12x65
(weak axis)
Concret e slab
1.2”
P, ∆
Wire mesh
φ0.21 4” @ 4”
W 18x35
W 18x35
90.55”
(d) Test setup and specimen (EJ-WC) - Lee 1987
Figure 2.9 Test setup and specimen for verification study problems.
47
3.5”
3”
Analytical and experimental results for the specimen by Bursi and Ballerini (1996) are
given in Figure 2.11. The specimen presented herein is the one with full shear
connection. It may be observed that the analytical model is able to capture the strength
reasonably except for large amplitudes of displacements where it underestimates the
strength (i.e., the lateral strength) in the positive moment direction by about 7.5%, while
it still predicts well the strength in the negative moment direction. Moreover, it can be
noticed that when the lateral applied load is pushing the specimen and the steel girder is
under compression, the experimental results show local buckling in the steel beam. This
phenomenon contributed to the sudden drop of the lateral strength that cannot be captured
by the present model.
The comparison with experimental results of specimen CG3 by Uang (1985), presented
in Figure 2.12, also shows reasonable agreement until local buckling occurs at the bottom
flange.
Figure 2.13 shows the comparison of experimental and analytical results of specimen EJWC by Lee (1987). The specimen is an exterior joint assemblage with the column acting
in its weak axis of bending, and the beam is connected to the column web by connected
plates. The comparison shows quite reasonable agreement until the bottom flange of the
steel beam develops severe local buckling.
The verification study presented in this section shows that the proposed composite beam
element can reasonably model the main behavioral issues of a composite beam. The
analytical model can capture to a good extent strength, stiffness, and stiffness degradation
until local buckling phenomenon of the bottom flange of the steel beam occurs. In the
verification study, the calibration parameters are fixed and same method is used for
calculating the member properties of all specimens.
48
500
Horizontal Load, Q [kN]
375
250
125
0
-125
-250
Experimental
Analytical
-375
-500
-100
-75
-50
-25
0
25
50
75
100
Displacement, δ [mm]
(a) Load-displacement relationship.
4000
Beam Moment [kips-in]
3000
2000
1000
0
-1000
-2000
Experimental
Analytical
-3000
-4000
-0.04
-0.03
-0.02
-0.01
0.00
Rotation [rad.]
0.01
0.02
(b) Moment-rotation relationship.
Figure 2.10 Experimental and analytical results - specimen Tagawa (1989).
49
400
200
100
0
-100
-200
Experimental
Analytical
-300
-400
-120
-80
-40
0
40
80
120
Horizontal Displacement, δ [mm]
Figure 2.11 Experimental and analytical results - Bursi and Ballerini (1996).
(Specimen with full shear connection)
6
4
Tip Load [kips]
Horizontal Load, Q [kN]
300
2
0
-2
Experimental
Analytical
-4
-4
-3
-2
-1
0
1
2
3
4
Tip Displacement, ∆ [in.]
Figure 2.12 Experimental and analytical results for specimen CG3 - Uang (1985).
50
60
Tip Load [kips]
40
20
0
-20
-40
Experimental
Analytical
-60
-3
-2
-1
0
1
2
3
Tip Displacement, ∆ [in.]
Figure 2.13 Experimental and analytical results for specimen EJ-WC - Lee (1987).
2.5 Composite Joint Panel Model
In design of moment frames using elastic analysis, it is generally acceptable to use
approximate techniques for modeling the behavior of joints. For instance, the combined
effect of the finite joint size and its flexibility is often modeled by considering a reduced
size of the joint as fully rigid. However, when inelastic analysis is used, either for design
practice or research, it is important to accurately represent both panel zone deformations
and finite joint size effects. This is particularly critical for analyses involving lateral
seismic loads where inelastic behavior often concentrates in, or adjacent to, beam-tocolumn joints. In composite RCS frames, modeling of joint response is complicated by
internal force transfer mechanisms that involve composite action between the steel and
concrete and exhibit strength and stiffness degradation under cyclic loading.
51
As shown in Figure 2.14, previous research (Sheikh et al., 1989 and Kanno and Deierlein,
1996) has identified two basic failure modes in the joints: a) panel shear, and b) bearing
of steel against concrete. Panel shear failure is similar in some respects to that observed
in steel or reinforced concrete joints, except that in mixed steel-concrete joints both
structural steel and reinforced concrete elements participate. Bearing failure occurs at
locations of high compressive stress and permits rigid rotation of the steel beam within
the concrete column. As discussed by Kanno (1993), the actual behavior usually involves
deformations associated with both failure modes. However, he observed that, whereas
cases with panel shear failures tend to have large bearing deformations, cases with
bearing failures do not have significant panel shear deformations. This behavior is
undoubtedly affected by the role of the steel beam web in helping to resist joint shear,
whereas the bearing strength is provided by concrete alone.
2.5.1 Joint Panel Kinematics
A mechanical idealization for the proposed joint model is shown in Figure 2.15a (Zhao,
1993, and El-Tawil et al., 1996). The model is comprised of a number of rigid bars
connected together by pins allowing panel shear distortions in each of the two vertical
planes, but not in the horizontal plane. At the center of the joint panel are two mutually
perpendicular rotational springs that represent the joint stiffness. Behavior is assumed to
be independent in the two orthogonal directions, and so each of these springs introduces
one additional degree of freedom into the structural analysis model. When the central
bars in each plane rotate with respect to one another, the rest of the rigid bars and the
connected members distort in the manner shown in Figure 2.15b.
Implementation of the model shown in Figure 2.15 in DYNAMIX involves the following
three basic steps: (1) degrees of freedom corresponding to the joint panel deformation are
added to the global model, (2) the stiffness matrices for beams and columns framing into
joints are modified via a transformation matrix to account for the finite panel size and to
link appropriate beam-column stiffness terms with the joint degrees of freedom, and (3)
52
the joint stiffness relationships are calculated and added to appropriate locations in the
global stiffness matrix. More details are provided by El-Tawil et al. (1996).
Concrete
Crushing
Gap
(a) Panel Shear Failure
(b) Bearing Failure
Figure 2.14 Panel shear and bearing modes of failure.
2.5.2 Joint Panel Moment-Distortion Hysteresis Models
Constitutive relations for the joint panel distortion are separated into two components,
one associated with panel shear distortion and the second with bearing deformations.
Moment-distortion hysteresis models are given for the two modes of failure in Figures
2.16 and 2.17. Panel shear distortion is modeled using a one-dimensional bounding
surface model. The joint bearing deformation model consists of semi-empirical equations
that account for the more severe pinching behavior observed in connections with bearing
failures. More details about the formulation of the models are presented by El-Tawil et al.
(1996). Both models include the effects of stiffness degradation which is based on the
total plastic energy accumulated during the loading history in a manner similar to that
discussed in Section 2.3.2. The two constitutive relationships are combined in a single
model that takes account of the kinematic relationships associated with the finite joint
size.
53
Pinned
connection
Bars rigidly
connected
RC Column
Beam
Beam
Steel
Beam
Rigid
Bar
Column
Connection
mechanism
Kinking
(a) 3-D joint panel model
(b) Kinematics of joint panels (undeformed
and deformed 2-D views)
Figure 2.15 Composite joint panel model.
Moment
Upper bounding line
Initiation of
yielding
Mns
din1
d1
Elastic
Elastic
Panel shear distortion
din2
d2
Mns
Reinitiation of
yielding
Lower bounding line
Figure 2.16 Constitutive model for joint panel shear.
54
Moment
Upper bounding line
Mnb
din1
Upper pinching line
θ br,2
d1
0.2Mnb
5Keb
Keb
θ br,1
Bearing distortion
Lower pinching line
closing gap
0.2Mnb
closing gap
5Keb
d2
Mnb
din2
Lower bounding line
Figure 2.17 Constitutive model for joint bearing.
2.6 Modeling of Geometric Nonlinearity
To consider geometric nonlinearity, DYNAMIX adopts a formulation based on work by
Powell (1969), Mahasuverachai and Powell (1982), Chen (1994), and Yang and Kuo
(1994), whereby the geometric stiffness matrix is derived to include spread-of-plasticity
effects. This involves the distinction between ‘internal’ and ‘external’ terms of the
geometric stiffness matrix that are related to the natural and rigid body deformations,
respectively. By definition, the internal part accounts for changes in equilibrium due to
member displacements relative to the member chord (Powell, 1969). Alternatively, it may
be thought of as that part of the geometric stiffness matrix that modifies the member
stiffness in the presence of initial stresses (Gattass and Abel, 1987). The external part
accounts for the change in orientation of member end forces as the member chord
undergoes rigid body motion (Powell, 1969 and Gattass and Abel, 1987).
55
It may be shown that the external portion of the geometric stiffness matrix is completely
independent of the assumed displaced shape and the effect of spread-of-plasticity. Hence,
it is only the internal part that is a function of the assumed member displacement fields.
Using the flexibility approach adopted by DYNAMIX, the actual inelastic displacement
fields can be calculated and updated as the analysis proceeds. The internal part of the
geometric stiffness matrix can therefore be formulated using these updated shape
functions to include spread-of-plasticity effects, and based on a virtual work formulation
described by Chen (1994). In general these shape functions derived by a flexibility
approach differ from the cubic Hermitian shape functions that are commonly used in the
development of elastic geometric stiffness terms.
2.6.1 Definitions, Assumptions and Limitations
Dealing with an idealized beam-column element undergoing incremental displacements,
three basic configurations can be defined. The first configuration represents the ‘initial’
undeformed state. The second configuration is assumed to have been previously
calculated to satisfy equilibrium and compatibility conditions (known as the ‘reference’
configuration), whereas the last configuration represents the ‘desired’, unknown
equilibrium state. Within the context of a virtual work formulation, the solution of a
geometrically nonlinear problem, is usually handled by either a total Lagrangian (TL), or
an updated Lagrangian (UL) approach. In the former, the incremental stiffness
characteristics are referred to the original (i.e., initial), undeformed configuration, during
all stages of an analysis. In the latter, the element stiffness relations are based on an
updated coordinate system referred to the deformed, or reference, configuration, at the
start of each load increment. It has been shown (Bathe and Bolourchi, 1979) that when
formulated properly, both approaches result in identical equilibrium relationships, and
that the UL method is computationally more efficient. The UL approach is adopted in
DYNAMIX as mentioned by El-Tawil and Deierlein (1996).
The major assumptions inherent in the virtual work formulation implemented in
DYNAMIX, as stated by El-Tawil and Deierlein (1996), are: (1) plane sections remain
56
plane after member deformation, and only doubly symmetric sections are considered, (2)
unrestrained warping behavior (i.e., St. Venant torsion) is implied, (3) only small strain
behavior is considered, and (4) loading is applied at the nodes. A common assumption of
great convenience used by many researchers, as well as in this work is that the
‘reference’ configuration is assumed to start out straight. Thus, member end forces are
assumed to act with respect to this straight configuration and are aligned in the same
direction. Among other things, this assumption permits the use of a single transformation
matrix to convert from local member coordinates to global coordinates. Implied in the
assumption is that rotations of the member ends with respect to the member chord
(natural end rotations) are small, although the total incremental end rotations (including
rigid body motion) can be moderate.
2.6.2 Total Geometric Stiffness Matrix Based on Hermitian Shape Functions
The inclusion of geometric nonlinearities in space frame analysis is complicated by the
non-vectorial nature of finite rotations. These quantities do not commute as vectors do.
The approach adopted in DYNAMIX, as implemented by El-Tawil and Deierlein (1996),
concerning this issue is based on Chen’s (1994) treatment, whereby he treats the nonvectorial nature of rotations in the basic kinematic relationships. Chen (1994) showed
that by using a proper rotation transformation matrix and consistently maintaining second
order accuracy in the virtual work expression, the so called ‘correction matrix’ derived by
other researchers using semi- and quasi-tangential moments results directly from the
virtual work principle.
For a given infinitesimal virtual displacement from the equilibrium state in the
‘reference’ configuration, the principle of virtual work can be written. Assuming plane
sections to remain plane after bending, making use of the approximate Green-Lagrange
strains, and making use of the orthogonality conditions for the principal axes of bisymmetrical cross sections, finite element procedures can be applied to extract the
member stiffness matrices. More details are given by Chen (1994) and El-Tawil and
57
Deierlein (1996). For a general case, the incremental member stiffness equation is written
as
F = [Km + Kgs + Kgr] d
(2.54)
where Km is the linear elastic tangent stiffness matrix, Kgs is the part of the geometric
stiffness matrix obtained without considering the effect of finite rotations, and Kgr is the
part of the geometric stiffness matrix that accounts for finite rotations. It is sometimes
referred to as the ‘correction’ matrix in the literature. F are the incremental member end
forces, and d are the incremental end displacements.
Notice that Kg = [Kgs + Kgr] represents the total geometric stiffness matrix which
correctly handles the effects of finite rotations. The distinction between Kgs and Kgr
should not be confused with the previously mentioned separation of Kg into its external
and internal parts. The total geometric stiffness matrix, Kg, based on Hermitian
displacement fields, for three-dimensional analysis is given by Chen (1994).
2.6.3 Geometric Stiffness Matrix as a Function of Spread-of-Plasticity
As shown in Figure 2.18, member plastification may have a significant effect on the
member displaced shape. As previously mentioned in this chapter, this is the primary
reason for adopting a flexibility approach for determining the inelastic stiffness matrix
since such an approach does not require the assumption of predefined displacement
fields. For geometric nonlinear analysis, the use of a geometric stiffness matrix based on
Hermitian fields gives rise to an inconsistency in the formulation. As noted earlier, this
inconsistency has been recognized in the literature (Attalla et al., 1994) and is generally
thought to not be of much consequence. However, the evidence supporting this position
is not conclusive, mainly incidental observations made of a few example problems.
To evaluate the differences between using elastic shape functions and the actual
displacement fields in the presence of inelasticity, a new formulation is developed and
58
implemented by El-Tawil and Deierlein (1996) whereby the geometric stiffness matrix is
calculated to include the spread-of-plasticity effects on the shape of the member. The
formulation relies upon flexibility based shape functions (FBSFs) that are continuously
updated as the analysis progresses. These shape functions are calculated based on the
inelastic sectional properties along the member length. The calculated displacement fields
are then used in a virtual work derivation to determine the geometric stiffness matrix.
Given that the shape functions are changing during the analysis, it is useful to make use
of the distinction between the internal and external geometric stiffness components. The
external geometric stiffness matrix is shown to be independent of the internal
displacement fields, and so it is only the internal geometric matrix that is affected by the
spread of plasticity. It should be mentioned that the derivation is carried out by El-Tawil
and Deierlein (1996) assuming that the FBSFs are calculated at the ‘reference’
configuration and are assumed to be constant throughout the step. It is therefore
implicitly implied that load increments are small enough that changes in the geometry of
the FBSFs during the step of the analysis are negligibly small.
2.6.4 General Comments
The term ‘P-δ effect’ implies the modification of local moments due to the interaction
between the axial load and the natural deformations. The term ‘P-∆ effect’ implies the
force modification associated with rigid body motion. To correctly model the
geometrically nonlinear behavior of members undergoing substantial plastic excursions,
the total element stiffness matrix should include full coupling between inelastic effects,
P-δ and P-∆ effects. The formulation adopted in DYNAMIX captures some but not all of
this coupling.
In DYNAMIX, the P-∆ effect is properly included in both the tangent geometric stiffness
matrix used in the predictor step of the analysis, and in the recovery of member end
forces. This is taken care of by the external part of the geometric stiffness matrix. On the
other hand, the inelastic member stiffnesses only partially include the P-δ effect in the
predictor step of the analysis through the effect of the internal part of the geometric
59
stiffness matrix. Because of the assumption of members being straight in the ‘reference’
configuration, the local amplification of moments due to P-δ effects is not calculated in
the analysis, and hence the inelastic stiffness terms calculated at Gauss points are based
on moments that are different than the true second-order moments. In other words,
although a flexibility approach is being used, the exact P-δ effects in the derivation of the
geometric stiffness matrix are not being completely considered.
Elastic field
Fully elastic member
Elastic field
Inelastic field
Plastified region
Fully elastic
Inelastic field
Elastic field
Fully elastic
Plastified region
Figure 2.18 Comparison between FBSFs and Hermitian shape functions in the presence
of spread of plasticity. (El-Tawil and Deierlein, 1996)
60
In addition, the accuracy involved in the partial handling of the P-δ effects is limited by
the accuracy of the assumed shape functions. As mentioned by El-Tawil and Deierlein
(1996), the FBSFs used in DYNAMIX are more accurate in this respect than the
Hermitian shape functions generally used. For the reason mentioned above, even under
elastic conditions, the Hermitian shape functions do not represent the exact shape for
axial load plus bending. Neither do the flexibility based shape functions used herein
capture this effect. In addition, ‘higher order’ effects of both P-δ and P-∆ are not included
by virtue of the fact that the highly nonlinear components of the Green-Lagrange strains
are neglected, i.e., terms that are greater than second-order in the virtual work equation.
The effect of higher order terms have been examined for simplified two-dimensional
cases by Powell (1969), and Yang and Kuo (1994).
The geometric nonlinearity as described above has been implemented in DYNAMIX by
El-Tawil and Deierlein (1996). The type of geometric nonlinearity that might be
considered in an analysis can be specified by the user as one of four options:
1. NONE: turns off geometric nonlinear features, i.e., nodal coordinate updating and
geometric stiffness terms are not included.
2. ELASTIC: includes nodal coordinate updating and geometric stiffness matrix based
on Hermitian displacement fields.
3. INELASTIC: includes nodal coordinate updating and geometric stiffness matrix based
on flexibility based shape functions, FBSFs.
4. EXTERNAL: includes nodal coordinate updating and geometric stiffness terms that
only consider external components.
Based on several examples problems (El-Tawil and Deierlein, 1996 – Chapter 5), it
appears that the inaccuracies associated with using elastic Hermitian shape functions for
the formulation of a geometric stiffness matrix are practically negligible. However, the
alternate inelastic geometric stiffness formulation seemed to alleviate the unloadingreloading problem that can affect the analysis near inelastic limit points. Moreover, it has
61
been shown through some parametric studies that when a discretization of more than two
elements/member is used, the external matrix may be used without compromising
solution accuracy, yet at the same time reducing the unloading-reloading problem. For
the analysis work done as part of this thesis, geometric nonlinearity is included by nodal
coordinate updating and geometric stiffness matrices based on Hermitian displacement
fields for all beam-column elements. For composite beam elements, the external
geometric matrix is solely used in the implementation. Discretization of composite beam
members in four elements per member as done for the structures investigated in Chapters
5 to 7 of this thesis guarantees using the external matrix alone without compromising
solution accuracy.
2.7 Overview of the Scheme of the Numerical Integration of the Equation of Motion
for Time History Analysis
The dynamic analysis of a given structure is based on finding a solution to the following
differential equation of motion:
M &x& + C x& + Kt x = Pa – M &x& s = P
(2.55)
where M is the diagonal mass matrix of the structure; C is the viscous damping matrix
that models energy dissipation; Kt represents the instantaneous tangent stiffness of the
structure at a given point in time. &x& , x& , and x are the acceleration, velocity and
displacement of each degree of freedom, measured with respect to the supports, and &x& s is
the absolute ground acceleration. Pa is the vector of external forces applied to the
structure, and P is the equivalent applied load vector. For analysis of three-dimensional
frames, the displacement, velocity and acceleration vectors each contain six degrees of
freedom per node, three translational and three rotational.
DYNAMIX uses numerical integration to solve the equation of motion in the time
domain. The Newmark Beta method which is a form of implicit numerical integration is
62
employed. Implicit integration involves the use of known quantities such as
displacement, velocity, and acceleration of a structure at a given time, i, and assumed
values for initially unknown quantities, such as acceleration of the structure at a future
time, i+1, to calculate more accurate values for the unknowns. In DYNAMIX, the
Newmark Beta method produces a set of simultaneous equations that, when solved, yield
the future values of displacement, velocity, and acceleration.
The constant acceleration version of the Newmark Beta method used in DYNAMIX,
which is unconditionally stable, assumes the acceleration to be constant between time
steps i and i+1. The value of the acceleration is considered to be an average of its values
at the beginning and end of a time step. Thus, for constant acceleration within a time step,
the relationship between stiffness, displacement, and force can be represented as:
K ∆xi = ∆ Pi
(2.56)
where
K = Kt +
4
2
M+ C
2
h
h
(2.57)
and
4

∆ Pi = Pi+1 + M  x& i + &x& i  + C x& i - Fi
h

(2.58)
where h is the time step size and Fi is the internal forces vector which can be calculated
as Fi = Kt xi. Then, change in displacement ∆xi, taking place between time steps i and
i+1, is calculated using Equation 2.56 and hence the displacement at the future step i+1 is
obtained by
xi+1 = xi + ∆xi
(2.59)
63
Future velocities and accelerations at i+1 can be then computed
x& i +1 = x& i +
&x& i +1 =
h
(&x& i + &x& i +1
2
)
(2.60)
4
(x i +1 - x i ) - 4 x& i - &x& i
2
h
h
(2.61)
The incremental displacements are used to find the incremental forces in each structural
component, and hence the whole system is updated to the new time step and the
procedure can be started all over again for the next time increment. For linear systems,
the Newmark Beta method can yield an exact solution. However, for path dependent
nonlinear problems, the Newmark Beta method can only approximate the correct answer.
In DYNAMIX, since material and geometric effects may impart nonlinearities in the
stiffness matrix, the tangent matrix also has to be updated at the beginning of each new
step. The implementation in DYNAMIX assumes the stiffness to be constant throughout
the time step.
It is also important to mention that an adaptive time step scheme is adopted by
DYNAMIX (Searer, 1994). The time step used throughout the numerical solution process
of the equation of motion is variable and is controlled via user-defined maximum and
minimum values. When the structure is completely elastic, the maximum time step is
used. As members begin to plastify, the time step is reduced linearly from the maximum
to the minimum, where the most plastified cross-section along the most critical member
controls the time step. However, user control over the minimum time step is preempted
when there is either (a) a change in a load history, (b) unloading of a member, (c)
breaching of the yield surface, or (d) breaching of the bounding surface.
64
2.8 Summary
A computer program (DYNAMIX- DYNamic Analysis of MIXed systems) for threedimensional inelastic second-order dynamic frame analysis is presented in this chapter.
The program, as described, employs a bounding surface stress-resultant plasticity model
that accounts for the interaction between axial loads and bi-axial bending moments of bisymmetric sections in a flexibility-based formulation including the effects of spread-ofplasticity, geometric nonlinearities, and cyclic stiffness degradation. The program has
been ported from DEC VMS to run on a DEC UNIX platform as part of this research.
First, a brief overview of available inelastic analysis models in the literature is presented.
Then, a quick review of the general concepts of the bounding surface model is given.
Formulation and main features of the general bi-symmetric three-dimensional beamcolumn element are discussed. Implementation of a two-dimensional beam element to
model composite beams (i.e., concrete floor slab with steel deck and steel beam) is also
presented. The composite beam element is a 1-D version of the bounding surface model
used for the general beam-column element including kinematic hardening and stiffness
degradation as a function of the accumulated plastic energy in the member. It aims to
capture the overall behavior of a composite beam in a computationally efficient manner,
particularly differences in the member’s stiffness and strength under positive versus
negative bending. Example problems are then analyzed to verify the accuracy of the
implemented composite beam model. An element for modeling inelastic behavior of
composite joint panels is also described. It considers both the finite joint size and the two
major deformation components of the composite joint, namely: panel shear and bearing
modes. The joint panel model also accounts for stiffness degradation and pinching
behavior under cyclic loading.
Different strategies implemented in DYNAMIX to model geometric nonlinearities are
also briefly presented. Comparison between using Hermitian shape functions versus
flexibility based shape functions is pointed out. Finally, a brief overview of the
65
integration scheme implemented in DYNAMIX for the numerical solution of the general
equation of motion for time history analysis of a structural system is provided.
66
Chapter 3
Stiffness Modeling of Reinforced Concrete
Beam-Columns
Accurate stiffness properties of beam-columns are necessary to reliably calculate deflections,
destabilizing second-order (P-∆) effects, and dynamic response characteristics of systems with
reinforced concrete structural elements such as the RCS moment resisting frames addressed
throughout this thesis. Routine distinctions of stiffness properties between “beams” or “columns”
for reinforced concrete structures can be overly simplistic, particularly for frames designed for
earthquakes or large wind loads where column compression forces are small relative to their
axial capacity. In this chapter, factors influencing beam-column stiffness in frame analysis are
reviewed, and simple formulae are proposed to determine effective flexural and shear stiffness
coefficients of beam-columns as a function of the applied axial compression. The proposed
stiffness coefficients represent conditions at incipient yield and are applicable for linear (elastic)
analyses and the linear pre-yield region of nonlinear (inelastic) analyses. Stiffness coefficients for
the proposed model are compared to test data and alternative recommendations in several
sources, including a CEB state-of-the-art report on seismic analysis of reinforced concrete
67
frames and several design codes (ACI-318, New Zealand Standard, Architecture Institute of
Japan Standard).
3.1 Introduction
For building design, it is commonly accepted to use rough estimates of the stiffness properties
for reinforced concrete structures given the many necessary simplifications employed for
analysis and a presumption that modest variations in the member stiffness coefficients will not
appreciably change the resulting member sizes. Consequently, traditional rules-of-thumb, such
as using one-half the gross moment of inertia for beams and the full moment of inertia for
columns, are widely employed, even though they are known to be quite approximate.
Consequences of this are seen, for example, in comparative studies of building analyses with
recorded earthquake motions that often resort to ad-hoc selection of stiffness and modeling
parameters (e.g, Browning et al. 1997, Hart et al. 1998). Advanced computer analysis
technologies, improved knowledge about structural behavior and loads, and initiatives to
develop multi-level performance-based design and analysis methods suggest a re-evaluation of
stiffness properties used in design. This chapter specifically addresses one of many issues,
determination of effective flexural and shear stiffness coefficients for reinforced concrete beamcolumns under combined flexure and axial effects. This issue is of great importance for the
accurate modeling of RCS moment frames investigated within this research.
As a brief review, recall how calculated stiffness properties affect structural design:
•
Deflections. Assumed stiffness coefficients will directly impact the design of structures
controlled by deflection criteria or slender structures sensitive to second-order (P-∆)
effects. The 1995 edition of the ACI-318 Building Code incorporated for the first time
explicit recommendations for stiffness parameters to use in the second-order analysis of
slender columns, but this does not address the broader set (i.e., at different limit states) of
68
deflection related issues in design. For example, in studying seismic requirements for
composite steel-concrete (RCS) frames, the structures are usually controlled by drift limits
and thereby sensitive to stiffness modeling assumptions.
•
Internal Force Distributions. For structures with conventional framing systems and regular
geometry that are designed based on elastic analysis, the internal force distribution is usually
not sensitive to the assumed stiffness coefficients. However, this is not generally true for all
cases. For example, where structures are inelastically designed to resist earthquakes and
structural components are distinguished as either “force” or “deformation” controlled,
accurate calculation of stiffness properties and the resulting internal forces become more
important. Alternatively, in non-conventional systems such as hybrid wall and frame systems
or mixed steel-concrete structures, the calculated internal force distribution can significantly
vary depending on the assumed stiffness properties. The degree to which the calculated
force distribution is inconsistent with the actual distribution can lead to larger than
anticipated inelastic force redistribution and deformations.
•
Dynamic Response. Given that the natural vibration frequencies of a structure are
proportional to the square root of its stiffness, the stiffness coefficients will affect dynamic
effects induced by earthquakes or wind effects in flexible structures. Depending on the
loading characteristics, and whether or not inelastic effects are modeled in the analysis,
changes in the stiffness can have either a positive or negative effect on structural
performance.
Aside from the fact that modern computer technologies enable more refined analyses, emerging
trends in engineering practice create incentives to utilize such analyses. Among these, structural
evaluations made for seismic rehabilitation or renovation often warrant more refined analyses to
achieve economical solutions. Performance-based engineering is another area where more
accurate analysis techniques are warranted. Refinements applied to frame analyses would
include more explicit modeling of (1) basic geometric features such as finite joint sizes, wall and
69
foundation elements, etc., (2) second-order geometric effects, e.g., P-∆, and (3) inelastic
behavior of members and connections associated with concrete cracking, steel yielding,
bond/slip, and nonlinear concrete compression behavior. Generally, inelastic behavior due to
concrete cracking prior to significant yielding or concrete crushing can be modeled fairly well by
linear analyses with appropriate stiffness coefficients. On the other hand, modeling of post-yield
behavior requires nonlinear inelastic analyses, technologies for which are becoming increasingly
accessible to design engineers
.
3.2 Basic Behavior and Design Issues
For beam-columns subjected to a given magnitude and distribution of internal forces, the
reduction in stiffness due to pre-yield load cracking is often modeled through secant stiffness
coefficients determined by calibration to tests or detailed analytical models. Accurate
establishment of these coefficients is, however, complicated by the nonlinear interaction of many
factors including loading magnitude and distribution, indeterminacy among structural elements
(beam-columns, joints, slabs, walls, etc.), creep and shrinkage, and foundation settlement. To
sort out the underlying behavior, consider first the response of an isolated beam-column
subjected to combined bending and axial load, and then, the integration of individual element
behavior in the analysis and design of overall framing systems.
3.2.1 Beam-Column Behavior
Shown in Fig. 3.1 is a reinforced concrete cantilever beam subjected to flexure. The spacing of
transverse flexural cracks decreases with increasing bending moment until it reaches a constant
minimum value that depends on the concrete tension strength and reinforcing bar bond transfer.
The overall member stiffness reflects the integration of properties for cracked and uncracked
sections along the member. While it is straightforward to determine the behavior of idealized
cracked and uncracked sections (Fig. 3.1b), the real member behavior is complicated by bond
70
slip and tension stiffening. The addition of axial compression considerably stiffens the member
by delaying the onset of cracking, and at high compression loads above the balance point there
is little stiffness reduction prior to the member reaching its design moment capacity.
Overall load-deformation response of the member (Fig. 3.1c), typically reflects a distinct loss in
stiffness at the cracking and the yield moments. Assuming that the limit state of interest is at the
design strength (roughly equivalent to the onset of significant nonlinearity due to excessive steel
yielding or concrete crushing), the load-deformation behavior can be linearized by a secant
stiffness to match the deformations near the yield load. The effective flexural stiffness, EIeff,
corresponding to this point can be back-calculated from displacements measured from tests or
determined from more detailed analyses. Referring back to Fig. 3.1b, one should expect EIeff to
lie between the response for the idealized cracked and uncracked section.
F
Uncracked
Section
Cracked
Section
M
F
Mn
My
Fn
Fy
Mcr
Cracked Section
Semi-cracked Section
Uncracked Section
Idealized Behavior
F cr
Actual Behavior
Φ
a) RC cantilever subjected b) behavior at the section
to lateral load.
level.
∆
c) Load-displacement overall
behavior of the member.
Figure 3.1 Behavior of reinforced concrete element in flexure (a) member subjected to lateral
load, (b) moment-curvature response, (c) load-deformation response.
3.2.2 Frame Behavior and Design
71
As shown in Figure 3.2, the overall load-deformation behavior of a frame resembles the
individual member response (Fig. 3.1c), except that stiffness changes due to cracking and
yielding occur more gradually due to structural indeterminacy. Selecting member stiffness
coefficients to approximate the system response is complicated by the loading conditions and
criteria for which the structure is designed. For example, columns in buildings governed by
gravity load in regions of low-seismicity are likely to be heavily stressed in compression and
bending, whereas in high-seismic regions, more stringent drift criteria and other requirements,
such as limits on the column-to-beam strength ratio, will result in lightly stressed columns. Other
differences between member and system response arise due to variations in heights, spans, and
other geometric and loading characteristics of the structure.
For behavior under pseudo-static monotonic loads (Fig. 3.2), distinct limit-states can be defined
at the service, factored, and ultimate load levels. Behavior for the first two of these, service and
factored loads, can be modeled fairly well by linear analyses where the element stiffness
coefficients are selected to reflect the displacements and force distribution at the prescribed
load. Assuming that a structure is “optimally” designed for strength, the factored load level
roughly corresponds to the onset of significant yielding in most members. However, many
structures are not optimally designed in this sense, and at the full factored load level only a few
members will have reached a yielding condition. In such cases, the average secant stiffness
coefficient of all members will be more than that at the onset of yield in an isolated member.
72
Lateral
Load
Ultimate
Yield
Factored
Service
Actual Behavior
Second-Order
Analysis with EIeff
Displacement
Figure 3.2 Load versus deflection behavior of a reinforced concrete frame.
Conditions beyond the factored load and approaching the ultimate load, where significant steel
yielding and/or concrete crushing occur, can only be accurately calculated by nonlinear
(inelastic) analysis. For design purposes, however, some quantities in the post-yield region, such
as lateral drift and associated P-∆ effects, are often estimated using semi-empirical adjustments
to the elastic analysis. One such approximation, employed in seismic design, is the estimation of
inelastic drifts based on an elastic analysis. For example, using the equivalent lateral force
method of the 1997 NEHRP Recommended Provisions (BSSC 1997) or the proposed
International Building Code 2000 (IBC 1998), the predicted inelastic seismic drift is inferred
from the elastic deflections through the seismic response parameters Cd/R where ∆inelastic
≈ ∆elastic (C d/R). Definitions of and rationale behind these parameters are given in Chapter 5.
Typical values of Cd and R for moment frames imply a ratio of inelastic to elastic drifts of
∆inelastic/∆elastic = Cd/R = 0.7 to 0.9. On the other hand, the so-called “equal displacement rule”
73
would predict ∆inelastic ≈ ∆elastic. Recent research (e.g., Nassar and Krawinkler 1991, Miranda
and Bertero 1994) indicates that the relationship depends on the building period, and based on
this the FEMA 273 Guidelines recommend that ∆inelastic/∆elastic =1.5 for short period structures,
reducing to ∆inelastic/∆elastic =1.0 for longer period structures.
While the accuracy of semi-empirical methods for estimating inelastic seismic deformations are
debatable, they all use a pseudo-elastic analysis as the basis for calculating drift and other
quantities such as induced base shear, internal forces, etc. The recently published FEMA 273
Guidelines indicate that frame analyses should approximate conditions “near the yield point”
and, further, recommend using EIeff = 0.5EcIg for beams and EIeff = 0.7EcIg for columns.
However, these values are not rigorously substantiated, nor is there general agreement between
recommendations made in ACI-318 (1995) and other standards.
3.3 Inelastic Frame Analysis
Aside from their use for elastic (linear) analyses, effective stiffness coefficients corresponding to
initial yield conditions are often used to model the initial loading regions for inelastic analyses.
Shown, in Figure 3.3, are two general categories of nonlinear analysis methods for frame
structures, referred to herein as concentrated-hinge and spread-of-plasticity methods. The
concentrated-hinge type (Fig. 3.3a) can employ several strategies to model bi-linear, multi-linear
or nonlinear response. One is through a mathematical assembly of multiple parallel elastic
elements connected through rigid-plastic hinges that capture abrupt change of stiffness at various
load levels. Alternatively, linear or nonlinear elastic-plastic springs can be devised to achieve
similar behavior (Powell and Chen, 1986). Bi-linear models are often used where the first
break-point occurs at member yielding, and in such cases the initial member stiffness would be
the same as that used to model behavior up to the onset of yielding in an elastic analysis. For
beam members, the hinges can be controlled by bending moments at the member ends, whereas
74
for beam-columns, the hinges should take into account the interaction of moments and axial
load.
The spread-of-plasticity approach (Fig. 3.3b) is more accurate than the concentrated-hinge
method in that it directly models the distribution of nonlinear behavior through the cross section
and along the member length. There are many variations on spread-of-plasticity
implementations, but most rely on modeling inelastic cross section behavior at discrete sampling
points along the member, either by explicitly integrating stresses and strains or through a stressresultant yield-surface approach as the one presented in Chapter 2 and used throughout this
research. The yield-surface implementation for reinforced concrete beam-columns (used within
DYNAMIX as explained in Chapter 2) is given herein (Fig. 3.4) for completeness. It employs
an inner loading and outer bounding surface to model moment-curvature and axial force-strain
behavior of the cross section under combined axial load and biaxial bending. Comparing this
model (Fig. 3.4) to the cross section behavior in Fig. 3.1, the elastic region inside the loading
surface employs an effective stiffness bounded between the cracked and uncracked section
properties that depends on the level of axial load. The rapid nonlinear reduction in stiffness
outside the loading surface is represented by the kinematic hardening model, presented in
Chapter 2, that involves tracking the movement and proximity of the two surfaces to one
another.
3.4 Review of Stiffness Guidelines
Existing guidelines to estimate the effective stiffness for analysis range from general ad-hoc
approaches to more theoretical ones. A number of these are reviewed below, followed by a
proposed effective stiffness model that seeks to balance practicality and simplicity with accurate
modeling of the governing behavior. A key aspect of the proposed approach, compared to
other commonly employed methods, is the explicit consideration of the variable axial
compression levels in beam-columns.
75
M
α EI
( 1−α)EI
Rigid-Plastic Hinge
α EI
My
Elastic
EI
EI
Inelastic Hinge Spring
θy
End rotation, θ
a) Concentrated-Hinge Models
M
Integration Points
My
K = ∫ BT k S B dl
L
or
EI
F = ∫ bT f S b dl
L
Curvature, Φ
b) Spread-of-plasticity Model
Figure 3.3 Nonlinear beam-column element models for frame analysis (a) concentratedhinge type, (b) spread-of-plasticity type.
P
M
Bounding Surface
Loading Surface
α EIeff
2
2
1
1
M
EIeff
Φ
Figure 3.4 Stress-resultant yield surface model and idealized moment-curvature response.
76
3.4.1 ACI-318 Building Code (1995)
The 1995 edition of ACI-318 introduced specific recommendations for effective stiffness
coefficients for second-order frame analysis, primarily based on the work of Hage and
MacGregor (1974) and MacGregor (1993). As explained by MacGregor (1993), the specified
stiffness coefficients of EIeff = 0.35EcIg and 0.70EcIg for beams and columns, respectively, were
obtained by reducing mean values of 0.4 EcIg and 0.8 EcIg by a resistance factor φ = 0.875.
This resistance factor is chosen to reflect average conditions in a frame, calculated as the
mathematical average of the lower bound φ = 0.75 for a single member and the upper bound of
φ = 1.0. The unreduced stiffness coefficients of 0.4 EcIg and 0.8 EcIg for beams and columns,
respectively, date back to earlier studies by Hage et al. (1974) and Kordina (1972).
MacGregor (1993) also cites coefficients recommended by other researchers of 0.5 EcIg for
beams and 0.3 EcIg to 0.9 EcIg for lightly to heavily loaded columns.
The lower-bound stiffness values in ACI-318 (1995) are intended for second-order analyses
under factored loads to evaluate strength (stability) effects. MacGregor (1993) notes that for
analysis of service-load deflections, φ should generally be taken equal to 1.0 and that the
stiffness coefficients would be 1.25 times the values given above. Applying these adjustments to
the average values of 0.4 EcIg and 0.8 EcIg, the EI values for computing service-load deflections
revert to the common rule-of-thumb values of 0.50EcIg and 1.0EcIg for beams and columns,
respectively.
Outside of the recommendations in the slender column provisions, ACI-318 (1995) does not
include more general information or recommendations for stiffness values to apply for dynamic
or other analyses where the lower-bound values applied to assess static second-order stability
effects at factored loads may not be appropriate. Moreover, the single stiffness designation for
columns does not account for the large variability in axial compression that may occur where
columns sizes are governed by lateral drift limits or other criteria besides compression capacity.
77
3.4.2 FEMA 273
As noted previously, the NEHRP Guidelines for the Seismic Rehabilitation of Buildings
(FEMA 273, 1997) recommend using 0.5EcIg and 0.7 EcIg for beams and columns,
respectively, in static and dynamic elastic analyses under earthquake loads. Comparing these to
the averages of 0.4EcIg and 0.8 EcIg described above, the FEMA 273 values tend to
acknowledge the likelihood that columns in seismically designed frames will have lower axial
compression, and therefore, smaller effective stiffness than columns governed by gravity loads.
However, since the corresponding beam stiffness is larger in FEMA 273, the net difference in
overall frame stiffness between the two sets of recommendations is probably negligible.
3.4.3 New Zealand Standard (1995)
Summarized in Table 3.1 are stiffness properties recommended by the New Zealand Concrete
Structures Standard (1995) for the elastic seismic analysis of frames. Here, the variation of
column axial load, P, and the expected inelastic ductility demand are explicitly considered.
Because seismic actions at the serviceability limit state for structures with ductility demands less
than µ=5 may be significantly less than those for the ultimate limit state, a reduced extent of
cracking and correspondingly increased structural stiffness of members may be expected under
serviceability limit state conditions. Accordingly, the stiffness under serviceability limit state
actions of structures designed for elastic response at the ultimate limit state may be based on
uncracked member sections using Ig. For the estimation of actions under serviceability limit state
conditions, particularly deflections, of structures with ductility demands between µ = 1.25 and µ
= 6, effective section properties may be interpolated between values based on gross concrete
areas and those corresponding to ultimate limit state conditions. For seismic design,
differentiating stiffness properties based on ductility demand helps to ensure a conservative
78
calculation of maximum forces in less ductile elements of the structure, i.e., elements that FEMA
273 refers to as “force controlled” elements.
Table 3.1 Effective section properties, Ieff, per New Zealand Standard (NZS 1995).
Type of Member
Checks at
Checks at Serviceability Limit
Ultimate Limit
State
State
µ = 1.25
µ=3
µ=6
1. Beams
Rectang. Beams
0.40 Ig
Ig
0.70 Ig
0.40 Ig
T-, L- beams
0.35 Ig
Ig
0.60 Ig
0.35 Ig
2. Columns
P / fc’ Ag > 0.5
P / fc’ Ag = 0.2
P / fc’ Ag =-0.05
0.80 Ig
0.60 Ig
0.40 Ig
Ig
Ig
Ig
0.90 Ig
0.80 Ig
0.70 Ig
0.80 Ig
0.60 Ig
0.40 Ig
3.4.4 CEB State-of-the-Art Report (CEB 1996)
In a CEB state-of-the-art report of seismic analysis and design, Filippou and Fardis (1996)
present a more theoretical model for effective stiffness derived from an inelastic cross-section
analysis. While they acknowledge the three distinct stiffness regions discussed earlier - the initial
uncracked state, the post-cracking response up to yielding of the tension steel, and the postyield behavior up to ultimate strength – they contend that for ultimate strength design the
distinction between the pre- and post-cracking state can be ignored. They justify this on the
basis that nonlinear response analysis is dominated by post-yield behavior, and the frame
members would be cracked prior to an earthquake due to gravity loads, concrete shrinkage,
temperature effects, etc. Accordingly, as shown in Figure 3.5, they recommend that the moment
versus curvature response can be approximated as bilinear, with the corner point between the
elastic and the post-yield branches defined as the effective yield point.
79
Moment
My
Suggested Bilinear Behavior
Mcr
EIeff
Φ cr
Idealized Trilinear Behavior
Φy
Curvature
Figure 3.5 Effective secant flexural stiffness per CEB (Filippou and Fardis, 1996).
Applying the assumption of plane-sections remaining plane, elastic stress-strain models for
concrete and steel (neglecting the tension-stiffening effect), and the yield condition for the
tension reinforcement, Filippou and Fardis calculate the yield moment My and curvature Φ y as:
d 

ξy − 1 

ξ
ξ




d
d
d


y
y
2 '
h
M y = bh f c  (ω 1 + υ ) 1 − 1 −  − υ  0.5 − 1  + ω 2  − 1 

h
3
h
3
h


 

1 − ξ − d1 

y

h 
Φy =
εy
(3.1)
(3.2)
d1 

1 − ξy −  h

h
with, ξ y, the normalized depth of the compression zone at yield, given as:
80

Ec ε y
f'
2
ξy = c
− (ω1 + ω2 + υ ) + (ω1 + ω 2 + υ ) + 2 '
Ec ε y
fc

1
 d1 
d1  2
(
)

1
−

υ
+
ω
+
ω


1
2
h
h 

(3.3)
where h and b are the section height and width; f c' is the concrete compressive strength; Ec is
the concrete elastic modulus; ε y is the yield strain of the reinforcement; d1 is the distance of the
reinforcement from the nearest extreme fiber; ω 1 = Ast Fy / Ag f c' and ω 2 = Asc Fy / Ag f c' ,
where Ast and Asc are the areas of the tension and compression reinforcement, respectively; Fy
is the yield strength of the reinforcement; Ag is the gross column area; and υ = P / Ag f c' is the
normalized axial force (+ compression, - tension).
Filippou and Fardis (1996) further suggest that based on the work done by Park and Ang
(1985), Φ y can be refined as follows to account for nonlinearity of concrete in compression

εy

 υ 
0.45
Φ y = 1.05 + 
− 0.05

d 
 0.84 + 2ω 1 − ω 2
 0.3  

1 − ξy0 − 1  h

h
(3.4)
where ξ y o is determined from Equation 3.3 with υ = 0.
Finally, as noted by Filippou and Fardis, these equations are derived exclusively from flexure
theory, and do not account neither for shear deformations nor for rotations due to slippage of
the longitudinal reinforcement from its anchorage into the joint of the frame.
3.4.5 Architectural Institute of Japan Standard (1991)
81
The Architectural Institute of Japan Standard (AIJ, 1991) suggests using the gross uncracked
stiffness, EcIg, for service load calculations and an effective stiffness, EIeff = αEcIg at factored
loads. The coefficient α is determined by the following equations based on Sugano (1970):
1
1
1 − Mc / M
= 1 + ( − 1)
α
αy
1− Mc / My
(3.5)
α y = (0.043 + 1.64 n ρt + 0.043 a/h + 0.33 ν) (dr/h)2
(3.6)
Mc = γ√ f c' .Ze + Ph/6
(3.7)
d
My = [ ω 1 + 0.5 ν (1-ν)] f c' b h2
h
(3.8)
where M is the target moment of interest, n is the modular ratio, Es/Ec; ρt is the ratio of tensile
reinforcement, Ast/bh; a is the shear span; dr is the distance from cross section edge to tensile
reinforcement; Mc is the cracking moment; Ze is the section modulus considering reinforcing
bars; d is the distance between compression and tension reinforcement; and the other terms as
defined in the previous section. γ=0.56 for f c' and √ f c' in MPa. For consistent comparison with
the other models, M is assumed equal to My in the verification study below, such that α=α y
from Equation 3.5.
3.5 Proposed Stiffness Coefficients
As a compromise between the practicality of the simple guidelines such as in ACI-318 (1995)
and the more theoretical approaches such as the CEB equations, the following linear relationship
is proposed to estimate the effective stiffness of beam-columns corresponding to the yield point:
82
EIeff /EIg,tr = (0.4 + P/2.4Pb) ≤ 0.9
(3.9)
where
EIg,tr
= gross transformed bending stiffness
P
= applied axial compression
Pb
= axial compression at balanced failure condition
Two significant distinctions between this equation and the ACI-318 (95) and FEMA 273
(1997) provisions are that it accounts for (1) variations in axial compression relative to the
balance point on the strength interaction surface and (2) variations in the steel reinforcement
through use of the gross transformed section properties. Considering the limiting values of 0.4
EIg,tr for P = 0 and 0.9 EIg,tr for P > 1.2Pb, the suggested stiffnesses per Equation 3.9 are
slightly larger than those of 0.4 EcIg and 0.8 EcIg previously suggested by Hage and MacGregor
(1974). Comparisons with test data shown below suggest that Equation 3.9 provides a more
accurate measure of the average secant stiffness properties at yield.
3.6 Verification Study
Data from reinforced concrete sub-assemblage tests from three sources (Watson and Park
1994, Azizinamini et al. 1992, and Kuramoto et al. 1994) are compared with calculated
response using the proposed model, Eq. 3.9. The tests are chosen to represent a range of axial
load levels varying from P/Pb=0.0 to 1.72 (P/Agf’c = 0 to 0.7). Comparisons are also made with
the other models reviewed above, and for two of the test specimens, comparisons are made to
data from nonlinear analyses.
83
3.6.1 Description of Test Specimens
The first set of data is by Watson and Park (1994) who conducted cyclic load tests on nine
reinforced concrete columns. The columns are 400-mm square sections with twelve 16-mm
diameter longitudinal bars equally spaced along the perimeter (ρs=1.5%) and various quantities
of transverse reinforcement. Concrete strengths range from f c' = 39 to 47MPa (5.7 to 6.8ksi).
The columns are first loaded with constant axial compressive loads (P/Pb=0.26 to 1.72) and
then cycled with quasi-static lateral loads.
The second series is by Azizinamini et al. (1992) who tested eleven columns, measuring 18 inch
(457 mm) square with eight #8 (25-mm diameter) bars equally spaced along the perimeter
(ρs=1.9%) and variable transverse reinforcement. Concrete strength is f c' = 6ksi (41.4MPa).
Load deformation data from two of these tests, NC2 (P/P b=0.54) and NC4 (P/P b=0.90), were
available in the published literature and used herein. Loading was applied in a similar manner to
Watson and Park’s tests.
Finally, the last test is of a reinforced concrete beam by Kuramoto et al. (1994). Tested under
reversed cyclic loading with no axial load, the beam has a rectangular cross-section of 300x400
mm and is doubly reinforced with six 19-mm diameter bars placed at top and bottom of the
longer dimension of the cross-section (ρs=1.4%) with Fy=342MPa (49.6ksi). Concrete strength
is f c' = 71.7MPa (10.4ksi). Transverse reinforcement consists of 6-mm diameter closed ties
spaced at 40 mm along the beam length with Fy=992.7MPa (144ksi).
3.6.2 Comparisons and Discussions
Measured and calculated stiffnesses (EIeff/EIg,tr) for the proposed model (Eq. 3.9) and other
models and guidelines are compared in Figs. 3.6 and 3.7 and Table 3.2. Measured EIeff values
from the tests are back-calculated from measured deflections at the yield point. The yield point
84
is typically defined at a load equal to 85% of the nominal member strength, calculated according
to the ACI-318 (1995) stress block procedure using measured material properties. The
measured deflections were determined from the envelope curve of the cyclic load-deformation
response.
1.0
0.8
EIeff / EIg,tr
Proposed Model
ACI-318, C
0.6
FEMA 273, C
FEMA 273, B
0.4
NZS, 1995 (µ=6)
ACI-318, B
WP tests, 1994
NC tests, 1992
K test, 1994
0.2
0.0
0.0
0.4
0.8
1.2
1.6
2.0
P / Pb
Figure 3.6 Proposed EI eff model compared to test data and other models.
As shown in Fig. 3.6, the proposed equation simulates the measured response fairly well,
particularly in capturing the change in stiffness with axial load. Shown for reference in Fig. 3.6
are EIeff from ACI-318 (with φ = 1), FEMA 273, and NZS. Note that for relating the code
values, based on EIg and P/f’cAg, to the proposed model in Fig. 3.6, the following equalities are
assumed: EIg = 0.85EIg,tr and Pb = 0.35 f c' Ag. These are average values of properties that can
range from about EIg = 0.75 to 0.95 EIg,tr and Pb/f’cAg = 0.3 to 0.45 for typical cross sections.
Comparisons of measured versus calculated values in Fig. 3.7 show that the two CEB models
tend to underestimate the stiffness, perhaps because they are based on modeling behavior of the
85
cracked cross section. Of the two CEB models, the first one with Φ y given by Eq. 3.2 appears
more accurate. The AIJ values are very low, but the reasons for this are not clear.
1.0
EIcalc. / EIg,tr
0.8
Proposed Eq. 3.9
CEB Eqs 3.1 and 3.2
CEB-M Eqs. 3.1 and 3.4
AIJ Eqs 3.5 to 3.8
0.6
0.4
0.2
0.0
0.0
0.2
0.4
0.6
0.8
1.0
EImeas. / EIg,tr
Figure 3.7 Comparison of effective stiffness coefficients with test data.
Behavior of two test specimens (WP-9 and K-1) is examined further in Figs. 3.8 and 3.9 where
theoretical moment versus curvature response plots and stiffness coefficients are compared. The
nonlinear moment versus curvature response curves are obtained from a fiber cross section
analysis based on a nonlinear stress-strain models for confined and unconfined concrete and
steel, including tension stiffening behavior. Details regarding the fiber analysis program and
stress-strain models are reported by El-Tawil and Deierlein (1996). Shown in the upper plots of
each figure are EIeff as a function of axial load for the various models. Shown in the lower plots
are fiber analysis moment versus curvature response curves at three axial loads, superimposed
on which are the secant stiffness lines. These plots tend to substantiate the observations made
86
earlier concerning trends between the alternative models and the relationship of the secant
stiffness values relative to cracked section behavior.
Table 3.2 Comparison of measured versus predicted stiffness.
ID
WP-1
WP-2
WP-3
WP-4
WP-5
WP-6
WP-7
WP-8
WP-9
NC-2
NC-4
K-1
Test Information
P/Agf’c P/Pb
ωs
0.1
0.26
14.3%
0.3
0.61
15.3%
0.3
0.79
15.3%
0.3
0.65
16.8%
0.5
1.24
17.4%
0.5
1.37
17.9%
0.7
1.56
17.0%
0.7
1.72
18.3%
0.7
1.02
17.9%
0.2
0.54
19.5%
0.3
0.90
19.5%
0.0
0.00
13.5%
ρv
0.45%
0.64%
0.42%
0.30%
0.62%
0.29%
1.18%
0.65%
2.17%
0.94%
0.53%
0.47%
Meas.
0.49
0.77
0.65
0.82
0.92
0.90
0.94
0.99
0.91
0.62
0.67
0.27
Eq. 9
0.51
0.65
0.73
0.67
0.90
0.90
0.90
0.90
0.83
0.63
0.78
0.40
EIeff/EIg,tr
CEB CEB-M
0.41
0.40
0.54
0.52
0.56
0.54
0.54
0.52
0.63
0.57
0.63
0.57
0.69
0.61
0.69
0.60
0.66
0.58
0.42
0.42
0.48
0.47
0.37
0.36
AIJ
0.27
0.31
0.32
0.31
0.37
0.37
0.41
0.42
0.40
0.22
0.24
0.15
Notes: (1) P = axial compression, (2) Pb = balanced compression load, (3) ωs = AsFy /A gf’c , (4) ρ v = A v/sb as
given in Eq. 3.11c, (5) CEB per Eqs. 3.1 and 3.2, and (6) CEB-M per Eqs. 3.1 and 3.4.
3.6.3 Cyclic Behavior
For the sake of completeness, cyclic loading test and analysis data are compared for specimens
K-1 and WP-9 in Figs. 3.10 and 3.11. The analytical results were developed using
DYNAMIX. As mentioned previously herein as well as explained in Chapter 2, the analytical
models used for these analyses are similar to that shown in Figs. 3.3b and 3.4 where the initial
elastic region is modeled per Eq. 3.9. The main point of including these examples here is to
demonstrate that the pseudo-elastic model, given by Eq. 3.9, has applications to the cyclic
nonlinear analysis of structures as well as linear static analysis. This proposed flexural stiffness
model is then used in modeling RC beam-columns of the composite RCS frames investigated
within this research. Detailed results for both static and dynamic nonlinear inelastic analyses will
be presented in Chapters 5 to 7.
87
88
1.0
EIeff/EIg,tr
0.8
0.6
0.4
Prop. Model
CEB
CEB-M
AIJ-Stand.
Experiment
0.2
0.0
0.0
0.4
0.8
1.2
1.6
2.0
P/Pb
2.5
2.5
P=0.0P b
P=0.5Pb
M / Mno
2.0
1.5
1.0
0.5
1.5
Fiber Anal.
Prop. Model
CEB
CEB-M
AIJ Stand.
1.0
0.5
0.0
0
2e-4
4e-4
6e-4
8e-4
0.0
1e-3
0
2e-4
Curvature (rad/inch)
4e-4
6e-4
8e-4
1e-3
Curvature (rad/inch)
2.5
2.0
M / Mno
M / Mno
2.0
1.5
1.0
0.5
P=1.2Pb
0.0
0
2e-4
4e-4
6e-4
8e-4
1e-3
Curvature (rad/inch)
Figure 3.8 Test specimen WP-9 by Watson and Park (a) variation in EIeff with axial load
(b) moment-curvature response.
89
1.0
Prop. Model
CEB
CEB-M
AIJ-Stand.
Experiment
EIeff/EIg,tr
0.8
0.6
0.4
0.2
0.0
0.0
0.4
0.8
1.2
1.6
2.0
P/Pb
2.5
2.5
P=0.0P b
P=0.5Pb
M / M no
2.0
1.5
1.0
0.5
1.5
Fiber Anal.
Prop. Model
CEB
CEB-M
AIJ Stand.
1.0
0.5
0.0
0
2e-4
4e-4
6e-4
8e-4
0.0
1e-3
0
2e-4
Curvature (rad/inch)
4e-4
6e-4
8e-4
Curvature (rad/inch)
2.5
P=1.2P b
2.0
M / Mno
M / Mno
2.0
1.5
1.0
0.5
0.0
0
2e-4
4e-4
6e-4
8e-4
1e-3
Curvature (rad/inch)
Figure 3.9 Test specimen by Kuramoto (a) variation in EIeff with axial load
(b) moment-curvature reponse.
90
1e-3
(a) experiment
675
Lateral Load (kN)
450
225
0
-225
-450
(b) analysis
-675
-60
-40
-20
0
20
40
60
Displacement (mm)
Figure 3.10 Comparison of cyclic load behavior for WP-9 specimen (a) experimental,
(b) DYNAMIX analysis.
91
(a) experiment
35
30
25
20
15
Shear (tonf)
10
5
0
-5
-10
-15
-20
-25
-30
(b) analysis
-35
-40
-32
-24
-16
-8
0
8
16
24
32
40
Displacement (mm)
Figure 3.11 Comparison of cyclic load behavior for Kuramoto specimen (a) experimental,
(b) DYNAMIX analysis.
92
3.7 Effective Shear Stiffness (GAeff)
Over the course of preparing the verification studies, it has been found that shear deformations
in some of the specimens were not negligible and should be separately accounted for in backcalculating the flexural stiffness coefficients. Moreover, shear deformations can be significant in
seismically designed frame structures with large columns with short span to depth ratios. These
observations are substantiated by Vecchio and Emara (1992) who reported that the shearrelated deformations contributed up to 20% of the total drift in frames that are otherwise
governed by flexural effects.
Effective Shear
Stiffness
GAuncr
GAeff
GAcr
Vc
V
5Vc Applied Shear
Figure 3.12 Proposed shear stiffness model.
Given the lack of test data on shear deformations and the fact that, even when significant they
are much smaller than flexural deformations, the proposed shear stiffness coefficients are fairly
simple. Assuming that the upper and lower bound stiffness are roughly equal to the uncracked
and cracked shear stiffness, the proposed shear coefficient, GAeff, is given by the following
equations and shown in Fig. 3.12:
93
GAeff = GAuncr
for V < Vc
(3.10a)
GAeff = GAuncr – (GAuncr – GAcr) (V – Vc) / 4 Vc
for Vc < V < 5Vc
(3.10b)
GAeff = GAcr
for V > 5Vc
(3.10c)
For a rectangular cross section and per Park and Paulay 1975,
GAuncr = 0.4Ec bh / 1.2
(3.11a)
GAcr = ρv Es bh / (1 + 4nρv)
(3.11b)
ρv = Av / s b
(3.11c)
where Av is the area of shear reinforcement; s is the spacing of the shear reinforcement; and the
other terms are same as defined previously. Per ACI-318 (95), the shear cracking strength of
the concrete, Vc, is calculated as:
Vc = 2(1 + P / 2000 Ag) √ f c' b d
(3.12)
where P/Ag should be expressed in psi.
The model was developed with the following principles in mind: (1) the effective shear stiffness
transitions from the uncracked to the fully cracked condition as a function of the ratio of the
applied shear force V to the concrete shear strength Vc, (2) the total applied shear V is less than
the shear strength of the transversally reinforced member which would usually not exceed 5Vc,
and (3) the stiffening effect of axial load is implicitly included through the calculation of Vc per
Eq. 3.12. For analysis purposes when one is computing response at the yield load, V can be
determined based on the flexural yield strengths of the sections. Using this model, the
contribution of shear to the total deformation in the verification examples was on the order of 5
to 20% for all beam-column specimens whereas it was about 45% for the beam specimen K-1.
94
3.8 Summary and Concluding Remarks
In this chapter, flexural and shear stiffness coefficients geared for representing conditions at the
onset of significant yielding have been proposed. The flexural stiffness model has been verified
by test results from several beam-column specimens for a wide range of axial load ratios.
Additionally, stiffness modeling recommendations from several existing codes and standards are
reviewed and compared to the proposed flexural stiffness model. A key advantage of the
proposed models is that they provide simple yet accurate formulae to account for the stiffening
effect of axial compression in beam-columns. The proposed models are used for modeling RC
beam-columns of RCS composite frames that are investigated in Chapters 5 to 7.
It has also been shown that some existing approaches, such as in ACI-318 (1995) and FEMA
273 (1997) are a bit over simplistic in that they do not distinguish between various levels of axial
load, reinforcing bar ratios, and other variables that affect member stiffnesses. On the other
hand, comparisons with test data show that some more theoretical approaches (e.g., Filippou
and Fardis 1996 and AIJ Standards 1991), do not yield substantially more accurate results. In
particular, since the overall member response reflects the integration of cracked, partially
cracked, and uncracked sections, solutions based on cracked section analyses tend to
underestimate the member stiffness. Compared to the proposed coefficients, the ACI-318
(1995) also underestimates the stiffness at initial yield, but this is conservative for second-order
static analyses to assess slenderness effects.
The topic addressed in this chapter is just one of many issues affecting the analysis and design of
reinforced concrete or composite steel-concrete structures. The models proposed herein are
not fundamentally different or new, and nor do they yield dramatically different solutions from
other models. Nonetheless, the proposed models are substantiated by test data and provide a
modest but important step towards improving the accuracy of practical analysis methods for
95
design. Given the availability of modern computing technologies, refinements of this sort are
appropriate modifications to incorporate along with other modifications to faithfully capture
important aspects of structural behavior in design.
96
Chapter 4
Seismic Damage Indices
This chapter will focus on the review of different types of damage indices proposed
throughout the literature with a special emphasis on trying to categorize the indices
according to different useful aspects governing the procedure of their evaluation (i.e.,
considering peak versus cumulative response type, local versus global, ductility versus
energy measures, etc.). Two new damage indices; a ductility-based and energy-based
index, are proposed and calibrated against available experimental data including
reinforced concrete columns, steel and composite beams, and composite reinforced
concrete-steel joint panel sub-assemblages. Although the number of data points is
limited, the new damage indices give promising results in predicting the evolution of
damage up to the state of failure at the local level, i.e., at the level of individual members
or sub-assemblages. A new technique for global damage (i.e., at the system level rather
than at the member level) determination is introduced in Chapter 6. This technique
integrates the effect of the local damage introduced in this chapter to reflect the overall
damaged condition of the structure.
97
4.1 Introduction
The earthquake response of frame-type structures (Reinforced Concrete, Steel, or
Composite) is a complex problem that has been researched for many years. Methods are
required to describe the amount of potential damage of such structures subjected to
earthquake loading. These methods are useful to check designs and to assess the behavior
of such frames if they are subjected to strong ground motions. An economical design
must allow some damage, but irrepairable damage should be avoided and collapse must
be prevented. In general, for large earthquakes that have long return periods, a certain
level of damage may be allowed, but this damage should be limited to the repairable
range. On the other hand, damage should be kept to a minimal level for frequently
occurring smaller earthquakes.
Most of modern seismic codes, especially those adopting performance based design
concepts, specify two fundamental performance criteria for earthquake-resistant
structures:
•
No collapse and no excessive damage under the design earthquake
•
Limitation of damage under an earthquake with higher probability of occurrence
than the design one.
The main goal of a performance-based design is to produce structures that have
predictable seismic performance within prescribed bounds under a specified level of
ground motion. For instance, FEMA 273 classifies structural performance levels in three
categories: 1) Immediate Occupancy, 2) Life Safety, and 3) Near Collapse. These
performance levels (or objectives) are to be checked against earthquake hazards
expressed in terms of their frequency of occurrence. Accordingly, for basic structures,
immediate occupancy should be guaranteed under occasional events with a 50%
probability of exceedance in 50 years, life safety under rare events with a 10%
probability of exceedance in 50 years (also known as Basic Safety Earthquake 1, BSE-1),
98
and finally, collapse prevention under very rare events with a 2% probability of
exceedance in 50 years, known as BSE-2. Many response parameters including ductility
demands, damage indices, and story drifts among others, can be used to measure
performance in the seismic design/evaluation process.
It is generally recognized that specific criteria for the implementation of the above
principles vary from code to code, and inconsistencies are not uncommon. However, the
terms collapse (or failure) and damage are more or less common in all codes and some
correlation with the ultimate and serviceability limit states is usually the objective. To
quantify seismic performance criteria, it is necessary to express damage in a quantitative
form, with failure corresponding to the maximum degree of damage a structure can
sustain. This is achieved through the use of damage indices (or indicators).
4.2 When Do We Need Damage Indices?
As summarized by Kappos (1997), typical situations where practical use of damage
indicators can be made include the following:
•
Seismic design checks of structures, in order to come up with an economical design
allowing some damage under large, less frequent earthquakes but still within the
repairable range as mentioned before in this chapter.
•
Post-earthquake damage assessment, in particular, its second and more detailed
stage, during which the required measures for repair and/or strengthening have to be
defined.
•
Reliability studies of existing structures and earthquake damage scenarios, on which
a decision can be made as to whether a structure should be strengthened or not (preearthquake strengthening).
99
•
Seismic performance predictions for novel types of structures, especially those of
great importance, may serve as a valuable aid in the seismic design of these
structures.
The need for empirical indices to quantify damage and to predict failure is less
accentuated if the available analysis programs are sophisticated enough to capture real
failure of the structures. By the word failure, one can point to either a global failure of the
whole structure (e.g., frame) through a mechanism including a soft story or several
stories, or a local failure involving any component(s) of the structure sub-assemblages.
Local failure may include crushing of concrete in reinforced concrete components, first
of the cover (i.e., spalling) and later of the confined core. Other local failure modes may
include buckling and possibly fracture of longitudinal bars, fracture of transverse
reinforcement, loss of anchorage (i.e., bond failure), etc… For structural steel
components, failure can involve severe local buckling of the cross-section components
(web and/or flanges), lateral buckling of the whole cross-section, fracture of the main
cross-section material or of the weld in welded connections, slip and total separation of
the steel cross-section and the reinforced concrete slab in composite beams, etc.
4.3 Definition of Damage Function and Damage Index
Damage functions are mathematical models (or formulae) involving some representative
variables, or state variables, quantifying the state of structural damage of a structure or of
its components. These state variables are generally related to irrecoverable (i.e., inelastic)
deformations such as strains, curvatures, rotations, or even displacements to depict either
local or global types of damage. These damage variables sometimes include notion about
forces (e.g. base shears, member resistances, etc…), or energy dissipated during inelastic
reversed cyclic loading. The values that these damage functions take at different stages of
loading are considered as damage indices and are used as a scale quantifying the level of
damage of the structure under consideration.
100
As pointed out in the CEB state-of-the-art report on RC frames under earthquake loading
(1996), the state variables can be defined as the variables which have the ability to
describe the evolution of the real state of degradation of a structure during an imposed
loading history. A damage model operates on specific state variables and permits
measures or indicators to be obtained which effectively indicate, during the complete
loading process, the proximity of some limit state in the structure, such as failure.
In order to describe suitably the evolution of the damage state of a structure, the damage
functions referred to should satisfy the following conditions (Capechi and Vestroni, 1986,
Carvalho, 1991):
•
be a monotonic function and not decrease with time
•
exhibit a significance invariance along time, so that two identical loading histories
may lead to equivalent damage increments
•
be non-dimensional and present values basically varying between two limits, 0
and 1 (or 100%), representing initial undamaged conditions and the final limit or
failure state.
Two types of damage indicators can be distinguished, namely the damage parameters and
the damage indices. In a damage model the former play the role of the state variables, and
the latter have essentially the character of damage function in the above-mentioned
context. A damage parameter may be defined as a physical property of the structural
response, the value of which is indicative of the state of the structure. Examples of this
type of variable are the interstory displacements, the deformation at member and section
levels, the ductility demand, the stiffness, the dissipated energy, etc. Alternatively, a
damage index is a variable that is capable itself of quantifying the amount of damage,
thus constituting a direct measure of structural damage. This measure may be considered
at the level of a cross-section, a member or a substructure, or at the global structure level.
101
Generally speaking, the numerous structural damage indices proposed in the literature are
typically based on one of the following approaches, either:
•
supply-demand approach, where the demand imposed by the earthquake with
respect to a certain structural quantity (e.g. deformation or energy) is related to
the corresponding capacity of the structural component or the structure as a
whole, or
•
state evolution approach, where the degradation of a certain seismic variable
(strength, stiffness, energy dissipation, fundamental period) is compared with a
pre-determined critical value, usually expressed as a percentage of the initial
value corresponding to the undamaged state.
4.4 Classification Schemes of Damage Indices and Categorization of Damage
Detailed discussions of the damage indices proposed in the literature and different
approaches used in their categorization can be found in several state-of-the-art papers and
reports (Williams and Sexsmith, 1995, Chung et al., 1987, CEB report on RC frames
under earthquake loading, 1996, etc...). A brief overview of the different types of damage
indices is given below to classify different indices and point out some of their features.
The most general classification of the damage indices is whether they are local or global
indices. This categorization can include all the others. Other ways of classifying damage
indices as suggested in the literature, as mentioned by Kappos (1997), are whether they
are deterministic or probabilistic indices (Banon and Veneziano, 1982, Ciampoli et al.,
1989, DiPasquale and Cakmak, 1989, and others), structural or economic indices (Dolce
et al., 1994, Kappos et al., 1996, Gunturi and Shah, 1992, Park and Ang, 1985, and
others), capacity-demand or state evolution indices (as discussed in Section 4.3),
structural or non-structural indices (e.g., Gunturi and Shah, 1992, where they derived loss
curves for non-structural elements and the building contents as functions of the interstory
102
drift and the story acceleration). Other sub-classifications may include deformation
based, stiffness based or energy based indices or even a combination of two or all of
them, also non-cumulative (i.e., peak response values) versus cumulative indices, lowcycle versus high-cycle fatigue indices, etc. One might also classify global indices as
weighted average local indices or modal parameters indices, as discussed later in this
section.
4.4.1 Local Versus Global Indices
Local damage indices refer to the damage state of a single member of a structure, or a
specific cross-section of that member, or even of a sub-assemblage of the structure (e.g.,
story in a building). On the other hand, global indices deal with the whole entity of the
structure. Depending on one’s perspective, the damage index calculated at a story level is
considered as a global damage indicator when compared to the member level of the
different structural members constituting this story. As pointed out by Kappos (1997), it
is easy to understand that the determination of the damage index becomes less accurate as
one shifts from a critical region (cross-section, or member) to the structure in its entity.
Furthermore, some damage indices (e.g. displacement ductilities) can be used both as
local and global indices, while others (e.g. interstory drifts or curvature ductilities) can be
used as either global or local.
(a) Local Indices may involve a single damage parameter (i.e., variable), such as
maximum deformation (curvature or rotation) or dissipated energy, or two or
more variables. Different types of local damage indices are given in Table 4.1.
For example, Banon et al. (1981) used a normalized cumulative rotation which
has some similarity to the common monotonic rotational ductility but the values
of the index at failure showed considerable scatter. Also, Banon and Veneziano
(1982) have used the flexural damage ratio (flexural stiffness divided by the
reduced secant stiffness at maximum displacement) as well as a normalized
dissipated energy ratio. Roufaiel and Meyer (1987) suggested a modified form of
the flexural damage ratio mentioned above and their index showed a good
103
correlation with the residual strength and stiffness of test specimens in flexure
with some significant shear and axial loads.
Park and Ang (1985) have used ductility and dissipated energy and their index has
been the most widely used among researchers. The first term in their index is a
simple, pseudo-static (peak) displacement measure. It takes no account of
cumulative damage, which is accounted for solely by the energy term (second
term). The advantages of this model are its simplicity, and the fact that it has been
calibrated against a significant amount of observed seismic damage of reinforced
concrete structures (but to a much less extent for steel structures). Among the
drawbacks of this index are 1) its weak cumulative component for practical cases
given the typical dominance of the peak displacement term over the accumulated
energy term, 2) its format using a simple linear combination of deformation and
energy in spite of the obvious non-linearity of the problem and the interdependence of the two quantities, and finally 3) its lack of considering the loading
sequence effect in the cumulative energy term.
Daali and Korol (1996) suggested two damage indices as a linear combination of
maximum response and either repeated effects in the form of low-cycle fatigue or
dissipated energy; the latter being a modification of Park and Ang’s damage
assessment model. Chung et al. (1989) have used the number of load cycles
together with a damage-based hysteresis model in a low-cycle fatigue type of
formulation. Stephens and Yao (1987) developed a cumulative damage index
based on the displacement ductility making use of positive and negative
displacement increments separately. McCabe and Hall (1989) developed a
damage index representing a second degree hysteretic energy ratio of the actual
hysteretic energy dissipated to the hysteretic energy corresponding to complete
damage. An extra term was introduced in the damage index definition to provide
for the additional damage arising from nonsymmetrical response which in turn
can lead to a residual offset and further damage. Bracci et al. (1989) have
suggested a damage index equal to the ratio of ‘damage consumption’ (loss in
104
damage capacity) to ‘damage potential’ (capacity), defined as appropriate areas
under the monotonic and the low-cycle fatigue envelopes.
As mentioned by Williams and Sexsmith (1995), a major problem of nearly all of
the formulations mentioned above is the need for weighting factors or exponents
which must either be derived by regressions performed on experimental data, or
assigned arbitrarily which leads to less confidence of their broad applicability to
different types of structures. Another issue that has been pointed out by the same
authors is that the combined models such as that by Park and Ang (1985) use a
simple linear combination of deformation and energy terms in spite of the obvious
non-linearity of the problem and the inter-dependence of the two terms. McCabe
and Hall (1989) discussed this issue and tried to overcome this drawback in their
damage model by using a second order hysteretic energy ratio as mentioned
above. A further problem with the practical applications of many of the models is
that, while the coefficients have been chosen as to give a value of 1.0 at failure,
no attempt has been made to calibrate lower values of the damage index against
observations of limited damage. In this respect, the combined model of Bracci et
al. (1989) and the work by Kanno (1993) appear to show a good correlation with
observed evolution of damage.
(b) Global Indices can be defined in terms of a global parameter, for instance global
ductility factors (based on story displacements), such as the one based on roof
displacement (Roufaiel and Meyer, 1987), or softening indices relating the
fundamental period of the structure to the final one (DiPasquale and Cakmak,
1989, and Rodriguez-Gomez and Cakmak, 1990); the latter approach can be used
with two modes (Mork, 1992) or more, in order to detect concentration of damage
in the top or the bottom of the structure. The approaches outlined above are likely
to provide reasonable estimates of the overall level of damage to a structure when
that damage is quite severe and evenly distributed. However, when localized or
relatively minor damage occurs, it is likely that it will have a significant effect
only on the higher modes of vibration, and that the uneven distribution of damage
105
will result in changes in the mode shapes. Under these circumstances,
examination of the higher mode shapes, or of flexibility coefficients, can be used
as a method of identifying both the magnitude and the location of the structural
damage (Raghavendrachar and Aktan, 1992).
Global indices can also be defined as weighted averages of individual member
indices (taken at each story of a building or for the entire structure). The
weighting factors may involve the energy dissipated by a member (Park et al.,
1985, Chung et al., 1987, and others), or the tributary gravity load of a member
(Bracci et al., 1989); both approaches generally tend to give more weight to the
members of the lower stories, which is conceptually correct, but they fail to
recognize that failure of a (soft) story typically means failure of the structure.
Another limitation of the use of weighted averages is that the resulting global
index can only be as reliable as the local values from which it is derived.
Furthermore, It has recently been shown by Ghobarah et al. (1999) that averaging
procedures of local indices used in the literature to calculate a global damage
estimator may give incorrect, and sometimes physically impossible, results in
some cases. Table 4.2 gives a brief summary of the different available approaches
of global damage indices.
106
Table 4.1: Summary of Selected Local Damage Indices.
Deformation Based
Non-
Cumulative
Stiffness
Based
Energy
Based
(Deformation
Repeated
Max. Response +
+Energy)
Cumul. Effect
Cumul. Effect
*Castiglioni and
Calado (1996)
*Daali and Korol
(1996)
Cumulative
-Maximum value
Fajfar (1992)
µθ =
µφ =
µ
d
=
θp
θu − θ y
φp
φu − φ y
δp
δu − δ y
-Range (peak to peak)
+
−
θ p,max + θ p, max
θu − θ y
*Banon et al. (1981)
NCR =
*Banon et al. (1981)
∑ θp
θy
*McCabe & Hall
(1989)
Equiv. hyst.cycles
N = Ht / Ry ∆U
∆U = Uwt - Uy
U wt =
Low-Cycle Fatigue
Combined
∑ H i ∆U i
Ht
*Stephens & Yao
(1987)
∆di = (∆δpt/∆δpf)α
α = 1 – (b*rl)
n
D = ∑ ∆d i
i=1
FDR =
ko
km
*Roufaiel & Meyer
(1987)
FDR =
kf
km
(k m − k o )
(k
f
− ko )
*Gosain et al. (1977)
Fi δ i
De = ∑
i Fy δ y
*Banon & Veneziano
(1982)
t
∫ M (τ )θ(dτ)
0
En(t) =
0. 5M y θ y
*Park and Ang
(1985)
D=
µ
µu
+β
EH
Fy D y µ u
*Bracci et al (1989)
D=
Ds + Dd
Dp
D+ = D+ + D- - D+ D*Fajfar (1992)
DM=
EH
F y D y (µ u − 1)
(
)
DM = ∆M / M y
 H p + Hn 
D=

 Ht

Dφ =
φm − φ y
φ f − φy
H p − Hn 
+

 Ht

2
µ max − 1
µm −1
1. 15
 µi − 1 

+ β1 ∑ 
 µm − 1
For C = c = 1, a = 0
⇒ NCR (Banon et
al., 1981)
= DM+Dφ -DM Dφ
∆M = c ∫ dE / φ y
-D =
L a
c
C ∑ i ∆ ξ pi
i=1
*McCabe and Hall
(1989)
*Kratzig et al. (1989)
+
+
∑ E p,i + ∑ E i
+
D =
+
+
E f + ∑ Ei
−
D (similar)
D=
-D = µ max / µ m
2
+ β2
(
)
∑ µi −1
µm
*Chung et al (1989)
D=
 α +i ni+ α i− ni− 
∑
+ +
− 
i
Ni 
 Ni
Where,
Ni = (M i – M f,i)/∆M i
Table 4.2: Selected Global Damage Indices.
Weighted Average Indices
Deformation Based
*Roufaiel and Meyer
(1987)
GDP =
d R − dY
d F − dY
(based on drift ratio)
d F = 0.06 H (building
height)
Energy Dissipated
*Park, Ang and Wen (1985)
*Chung et al. (1987)
*Kunnath et al. (1992)
*IDARC 2D (Version 4.0)
∑ Di Ei
Dstory = i
∑ Ei
i
(i refers to members)
Similarly,
*FEMA 273
Residual “permanent”
displacement
Dstructure =
story story
Ei
∑ Di
i
story
∑ Ei
i
Modal Parameters Based
Gravity Loads
*Bracci et al. (1989)
(b+1)
∑ Wi Di
i
Dstory =
b
∑ Wi Di
i
Softening Indices
Mode Shapes
*Roufaiel and Meyer (1987)

14 .2δ y 


D global =
f

und − 1

f dam

δ f − δy
*DiPasquale and Cakmak (1989)
Special Case: b=1, Wi =1
D
= ∑ D2 / ∑ D
story
i
i
•
•
•
Tund
Tm
2
Tdam
Plas. Soft.: Dpl = 1 –
2
Tm
2
Tund
Final Soft.: DF = 1 –
2
Tdam
Max. Soft.: Dm = 1 –
*Mork (1992)
extended Dm to include 2nd mode eff.
D1 = 1 −
k1, m
k1,und
, D2 = 1 −
k 2, m
k 2,und
*Nielsen (1992) related D1 , D2 and the
overall damage index Dm.
*Raghavendrachar and
Aktan (1992)
Examination of the
higher mode shapes, or
of flexibility coeff., can
be used as a method of
identifying both the
magnitude and location
of the structural
damage.
4.4.2 Categorization of Damage
There is very little published information on the methods to classify seismic damage and
to relate damage index values to the actual state of damage for the full range of damage
evolution from the virgin state up to the ultimate state of failure.
Many attempts to correlate damage indices with observed damage use a simple
classification based on visual signs of damage. For example, Reinhorn, Kunnath, and
Mander (1992) use the following for reinforced concrete structures:
•
None to slight: undeformed / uncracked (or localized minor cracking at worst),
corresponds to damage index value, D, in the range of 0.0 to 0.30.
•
Minor: moderate cracking, and steel tie yielding, D in the range of 0.30 to 0.50.
•
Moderate: severe cracking, localized spalling of concrete, and main rebar yielding,
corresponding to a D value of 0.50 to 0.60.
•
Severe: exposure and buckling of reinforcing bars, and crushing of concrete core with
a D value between 0.60 and 1.0.
•
Collapse: with a D value equal or greater to 1.0.
Kanno (1993) defined a parallel guidelines for steel structures describing the slight
damage by onset of steel yielding, minor damage by initiation of local buckling,
moderate damage by larger local buckling, severe damage by lateral torsional buckling,
and finally collapse by fracture of structural steel or weld and loss of capacity.
Another approach to damage classification is to relate it to the repairability of the
building. Bracci et al. (1989) and Stone and Taylor (1993) use the categorization:
undamaged or minor damage, serviceable, repairable, irrepairable, and collapsed. As
pointed out by Williams and Sexsmith (1995), while this scale may be harder to apply in
practice, it is perhaps more helpful as a tool for retrofit decision-making, or for outline
planning and costing of post-earthquake reconstruction. Again, Kanno (1993) correlated
this categorization to the former classification based on visual signs of damage: he
108
considered undamaged or serviceable state as corresponding to slight degree of damage,
repairable state corresponding to minor to moderate damage, irrepairable state
corresponding to moderate to severe damage, while collapse state is the same which
means complete failure and loss of capacity of the structure or of its components. EERI
(1994) adopts a scale which includes consideration of non-structural damage, the likely
duration of loss of function and risk of casualties to building occupants.
An alternative approach to the assessment of damage is a consideration of the
survivability of the structure under a second earthquake or aftershock (Rodriguez-Gomez
and Cakmak, 1990). This is likely to be the first concern for moderately or severely
damaged structures in the immediate aftermath of an earthquake, when the risk of
aftershocks (usually considerably smaller than the main shock) is high. This measure, as
pointed out by Williams and Sexsmith (1995), has the best potential of correlating with
fatalities and loss of use.
4.5 Proposed Damage Indices
It has been recognized through experience and analyses that seismically induced forces
cannot themselves cause the total collapse of a structure if the structure has adequate
deformation capacity. Therefore, both the strength and deformation characteristics need
to be considered to properly evaluate structural resistance against earthquake forces.
Moreover, seismic damage of any structure (or of its components) is related to a large
extent to irrecoverable deformation. Accordingly, many researchers have proposed
parameters through definitions of different damage functions (i.e., damage indices) to
evaluate seismic resistance that consider strength and deformation characteristics in
different ways. One example of these indices is energy dissipation capacity which is the
product of force and deformation. However, there is currently no consensus as to a single
parameter to evaluate the seismic resistance (or the state of damage) for structures.
109
Through this research, two new local damage indices are proposed. The first one draws
on a damage index suggested by Kratzig et al. (1989) based on dissipated energy. The
second is a ductility index that is based on the notion that peak and cumulative (inelastic)
deformations really constitute the main cause of damage and failure for many types of
structures. Definition of these two proposed damage indices along with their advantages
are given below. Additionally, criteria for defining failure for different types of structural
components as needed by these indices are described, and the two indices are applied to
experimental data to assess their ability to predict failure as well as its evolution at the
structural component (i.e., local) level. This experimental data only focused on
components and sub-assemblages useful to the main theme of this thesis; which includes
tests of reinforced concrete columns, steel and composite beams, and composite joints
sub-assemblages.
4.5.1 Energy-Based Damage Index
As mentioned above, the proposed energy damage index is largely based on one
suggested previously by Kratzig et al. (1989). As pointed out by Kratzig, the main
concern lies in deeper insight into cyclic damage accumulation processes of structural
members. Accordingly, the following two questions furnish useful insights for the
definition of meaningful and realistic damage indices:
1. Which physical entity mirrors the effects of both loading history and damage
accumulation process?
2. How can cyclic damage effects be normalized with respect to the ultimate failure
mode?
Regarding the first question, one would recall the dissipated energy, which is positive for
all inelastic processes and is being accumulated during cyclic loading parallel to the
damage evolution. As reported by Kratzig et al. (1989), experimental evidence by Muller
(1983) supports the idea that failure modes under cyclic loading correspond to those for
monotonically increasing loads. Therefore, failure modes for monotonic loads can be
110
employed for the determination of values of dissipated energy for normalizing purposes
which are independent of the loading history. This can serve as an explanation for the
second question.
The proposed damage index may act on sectional and member level, and is given as
follows:
N + 
 ∑ E PHC, i 


 i=1

+
+
DE =
α
−
DE =
+
(E )
α
N − 
 ∑ E PHC, i 


 i=1

(E )
− α
f
+
n + 

+  ∑ E FHC
,i 
 i=1

+ α
f
−
n + 

+  ∑ E FHC
,i 

 i=1

n − 

+  ∑ E FHC
,i 
i
=
1


−
( ) + (D )
DE = γ DE+
γ
β
n − 

+  ∑ E FHC
,i 

i
=
1


−
− γ
E
β
(for positive deformations)
(4.1)
(for negative deformations)
(4.2)
( ≥ 1.0 means failure)
(4.3)
β
β
where, N+ and n+ are number of positive Primary Half Cycles (PHC) and Follower
Half Cycles (FHC), respectively
+
E PHC
,i
plastic (i.e., dissipated) energy corresponding to positive PHC
number i
+
E FHC
,i
plastic (i.e., dissipated) energy corresponding to positive FHC
number i
E +f
normalizing energy for positive deformations (or in other
words, energy absorbed up to failure for monotonic loading)
α, β and γ calibration parameters
111
Similar
definitions
apply
to
Equation
4.2
for
values
associated
with
negative
deformations.
As it is clear from the equations and definitions of different terms shown above, the
damage index is based on primary and secondary (or follower) half cycles of the loading
process, an essential distinction introduced by Otes (1985), employing experimental
insight into cyclic behavior of reinforced concrete members. As shown in Figure 4.1,
PHC is the name for any half cycle with maximum amplitude, followed by a certain
number of follower half cycles of smaller amplitude. Whenever a certain deformation
maximum, φ i, corresponding to the primary half cycle PHCi is exceeded, a new primary
half cycle PHCi+1 is established, otherwise, one is dealing with consecutive FHCs.
E PHC ,i+1 term used in either Equation 4.1 or 4.2 is calculated as the energy dissipated
between the previous deformation maximum, φ i, and the current deformation maximum,
φ i+1 . Thus, basically, for the step corresponding to E PHC ,i+1 (i.e., φ i+1 ), the contribution of
this quantity is split into two parts: E PHC,i which is added to Equation 4.1 or 4.2 as a
FHC, and ( E PHC ,i+1 - E PHC,i ) which is added as the PHC contribution associated with
primary half cycle number i+1. Every PHC corresponds to a certain damage degree.
θp+
Inelastic Deformation
PHC4
PHC3
PHC2
PHC2
PHC1
FHC
FHC
PHC1
FHC
FHC
FHC
Cycles
PHC1
PHC2
FHC
FHC
PHC1
PHC3
θp−
PHC4
CASE (A)
FHC
FHC
FHC
PHC2
CASE (B)
PHC1
CASE (C)
Figure 4.1 Definition of PHCs and FHCs and load sequence effects.
112
Mathematically, the combined damage index, DE, is expressed through the variables DE+
and DE− per Equation 4.3. One might think of DE as a point in the damage plane with
coordinates ( DE− , DE+ ). The damage plane is a two-dimensional plane say with the
horizontal axis, DE− , corresponding to negative deformations, and the vertical axis, DE+ ,
corresponding to positive deformations. This point moves in the damage plane describing
the evolution of damage of the structural member and showing failure once it reaches a
certain pre-defined full damage surface given by
(D ) + (D )
− γ
E
+ γ
E
= 1.0
(4.4)
Damage Index for positive deformations, D+
1.0
0.8
0.6
0.4
0.2
- 2
+ 2
(D ) + (D ) = 1
(D- )3 + (D + )3 = 1
(D- )4 + (D + )4 = 1
(D- )5 + (D + )5 = 1
(D- )6 + (D + )6 = 1
0.0
0.0
0.2
0.4
0.6
0.8
1.0
Damage Index for negative deformations, D
Figure 4.2 Different failure surfaces for different values of γ.
Figure 4.2 shows different damage (failure) surfaces resulting from different values of the
exponent γ. This variable is essentially determined through calibration with experimental
data of tests under cyclic loading scheme with amplitudes varying between positive and
113
negative deformations. In Equation 4.4, a large value of γ implies a damage surface with
weak interaction between damage in the positive and negative deformation directions,
whereas a small value implies high interaction.
4.5.1.1 Some details and advantages of the energy-based damage model
Properties of this damage model can be summarized as follows:
(a) The energy dissipated during monotonic loading to failure (corresponding to a simple
primary half cycle) has to be furnished by a certain number of primary half cycles in
the case of cyclic loading. This energy content is mainly due to cracking and straining
of steel.
(b) The inclusion of FHC terms in both the denominator and the numerator of Equations
4.1 and 4.2 means that they contribute considerably less to the damage index than the
primary terms. This enables the index to account for cumulative-type damage, since a
high value of the index can be generated either by a single high amplitude cycle or by
repeated cycling at lower amplitude. An infinite number of FHCs makes numerator
and denominator equal and consequently leads to failure condition. To summarize,
PHCs have strong influence on the damage evolution, while FHCs contribute
relatively less to damage, but certainly not negligibly on the long term.
(c) The way of treating positive and negative deformations separately is useful to handle
elements with different values of dissipated energy up to failure, Ef, for positive and
negative loading. Example of where this is useful would include unsymmetrical
sections such as un-symmetrically reinforced concrete sections, composite steel
beams, etc.
(d) For monotonic loading up to failure (i.e., only one simple half cycle to failure), either
Equation 4.1 or Equation 4.2 gives a value of 1.0. While for elastic loading
114
(undamaged case), both equations should give a value of zero. This satisfies the
requirements for a damage index as discussed before in Section 4.3.
(e) Using the idea of PHCs and FHCs, the present index reflects the “temporal”
sequence of loading cycles and its effect on the damage evolution. To appreciate the
importance of accurately accounting for load sequence effects, one might notice that
the proposed damage index is able to recognize the fact that the inelastic deformation
histories of Cases (A) through (C) in Figure 4.1 will cause same value of DE at the
end but with different damage evolution paths. For example, a large deformation
amplitude associated with the first PHC of Case (C) will cause most of the damage
after the first half cycle of loading followed by smaller effects from subsequent
FHCs. On the other hand, damage due to Case (A) is nearly equally furnished through
a set of PHCs throughout the loading history. Accurately tracing the evolution of
damage is crucial for performance assessment of structural components under seismic
type of loading.
This proposed index, as defined above through Equations 4.1 and 4.2, is close to the one
suggested by Kratzig et al. (1989). Among differences between the two indices are the
exponents assigned to the PHCs and the FHCs terms to describe different behavior of
different structural materials or components, and the way of combining the intermediate
damage functions DE+ and DE− to get the total damage index, DE. In Kratzig model, DE is
computed as follows:
DE = DE+ + DE− - DE− DE+
(4.5)
which essentially ignores any type of interaction between the damage due to positive and
negative deformations; this is clear since DE will never reach a value of 1.0, which
defines failure, unless either DE− or DE+ reaches 1.0 which means failure due to either
negative or positive deformations. The combination scheme used in this research (Eq.
4.3) is proposed and checked versus experimental data to remedy this drawback in the
Kratzig et al. model.
115
4.5.2 Ductility-Based Damage Index
A cumulative ductility-based damage index using the same idea of PHCs and FHCs as
the energy-based index described in the previous section is proposed. This index deals
with inelastic (i.e., irrecoverable) deformation (e.g. plastic rotation at member ends)
which constitutes a major cause of local damage of structural components.
This cumulative damage index is proposed and tested since its evaluation is much easier
than the energy-based one suggested in the previous section. Although, this ductilitybased index deals only with one aspect of the damage problem (deformation) and ignores
the second aspect (force or resistance), it might be useful and more practical in damage
assessment because of its straightforward application and less complicated calculation,
provided it gives good results when compared with experimental data.
The cumulative ductility-based damage index is given as follows:
(θ
+
θ
D =
((θ
(θ
−
θ
D =
Dθ =
)
−θ y )
)
+
f
α
current PHC
)
−θ y )
)
α
−
p
((θ
γ
current PHC
α
+
p
−
f
α
n
+  ∑θ p+

 i=1
+


FHC ,i 


+  ∑ θ +p
 i=1
β
n+
n
+  ∑ θ p−

 i=1
−
FHC, i


FHC ,i 

n
+  ∑ θ p−
 i =1
−




(for positive deformations)
(4.6)
(for negative deformations)
(4.7)
β


FHC, i 

(D ) + (D )
+ γ
θ
β
− γ
θ
β
(4.8)
116
where θ +p
current PHC
is the current maximum positive plastic rotation corresponding to
the latest PHC; once a new PHC is established, this term takes the
new value, otherwise it keeps its old value.
θ +p
(θ
max. positive plastic rotation corresponding to FHC number i
FHC, i
− θy )
+
f
plastic rotation capacity of the member up to failure under
monotonic loading in the positive deformation direction (method
of calculation will be discussed later)
α, β and γ
calibration parameters.
Similar definitions apply to Equation 4.7 for negative deformations. Note that values of
the variables corresponding to negative deformation (i.e., negative plastic rotation) are
taken as absolute values. Another important note is that the method of computing and
counting the effect of PHCs and FHCs is similar to what has been done for the energybased index. Furthermore, if two consecutive follower have cycles θ +p
FHC, i
and θ p+
FHC, i +1
(i.e., with the same sign) take place, the value used for the FHC’s term in Equation 4.6 is
the difference: (θ p+
FHC, i +1
- θ +p
FHC, i
).
4.5.2.1 Some details of the ductility-based damage index
This index has characteristics of both a ‘peak ductility’ damage measure (in the sense of
ATC 40 and FEMA 273) and a ‘cumulative ductility’ damage measure. Each time this
damage model is applied to a certain irrecoverable deformation history, a check is made
to see if for any PHC the ratio θ p /(θf-θy) ≥ 1.0 which would imply a peak ductility type
of failure. Otherwise, Equations 4.6 through 4.8 are computed, thus constituting a
cumulative type of damage, and failure is reached when Dθ in Equation 4.8 equals 1.0
based on a certain full damage surface (defined by the calibration parameter γ)
determined according to experimental data. PHCs and FHCs play the same role and offer
117
the same advantages as those cited in Section 4.5.1.1 for the cumulative energy-based
damage index.
Generally speaking, the proposed ductility index relates to simpler ductility acceptance
criteria used in ATC 40 and FEMA 273, but still captures many cumulative effects
related to energy measures and further includes loading sequence effects that can be
significant in the damage calculation/prediction process.
4.6 Identification of Deformation and Energy Values Corresponding to Failure
Failure (or ultimate) dissipated energy and inelastic deformations under monotonic
loading serve as normalizing terms in the damage functions (Equations 4.1, 4.2, 4.6 and
4.7), describing the energy- or ductility-based failure indices that need to be identified.
Generally, failure (or ultimate) conditions are not easy to define even under simple types
of loading and thus they are considered as the most challenging part of the proposed
damage indices. In this section, criteria and procedures for computing failure values are
discussed. Since this research mainly focuses on seismic behavior of composite RCS
frames, failure criteria are only presented for reinforced concrete columns, steel and
composite beams, and composite joint panels.
4.6.1 Reinforced Concrete Columns
The following are a number of failure criteria for reinforced concrete columns:
•
A more or less arbitrary strength drop (values ranging from 10% to 30% may be
reasonable), observed in the load-deflection or the moment-rotation curve. This
approach is fairly arbitrary and perhaps in some situations inappropriate.
118
•
Failure of confinement, corresponding to fracture of at least one hoop or spiral
which causes the onset of cyclic strength degradation leading to progressive
failure.
•
Attainment of an ultimate tensile strain, ε su , in longitudinal reinforcement which
is a measure of the likelihood of reinforcing bar rupture.
•
Onset of buckling of longitudinal reinforcement either between two consecutive
layers of transverse reinforcement or over a series of transverse reinforcement
bars. This is followed within a few cycles by fracture of longitudinal
reinforcement and rapid strength degradation.
•
Attainment of an ultimate (or failure) compressive strain, ε cu, of confined core
concrete causing crushing and loss of capacity.
One should make use of these criteria to define available capacities (failure points) as
expressed by the plastic rotation, (θ f − θ y ) , or plastic energy, Ef, used as normalizing
values in the proposed damage indices. For reinforced concrete columns, the most
promising variable that can be used to quantify limiting values describing failure (or
available capacity) is found to be the attainment of an ultimate compressive strain, ε cu, of
confined core concrete. This is more likely to happen in columns before reaching an
ultimate tensile strain of longitudinal reinforcement. Thus, a limiting value for ε cu is
adopted in this research to compute available capacity following Paulay and Priestly
(1992).
According to Paulay and Priestly, and as shown in Figure 4.3, the strain at peak stress,
ε cc, does not represent the maximum useful strain, as high compression stresses can be
maintained at strains several times larger. The useful limit occurs when transverse
confining steel fractures, which may be estimated by equating the strain energy capacity
of the transverse steel at fracture to the increase in energy absorbed by the concrete
119
shown shaded in Figure 4.3. A conservative estimate for ultimate (or failure)
compression strain is given by
ε cu = 0.004 +
1.4ρ s f yh ε sm
(4.9)
f cc'
where f yh is the yield strength of the transverse reinforcement, f cc' is the compression
strength of the confined concrete, ε sm is the steel strain at maximum tensile stress, and ρ s
is the volumetric ratio of confining steel. For rectangular sections ρ s = ρ x + ρ y . Typical
values for ε cu show a 4- to 16-fold increase over the traditionally assumed value for
unconfined concrete.
Compressive
stress, fc
Confined
concrete
First hoop
fracture
fcc'
Unconfined
concrete
fc'
Assumed for
cover concrete
Ec
εt
E sec
ε co 2 ε co ε sp
ε cc
ε cu
Compressive strain, ε c
ft'
Figure 4.3 Stress-strain model for monotonic loading of confined and unconfined
concrete in compression (Paulay and Priestley, 1992).
120
The compression strength of confined concrete, f cc' , is directly related to the effective
confining stress,
f l ' , that can be developed at yield of the transverse reinforcement,
which for rectangular sections is given by
f lx' = K e ρ x f yh
,
f ly' = K e ρ y f yh
(4.10)
where ρ x and ρ y are the effective section area ratios of transverse reinforcement to core
concrete cut by planes perpendicular to the x and y directions, and Ke is a confinement
effectiveness coefficient, relating the minimum area of the effectively confined core to
the nominal core area bounded by the centerline of the peripheral hoops. Ke is given by
Mander et al. (1988) as
n

2 
∑ wi 
'
'

s 
s 
1 −

1 − i =1 1 −


 6bc hc  2bc  2hc 

Ke = 
(1 − ρ cc )
(4.11)
where bc and hc are width and depth of confined concrete (centerline to centerline of
hoops) respectively, s’ is clear spacing between hoops, ρ cc is the ratio of area of
longitudinal steel to area of core section wi is the ith clear transverse spacing between
adjacent longitudinal bars, and n is the number of longitudinal bars. Typical values of Ke,
as given by Paulay and Priestly (1992), are 0.95 for circular sections, 0.75 for rectangular
column sections, and 0.6 for rectangular wall sections.
Then,
f cc'
for rectangular sections with equal effective confining stress
f l ' in the
orthogonal x and y directions is related to the unconfined strength, f c' , by the relationship
f
'
cc

7.94 f l ' 2 f l '  '

= − 1.254 + 2.254 1 +
− ' fc

f c'
f c 

121
(4.12)
For rectangular sections with unequal effective confining stresses f lx' and f ly' , f cc' may
be found from figures such as the one reported by Paulay and Priestly (1992, Fig. 3.6, pp.
102).
Once the limiting strain, ε cu , is determined, a fiber analysis can be carried out on the
cross-section resulting in the moment-curvature (M-φ) relationship. The available
curvature capacity of the cross-section may thus be defined as the value (φ f - φ y), where φ f
is the curvature corresponding to the attainment of ε cu , Equation 4.9, at the extreme
compression fiber of the confined core. φ y is the yield curvature corresponding to the
attainment of the steel yield stress, ε sy, in longitudinal reinforcement bars at one side of
the column. The available plastic rotation capacity, θ p = (θ f − θ y ) , needed in Equations
4.6 and 4.7 can thus be easily estimated as follows:
(θ
f
− θ y ) = (φ f − φ y ) l p
(4.13)
where lp is an assumed plastic hinge length over which plastic rotation is concentrated. A
good estimate of the plastic hinge length, as reported by Paulay and Priestly (1992),
considering the tensile strain penetration phenomenon, and spread of plasticity resulting
from inclined flexure-shear cracking, is given as follows:
l p = 0.08 l + 0.15d b f y
(ksi)
(4.14)
= 0.08 l + 0.022d b f y (MPa)
where l is the length from section of maximum moment to the point of inflection, db is the
bar diameter for the longitudinal reinforcement, and f y is the yield stress of the
longitudinal reinforcement.
122
Then, to compute the failure (or limiting) value for plastic energy, or in other words the
available plastic energy capacity up to failure, Ef, the area under M-θ curve is calculated
in the region between θy and θf.
4.6.2 Steel and Composite Beams
In this section, methods for calculating steel and composite beam’s inelastic rotation
capacity and plastic energy up to the limiting state (or failure) are presented. The main
criteria that define failure of steel and composite beams involve the interaction of local
and lateral buckling, unloading (or strain-weakening) mechanisms, crushing of concrete
slab, separation between concrete slab and steel cross-section which is a type of loss of
composite action, among other failure aspects.
In more details, one can differentiate between modes of failure of steel beams or
composite beams under negative (i.e., hogging) moment, and composite beams under
positive (i.e., sagging) moment. For the former, two modes of failure might be identified
as pointed out by Kemp and Dekker (1991): (1) lateral buckling dominant that induces
local flange buckling at higher lateral slenderness ratios, or (2) local flange buckling
dominant that induces lateral and local web buckling at lower lateral slenderness ratios.
For positive bending failure of composite beams, Ansourian (1982) reports that if
secondary failures are prevented, collapse is assumed to be reached when crushing failure
of concrete slab occurs. Composite beams under sagging bending are classified as ductile
if strain hardening of the lower flange occurs before crushing failure of the slab, while
they are considered brittle if crushing occurs before significant strain hardening is
developed.
As shown in Figure 4.4, the inelastic rotation capacity, θ p = (θ f − θ y ) , is the rotation
available beyond the elastic rotation capacity, θ e (or θ y ) , and prior to the moment falling
below the design resistance, Mp . This rotation capacity may be provided either by the end
connection or by the member over the length Li between the section of maximum
moment and adjacent point of inflection. The methods followed in this research for the
123
determination of this available rotation capacity are those based on the work by Kemp
and Dekker (1991), and Ansourian (1982). These methods are based on semi-empirical
formulations – briefly discussed herein in Sections 4.6.2.1 and 4.6.2.2 – and rely on
results from several tests on steel I-sections in plain steel and composite structures of
centrally loaded beams (i.e., under moment gradient). When a rigid frame is subjected to
horizontal loading such as seismic force or wind pressure, the constituent beams and
columns undergo double curvature bending which can be simulated by an assembly of
the configurations of cantilever beams. Also the rotation capacity of cantilever beams can
be compared to those of centrally loaded beams which are often used as test specimens.
Moment
θe
θp
Design moment resistance
Mp
Rotation Capacity,
r a = θ p /θ e
Li
Rotation θ
Maximum
Moment
1
Rotation ratio, θ/θ e
Figure 4.4 Moment-rotation relationship for steel beams.
Once the plastic rotation capacity, θp , is computed, an idealized moment-rotation (M-θ)
relationship is proposed in Figure 4.5 where the maximum moment is suggested as
approximately 1.3Mp based on data from tests. Moreover, as an approximation, according
to Kemp and Dekker, the point of maximum moment corresponds to roughly half the
rotation capacity. Thus, the available plastic energy capacity up to failure, Ef, can be
given by the shaded area in Figure 4.5 which can be written as
124
E f = 1.15M p θ p
(4.15)
Moment
1.3 Mp
Mp
Design moment resistance, Mp
θe
θp
End Rotation
Figure 4.5 Idealized moment-rotation relationship for Ef calculation for steel beams.
4.6.2.1 Case of steel beams and composite beams under hogging bending
The plastic rotation capacity proposed in this research is based on the work by Kemp and
Dekker (1991). The method is summarized as follows and it is important to point out that
it takes into consideration the interaction between local and lateral buckling as previously
mentioned.
First, an effective lateral slenderness ratio, λe, accounting for the different flange and web
slenderness, is given as follows
λ e = K f K w ( Li / i c ε f )
where (0.75<KfKw<1.30)
(4.16)
in which Li is the length of the beam from the section of maximum moment to the point
of inflection, ic is the radius of gyration about the minor axis for the part of the cross125
section in compression, ε f is
235 / f yf
(where f yf is the yield stress of the flange in
MPa), and Kf and Kw are empirical factors to allow for the actual flange and web
slendernesses, respectively. The following expressions for Kf and Kw are given
Kf =
Kw =
(b / t
f
)
εf
(4.17)
20
αd w
33t wε w
[460 − (Li / ic ε )]
Kw =
αd w
33t w ε w
400
(33 < (αd w / t wε w ) ≤ 40 )
(4.18)
((αd w / twε w ) ≤ 33)
(4.19)
where b is the flange width, t f is the flange thickness, α is the proportion of the depth of
the section in compression between the centers of the two flanges, dw is the web depth, t w
is the web thickness, and ε w is
235 / f yw (where f yw is the yield stress of the web in
MPa). It is worth pointing that Equation 4.19 reflects the benefit of distortional web
restraint as the lateral slenderness ratio increases.
Next, Kemp and Dekker (1991) proposed an empirical expression for the relationship
between λe and available rotation capacity ra (ra = θp /θe) for 20<(Li/icε f)<100, assuming
values of s=10 and e=50. Note that s is defined as the ratio of strain at onset of strain
hardening to yield strain, and e is the ratio of initial modulus of elasticity to the strain
hardening modulus, E/E sh . The rotation capacity, ra , is given as
3(60 / λ e )
ra =
2α
1. 5
(4.20)
126
As reported by Kemp and Dekker, the predicted rotation capacity given by Equation 4.20
reflects a lower bound to the tests because it is based on a conservative stress-strain
relationship with s=10 and e=50.
For composite beams under hogging moment, the available rotation capacity given by
Equation 4.20, derived based on using the plastic moment resistance and flexural rigidity
of the steel section alone, should be adjusted. Provided account is taken of the axial
compression force applied to the steel section to balance the tension force in the
reinforcement of the slab of the composite beam, Kemp and Dekker report that it is
common practice to assume that local buckling of the steel section will give similar
inelastic rotations θp in steel and composite applications. This was illustrated in two pairs
of tests conducted by Climenhaga and Johnson (1972). Thus, the available rotation
capacity for hogging direction of composite beams should be adjusted by multiplying ra
(
)
from Equation 4.20 by the ratio M ps / M 'p (EI / EI s ) , in which Mps and M 'p are the
negative moment resistances of steel section and composite beam, respectively, and EI
and EIs are flexural rigidities of the uncracked composite and steel section, respectively.
Further adjustment is needed to account for the effect of axial compression balanced by
the slab reinforcement, and this is achieved by dividing the modified ra from the previous
step by the value 2α. The last modification of ra is to acknowledge cracking of concrete
adjacent to supports which allows more inelastic rotations to occur; thus, the additional
available rotation capacity due to cracking in a region of uniform moment gradient
adjacent to the support is given by (EI/EIs)-1. The adjusted available plastic rotation
capacity of composite beams in hogging bending direction can be finally given as
 M ps  EI
ramod =  ' 
 M p  EI s


 1 
 EI

  ra + 
− 1
 2α 
 EI s

(4.21)
For cases of linear moment gradients as is the case in most tests, the elastic rotation might
be calculated as follows
127
θe =
0.5M p Li
(4.22)
EI
in which Mp might be M 'p which is the hogging moment capacity for composite beams.
Thus, θp is easily computed by multiplying Equation 4.20 or Equation 4.21, for the case
of steel beam or composite beam under negative bending respectively, by the value of θe.
4.6.2.2 Case of composite beams under sagging bending
The plastic rotation capacity up to failure for the case of composite beams under positive
bending follows ideas proposed by Ansourian (1982). As mentioned before, composite
beams under sagging bending can be classified into two different modes as either brittle
or ductile. These two modes of behavior are identified by comparing the actual depth to
the neutral axis of the cross-section with a limiting depth defined by the strain hardening
strain, ε sh , at the lower flange and the crushing strain, ε cu, at the top of the slab. It has
been shown based on comparisons with about 60 tests of different beams that a certain
ductility parameter χ is a reliable index of the shape of the moment-curvature
relationship which is crucial in the determination of the available rotation capacity. This
parameter χ - neglecting the effect of longitudinal slab reinforcement - is given as
follows
χ=
0.72 f c' Bc ε cu ( D s + Dc )
As f y (ε cu + ε sh )
(4.23)
in which f c' is the concrete compressive strength, f y is the structural steel yield stress, As
is the steel beam cross-sectional area, Bc is the width of concrete slab, Dc is the thickness
of concrete slab, and Ds is the depth of steel beam.
The test specimens used to calibrate Equation 4.23 cover a wide range of steel beam sizes
and slab dimensions, yield strengths (varying between 29 and 72.5 ksi), and concrete
128
strengths ( f c' = 1.5 and 5.8 ksi). The resulting variation in parameter χ is from 0.7 to 3.5.
Values from test and analysis results are plotted by Ansourian (1982) as χ versus nondimensional plastic rotation, ra , defined as the ratio of the ultimate (or failure) plastic
rotation, θp , to the elastic rotation, θe, at the collapse load. The value of ra is therefore
independent of the span. Given herein is the mean regression line for ultimate plastic
rotation ratio ra in the range χ = 1-3.5 as computed by Ansourian (1982)
ra = 2.5χ − 1.6
(4.24)
When expressed in terms of the composite beam properties, and taking ε sh as 0.015, and
ε cu as 0.005 (for concrete slab), Equation 4.24 takes the form
0.45 f c' Bc ( D s + Dc )
ra =
− 1 .6
As f y
(4.25)
Then, the plastic rotation capacity at ultimate state as defined by Ansourian is computed
by multiplying ra of Equation 4.25 by the value of the elastic rotation θe calculated at the
collapse load level.
4.6.3 Composite – Reinforced Concrete-Steel – Joint Panels
Composite joint panel behavior is basically characterized by two failure modes: panel
shear failure and vertical bearing failure. Panel shear failure is similar to that typically
associated with structural steel or reinforced concrete joints; however, in composite
joints, both structural steel and reinforced concrete panel elements participate. Bearing
failure occurs at locations of high compressive stresses and may be associated with rigid
body rotation of the steel beam within the concrete column. It is worth pointing that,
based on experimental observation, shear failure is generally accompanied by bearing,
whereas bearing failure is not accompanied by shear strength deterioration (Kanno,
1993).
129
After this brief introduction about failure modes governing composite joints behavior,
one should make use of that in order to determine limiting, ultimate, (or failure) values
for variables associated with the joint behavior and useful for the damage indices
proposed in this chapter. The variables used in the ductility- and the energy-based
damage indices are total joint panel distortion (including both shear and bearing shares)
and hysteretic dissipated energy in the joint panel mechanism. The total joint distortion is
used and not its plastic component because of the interaction between joint shear and
bearing behavior and consequently the difficulty of extracting the plastic component out
of the total distortion from the constitutive models implemented in DYNAMIX - the
software used throughout this research - to model composite joint behavior. This
drawback might lead to a value of the ductility-based damage index under predominantly
elastic behavior (under elastic loading damage indices should have zero value), but
generally this value would be very small. In general, values of damage indices below
about 0.2∼0.3 on a scale ranging from 0.0 to 1.0 define case of slight or minor damage
which is really not crucial to the overall performance of the component.
At present, no models are available to analytically predict the ultimate deformation
capacity of composite joints. Therefore, the selection of suitable values for this damage
parameter is based on the results of the experimental work by Kanno (1993). Twelve
specimens are chosen including seven failing in shear and five failing in bearing. Total
distortion at failure is reported where the failure point is defined by Kanno as the point at
the end of the half cycle where the load first drops to 20% below the maximum strength.
Then, a least square fit is applied herein, see Figure 4.6, on the twelve values for the total
joint distortion at failure, and the equation of the best line fitting this data is given as
γ f , cyc =
1.96 − (M ns / M nb )
18.97
(4.26)
where γ f , cyc is the predicted total joint distortion at failure under cyclic loading with
varying sign amplitudes using least square fit, and Mns and Mnb are the nominal shear and
130
bearing moment capacities of the joint, respectively. Note that Mns and Mnb may be
calculated using the provisions for composite joints in the ASCE Guidelines (1994).
JP Distortion at failure, γf,cyc [rad.]
0.12
0.10
0.08
0.06
0.04
0.02
0.00
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
M ns / M nb
Figure 4.6 Values of cyclic joint panel distortion at failure by least square fit based on
results by Kanno (1993).
However, the failure value of the damage parameter used in the denominator of the
damage indices proposed in this chapter corresponds to monotonic loading up to failure.
Thus, since Kanno’s experiments are conducted under cyclic loading with varying sign
amplitudes, the value of γ f , cyc should be modified to get a suitable value corresponding
to failure limit state under monotonic loading, γ f , mon . A reasonable amplification factor
is found to be equal to 1.2, and thus one may write
γ f , mon = 1.2 γ f , cyc
(4.27)
An idealized joint panel moment versus joint panel distortion as suggested by Sheikh et
al. (1989) is shown in Figure 4.7. The area under this curve up to the point of γ f , mon as
defined by Equation 4.27 gives the monotonic energy capacity available up to failure
131
state, Ef, to be used as a normalizing factor in the energy-based damage index of
Equations 4.1 and 4.2. Ef can thus be given as
E f = (1.15γ f , mon − 0.00575)M n , ASCE
(4.28)
in which Mn,ASCE is the nominal moment capacity of the composite joint according to the
ASCE Guidelines (1994); note that Mn,ASCE is the smaller of Mns and Mnb defined above.
1.15Mn,ASCE
Joint Panel Moment
Mn,ASCE
0. 5Mn,ASCE
0.002
0.01
0.02
γf
Joint Panel Distortion, γ
Figure 4.7 Idealized moment-distortion for composite joint panels, Sheikh et al. (1989)
4.7 Calibration and Verification
The energy- and ductility-based local damage indices proposed in this chapter and given
in Equations 4.1 to 4.3 and 4.6 to 4.8 are calibrated and tested versus different
experimental test results. The experimental data include reinforced concrete columns,
steel and composite beams, and composite RCS joint sub-assemblages. For each type of
element, the two damage indices are assessed in terms of their ability to capture both the
132
failure point and the evolution of damage up to failure. In this section, an attempt to
correlate values of the damage indices, on a scale from 0.0 to 1.0, to the status of damage
of the component is also presented. Some limitations in that concern are faced because of
the small number of verification tests and the very limited description of the damage
evolution up to failure in the different experiments. The tests conducted by Kanno (1993)
are of great help since he provided detailed description of the damage history up to
failure for different specimens.
4.7.1 Reinforced Concrete Columns
The two proposed damage indices are applied to six reinforced concrete column subassemblages with different axial load levels. The columns are as follows: two specimens
(NC2 and NC4) by Azizinamini et al. (1992), one specimen (U4) by Ozcebe and
Saatcioglu (1987), and three specimens (WP2, WP4, and WP9) by Watson and Park
(1994). Relevant values to the damage indices calculations for these specimens are
computed as discussed in Section 4.6.1 and given in Table 4.3. All of the specimens but
one (WP9 by Watson and Park) failed during testing.
Table 4.3: Useful Values for Calculation of RC Columns Damage Indices.
Specimen
Ef
ε cu
θ p = (θ f − θ y )
(kips-in.-rad.)
(in./in.)
(rad.)
NC2
NC4
U4
WP2
WP4
WP9
0.0559
0.0372
0.0590
0.0305
0.0148
0.0666
0.1230
0.0590
0.1477
0.0567
0.0284
0.0779
461.3
218.0
350.0
178.6
85.2
278.2
Based on the results of the damage indices, the full-damage surface parameter, γ, defining
the interaction between the damage due to positive and negative deformations, is taken as
6.0 for both the plastic rotation (ductility-based) damage index and the plastic energy
(energy-based) damage index. The α calibration parameter, i.e., the exponent of the PHC
term in the indices, is taken as 1.0, while the β parameter, the exponent of the FHC term,
133
is taken as 1.5 and 0.95 for the ductility-based and energy-based damage index,
respectively. The values of both the ductility- and energy-based damage indices at failure
are summarized in Table 4.4.
As an example, detailed results of specimen WP4 are given in Figures 4.8 and 4.9 for the
evolution of the damage index. Figures 4.8a and 4.9a show the components of the
damage index history corresponding to positive and negative deformations and these
components are drawn on the two-dimensional damage plane. This type of plot clearly
shows the effect of the interaction between the positive and negative deformations on
causing total damage (i.e., failure) to be reached at values of either plastic rotation or
plastic energy less than their monotonic values at failure given in Table 4.3. On the other
hand, Figures 4.8b and 4.9b show the evolution of the total damage as described by the
combined indices (i.e., combined damage due to positive and negative deformations all
together); the final point in these plots is basically the point with the value given in Table
4.4. Statistical measures presented in Table 4.4 prove the good performance of the two
indices in adequately predicting failure. Values of the two indices for specimen WP9 also
capture the fact reported that the specimen did not fail at the end of the test although
some damage has taken place as mentioned by Watson and Park.
Table 4.4: Value of Damage Indices at Failure State for RC Columns.
Specimen
Damage Index DE
Damage Index Dθ
NC2
1.047
0.971
NC4
0.963
1.000
U4
1.048
0.989
WP2
1.039
1.030
WP4
0.986
0.999
Statistical
µ=1.015 σ=0.039 c.o.v.=3.8% µ=0.998 σ=0.021 c.o.v.=2.1%
Measures
(WP9)
0.694
0.771
µ = mean, σ = standard deviation, and c.o.v. = coefficient of variation.
134
1.2
Combined Damage Index, Dθ
Damage Index for positive deformations, Dθ+
1.0
0.8
0.6
0.4
0.2
0.0
0.0
1.0
0.8
0.6
0.4
0.2
0.0
0.2
0.4
0.6
0.8
1.0
0
9
18
Damage Index for negative deformations, Dθ-
(a) Components of ductility-based damage index
27
36
45
Time
(b) Combined ductility-based damage index
1.2
1.0
Combined Damage Index, DΕ
Damage Index for positive deformations, DΕ+
Figure 4.8 Ductility-based damage index - Watson and Park (1994), Unit WP4
0.8
0.6
0.4
0.2
0.0
0.0
1.0
0.8
0.6
0.4
0.2
0.0
0.2
0.4
0.6
0
0.8
1.0
Damage Index for negative deformations, DΕ-
9
18
27
36
45
Time
(a) Components of energy-based damage index
(b) Combined energy-based damage index
Figure 4.9 Energy-based damage index - Watson and Park (1994), Unit WP4
135
Figure 4.10a Load-displacement relationship - Watson and Park (1994), Unit WP2
1.2
I
1.0
G
0.8
E
J
Combined Damage Index, DΕ
Combined Damage Index, Dθ
1.2
H
F
0.6
D
C
0.4
0.2
A
B
0.0
0
9
18
27
36
I J
1.0
0.8
E
0.6
F
GH
D
C
0.4
B
0.2
A
0.0
45
0
Time
9
18
27
36
Time
Figure 4.10b Results for combined ductility- and energy-based damage indices
Watson and Park (1994), Unit WP2
136
45
Figure 4.11a Load-displacement relationship - Watson and Park (1994), Unit WP4
1.2
1.0
I
0.8
G
Combined Damage Index, DΕ
Combined Damage Index, Dθ
1.2
J
H
F
0.6
E
0.4
D
0.2
B
A
C
0.0
0
9
18
27
36
J
I
1.0
0.8
E
F
G
H
0.6
0.4
D
C
0.2
A
B
0.0
45
0
Time
9
18
27
36
Time
Figure 4.11b Results for combined ductility- and energy-based damage indices
Watson and Park (1994), Unit WP4
137
45
As mentioned earlier, an attempt is made herein to correlate the degree of damage to the
value of the damage index as much as allowed by the information about the damage
status reported during the experiments. Since this information is not readily available for
the column specimens presented herein, the best that can be done is to correlate each
point at the end of half cycles in the experimental response with its associated point in the
combined damage index history. Plots showing this correlation are given in Figures 4.10
and 4.11 for specimens WP2 and WP4 for both Dθ and DE damage indices. It is worth
pointing that, as reported by Watson and Park, unit WP2 failure is defined by fracture of
longitudinal bars, while unit WP4 failure is defined when buckling of the longitudinal
reinforcement occurs; both failure modes cause loss of capacity.
4.7.2 Steel and Composite Beams
In order to check the ability of the proposed damage indices to capture the damage up to
failure of steel and composite beams, they are applied to some experimental data. Tests
include two specimens for plain steel beam case which are tested by Kanno (1993) in his
work on composite reinforced concrete column-steel beam sub-assemblages; these two
specimens (OB1-1 and OBJS1-1) show beam failure rather than joint failure. For the case
of composite beams, three tests are considered: specimen CG3 tested by Uang (1985),
specimen EJ-WC tested by Lee (1987), and the specimen with full shear connection
tested and reported by Bursi and Ballerini (1996). Values necessary for the evaluation of
the proposed damage indices are given in Table 4.5.
Table 4.5: Values for Calculation of Damage Indices for Steel and Composite Beams.
Specimen
M +p
(θ − θ )+
E +f
M −p
(θ − θ )−
E −f
f
OB1-1
OBJS1-1
CG3
Bursi
EJ-WC
y
(kips-in)
(rad.)
1947.6
1947.6
230.0
4315.6
5386.5
0.061
0.061
0.085
0.038
0.022
f
(kips-in.rad.)
136.6
136.6
22.4
124.1
106.5
138
y
(kips-in)
(rad.)
1947.6
1947.6
145.0
2861.1
2785.6
0.061
0.061
0.060
0.065
0.037
(kips-in.rad.)
136.6
136.6
10.0
213.9
118.5
As for the case of reinforced concrete columns, calibration of the proposed damage
indices versus experiments shows that values of γ=6.0 and α=1.0 are again suitable for
both ductility- and energy-based damage indices. β assumes values of 1.5 and 0.95 for
ductility-based and energy-based indices, respectively. Values of the combined indices at
failure state as defined in tests are given in Table 4.6.
Table 4.6: Combined Damage Indices at Failure for Steel and Composite Beams.
Specimen
Damage Index DE
Damage Index Dθ
OB1-1
1.020
1.052
OBJS1-1
1.045
1.027
CG3
1.034
1.025
Bursi et al.
1.032
0.998
EJ-WC
0.990
0.948
Statistical
µ=1.024 σ=0.021 c.o.v.=2.1% µ=1.010 σ=0.040 c.o.v.=4.0%
Measures
Detailed results in terms of the components of damage indices corresponding to positive
and negative deformations, as well as combined indices, are given in Figures 4.12 to 4.15
for specimens OB1-1 and CG3. From statistical measures shown in Table 4.6, both
indices show good performance in capturing total failure of specimens defined in almost
all of the cases by loss of capacity (i.e. strength) due to severe buckling of the
compression flange.
In order to relate the damage index value to the state of damage, corresponding points in
both experimental response and evolution of damage index are highlighted. As an
example, this correlation is given in Figures 4.16 and 4.17 for specimens OB1-1 (steel
beam by Kanno, 1993) and CG3 (composite beam by Uang, 1985), for the two damage
indices.
139
1.2
Combined Damage Index, Dθ
Damage Index for positive deformations, D θ+
1.0
0.8
0.6
0.4
0.2
0.0
0.0
1.0
0.8
0.6
0.4
0.2
0.0
0.2
0.4
0.6
0.8
1.0
0
5
10
Damage Index for negative deformations, D θ-
15
20
25
Time
(a) Components of ductility-based damage index (b) Combined ductility-based damage index
1.0
1.2
0.8
1.0
Combined Damage Index, D Ε
Damage Index for positive deformations, D Ε+
Figure 4.12 Ductility-based damage index - Kanno (1993), Unit OB1-1.
0.6
0.4
0.2
0.0
0.0
0.2
0.4
0.6
0.8
0.6
0.4
0.2
0.0
0.8
1.0
Damage Index for negative deformations, DΕ-
0
5
10
15
20
25
Time
(a) Components of energy-based damage index
(b) Combined energy-based damage index
Figure 4.13 Energy-based damage index - Kanno (1993), Unit OB1-1.
140
1.2
Combined Damage Index, Dθ
Damage Index for positive deformations, D θ+
1.0
0.8
0.6
0.4
0.2
0.0
0.0
1.0
0.8
0.6
0.4
0.2
0.0
0.2
0.4
0.6
0.8
1.0
Damage Index for negative deformations, Dθ-
0
4
8
12
16
20
Time
(a) Components of ductility-based damage index (b) Combined ductility-based damage index
1.0
1.2
0.8
1.0
Combined Damage Index, DΕ
Damage Index for positive deformations, D Ε+
Figure 4.14 Ductility-based damage index - Uang (1985), Unit CG3.
0.6
0.4
0.2
0.0
0.0
0.2
0.4
0.6
0.8
0.8
0.6
0.4
0.2
0.0
1.0
0
Damage Index for negative deformations, DΕ-
4
8
12
16
20
Time
(a) Components of energy-based damage index
(b) Combined energy-based damage index
Figure 4.15 Energy-based damage index - Uang (1985), Unit CG3.
141
Figure 4.16a Beam shear-drift angle relationship - Kanno (1993), Unit OB1-1
1.2
I
Combined Damage Index, DΕ
Combined Damage Index, Dθ
1.2
1.0
H
G
0.8
E F
0.6
D
C
0.4
A
0.2
B
0.0
0
5
10
15
20
I
1.0
H
E
0.8
F
G
D
0.6
C
0.4
B
A
0.2
0.0
25
0
Time
5
10
15
20
Time
Figure 4.16b Results for combined ductility- and energy-based damage indices
Kanno (1993), Unit OB1-1
142
25
8
A
E
C
6
G
Tip Load [kips]
4
2
0
-2
H
D
-4
B
F
-6
-4
-3
-2
-1
0
1
2
3
4
Tip Displacement, ∆ [inches]
Figure 4.17a Load-displacement relationship - Uang (1985), Unit CG3
1.2
Combined Damage Index, DΕ
Combined Damage Index, Dθ
1.2
H
1.0
F
0.8
C
0.6
A
D
G
E
B
0.4
0.2
0.0
0
4
8
12
16
H
1.0
G
0.8
C
0.6
F
D
E
B
A
0.4
0.2
0.0
20
0
Time
4
8
12
16
Time
Figure 4.17b Results for combined ductility- and energy-based damage indices
Uang (1985), Unit CG3
143
20
4.7.3 Composite Reinforced Concrete-Steel Joints
In the course of this research, the two proposed damage indices are finally checked for
cases of composite joint panels of reinforced concrete columns and steel beams. The
experimental tests considered are those conducted by Kanno (1993); they comprise seven
specimens failing mainly in joint shear and five specimens failing in joint bearing as
mentioned before. Values necessary for the calculation of the damage indices are given in
Table 4.7; these values are computed based on the procedure presented in Section 4.6.3.
A value of γ of 5.0 is chosen for the ductility-based damage index relying on the total
joint distortion rather than its plastic component due to the reasons discussed earlier in
this chapter. While γ=2.0 is proposed for the energy-based damage index as suggested by
the results shown in Table 4.8. Calibration parameters α and β are taken as 0.75 and 3.0
for the ductility-based index, and 0.8 and 0.7 for the energy-based index. It is important
to mention that failure point for all specimens is defined as suggested by Kanno as the
point at the end of the half cycle where the load first drops to 20% below the maximum
strength.
Table 4.7: Values for Calculation of Damage Indices for Composite RCS Joints.
Specimen
Mns/Mnb
Mn,ASCE
Ef
γ f,cyc
γ f,mon
(kips-in.)
(kip-in-rad)
(rad.)
(rad.)
Joint Bearing Failure Mode
OJB1-0
1.190
0.043
0.051
6108.9
323.2
OJB2-0
1.170
0.042
0.050
7596.5
393.1
OJB4-0
1.230
0.038
0.046
6889.5
324.8
OJB5-0
1.030
0.049
0.059
6935.1
430.7
OJB6-1
1.210
0.039
0.047
6218.2
300.3
Joint Shear Failure Mode
OJS1-1
0.680
0.067
0.080
2933.4
253.0
OJS2-0
0.614
0.071
0.085
2834.5
260.8
OJS3-0
0.658
0.068
0.082
5519.2
488.7
OJS4-1
0.658
0.068
0.082
5519.2
488.7
OJS5-0
0.581
0.073
0.088
5658.2
540.1
OJS6-0
0.633
0.070
0.084
5280.8
479.8
Combined Beam and Joint Shear Failure
OBJS2-0
0.537
0.075
0.090
3952.2
386.3
144
There are some observations that should be pointed out from the results presented in
Table 4.8. First of all, one can notice that for the specimen OBJS2-0 failing in a
combined beam and joint shear failure mode, the use of parameters related to the
behavior of the joint panel alone might not be suitable to predict total failure of the
specimen; damage variables related to beam behavior should also be considered if the
state of total failure is required to be captured. Also, it is worth pointing out that the
prediction of failure through the two proposed damage indices is much better for joints
with predominantly shear failure mode than for joints with bearing failure mode. This
drawback is mitigated by the fact that seismically designed and detailed composite joint
panels should not be prone to bearing failure. Additionally, the ductility-based damage
index was found to be more successful in capturing total failure for bearing failure
specimens than was the energy-based damage index; one reason behind this is the lack of
the analytical model implemented in DYNAMIX for joint panel analysis in capturing the
actual strength and its degradation for this type of behavior.
Table 4.8: Combined Damage Indices at Failure for Composite RCS Joints.
Specimen
Damage Index DE
Damage Index Dγ
Joint Bearing Failure Mode
OJB1-0
1.126
0.977
OJB2-0
1.019
0.827
OJB4-0
1.077
0.928
OJB5-0
0.943
0.800
OJB6-1
0.925
0.776
Statistical
µ=1.018 σ=0.086 c.o.v.=8.4% µ=0.862 σ=0.087 c.o.v.=10%
Measures
Joint Shear Failure Mode
OJS1-1
1.032
1.011
OJS2-0
0.998
0.971
OJS3-0
1.054
1.027
OJS4-1
0.978
1.052
OJS5-0
0.990
1.031
OJS6-0
0.997
1.044
Statistical
µ=1.008 σ=0.029 c.o.v.=2.9% µ=1.023 σ=0.029 c.o.v.=2.8%
Measures
Combined Beam and Joint Shear Failure
OBJS2-0
0.824
0.901
145
1.2
0.8
1.0
Combined Damage Index, Dγ
Damage Index for positive deformations, Dγ+
1.0
0.6
0.4
0.2
0.0
0.0
0.2
0.4
0.6
0.8
0.6
0.4
0.2
0.0
0.8
1.0
Damage Index for negative deformations, Dγ -
0
6
12
18
24
30
Time
(a) Components of ductility-based damage index
(b) Combined ductility-based damage index
1.0
1.2
Combined Damage Index, DΕ
Damage Index for positive deformations, DΕ+
Figure 4.18 Ductility-based damage index - Kanno (1993), Unit OJS1-1.
0.8
0.6
0.4
0.2
0.0
0.0
0.2
0.4
0.6
0.8
1.0
Damage Index for negative deformations, DΕ-
1.0
0.8
0.6
0.4
0.2
0.0
0
6
12
18
24
30
Time
(a) Components of energy-based damage index
(b) Combined energy-based damage index
Figure 4.19 Energy-based damage index - Kanno (1993), Unit OJS1-1.
146
1.2
0.8
1.0
Combined Damage Index, Dγ
Damage Index for positive deformations, Dγ+
1.0
0.6
0.4
0.2
0.0
0.0
0.2
0.4
0.6
0.8
1.0
Damage Index for negative deformations, Dγ -
0.8
0.6
0.4
0.2
0.0
0
5
10
15
20
25
Time
(a) Components of ductility-based damage index
(b) Combined ductility-based damage index
1.0
1.2
0.8
1.0
Combined Damage Index, DΕ
Damage Index for positive deformations, DΕ+
Figure 4.20 Ductility-based damage index - Kanno (1993), Unit OJS4-1.
0.6
0.4
0.2
0.0
0.0
0.8
0.6
0.4
0.2
0.0
0.2
0.4
0.6
0.8
1.0
Damage Index for negative deformations, DΕ-
0
5
10
15
20
25
Time
(a) Components of energy-based damage index
(b) Combined energy-based damage index
Figure 4.21 Energy-based damage index - Kanno (1993), Unit OJS4-1.
147
Figure 4.22a Beam shear-drift angle relationship - Kanno (1993), Unit OJS1-1
1.2
L
1.0
Combined Damage Index, DΕ
Combined Damage Index, Dγ
1.2
K
I J
0.8
0.6
H
G
E F
C
0.4
D
AB
0.2
0.0
0
6
12
18
24
L
1.0
I J
0.8
H
G
EF
0.6
K
D
0.4
C
A
0.2
B
0.0
30
0
Time
6
12
18
24
Time
Figure 4.22b Results for combined ductility- and energy-based damage indices
Kanno (1993), Unit OJS1-1
148
30
Figure 4.23a Beam shear-drift angle relationship - Kanno (1993), Unit OJS4-1
1.2
I
H
1.0
Combined Damage Index, DΕ
Combined Damage Index, Dγ
1.2
G
0.8
E F
0.6
C
D
0.4
A B
0.2
0.0
0
5
10
15
20
H
1.0
I
G
E
0.8
F
D
0.6
C
0.4
A
B
0.2
0.0
25
0
Time
5
10
15
20
Time
Figure 4.23b Results for combined ductility- and energy-based damage indices
Kanno (1993), Unit OJS4-1
149
25
Detailed results in terms of the components of damage indices corresponding to positive
and negative deformations, as well as combined indices, are given in Figures 4.18 to 4.21
for specimens OJS1-1 and OJS4-1. Also, due to the fact that good information about
evolution of damage monitored during testing and reported by Kanno is available, a very
useful correlation can be realized between the evolution of damage as measured by the
proposed damage indices and the level of observable damage of specimens. Figures 4.22
and 4.23 present this information for the two specimens OJS1-1 and OJS4-1 chosen
before. Specimens failing in shear, rather than bearing, are more important in the course
of this research (seismic behavior of RCS composite frames) since bearing failure mode
is one that is generally avoided.
4.8 Useful Conclusions and Guidelines for Damage Categorization
Once the damage indices are calculated at the local level for various structural
components, it is useful to relate them to the level of damage attained by the component.
This information is quite important to assess the behavior of structures according to
performance criteria often expressed at the following structural performance levels:
immediate occupancy, life safety, and collapse prevention (FEMA 273). As an example,
the type of damage associated with each performance level is summarized in Table 4.9
for primary and secondary systems of concrete and steel moment frames as presented by
FEMA 273. This might be further related to the repairability level; i.e., whether this level
of damage leads to an irrepairable structure or not.
In this section, an attempt to relate values of the two proposed local damage indices to the
corresponding probable actual level of damage of the structural component is presented.
150
Elements
Concrete
Frames
Steel
Moment
Frames
Table 4.9: Structural Performance Levels and Damage.
Type
Structural Performance Levels
Immediate
Life
Collapse
Occupancy
Safety
Prevention
Primary
Minor hairline
Extensive damage
Extensive cracking
cracking. Limited
to beams. Spalling
and hinge form. in
yielding possible at
of cover and shear
ductile elements.
”
a few locations. No
cracking (<1/8
Limited cracking
crushing (strains
width) for ductile
and/or splice
below 0.003).
columns. Minor
failure in some
spalling in nonnon-ductile
ductile columns.
columns. Severe
Joint cracks <1/8”
damage in short
wide.
columns
Secondary Minor spalling in a
Extensive cracking
Extensive spalling
few places in
and hinge form. in
in columns
ductile columns and
ductile elements.
(limited
beams. Flexural
Limited cracking
shortening) and
cracking in beams
and/or splice failure
beams. Severe
and columns. Shear
in some non-ductile
joint damage.
cracking in joints
columns. Severe
Some reinforcing
<1/6” width.
damage in short
buckled.
columns.
Primary
Minor local yield.
Hinges form. Local
Extensive
at a few places. No
buckling of some
distortion of beams
observable fractures beam elements.
and column panels.
Minor buckling or
Severe joint
Many fractures at
observable perman.
distortion; isolated
connections.
distortion of
connection failures.
members.
A few elements
may experience
fracture.
Secondary Minor local
Extensive distortion
Same as primary.
yielding at a few
of beams and
places. No
column panels.
fractures. Minor
Many fractures at
buckling or
connections.
observable perman.
Distortion of
members.
First, looking at the reinforced concrete columns sub-assemblages presented in this
chapter, it is hard to draw some strong conclusions concerning relating damage indices
values to the actual damage due to the lack of observable damage information reported
151
during testing. However, one can relate the ductility level attained by each column
(displacement ductility as reported by experimentalists) to the damage index and
consequently to the corresponding structural performance level. A displacement ductility
in the range 1.0-2.0 and less (corresponding to a damage index value of approximately
0.25-0.3 and less) can be related to immediate occupancy structural performance level. A
damage index value in the range 0.3 to 0.6 which corresponds to a displacement ductility
ranging approximately from 2.0 to 3.0 (sometimes higher depending on the confinement
level) can be related to life safety level. Damage indices above 0.6 up to 0.95 (equivalent
to ductility level of about 3.0 to 4.0 or higher again depending on the confinement of the
columns) can be considered as near collapse while damage indices above 0.95 means
collapse or total failure. Note that the proposed limits are quite approximate due to the
limited number of test data and damage information reported in testing. Moreover, the
proposed damage indices are local indices and are not truly meaningful until they are
combined or integrated in a certain way for the different components of the structure to
be able to assess the overall damage of the structure. A summary of the suggested ranges
is given in Table 4.10. Taking a further step, one might also think of a value of the
damage index of about 0.6 as the limit for repairable damage.
Table 4.10: Correlation of Damage Index and Damage State.
Performance Level
D
Immediate Occupancy
0.25-0.3 and less
Life Safety
0.3-0.60
Near Collapse
0.60-0.95
Collapse
>0.95
Considering the behavior of steel and composite beams, one can propose the same ranges
as those for the RC columns. As an example, the specimen OB1-1 by Kanno (1993) that
has beam-type failure (Figure 4.16) shows minor damage defined by initial shear cracks
and beam yielding at a damage index value of about 0.3 (Immediate Occupancy
performance level). Some observable damage due to initial local buckling occurs at Dθ
and DE of around 0.5 to 0.6 (Life Safety performance level). Then, severe damage
manifested by large local buckling of both flanges takes place at a value of damage
152
indices of about 0.85 to 0.9 (Near Collapse level). Finally, failure (i.e., Collapse) occurs
at Dθ = 1.020 and DE = 1.052.
Finally, a careful observation of the cases of composite joints with joint shear failure type
(more relevant to seismic design) such as these given in Figures 4.22 and 4.23, reveals
the ranges as suggested in Table 4.10 to be assigned to the damage indices corresponding
to different performance levels.
4.9 Summary
In this chapter, a literature review of the seismic damage indices is presented with some
emphasis on their classification as local and global damage indices. Definition of the
damage function (or the damage index) is also discussed along with explanation of why
we need such indices. Information about categorization of damage as suggested by
different researchers is also given.
Two proposed local damage indices are presented; a ductility-based index as well as an
energy-based index. The two damage indices are based on the idea of primary and
follower half cycles in a formulation that take into consideration the ‘temporal’ effect of
loading (i.e., loading sequence or history) and cumulative damage. Identification of some
ultimate (i.e. limiting or failure) deformation and energy values to be used with the
proposed indices has been carried out. Procedures for calculating such values are
developed for reinforced concrete columns, steel and composite beams, and composite
joint panels.
The two proposed indices are then tested by applying them to selected experimental data
including reinforced concrete columns, steel and composite beams, and composite RCS
joint sub-assemblages. Results obtained concerning the values of the indices at total
failure as well as the evolution of damage up to failure show the ability of the proposed
indices in capturing to a good extent the evolution of damage up to failure of the
153
structural component (or sub-assemblage) under consideration. In spite of the small
number of data, statistical measures calculated show that the proposed damage indices
are promising measures of damage and failure under seismic type of loading.
Finally, an attempt is made to correlate the observable degree of damage to the value of
the damage index as much as allowed by the information about the damage status
reported during the experiments. This correlation may be useful in terms of its impact on
the performance based design adopted in new seismic codes which classifies the status of
the structure according to the consequences of its level of damage: immediate occupancy,
life safety, near collapse.
Finally, it is worth pointing that the proposed damage indices are local indices and thus
their values have to be combined in a certain scheme for the different members of the
structure to be able to assess the overall damage of the structure. This issue is discussed
in details in Chapter 6 proposing a new technique for global damage (and collapse)
determination by integrating information on local damage at the components level.
154
Chapter 5
Case Study Buildings Design and Selection of
Records
This chapter explains the design procedure for case study buildings investigated in this
research. A brief outline of seismic design methods and criteria proposed by recent
seismic codes is presented. Descriptions of the 6- and 12-story RCS-framed buildings and
6-story steel-framed building are given including the controlling design criteria and
member sizes. These case study buildings follow the general layout of the theme structure
proposed as part of the US-Japan program on hybrid structures, Phase 5. Finally,
selection of records for the time history analyses of the proposed designs is presented.
The records fall under two categories corresponding to general and near-fault conditions.
5.1 Overview of Different Seismic-Resistant Design Methods
Techniques to design and analyze structures for seismic loads according to recent seismic
codes and recommendations include (1) the equivalent lateral force static analysis
procedure, (2) modal response spectrum analysis, (3) dynamic linear and/or nonlinear
time history analysis, and (4) the static inelastic pushover analysis. The latter adopts
155
either the “capacity spectrum” method or the “displacement coefficient” method as per
FEMA 273. These design techniques are briefly discussed in this section.
5.1.1 Equivalent Lateral Force Static Procedure
Building codes have traditionally attempted to represent the dynamic earthquake effects
with an equivalent static lateral load distribution as an efficient and simple way for
seismic design and evaluation. The Equivalent Lateral Force (ELF) procedure is by far
the most widely used method and has been adopted by UBC 1997, NEHRP 1997,
ASCE7-95, IBC 2000, among other codes and standards. For instance, the recently
approved IBC 2000 provisions include an equivalent lateral load base shear, V, calculated
by the following equation
V=
S D1
W
R
 T
 I
≤
S DS
R
 
I 
(5.1)
W
where W is the effective seismic weight of the structure (dead load plus portions of other
relevant loads), T is the fundamental period of the structure, I is an occupancy importance
factor, SD1 is the design spectral response acceleration at a period of 1 second, and R is a
response modification factor. SDS is the design spectral response acceleration at short
period (taken as 0.2 second in seismic hazard maps). A minimum base shear of
0.044SDSW is also enforced to protect against excessively small values for long period
structures. Following these requirements, the IBC 2000 design spectrum is given in
Figure 5.1.
The spectral acceleration values SDS and SD1 are calculated by the following equations
SDS = 2/3 SMS = 2/3 Fa SS
(5.2a)
SD1 = 2/3 SM1 = 2/3 Fv S1
(5.2b)
156
Spectral Response
Acceleration (g)
SDS
SD1
T
SD1
0.044 SDS
0.2
SD1
SDS
1.0
Period, T (sec.)
Figure 5.1 IBC 2000 Design response spectrum.
where Fa and Fv are tabulated site coefficients, given as a function of the site class and
mapped spectral accelerations at short period and at a 1 second period, respectively. As
such, the design spectral accelerations, SDS and SD1, are computed by first modifying
mapped spectral accelerations, SS and S1 , according to site conditions to get maximum
considered earthquake spectral accelerations associated with the hazard at that specific
site, SMS and SM1. These are then multiplied by a factor of 2/3 which approximates the
relationship between the maximum considered earthquake (with a probability of
occurrence of 2%in50years) and the design level earthquake (with a probability of
occurrence of 10%in50years).
The base shear V is distributed up the height of the structure according to the following
parabolic distribution
Fx =
w x h kx
V
n
∑w
i =1
i
h
(5.3)
k
i
157
in which wi and wx are the portions of the total gravity load of the building, W, located at
level i or x, hi and hx are the heights from the base to level i or x, and k is a distribution
exponent related to the building period. The role of k is to guarantee a distribution of
forces up the height of the building that for example mimics the first mode shape for
short period structures (a value of k=1, i.e., triangular distribution for periods of 0.5
seconds or less) or mimics combination of first and higher modes for longer period
structures with a maximum value of k=2, i.e., parabolic distribution, for buildings with
periods of 2.5 seconds or more. For structures with rigid diaphragms, the IBC requires an
increase of the applied lateral load to account for accidental torsion. The accidental
torsion moments is calculated by assuming displacement of the center of mass each way
from its actual location by a distance equal to 5 percent of the dimension of the building
perpendicular to the direction of the applied forces. The torsion moments are then
distributed among the different systems constituting the lateral load resistance of the
building in each direction, and then distributed along the height according to the same
Equation 5.3.
For member design, the combined effect of horizontal (as reflected by the base shear V
mentioned above in Equation 5.1) and vertical earthquake-induced forces should be
considered. This combined effect, denoted by E and applied in the seismic load
combinations, is computed as follows
E = ρ QE ± 0.2 SDS D
(5.4)
where ρ is a reliability factor based on system redundancy, QE is the effect of the
horizontal seismic forces, D is the effect of dead load, and SDS is as defined before. The +
or - signs are to differentiate whether the effects of gravity aggravates or counteracts the
seismic load, respectively. In certain cases, so called “force controlled” elements are
required to be designed for the full capacity of the supported (or adjacent) elements. For
such components sensitive to effects of structural overstrength, the maximum combined
158
effect of horizontal and vertical earthquake-induced forces, Em, is set by IBC 2000 as
follows
Em = Ω o QE ± 0.2 SDS D
(5.5)
where Ω o is the system overstrength factor. Values of Ω o are tabulated for different types
of seismic-force-resisting systems. For instance, Ω o is taken as 3 for moment resisting
frame systems of any type and any material. The IBC further specifies that the term Ω o QE
need not exceed the maximum force that can be transferred to the element by the other
elements of the lateral force resisting system.
Base Shear
Elastic Response Force Level
Materials and Design
Overstrength
∆d = ∆e/R
∆in = (Cd/I) ∆e/R ∆e
Total
Design Force Level
Elastic Drift
Vd = Ve/R
Inelastic Drift
Ro
Overstrength
System
System Overstrength
(Redundancy)
Ductility
Rd
Ela
stic
Re
spo
nse
Ve
Lateral Drift
Figure 5.2 Elastic versus inelastic behavior as related by R and Cd factors.
IBC 2000 provisions also provide an approximation for the inelastic dynamic deflections
of the structure, ∆in , by amplifying the calculated lateral deflections at the design load
level, ∆d as follows
159
∆ in =
Cd
C
∆
∆d = d ∗ e
I
I
R
(5.6)
where the deflections at the design force level are calculated using the elastic stiffness of
the structure and the lateral force distribution, and Cd is an amplification factor tabulated
for different types of seismic-force-resisting systems. The relationship between R and Cd
factors is summarized in Figure 5.2, which shows the reduction of elastic base shear to
inelastic base shear, and the amplification of design deflections to predict inelastic
deflections.
The amplified story drifts as computed by Equation 5.6 must be less than the maximum
allowable story drift as specified in IBC 2000 or other governing codes, typically
between 1.5% and 2.5% of the story height depending on the building type and the
seismic use group of the building. When evaluating drift limits, the minimum design base
shear limits applied to Equation 5.1 need not apply. So, for example, for calculating drifts
the applied load may be determined using the calculated (actual) period T rather than the
upper limit of 1.2Ta allowed by IBC for design base shear calculation. This coefficient of
1.2 depends on the design spectral response acceleration at 1 second period, SD1, at the
site of the building. 1.2 corresponds to sites with SD1≥0.4g; higher values are proposed
for lower SD1 values. Note that Ta is the approximate fundamental period in seconds
given in IBC by the following formula
Ta = C T h 3/4
n
(5.7)
where CT is a building period coefficient that depends on the lateral load resisting system
and its material (i.e., steel versus reinforced concrete), and hn is the height (in feet) above
the base to the highest level of the building. IBC 2000 suggests CT values of 0.035 and
0.030 for steel and reinforced concrete moment resisting frame systems, respectively. The
value of 0.030 has been used for Ta calculation for the RCS frames studied in this thesis.
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Thus, there are two primary controls on the seismic design of a structure: minimum
strength as specified through the seismic response factor R, and minimum stiffness as
specified through the deflection limit and the seismic coefficients Cd/R. It is interesting to
note that these two requirements are interrelated and competing. For instance, if the
stiffness of a structure is increased so that it meets the drift requirements, then the period
will shorten, which may attract more forces (if we are on the descending branch of the
design response spectrum), and which may then increase the drift. Moreover, for a given
type of structural system and materials the strength and stiffness properties of the
building are not uncoupled.
The equivalent lateral force method is by far the most simple method to use and
understand. However, the major drawback of the method is the highly empirical nature of
the force reduction and displacement amplification factors, R and Cd. In brief, this design
method attempts to convert the inelastic dynamic behavior (or demand) of a structure to a
probable “worst scenario” earthquake to an equivalent static force, evaluated using an
elastic model of the structure.
5.1.1.1 Rationale of the R and Cd factors
Through the strength reduction factor R, the inelastic strength demands are determined
based on values of elastic strength demands. The value of R is dependent on how the
structure is expected to perform during an earthquake, and it represents the
approximation of inelastic response based on the elastic responses. As described below,
the reduction factor R is a simple device which attempts to account for many different
behavioral effects.
Effect of system ductility and damping: During an earthquake, ductility enables the
structure to dissipate kinetic energy induced by the earthquake ground motions. Ductility
allows indeterminate structures to develop their full strength and enables the structure to
move through large deflections at that strength. Different structures exhibit different
ductility and damping characteristics, so the R value must likewise depend on the
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structure. For instance, a concrete structure, which is more likely to experience a
degradation of the stiffness of its hysteresis loop during cyclic loading (also known as
pinching behavior), might experience less hysteretic damping during an earthquake than a
ductile steel frame which does not experience degradation of its hysteresis loops. The part
of the R value based on system ductility and damping effects, Rd, is shown in Figure 5.2.
Beside being a function of ductility, Rd accounts for other relevant dynamic phenomena
associated with period lengthening, pinching, etc… Bertero (1986) notes that the elastic
response spectrum, upon which the code is based, assumes a viscous damping ratio of
5%, which accounts for some of the hysteretic damping that occurs when the structure
experiences significant nonlinear behavior. However, it is not clear whether the Rd value
should only incorporate damping effects beyond those covered in the 5% viscous
damping. This is because the 5% viscous damping may account for some amount of
inelastic damping in the structure, but it may also pertain to inelastic response in the
foundation.
Effect of overstrength: Overstrength is the expected lateral load capacity of the structure
in excess of the minimum specified lateral seismic design forces. There are many sources
for overstrength associated with each structure. The first is material overstrength.
Nominal member strengths, determined using the nominal specified steel yield strengths,
do not account for the increase in ultimate strengths resulting from (a) differences
between the expected and nominal yield strengths, (b) strain hardening, (c) strain rate
effects, etc… For instance, Ellingwood et al. (1980) show that the actual average value of
the yield stress of steel is 5% greater than the nominal yield stress. They also mention
that earthquake induced strain rates increase the static yield stress by another 10%.
Furthermore, the ultimate stress at failure may be another 20% higher than the yield
stress. Of course, this increase in strength is usually accompanied by post-yielding
stiffness, which attracts increased internal forces, so it is not immediately obvious how
much the R value is affected. Uang (1991) suggests that all these increases be multiplied
to the calculated overstrength of the structure using nominal material properties to
account for the different material overstrength sources (i.e., Ω o x1.05x1.10). This
amplified overstrength value contributes to the value of Ro and accordingly to the final
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value of R as shown schematically in Figure 5.2. It is important to mention that the 1997
AISC Seismic Provisions introduced the notion of “expected yield strength, Fye” through
a multiplier, Ry , to the specified minimum yield strength, Fy, for calculating members or
connections capacity (i.e., strength).
The second contribution to overstrength is the one referred to as the design overstrength.
When a structure is designed, some (if not all) of the members will be slightly larger than
what is actually necessary to support the calculated loads. This is due to discrete sizing of
structural elements. Again, this will add additional strength, accounting for some further
increase in Ro . Both material and design overstrength are grouped together and shown in
Figure 5.2. Sometimes they are defined as the sources of overstrength causing the force
associated with the “first significant yield” level – a level beyond which the global
structural response starts to deviate significantly from the elastic response.
A third source of overstrength, termed system overstrength (or system redundancy) refers
to the additional strength resisted by indeterminate structures between the point at which
one or more structural elements first yield and the overall inelastic limit strength of the
system. Defining the point at which the first elements yield will depend upon the system
type; for ductile moment frames this point usually occurs when one or more members
first reach their plastic moment strength. Definition of the inelastic limit strength depends
upon the type of analysis being used to measure it. Typically, it would be defined based
on the peak load calculated using a second-order inelastic static (pushover) analysis that
takes into account destabilizing P-∆ effects.
In brief, overstrength is a result of a) the additional strength provided to limit structural
drifts, b) a greater than minimum member strength, c) a higher than minimum material
yield strength, and d) redistribution of forces due to redundancy. If each of these effects
is quantified and given a value (e.g., R1 through R4 ), the final overstrength component,
Ro , of the strength reduction factor is equal to their product, i.e., Ro =R1 R2 R3 R4 .
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The current seismic codes in the United States are combining all previously mentioned
overstrength effects, Ro , along with the effect of ductility and damping, Rd, to generate
different values of R for different types of lateral resisting structural systems. Thus, the
total R value for a specific system might be loosely given by R=Ro Rd. An important
factor implicitly considered in the R values given by codes is the past experience based
on the past performance of similar structures. If a type of structure performs well during
an earthquake, then its design has been proven to be effective, and it will probably be
used again. If a type of structure performs poorly, then that design is less likely to be built
in the future. This evolution has definitely formed the backbone of building codes.
Therefore, the R value is also a measure of the code committee’s confidence in a type of
structure. Besides the notion of satisfactory performance of real structures in past
earthquakes, seismic codes employ good engineering understanding of the basic
principles of structural mechanics and strength of materials and implicitly consider more
factors in their determination of the semi-empirical R values. Among these are the type of
structural materials used and the design process. Some materials fare better than others
during an earthquake. For instance, cyclic loading, large internal forces, and lateral
displacements have different effects on different structural materials. Also, the difference
in R values between ordinary (R=4) and special (R=8) moment steel frames is a good
example of how the design process affects R values. The ordinary steel moment frame is
designed with normal detailing of connections and splices, while a special moment frame
is designed with ductility and redundancy in mind. Extra attention is given to detailing at
connections so that the full moment capacities of the beams can be developed, and the
members will be sized to minimize plastification in the columns, prior to significant
yielding of the beams. These measures are intended to provide greater assurance that the
stability of the structure will remain intact during an earthquake. In addition, the
increased ductility and redundancy should provide significantly greater hysteretic
damping and thus justify an increase in R. Still missing important factors yet to be
considered in the R values suggested by codes are: the effect of the period of the
structure, the level of ductility it is designed for, and the type of soil in the site. Miranda
and Bertero (1994) studied the effects of ductility, period, and site characteristics on the
strength reduction factors, R, that together with an accurate estimate of the total
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overstrength of a given structure can lead to a more rational and transparent seismic
design approach than the approach currently used in seismic codes in the United States.
Like the R value, the Cd coefficient is decided on by semi-empirical reasoning. The Cd, as
defined in Figure 5.2, attempts to correlate the maximum elastic static drifts under the
code forces with inelastic dynamic drifts that occur during an earthquake. In the Mexican
building code and the Eurocode, the equivalent of the Cd is set equal to the reciprocal of
R, i.e. adopting the so-called “equal displacement” rule. This would physically correlate
to the structure deflecting the same amount, whether elastically under the unreduced code
forces, or inelastically under the reduced forces. This “equal displacement” rule seems to
have very little theoretical reasoning, yet it holds up well for the documented cases as
reported by Uang and Maarouf (1993). On the other hand, Newmark and Hall (1982)
state that, for structures with short periods, inelastic deformation can even be larger than
the elastic. However, to calculate expected inelastic deformations, UBC 1997, NEHRP
1997 and IBC 2000 recommend that the elastic deformations under the elastic forces be
scaled down by a ratio of 3/8 (UBC) or by a ratio (C d/R) that ranges from about 0.5 to
1.0, depending on the structural system (NEHRP and IBC), irrespective of the period of
the building.
5.1.2 Modal Response Spectrum Analysis
Modal analysis provides a more accurate approximation of the elastic dynamic response
than the equivalent lateral force procedure. However, modal analysis is also limited to
consideration of elastic response where superposition principle still holds. For modal
analysis, inelastic effects can be approximated using artificially large values of viscous
damping, although the scientific basis of this technique is questionable. By performing an
eigenvalue analysis on a structure, the natural frequencies of the structure can be
determined. The modal responses can then be calculated using a response spectrum curve
to find the maximum response for each mode. Definition of a design response spectrum
or selection of a specific earthquake response spectrum is one of the challenges of this
design method. The analysis should include a sufficient number of modes to obtain a
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combined modal mass participation of at least 90 percent of the actual building mass in
each of the two orthogonal directions, IBC 2000. IBC further suggests that the
combination, if using the response spectrum method, shall be carried out by taking the
square root of the sum of the squares of each of the modal values or by the complete
quadratic combination (CQC) technique.
Briefly, modal response analysis results in the maximum response for the elastic structure
for a given design spectrum or for a given earthquake. However, as with the ELF
procedure, the nonlinear response may be completely different. In general, it may be
possible to use the response (design) spectrum to approximately predict the response of
the structure to a maximum credible earthquake, but the theoretical validity of this
procedure is not thoroughly proven.
5.1.3 Time History Analysis
The IBC 2000 provisions require that time history analyses be performed with pairs of
appropriate horizontal ground-motion time-history components that should be selected
and scaled from not less than three recorded events. The IBC further specifies that time
histories should have magnitudes, fault distance and source mechanisms that are
consistent with those that control the maximum considered earthquake at the site. Using
the equation of motion, including mass, damping, stiffness matrices of the building, the
response of the structure due to the applied ground motion is calculated through a
stepwise numerical integration scheme. The analysis model may be as complicated or
simple as the designer desires, including both geometric and material nonlinearities as
desired. A more efficient way to conduct a full elastic time history analysis of a given
structure is to perform a modal time history analysis. Provided that one includes enough
modes (theoretically speaking all modes) in the modal time history analysis, the results
will be identical to a full time history analysis. However, while inelastic regular time
history analysis may be carried out, modal time history analysis is only limited to elastic
behavior. Nevertheless, the advantage of modal time history approach over modal
response spectrum analysis presented in the previous section is that the various modal
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responses can be superimposed directly in time, whereas in the modal response spectrum
method, various assumptions need to be made to superimpose the modal maximum
values.
Either process, full versus modal time history analysis, has the drawback that the
calculated response is only valid for a single specific earthquake. The behavior of the
structure to a different earthquake may be entirely different. To account for uncertainties
in response under different earthquakes, the IBC 2000 recommends that the parameter of
interest used for design should be chosen as the maximum response if three time history
analyses using three different appropriate records are performed. If seven or more time
history analyses are performed, then the average value of the response parameter of
interest may be used for design.
IBC 2000 further suggests that if either elastic (i.e., linear) or nonlinear time history
analysis is used, strength design should be used to determine member capacities.
Moreover, the responses computed from the nonlinear analysis should not be reduced by
R/I, where R and I are already defined in Section 5.1.1. Another issue that an engineer
performing a nonlinear dynamic analysis must resolve is the determination of when a
structure has reached its inelastic strength limit state (or failure state). An adequate and
efficient technique is presented in the following chapter as a solution to this problem.
5.1.4 Static Inelastic Pushover Analysis
The static pushover analysis is a simplified nonlinear analysis technique to estimate the
demands imposed on a structure by earthquake ground motions. Ideally, performance
evaluation of a structure should be based on nonlinear time history analyses utilizing a
suite of representative ground motions. However, the pushover analysis can identify
critical regions of high force and deformation demands and provide reasonable estimates
of overall structural behavior and expected damage. But one should always keep in mind
all the assumptions, simplifications and limitations of this method as pointed out by
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Saiidi and Sozen (1981), Fajfar and Fishinger (1988), Qi and Mohle (1991), Krawinkler
et al. (1997) among others.
The pushover analysis involves applying a predetermined lateral load pattern that
approximates the earthquake-induced inertia forces, and pushing the structure under this
load pattern to the level of deformation expected in a design earthquake. The level of
deformation might be calculated as suggested by FEMA-273 document through the socalled “displacement coefficient” method using a target displacement concept. The target
displacement, δ t , up to which the structure should be pushed, intended to represent the
maximum displacement likely to be experienced during the design earthquake, is given
by
δ t = C o C 1 C 2 C 3 Sa
Te2
4π 2
g
(5.8)
where Co is a factor relating spectral displacement and likely building roof displacement,
C1 is relating expected maximum inelastic displacements to displacement calculated for
linear elastic response, C2 is a factor to represent the effect of stiffness degradation and
strength deterioration on maximum displacement response, C3 is a factor to represent
increased displacements due to dynamic P-∆ effects, Te is the effective fundamental
period of the building in seconds, and Sa is the response spectrum acceleration at the
effective fundamental period and damping ratio of the building in terms of g. As such,
one might notice that the above equation is a way to relate the maximum MDOF inelastic
displacement to the maximum SDOF elastic displacement which is the spectral
displacement at the effective period of the structure, Sd, that might be written as
Sa( Te2 /4π 2 )g.
As explained above, it is obvious that the accuracy of the pushover analysis is dependent
on the distribution of the applied equivalent lateral forces (i.e., the lateral load pattern).
Moreover, it has been reported by different researchers (e.g. Lawson et. al., 1994) that the
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pushover results correlate well with dynamic results for short period structures whose
response is governed by first mode vibrations. On the other hand, for taller structures in
which higher mode effects become important, large differences in static and dynamic
results may occur and the use of the pushover procedure as an analysis and assessment
tool may be questionable. However, when applied with sound engineering judgment and
due regard to some of the method’s pitfalls, the pushover method can be an effective tool
for an approximate evaluation of deformation demands in critical elements within a
structure. For more confidence in results, multiple load patterns can be used to bracket
ranges of structural behavior. For more details about the pros and cons of a pushover
analysis, one may consult for instance Krawinkler and Seneviratna (1998).
In the “displacement coefficient” method adopting a pushover analysis up to a prespecified target displacement, in essence, the displacement demand is determined from
inelastic displacement spectra which are obtained from elastic displacement spectra by
using a number of correction factors based on statistical analyses. In principle, inelastic
spectra are expected to be more accurate than elastic spectra with equivalent damping
especially in the short-period range and for high ductilities.
An alternate approach to the displacement coefficient method is the capacity spectrum
method. It is the main method adopted by ATC 40 document for seismic evaluation and
retrofit of concrete buildings. Its popularity as a nonlinear static analysis procedure for
seismic design and evaluation is rapidly increasing as the structural engineering
community is now developing a new generation of design and rehabilitation procedures
that incorporate performance based engineering concepts.
The capacity spectrum method has been developed by Freeman (Freeman et al, 1975,
Freeman, 1998). By means of graphical procedure, this method compares the capacity of
a structure with the demands of earthquake ground motion on the structure. The graphical
presentation allows an intuitive explanation of how the structure will perform when
subjected to earthquake ground motion; i.e., whether or not the structure will survive the
event and, if it does survive, how damaged the structure will be. The capacity of the
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structure is represented by a force-displacement curve obtained by nonlinear static
(pushover) analysis as discussed previously. The base shear forces and roof
displacements are converted to spectral accelerations and spectral displacements of an
equivalent SDOF system, respectively. These spectral values define the capacity
spectrum. According to ATC 40, any point (Vi, ∆roof) on the capacity (or pushover) curve
is converted to the corresponding point (Sai, Sdi) on the capacity spectrum using the
equations:
V /W
Sai = i
α1
Sdi =
∆ roof
PF1 * φ1, roof
(
(5.9)
)
(5.10)
where α 1 and PF1 are respectively the modal mass coefficient and participation factors for
the first natural mode of the structure, and φ 1,roof is the roof level amplitude of the first
mode.
Demands of the earthquake ground motion are defined by highly damped elastic spectra
to simulate the damping experienced by a structure at different damaged states up to the
verge of collapse. These curves are represented in either the traditional AccelerationPeriod Response Spectrum (APRS) format or the Acceleration-Displacement Response
Spectrum (ADRS) format, in which spectral accelerations are plotted against spectral
displacements, with the periods T represented by radial lines as shown in Fig. 5.3. As the
structure is incrementally loaded, individual members begin to plastify. The structure will
then undergo progressive plastification under excessive loading, and as this damage
occurs, the stiffness of the structure will decrease, and the instantaneous period of the
structure will lengthen. This lengthening behavior is quite obvious in the APRS format.
The intersection of the capacity spectrum and demand spectrum estimates the inelastic
strength and displacement demand under the given earthquake.
170
Demand spectrum
Spectral Acceleration
Spectral Acceleration
Demand spectrum
Capacity spectrum
5% damped
B
A
15% damped
T1 T2
Period, T
T1
Capacity spectrum
T2
B
A
T3
T3
Spectral Displacement
Traditional APRS Spectrum
ADRS Spectrum
Figure 5.3 Capacity spectrum superimposed over demand response spectra.
A controversial aspect of the capacity spectrum method is the use of highly damped
elastic spectra for the determination of seismic demand. According to Krawinkler (1995),
“there are two fundamental flaws that render the quantitative use of the capacity spectrum
method questionable. First, there is no physical principle that justifies the existence of a
stable relationship between the hysteretic energy dissipation of the maximum excursion
and equivalent viscous damping, particularly for highly inelastic systems. The second
flaw is that the period associated with the intersection of the capacity curve with the
highly damped spectrum may have little to do with the dynamic response of the inelastic
system.”
The questionable representation of the seismic demand in the capacity spectrum method
by highly damped elastic spectra can be eliminated as proposed by Fajfar (1998) through
the use of inelastic (i.e., ductility-based) demand spectra. In principle, seismic demand at
different hazard (or intensity) levels can be defined by any inelastic spectra. However, the
specific ductility-based response spectra presented by Fajfar (1998) are based on
statistical analyses in which the near-fault impulsive type of ground motion has not been
included. Furthermore, the proposed spectra are not suitable (too conservative) in the
very long-period range. In this range, spectral displacements should be equal to the peak
ground displacement. Therefore, additional research on these topics is needed.
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5.2 Case Study Building Designs
The case study building investigated in this research is designed according to the general
layout of a theme structure (with an architecture plan as given in Figure 5.4) proposed as
part of the US-Japan program on hybrid structures. This theme structure is intended to
define a standard floor plan with framing and service core layouts representative of
typical office building construction. Using this layout, three case study buildings have
been designed for this research: (1) a 6-story composite RCS Special Moment Frame
(SMF) building, (2) a 12-story RCS SMF building, and (3) a 6-story STEEL SMF
building. The buildings are assumed to be located in high seismic region and are designed
according to appropriate portions of the following standards: IBC 2000, 1997 AISC
Seismic Provisions, ASCE7-95 Standards for Minimum Design Loads, AISC-LRFD
(1993), ACI-318 (1995), and the ASCE Design Guidelines for Moment Connections
Between Steel Beams and Reinforced Concrete Columns (1994). Note that the seismic
31.5’ (9.60m)
42’ (12.80m)
115.5’ (35.20m)
42’ (12.80m)
provisions of the IBC 2000 are very similar to those in the 1997 NEHRP.
6 @ 21’ = 126’ (6 @ 6.40m = 38.40m)
Figure 5.4 Architecture Plan of US-Japan Theme Structure.
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Seismic design forces are based on mapped spectral accelerations Ss=1.5g and S1 =0.72g
using IBC 2000. These values result in similar design base shears to ones obtained per the
1994 NEHRP with Aa=Av =0.40g, comparable to what has traditionally been termed
“seismic zone 4”. The soil condition at the buildings location is assumed to be that of the
site class D as per IBC 2000. The buildings are assigned a Seismic Use Group I and a
Seismic Design Category D according to the previously mentioned seismic provisions.
The equivalent lateral force static procedure as outlined in Section 5.1.1 is used for the
design process of the buildings. In all designs, a space frame concept has been used, i.e.,
the lateral load carrying systems consist of seven Special Moment Frames (SMF) with
three unequal bays in the short direction, and four SMFs with six equal bays in the long
direction. A structural plan and structural elevations for the 6-story RCS building are
shown in Figs. 5.5 and 5.6. Member dimensions and properties for all three building
designs (6- and 12-story RCS and 6-story steel) are given in Tables 5.1 through 5.3.
Rolled W shapes (Grade 50 steel) are used for beams and 6ksi normal weight concrete is
used for RC columns. Longitudinal and transverse steel reinforcement of the columns is
designed according to seismic details and recommendations as given in ACI-318 Chapter
21 with a nominal yield strength of 60ksi. Transverse column reinforcement consists of
#4 closed hoops plus #4 single ties (total of 4 branches) every 3 inches with detailing as
shown in Figures 5.7 and 5.8.
Table 5.1 Main design details and cross-sections dimensions of 6-story RCS building.
Floor #
COLUMNS
BEAMS
Outer Columns
Inner Columns
Short Direction
Long Direction
(short direction
(short direction
Frames
Frames
fr.)
fr.)
1-4
25.6”x25.6”
25.6”x25.6”
W 24x68
W 18x60
(650x650 mm)
(650x650 mm)
12#9 bars
12#10 bars
5-6
23.6”x23.6”
23.6”x23.6”
W 21x62
W 16x40
(600x600 mm)
(600x600 mm)
12#8 bars
12#9 bars
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Table 5.2 Main design details and cross-sections dimensions of 12-story RCS building.
Floor #
COLUMNS
BEAMS
Outer Columns
Inner Columns
Short Direction
Long Direction
(short direction
(short direction
Frames
Frames
fr.)
fr.)
1-3
33.5”x33.5”
33.5”x33.5”
W 27x94
W 21x62
(850x850 mm)
(850x850 mm)
12#9 bars
12#9 bars
4-6
31.5”x31.5”
31.5”x31.5”
W 27x94
W 21x62
(800x800 mm)
(800x800 mm)
12#9 bars
12#9 bars
7-9
29.5”x29.5”
29.5”x29.5”
W 24x84
W 18x60
(750x750 mm)
(750x750 mm)
12#8 bars
12#8 bars
10 - 12
25.6”x25.6”
25.6”x25.6”
W 24x68
W 18x50
(650x650 mm)
(650x650 mm)
12#8 bars
12#8 bars
Table 5.3 Main design details and cross-sections of 6-story STEEL building.
Floor #
COLUMNS
BEAMS
Outer Columns
Inner Columns
Short Direction
Long Direction
(short direction
(short direction
Frames
Frames
fr.)
fr.)
1-4
W 14x370
W 14x370
W 24x68
W 18x60
(strong axis)
(strong axis)
5-6
W 14x311
W 14x311
W 21x62
W 16x40
(strong axis)
(strong axis)
As shown in Figures 5.7 and 5.8, alternate concrete column details are provided based on
a cast-in-place and precast construction method, again for the case of the 6-story RCS
space frame design. The cast-in-place method involves placing the column concrete after
erection of structural steel. As shown in Figure 5.7, this requires the use of small steel
erection columns (W10 shapes) that are later encased in concrete. The alternate
construction method (Figure 5.8) involves precasting the columns and connecting them in
the field using grouted sleeve connectors. Depending on which construction method is
used, the details of the beam-column joint will vary somewhat. However, as shown in
Figure 5.9, the basic features of the joint detail for either method of construction are
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similar. The joint detail shown in Figure 5.9 is similar to details used in high-rise
buildings and to joint subassemblies that have been seismically designed, detailed and
tested at the University of Texas at Austin and Cornell University (Kanno and Deierlein
1994, 1996).
Beams for
Lat. Bracing
B
B
Bolted
Field
Splice
3-1/4” slab
2” deck
W18
or W 16
26” x 26”
(Typical)
W 21
W 14 (Typ.)
W 24 or
31.5’ (9.60m)
42’ (12.80m)
42’ (12.80m)
11 @ 10.5’ (11 @ 3.20m)
115.5’ (35.20m)
A
6 @ 21’ = 126’ (6 @ 6.40m = 38.40m)
Figure 5.5 Typical structural plan for 6-story RCS building.
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A
6
79’ (24.00m)
6 @ 13’ (6 @ 4.00m)
W 21x62
Beam
Splice
5’
5
23.6” x 23.6”
5’
W 24x68
4
25.6” x 25.6”
W 24x68
3
2
1
42’ (1 2.80m)
31.5’ (9.60m)
115.5’ (35.20m)
42’ (12.80m)
Frame Elevation A
6
6 @ 13’ (6 @ 4.00m)
79’ (24.00m)
W 16x40
5
23.6” x 23.6”
W 18x60
4
25.6” x 25.6”
W 18x60
3
2
1
6 @ 21’ (6 @ 6.40m)
126’ (38.40m)
Frame Elevation B
Figure 5.6 Elevation of typical frames in both directions – 6-story RCS building.
176
12 # 9 or # 10 bars
25.6”
# 4 bars
W 10
Erection
Col.
25.6”
L = 27”
Ties #4 @ 6”Ties #4 @ 3”
X
Mid-height
of
Column
Ties #4 @ 3”
L = 27”
Lap Splice
L = 37.5”
Section X - X
See Fig. 5.8 for
Joint Detail
W 18x60
W 24x68
Figure 5.7 Cast-in-place RC column details.
177
X
12 # 9 or # 10 bars
25.6”
# 4 bars
25.6”
X
L = 48” Typical
Leveled and Grouted
Field Splice
Grouted Sleeve
Connection
Ties #4 @ 3”
Ties #4 @ 6”
Section X - X
See Fig. 5.8 for
Joint Detail
W 18x60
W 24x68
Figure 5.8 Precast RC column details.
178
X
B
W18x60
A
W24x68
25.6”
B
Section 1 - 1
W18x60
FBP
25.6”
A
25.6”
Band
Plate
W24x68
25.6”
Section 2 - 2
Figure 5.9 Joint details for 6-story RCS building.
179
Band Plate
1
1
W 18x60
#9 or #10 bars
2
FBP
W 24x68
25.6”
Section A - A
Band Plate
W 24x68
#9 or #10 bars
FBP
W 18x60
E-FBP
25.6”
Section B - B
Figure 5.9 Joint details for 6-story RCS building. (Continued)
180
2
The following are a few key aspects of the joint detail of RCS buildings and their
implications on construction:
•
Ideally, the beams should run continuous through the joint so that they are not
interrupted at the column face where moment forces are maximum. As shown in Fig.
5.9, in the two-way space frame the choice is made to have the deeper W24x68
beams continuous with the smaller W18x62 beams spliced as shown.
•
The W24 beams that run continuous through the joint will be field spliced using
bolted moment connections several feet away from the column in a region of lower
moment.
•
Detailing and installation of transverse column reinforcement is a critical aspect of
the joint design. Congestion for transverse ties can be reduced by using the external
steel bands as shown above and below the joint. Alternate detailing measures that
could be considered to improve constructability might involve using welded rebar
anchors, fiber reinforced concrete, etc…
•
Reliable concrete placement in the joint will probably require the use of high slump
concrete made with superplasticizers.
Accordingly, the following are a few additional points to note regarding differences in the
connection and member details for the two methods of construction:
•
Pre-cast Column Method: Where the columns are precast, short sections of steel
beams will be cast into the column in the two framing directions. Therefore, bolted
field splices will be required for beams framing in both directions. Systems similar to
this precast type of construction have recently been used in Japan for the construction
of low-rise office and retail buildings.
•
Cast-in-place Method: As shown in Fig. 5.7, the steel erection columns required for
this method of construction will be interrupted at the joints to allow the larger steel
beams to pass continuous through the joints. Since the steel beams and columns will
be erected first, the longitudinal and transverse steel for the reinforced concrete
columns will need to be installed around the steel (erection) column. Field
connections for the smaller W18 beams can probably be done right in the beam-
181
column joint without any additional field splices. The deeper W24 beams will still be
spliced outside the joint, however, since it is not essential to have splices on both
sides of the joint, this scheme may require fewer bolted moment splices than the
precast scheme.
5.2.1 Overview of the ASCE Design Criteria for Composite Beam-Column Joints
This section presents a brief summary of the ASCE design guidelines (1994) for
composite moment connections between steel beams and reinforced concrete columns.
The guidelines are based on early work on composite joints conducted by Sheikh et al.
(1989) and Deierlein et al. (1989). These recommendations address the proportioning and
detailing of these joints, taking into account the interaction of the structural steel and
reinforced concrete components. The recommendations are based primarily on tests of
cruciform-shaped specimens of typical joints where the steel beams are continuous
through the reinforced concrete column. Calculating the joint strength is the main design
aspect presented in this section. For other detailing considerations including stiffeners
and reinforcement the reader should refer to the ASCE guidelines. Although the
guidelines mention that use of composite joints is limited to regions of low-to-moderate
seismic zones, more recent work by Kanno et al. (1994) shows that composite joints are
equally effective for regions of high seismicity.
The joint strength should be checked using the AISC-LRFD method. Using the design
guidelines, the connection strength is determined by considering several individual
failure modes such as steel yielding or concrete crushing. As discussed in Chapter 2,
composite joint behavior is characterized by two modes of failure: (1) panel shear failure
involving both structural steel and reinforced concrete panel elements, and (2) bearing
failure occurring at locations of high compressive stresses and associated with rigid body
rotation of the steel beam within the concrete column. Addition of vertical joint
reinforcement is sometimes used as one means of strengthening against bearing failure.
Accordingly, the joint strength should be checked for these two basic failure modes.
182
Basically, joint design strength is obtained by multiplying the nominal strength by a
resistance factor, φ. ASCE guidelines suggest a value of φ=0.7 due to lack of
experimental data and to provide a conservative value that is approximately 20% below
the value of φ=0.85 used for composite members in the AISC-LRFD Specification
(1993). This lower value of φ reflects the guidelines philosophy of providing a greater
reliability index for composite connections. However, based on more recent research
conducted on composite joints, Kanno (1993) found that the ASCE method provides
conservative joint strength when compared to test results. Accordingly, the φ value can be
relaxed and a similar value to that used for composite members by AISC-LRFD is
suggested (i.e., φ=0.85). Due to the interaction between steel and concrete mechanisms, a
single φ factor is used throughout the design (regardless of the individual modes of
failure).
Following same notation given by ASCE guidelines, the vertical bearing nominal
moment strength, Mbr, of the composite joint is given by
Mbr = 0.7 h Ccn + hvr (Tvrn + Cvrn )
(5.11)
where Ccn is the nominal compression strength of bearing zone and h is the depth of
concrete column measured parallel to beam. Ccn is given as 0.6f’cbjh where bj is the
effective width of the joint panel. Ccn is calculated using a bearing stress of 2f’c over the
bearing area (0.3h long and bj wide). The maximum bearing stress 2f’c reflects
confinement of the concrete by reinforcement and the surrounding concrete based on test
data by Sheikh et al. (1989) and Deierlein et al. (1989). Tvrn and Cvrn are the nominal
strengths in tension and compression, respectively, of the vertical joint reinforcement, if
any, which is attached directly to the steel beam, and hvr is the distance between the bars.
This detailing is not used for the RCS case study buildings in this thesis.
On the other hand, the nominal moment strength of the joint due to shear behavior, Msh ,
is the sum of the nominal moment resistance of the following components: (1) the steel
panel (i.e., the web of the continuous beam running through the column), Msn ; (2) the
183
inner concrete compression strut, Mcsn ; and (3) the outer concrete compression field, Mcfn.
Msh may be thus given as
Msh = Vsn df + 0.75Vcsn dw + Vcfn (d + do )
(5.12)
where df is the center-to-center distance between the beam flanges, dw is the depth of the
steel web, do is the additional effective joint depth provided by attachments to beam
flanges such as extended Face Bearing Plates, E-FBP (refer to Fig. 5.9), and d is the
depth of steel beam measured parallel to column. Tests have shown that the contributions
of the three joint shear mechanisms are additive as manifested by Equation 5.12. The
concrete contribution comes from the concrete compression strut that forms within the
inner panel width, bi (taken equal to the greater of the FBP width, bp , or the beam flange
width, bf), and the compression field that forms in the outer panel width, bo . The concrete
compression strut is mobilized through bearing against the FBPs within the beam depth.
The compression field is mobilized through a horizontal strut and tie mechanism that
forms through bearing against either a steel column (e.g., the erection column) above and
below the beam and/or extended FBPs.
In Equation 5.12, Vsn is calculated as 0.6Fysp t sp jh, where Fysp and t sp are the yield strength
and thickness of the steel panel respectively. jh is the horizontal distance between bearing
force resultants given by Equation (16) of the ASCE guidelines. Vcsn is given as
1.7
f c' bp h provided it is less than or equal 0.5f’cbp dw. All terms are as defined before;
f c' and f’c are in MPa. Vcfn is limited by the sum of forces resisted by the horizontal
column ties, Vs’, and the concrete, Vc’, provided this sum is less than 1.7 f c' bo h. Vc’ is
given as 0.4 f c' bo h except where the column is in tension, in which case, Vc’ = 0. Vs’ is
calculated as Ash Fysh 0.9h/sh , where Ash is the cross-sectional area of reinforcing bars in
each layer of ties spaced at sh through the beam depth, and Fysh is the yield strength of the
reinforcement. The ASCE guidelines recommend that within the beam depth, one pair of
cap ties in each layer should pass through holes in the beam web (refer to Fig. 5.9) to
184
provide continuous confinement around the joint. Tests have shown that the holes in the
beam web do not reduce the web shear capacity, provided that: (1) the holes are located
within 0.15h of the face of the concrete column, and (2) the ratio of the net area to the
gross area of the web, measured at the holes, is greater than 0.7. The FBPs provide
confinement in the center of the column which enhances the anchorage and development
of the cap ties.
5.2.2 Summary of Design Values and Governing Criteria
Unit floor loads assumed in the design are 76psf and 50psf for dead and live loads,
respectively, for typical floors. For the roof, the assumed dead and live loads are 67psf
and 50psf, respectively. The live load of 50psf is based on the office floor load specified
in the ASCE 7-95 standards. Composite steel/concrete floor deck (VULCRAFT type) is
used with 5.25/3.25 inch slab/deck thickness. 10psf partition loads are also included in
the unit floor dead loads given above. At the perimeter of the building, a concentrated
wall load based on a 20psf wall weight is also assumed. While the design meets all
relevant load combinations as specified by the IBC (2000), the controlling load
combination was the one combining dead and live loads with the earthquake effects as
given by 1.2D+0.5L+1.0E, where E is computed per Equation 5.4.
Based on the above assumed dead loads including the 10psf partition loads required by
code, seismic masses used in period calculation as well as in the time history analyses are
determined. These masses are given (in kips.sec2 /ft) in Table 5.4 for the roof and typical
floors for the three case study frames.
Floor Type
Typical
Roof
Table 5.4 Seismic masses for case study frames.
6-Strory RCS
12-Story RCS
6.49
7.37
5.65
5.58
185
6-Story STEEL
6.12
5.39
For completeness, Figs. 5.10 through 5.12 show gravity and design lateral loads for an
inner frame in the short direction for the three case study buildings: 6-story RCS, 12story RCS, and 6-story STEEL, respectively, again based on the above assumed loads.
Roof Load:
WDL = 1.426 k/ft
WLL = 1.054 k/ft
Typical Floor Load:
WDL = 2.121 k/ft
WLL = 1.054 k/ft
51.6 k
41.1 k
26.3 k
14.8 k
6.6 k
1.6 k
Figure 5.10 Gravity and design lateral loads for the 6-story RCS frame.
186
Roof Load:
WDL = 1.426 k/ft
WLL = 1.054 k/ft
Typical Floor Load:
WDL = 2.121 k/ft
WLL = 1.054 k/ft
32.8 k
36.4 k
30.1 k
24.4 k
19.2 k
14.7 k
10.8 k
7.5 k
4.8 k
2.7 k
1.2 k
0.3 k
Figure 5.11 Gravity and design lateral loads for the 12-story RCS frame.
187
Roof Load:
WDL = 1.473 k/ft
WLL = 1.054 k/ft
Typical Floor Load:
WDL = 1.844 k/ft
WLL = 1.054 k/ft
38.8 k
30.8 k
19.7 k
11.1 k
4.9 k
1.2 k
Figure 5.12 Gravity and design lateral loads for the 6-story STEEL frame.
Main seismic relevant properties of the different designs are given in Table 5.5 for an
inner frame in the short direction for each building. Note that the weight shown in Table
5.5 is for the whole building. Also, note that the design base shear values shown include
the effect of accidental torsion. They are further based on the upper limit, 1.2Ta, allowed
by code for the period calculation, where Ta is as per Equation 5.7. Ta is calculated for the
6- and 12-story RCS frames as 0.79sec. and 1.33sec., respectively (based on CT =0.030),
while it is 0.93sec. for the 6-story STEEL frame (with CT =0.035). It is obvious that the
approximate code Equation 5.7 is underestimating the fundamental period of the three
frames. Note that the period values given in Table 5.5 are calculated using DYNAMIX
and considering composite beams and joint panel size and flexibility effects.
188
Table 5.5 Summary of design parameters for case study buildings.
Item
6-Strory RCS
12-Story RCS
6-Story STEEL
Weight W, kips
8569
18880
7518
Vdesign / W
0.116
0.069
0.099
Period To , sec.
1.25
2.07
1.26
For comparison purposes, Table 5.6 gives Vdesign /W ratio for the case study frames for
four different cases: (1) a lower bound where Vdesign is calculated based on the actual
period (per Table 5.5) and not accounting for accidental torsion, (2) Vdesign is determined
as in (1) but considering accidental torsion, (3) Vdesign is based on the code limit on the
period (1.2Ta) and again ignoring accidental torsion effects, and (4) an upper bound of
the design base shear where Vdesign is as given in Table 5.5, i.e., based on 1.2Ta, and
accounting for accidental torsion. Note that values corresponding to Cases (3) and (4) can
be calculated at the early stages of the design, as it is meant to be. On the other hand,
values of Vdesign /W for Cases (1) and (2) are only determined at the last phase of the
design process (i.e., once we have the final design configuration with accurate values for
all member properties). Moreover, overstrength values, Ω, given in the next two chapters
for the case study frames are obviously directly related to Vdesign /W ratio. Although,
according to code, Ω should be based on the value of Vdesign /W associated with Case (4),
actual overstrength of the frame (corresponding to Case (2)) is significantly larger.
Table 5.6 Comparisons of different Vdesign /W ratios for the case study frames.
Case #
6-Strory RCS
12-Story RCS
6-Story STEEL
(1)
0.072
0.044
0.071
(2)
0.088
0.053
0.087
(3)
0.095
0.056
0.081
(4)
0.116
0.069
0.099
All of the building designs satisfy the following major criteria for seismic design: drift
requirements, strength requirements, and strong column-weak beam criterion (SCWB).
For RCS buildings design, all composite joints satisfy the following criterion:
Mjoint / (∑ 1.1 Ry Mp,beam) ≥
1.0
(5.13)
189
where Ry (taken herein as 1.15) is the multiplier specified by 1997 AISC Seismic
Provisions to consider the effect of the “expected yield strength, Fye”, Mp,beam is the beam
plastic moment capacity (calculated based on the specified minimum yield strength, Fy),
and Mjoint is the joint moment strength. Mjoint is calculated as the minimum of Mbr and Msh
per Equations 5.11 and 5.12, respectively. If this limitation is satisfied, it is considered
that failure is fully controlled by beam hinging of the beam running continuous through
the joint, and sufficient seismic resistance will be provided as long as the beam is
designed following proper seismic codes and specifications (1997 AISC Seismic
Provisions).
Concerning drift requirements, the inelastic story drift for all case study buildings
satisfies the maximum limit of ∆sx,in < 0.02 hsx , where ∆sx,in is the inelastic interstory drift
calculated per Eq. 5.6 and hsx is the story height. The limiting ratio of 0.02 applies for all
buildings with seismic use group I, higher than four stories, and other than masonry shear
wall or masonry wall frame buildings. As permitted by seismic codes (e.g., IBC 2000),
for purposes of this drift analysis, the redundancy coefficient ρ in Equation 5.4 is taken as
1.0, and upper bound limitation on the computed fundamental period, T, of the building
used for the determination of the design base shear (as per Equation 5.1) is ignored.
It is useful to note that drift requirements control the design of the 6-story steel frames,
while both drift and SCWB criteria control the design of the 6- and 12-story RCS frames.
For RCS frames design, the SCWB criterion is imposed as follows:
∑ M p,column / ∑ (1.1 R y M p,beam ) ≥ 1.0
(5.14)
The numerator presents the sum of moments, at the center of the joint, corresponding to
the nominal flexural strength of the reinforced concrete columns framing into the joint.
As suggested by ACI 318-95, Chapter 21, column flexural strength is calculated for the
factored axial force, consistent with the direction of the lateral forces considered,
resulting in the lowest flexural strength. The denominator is the sum of the moments in
the steel (or composite) beams at the intersection of the beam and column centerlines. All
190
terms are defined before for Equation 5.13. To satisfy Equation 5.14 at all column-beam
connections, the longitudinal reinforcement in columns is sometimes increased than the
amount
needed
for
strength
requirements,
thus
avoiding
changing
the
columns
dimensions already satisfying drift requirements. The SCWB concept represents more of
a global frame concern than a concern at the interconnections of individual beams and
columns. The real benefit of satisfying the SCWB criterion is that the columns are
generally strong enough to force flexural yielding in beams in multiple levels of the
frame, thereby achieving a higher level of energy dissipation. However, it should be
noted that compliance with the SCWB concept and Eq. 5.14 gives no assurance that
individual columns will not yield, even when all connection locations in the frame
comply. Nonetheless, it is believed that yielding of the beams rather than columns will
predominate and the desired inelastic performance will be achieved in frames composed
of members that meet the requirement in Eq. 5.14.
5.3 Selection of Ground Motion Records
Time history dynamic analyses (linear or nonlinear) require the availability of
earthquake records compatible with the site seismic characteristics. Moreover, the chosen
records should be consistent and representative of a given level of earthquake depicting a
specific prescribed hazard level at the site under consideration. The seismic hazard at a
site is characterized by potential earthquakes that may occur at the site during the lifetime
of the building, typically described by the magnitude of potential earthquakes and their
proximity to the site.
In brief, once a site is investigated, one should select a set of records compatible with the
site seismicity. Then, the hazard is identified in terms of a response spectrum (or a time
history acceleration) with a given probability of exceedance (or a given recurrence
period) to which the chosen records are scaled up and/or down to represent the hazard at
the site. Finally, the building located in that site is subjected to this suite of scaled records
and a time history analysis is carried out.
191
By scaling of ground-motion records, we mean to increase or decrease each of the
ground-motion records by a constant factor so that the spectral acceleration at a given
frequency and damping is equal to the target spectral acceleration. In this process, the
spectral shape, relative phases, and duration of the ground motion remain unchanged. The
advantage of scaling of records (demonstrating magnitude, M, and distance, R,
conditional independence of response given spectral acceleration) is that when we are
given a target ground motion intensity we need not be overly concerned with what is the
M and R of the ground motion records that we use for structural analysis.
There is, in fact, a wide-spread concern in the engineering community regarding the
practice of scaling records. For example in Han and Wen (1994) it is stated that “scaling
an earthquake to attain a target damage level of different intensity is questionable since
scaling a ground motion does not account for variations in ground motion characteristics
(e.g., frequency content) which change with intensity”. Many researchers have stated that
scaling procedures based on a single parameter (frequently Peak Ground Acceleration,
PGA) do not work well across the entire spectrum of structural oscillator frequencies and
that they have to be discouraged. The dependency of ground motion spectral
characteristics on M and R (and therefore “intensity”) has been recognized by
seismologists for many years. As stated by Bazzurro et al. (1998) “a M=5 record scaled
to match the PGA of a M=7 record will certainly be deficient in the frequency content
below 1Hz, and its use would therefore underestimate the response of a long period
building”. Shome et al. (1999) thus suggested a scaling strategy which is structurefrequency-specific in which the record is scaled to match a target spectral acceleration at
the fundamental frequency (period) of the structure, unlike the single-parameter allstructures scaling procedures proposed in the past. This scaling strategy is the one used
throughout this research. Accordingly, if the seismic response of two structures with
same damping, ξ, but different fundamental periods, T1 and T2 , were to be analyzed, the
same record would be scaled differently to match the different values of the target
Sa(T1 ,ξ) and Sa(T2 ,ξ). It has been shown (Shome et al., 1999) that within reasonable
limits, scaling records using this frequency-specific approach not only does not
192
“significantly” alter the median structural response displacements but also considerably
reduces the “conditional” variability in the response itself. This last property as will be
discussed in details in Chapter 6 enables one to run many fewer analyses to attain the
same desired level of accuracy in the response estimation. Among other scaling strategies
used are scaling to a weighted average spectral acceleration over a range of periods
including higher mode effects and period lengthening due to damage, or scaling to a
spectral acceleration level averaged over a period band (e.g., ±15%) around the
fundamental period of the structure, etc… Scaling to the spectral acceleration at high
level of damping might introduce some reduction in the conditional dispersion of the
response mainly as a result of smoothing variations in the acceleration response spectra
values. However, the use of 5% damping is suggested to be able to use widely available
attenuation laws and seismic hazard information. A detailed discussion is given in
Chapter 6 concerning this point.
For the research carried out in this thesis, two sets (or bins) of records are selected for
buildings assessment. Each bin is composed of eight recorded ground motions. As such,
one can investigate the appropriateness of the code suggestion of basing the response
quantity of interest on the average value rather than on maximum values. The two bins of
records represent a suite of eight general records and a suite of eight near-fault records,
respectively. Main characteristics of each bin are given below. We have to keep in mind
that for the purpose of this research, we are dealing with a set of buildings located at a
generic site (with high seismicity) rather than at specific sites governed by specific
ground motion characteristics in terms of faulting mechanisms, magnitudes, and
distances. In other words, the study of the dependency of different response parameters
on magnitude, distance or faulting mechanisms is not considered in this thesis. We are
rather looking at the response, at different hazard levels, to general types of earthquake
events as well as to near-fault or (near-source) events characterized by their impulsive
effects as will be discussed in the following sections.
193
5.3.1 General Records
For the purpose of this study, general records are defined as those recorded at moderate to
large distances (above 10 to 15 km) from earthquake faults that do not exhibit directivity
effects of “near-fault” records. Six out of the eight accelerograms considered are recorded
in California on stiff soil (site class D as per NEHRP 1997 and IBC 2000). The other two
recorded motions are the magnitude 8 Valparaiso (1985) earthquake at Llolleo station in
Chile also derived from recording on soil category D, and the magnitude 7.4 Miyagi-oki
(1978) earthquake at Ofuna station in Japan. The latter record represents a recorded event
on rock converted to stiff soil (site class D) by Somerville(1997) as part of the SAC Steel
Project. Main characteristics of the bin of general records are given in Table 5.7.
The eight records given in Table 5.7 might be considered as representing two possible
scenario events: one representing closer, smaller magnitude events and the other more
distant, larger magnitude events. The larger events are represented by the Valparaiso
(1985) and the Miyagi-oki (1978) records, and the moderate events are the other six
records in the bin. A controlling criterion in our selection of general records is their
spectral acceleration at the fundamental period of the case study buildings. A reasonable
value (not very low) is targeted to avoid large scaling needed to reach different hazard
levels (e.g., 2%in50years level representing near collapse state or Basic Safety
Earthquake 2, BSE-2, as per FEMA 273) for performance assessment of buildings. For
comparison purposes, acceleration response spectra of the selected ground motions are
shown in Figure 5.13 along with the target 2%in50years response spectrum for the site of
the buildings derived according to IBC 2000 provisions.
Among data given in Table 5.7 are (1) the strong motion duration, tSM, as proposed by
Trifunac and Brady (1975) and defined in Chapter 6, (2) the moment magnitude M, (3)
the distance R to the rupture zone (not the epicentral distance), and (4) the peak ground
acceleration, PGA. Acceleration, velocity, and displacement time histories and response
spectra of the records of the bin are given in Appendix A.
194
2.0
Miyagi
Valparaiso
LP89-HCA
LP89-HSP
LP89-WAHO
CM92-RIO
LA92-YER
Mendocino
2%in50yrs,
IBC 2000
Sa [g]
1.5
1.0
RCS 12-story
T1=2.07sec
0.5
RCS 6-story STEEL 6-story
T1=1.25sec T1=1.26sec
0.0
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
Period, T [sec.]
Figure 5.13 Comparison of acceleration response spectra of general records and the
2%in50years site response spectrum (IBC 2000).
Table 5.7 Main characteristics of general records.
M
R
PGA
tSM
Station
[km]
[g]
[sec]
Miyagi-oki (1978)
7.4
66.0
0.44
17.72
Ofuna - Japan
Valparaiso (1985)
8.0
42.0
0.54
37.55
Llolleo - Chile
Loma Prieta (1989)
6.9
28.2
0.27
17.405
Hollister City Hall
Loma Prieta (1989)
6.9
28.8
0.37
16.395
Hollister South & Pine
Loma Prieta (1989)
6.9
16.9
0.37
10.465
WAHO
Cape Mendocino (1992)
7.1
18.5
0.39
15.36
Rio Dell Overpass
Mendocino (1992)
7.1
8.5
0.50
17.9
Petrolia
Landers (1992)
7.3
24.9
0.24
17.6
Yermo Fire Station
Earthquake
5.3.2 Near-Fault Records and Directivity Effects
Sites located near the rupture zone are naturally more affected than distant sites. Ground
motions recorded at these sites have shown the following particularities: 1) richness in
high frequency, 2) enhanced long period spectral content, 3) high PGV and PGD, and 4)
pulse-like time histories (Mahin and Bertero, 1978, Anderson and Bertero, 1987, and
Somerville et al., 1997). As pointed out by Krawinkler and Alavi (1998) among others,
195
the presence of pulses in earthquake records is a strong indicator of potential
destructiveness of that given record. Pulses, likely to occur in near-fault regions as a
result of fault rupture, are particularly dangerous to structures responding inelastically
since they put high demand on the lower floors of a building increasing their vulnerability
to P-∆ effects (Anderson and Bertero, 1987). However, structures designed to remain
elastic should not be affected by pulses (Mahin and Bertero, 1978).
Not all buildings near the epicenter are affected in the same way. It all depends on the
location and direction of the building with respect to the direction of seismic waves and
the closeness to the rupturing fault. If the angle between the source-to-recording site
vector and the direction of rupture propagation is small, the recorded ground motion may
be substantially increased in amplitude (Joyner and Boore, 1988). This phenomenon is
called directivity and has been observed in sites near fault rupture zones away from the
epicenter in case of strike-slip faults and updip in the case of dip-slip faults (Somerville et
al., 1997). The effects of directivity can also be observed in building damage where, for
example, many structures damaged by the Northridge earthquake exhibited northward
directivity and structures damaged in Kobe showed northwest directivity.
Directivity, as pointed out by Somerville et al. (1997), is the result of rupture propagation
toward a given site at a velocity close to the shear wave velocity of the rock. This causes
most of the rupture energy to arrive in a single large long-duration pulse that occurs in the
beginning of the record in the direction perpendicular to the fault. Moreover, due to the
nature of the phenomenon, directivity is mainly felt by sites close to the rupture zone but
not too close to the epicenter. Among the first earthquakes where directivity was
identified, was the 1979 Imperial Valley earthquake.
Generally, fault rupture occurs at an initial point and then propagates in one or two
(opposite) directions. In case of propagation in one direction only, records at sites in the
rupture and slip direction and close to the fault will be short and impulsive,
corresponding to forward directivity, whereas records at sites in the direction opposite to
propagation will be longer and pulse-less, corresponding to backward directivity (Paulay
196
and Priestly, 1992, Somerville and Graves, 1993, and Somerville et al., 1997). Moreover,
if rupture propagation direction is known, it is possible to map the recorded motion into a
set of axes parallel and perpendicular to the rupture direction. This has been implemented
by Somerville (1997) for the SAC project where he resolved the records into strikenormal and strike-parallel components. Fault normal components are much more
destructive to structures than fault parallel components. This has been revealed through
the near-fault records processed by Somerville (1997) as reflected by the difference in the
response spectra (acceleration, velocity, and displacement) of fault normal and fault
parallel components of a given near-fault record with forward directivity, with the former
considerably larger than the latter.
In the present work, eight near-fault records with forward directivity are selected. The
basic properties of the recorded motions are given in Table 5.8. The eight records span
the magnitude range of 6.5 to 7, and the distance range of 1.2 to 7.5km. All eight nearfault records represent motions in soil type D as per IBC 2000 (either recorded on those
conditions,
or
modified
for
those
conditions,
Somerville,
1997).
Among
given
information is the so-called pulse period, Tp , as defined by Krawinkler and Alavi (1998)
as the period at the peak of the velocity response spectrum. It is considered as one of the
major characteristics of a near-fault ground record that might be related to the period of a
given structure in order to predict the structure’s performance (or the severity of the
damage). For the present work, only the fault-normal components are utilized as we are
after capturing the effect of the worst case scenario on the structure. Acceleration,
velocity, and displacement time histories and response spectra of the records of the bin
are given in Appendix A. Acceleration response spectra of the eight selected ground
motions are shown in Figure 5.14 superimposed on the target 2%in50years response
spectrum for the site of the buildings derived according to IBC 2000 provisions.
197
Table 5.8 Main characteristics of near-fault records.
M
R
PGA
Tp
tSM
Station
[km]
[g]
[sec]
[sec]
Imperial Valley (1979)
6.5
1.2
0.43
3.4
8.17
Array 06
Loma Prieta (1989)
7.0
3.5
0.72
3.0
9.52
Los Gatos
Loma Prieta (1989)
7.0
6.3
0.69
1.0
3.26
Lexington
Erzincan (1992)
6.7
2.0
0.43
2.3
7.135
Erzincan - Turkey
Northridge (1994)
6.7
7.1
0.72
1.3
5.54
Newhall
Northridge (1994)
6.7
7.5
0.89
1.0
7.01
Rinaldi
Northridge (1994)
6.7
6.4
0.73
2.4
6.83
Sylmar
Kobe (1995)
6.9
3.4
1.09
0.9
7.1
JMA – Japan
Earthquake
4
IV79-A6
LP89-LG
LP89-LX
EZ92-EZ
NR94-NH
NR94-RS
NR94-SY
KB95-JM
2%in50yrs,
IBC 2000
Sa [g]
3
2
1
RCS 12-story
T1=2.07sec
RCS 6-story
T1=1.25sec
0
0.0
0.5
1.0
STEEL 6-story
T1=1.26sec
1.5
2.0
2.5
3.0
3.5
4.0
Period, T [sec.]
Figure 5.14 Comparison of acceleration response spectra of near-fault records and the
2%in50years site response spectrum (IBC 2000).
5.4 Summary
A brief overview of different earthquake-resistant design methods proposed by recent
seismic codes is presented in this chapter. Some of the main design factors and criteria
and the rationale and concepts behind them are discussed. A full description of the design
198
of three buildings investigated throughout this thesis: 6-story RCS, 12-story RCS, and 6story Steel is then provided. The structural designs are according to the general
architectural layout of a theme structure proposed as part of the US-Japan program on
hybrid structures. All designs are carried out according to appropriate recommendations
and guidelines given by relevant recent seismic codes and standards in the United States.
The Equivalent Lateral Force Static Procedure summarized in Section 5.1.1 is the main
design method adopted herein. Main design details and properties of the three buildings
are also given.
Earthquake records considered throughout this research for the time history analyses of
the buildings for seismic assessment and performance studies are presented. Two sets (or
bins) of records are chosen. Each set consists of a suite of eight recorded ground motions.
The two bins are designated as: bin of general records and bin of near-fault records with
forward directivity. Main factors behind the selection of records are discussed. General
characteristics and seismic properties of the records relevant to their likely effect on the
buildings are also presented.
199
Chapter 6
Detailed Performance Study of 6-Story RCS
Frame
This chapter presents a detailed study of the behavior of a 6-story composite RCS frame
in a Performance Based Design (PBD) context. The frame is one of the seven framing
bents in the short direction of the 6-story RCS case study building presented in the
previous chapter. The chapter organization is as follows. First, modeling and analysis
assumptions are discussed. Second, results from an exploratory nonlinear static pushover
analysis are presented. In the following section, Incremental Dynamic Analyses (IDA) as
proposed by Cornell and his co-workers (1998) are summarized. Next, one of the main
thrusts of this chapter dealing with identifying a technique for capturing a global failure
state based on monitoring cumulative damage effects is described. Then, a detailed
statistical study relating demand to global and local response parameters is presented.
Relationships among local and global response measures (or acceptance criteria) as well
as input demand parameters (such as spectral acceleration Sa, etc.) are discussed and tied
to a criterion to assess the point at which the frame becomes unstable. Correlations
between different input parameters characterizing the intensity or destructiveness of a
ground record and different response parameters are studied. Characteristic parameters of
200
the input include: Sa at the fundamental period of the structure (or multiples of the
fundamental period), duration of the strong motion of the record, pulse period for nearfault ground motions, etc…
6.1 Modeling and Analysis Assumptions
Summarized in this section are the main considerations for the modeling and analysis of
the frame under investigation. Please refer to Tables 5.1 and 5.4 and Figures 5.6 and 5.10
for all relevant details of the frame including: dimensions, member sizes, boundary
conditions,
seismic
masses,
gravity
loads,
design
lateral
loads,
and
members
discretization. Further details about loading and mass characteristics used throughout the
analysis, numerical models, and modeling of damping are briefly discussed. More details
on these general topics can be found in Chapter 2.
6.1.1 Frame Loading and Mass Characteristics
Full dead load and 25% of the live load were first applied to the frame, followed by the
earthquake load – applied either through a static nonlinear analysis or a time history
nonlinear analysis. As DYNAMIX cannot handle distributed loads, the live and dead
loads were lumped onto the beam nodes (outer beams are discretized in four members
each while inner beam in three members) according to tributary area. Similarly, seismic
masses were lumped at the nodes.
6.1.2 Numerical Models
The
columns
are
modeled
with
reinforced
concrete
beam-column elements of
DYNAMIX with the strength and stiffness values determined according to the principles
in Chapters 2 and 3. The outer beams are modeled with the composite beam element,
while the inner beams are modeled as plain steel beam elements since the slab is not
present at the elevator core region of the building. All analyses included geometric
201
nonlinearity (P- ∆ effects) as well as material nonlinearity following a stress-resultant
plasticity bounding surface model with kinematic strain hardening. Stiffness and strength
degradation under cycling loading are modeled based on the dissipated accumulated
plastic energy. All analyses include the joint panel effects, including both the joint panel
size and the joint flexibility, and full column base fixity is assumed. Tables 6.1 through
6.3 give stiffness and strength properties for members (columns and beams) and joint
panels of the frame as modeled in DYNAMIX.
Story #
1-2
Outer
1-2
Inner
3-4
Outer
3-4
Inner
5-6
Outer
5-6
Inner
Table 6.1 Stiffness and strength values of RC columns.
Axial Properties
Bending Properties
Squash Balance Tensile
EA
Strength
EI
Load
Load
Strength
(kips)
at P=Pbal (kips.in2 )
(kips)
(kips)
(kips)
(kips.in)
6
5662
1212
1008
2.89x10
17390
9.35x107
Shear
GA
(kips)
2.82x105
5816
1134
1204
2.89x106
18620
1.01x108
3.19x105
5662
1212
1008
2.89x106
17390
9.17x107
2.67x105
5816
1134
1204
2.89x106
18620
9.68x107
2.83x105
4819
1065
810
2.46x106
13260
6.47x107
2.18x105
4953
996
981
2.46x106
14270
6.70x107
2.22x105
Reasonable accuracy for the solution process is achieved in DYNAMIX through a set of
analysis control parameters. These parameters include yield surface tolerance criterion
(tolerance value set to 0.0005), force point deviation control, unload-reload detection
scheme (tolerance value = 0.05), as well as adjustable (i.e. adaptive) time step. Newmark
Beta method is used for numerical integration with a δ = 0.50 and α = 0.25. These
Newmark Beta parameters prescribe constant average acceleration over each time step,
which guarantees second order accuracy of the solution. For more details please refer to
Mehanny et al. (1999).
202
Floor #
1-4
5-6
1-4
5-6
Floor #
1-2
Outer
1-2
Inner
3-4
Outer
3-4
Inner
5
Outer
5
Inner
6
O&I
Table 6.2 Stiffness and strength values of composite and steel beams.
Flexural Strength
Flexural Stiffness, EI
Shear
(kips.in)
(kips.in2 )
Stiffness, GA
(kips)
Positive
Negative
Positive
Negative
COMPOSITE BEAMS
14920
10200
1.31x108
5.31x107
1.10x105
7
7
12230
8294
9.98x10
3.86x10
9.37x104
STEEL BEAMS
10200
10200
5.31x107
5.31x107
1.10x105
7
7
8294
8294
3.86x10
3.86x10
9.37x104
Table 6.3 Properties of composite joint panels.
Dimensions
Strength, M joint
Stiffness
(inches)
(kips.in)
(kips.in)
Horizontal
Vertical
Shear
Bearing
Shear
Bearing
6
23.0
29.7
22470
29800
5.20x10
7.10x106
23.0
29.7
22470
29800
5.38x106
7.41x106
23.0
29.7
22470
29800
4.98x106
6.73x106
23.0
29.7
22470
29800
5.10x106
6.95x106
21.3
26.2
17010
23340
3.61x106
5.01x106
21.3
26.2
17010
23340
3.68x106
5.13x106
21.3
21.0
17010
23340
4.67x106
7.10x106
6.1.3 Modeling of Damping
All structures, even structures that are undamaged and within the elastic range, exhibit
damping. In general, elastic structures have small damping caused by connection
slippage, microcracking of concrete and nonstructural elements, and several other nonconservative events. Damping in elastic structures is usually modeled as viscous
damping, i.e., where the damping force is proportional to the velocity of the structure. On
the other hand, when the structure is loaded into its inelastic range, hysteretic damping,
caused by inelastic deformations (yielding of steel, cracking and crushing of concrete,
203
etc.), dominates the behavior. Hysteretic nonlinear behavior allows energy to be
dissipated during cyclic loading. Accordingly, while damping in elastic structures is
modeled as viscous damping, hysteretic damping due to inelastic behavior is modeled
through a nonlinear material model.
Viscous damping is modeled through proportional (Rayleigh) damping. The viscous
damping matrix, C, is formed as a linear combination of the diagonal mass matrix, M,
and the elastic stiffness matrix, K, as
C = α1 M + α2 K
(6.1)
The percentage of critical damping, ξ, for a specific mode, n, depends on α 1 , α 2 , as well
as on the frequency, ω n , and is expressed as
ξn =

1  α1

+ α 2ω n 
2  ωn

(6.2)
Thus, the two coefficients, α 1 and α 2 , allow the specification of the percentage of critical
damping for any two modes, i and j, where the coefficients are computed as follows,
based on the natural frequencies and the desired percentage of damping associated with
these two modes
α 1 = 2ξ iω i − α 2ω i2
α2 = 2
(6.3a)
ω jξ j − ω iξ i
(6.3b)
ω 2j − ω i2
For the RCS 6-story frame, 2% of critical damping in the first and third modes are
assumed. This low percentage of damping is chosen since the composite frame is
designed to have more plastification and damage within the steel and composite beams
204
rather than within the reinforced concrete columns. Hence, it is considered to behave
more like a steel frame than a reinforced concrete frame. As shown in Table 6.4, the
effective modal masses of the first three modes of the structure constitute about 95.8% of
the total mass. Periods of the frame from an eigen-value analysis as well as percentage of
critical damping, ξ, at different modes are also given in Table 6.4, showing the smallest
critical damping value of 1.4% for the second mode and the largest critical value of 5%
for the sixth mode. These values calculated according to Equation 6.2 are believed to be
reasonable, encompassing adequate range of damping for the frame under investigation.
Mode
1
2
3
4
5
6
Table 6.4 Modal properties of the 6-story RCS frame.
Period
% of Effective
Modal
(sec.)
Modal Mass
Participation
Factor, Γ
1.25
79.4
1.59
0.40
11.4
0.60
0.21
5.0
0.40
0.12
2.5
0.28
0.09
1.4
0.21
0.07
0.3
0.21
% of Critical
Damping
ξ
2.0
1.4
2.0
3.0
4.0
5.0
6.2 Static Inelastic (Push-Over) Analysis
In the present work, a static pushover analysis with geometric nonlinearity (P-∆ effects)
is performed using the IBC 2000 equivalent lateral force distribution (Fig. 5.10, Chapter
5). The base shear/weight ratio versus total roof drift is shown in Figure 6.1 for the 6story RCS frame where the full dead load and 25% of the live load were applied first
before ramping up the lateral loading. The figure reveals that the static lateral
overstrength, Ω, of the frame is about 3.9, i.e., Ω = Vu/Vd ≅ 0.46/0.12 ≅ 3.9, where Vu is
the ultimate base shear under the code lateral load pattern, and Vd is the design lateral
load considering accidental torsion and based on the upper cap (1.2Ta) on the period
proposed by IBC for design base shear calculation. However, as presented in Table 5.6
(Chapter 5), the actual overstrength of the frame is considerably higher than 3.9. For
205
instance, ignoring accidental torsion effect and with the calculated period, T1 =1.25sec,
the actual Ω is in the order of 6.3 (=3.9x(0.116/0.072)). This actual overstrength is the
one affecting the response since all assessment time history analyses will be based on a 2dimensional configuration with an actual period of 1.25 second and not 1.2Ta.
The target displacement, δ t , for the frame calculated according to Equation 5.8 and a
2%in50years hazard level (reflected in the value of Sa(Te,ξ)) is about 24.2 inches,
corresponding to a total roof drift ratio, ∆r/H, of 0.026, where H is the height of the
building. At this pre-specified target displacement the structure has not yet reached its
maximum lateral capacity of Vu = 0.46W with a corresponding roof drift ratio, ∆r/H, of
about 0.039.
Distributions of interstory drift ratios, IDR, up the height of the frame are given in Figure
6.2 at δ t and other deformation levels. Interstory drift distributions reveal that the frame
fails in a multiple story mechanism involving the first three stories. It follows that
elements with the highest deformation demands are the base of the ground floor columns
and beams of the first three floors. These findings are one of the merits of carrying out
the pushover analysis as an exploratory step in the analysis to identify the critical regions
of a structure and probable overall behavior under a real earthquake record. But one
should keep in mind that this behavior is also ground record dependent since a certain
record with a specific frequency and energy content might trigger higher modes of the
structure even with a short period structure supposed to behave in a first mode fashion.
Accordingly, the pushover results presented above should be looked at with great care.
Finally, an important note is that both the early formation of base hinges as compared to
other locations within the frame, and the high lateral overstrength, as observed herein, are
the direct consequences of the inconsistency between the prescribed strength and stiffness
(i.e., drift limitations) imposed by current design codes, as reported by Leelataviwat et al.
(1998). Figure 6.3 shows the progression of damage under applied lateral loads at
different stages throughout the static pushover process. The lateral load is applied from
left to right causing the left ends of the outer beams to behave in composite action.
206
Base Shear-Weight Ratio, V/W
0.5
0.4
0.3
Static POC
Design Load
Target Disp., δt
0.2
Max. Strength
∆r /H = 0.06
0.1
∆r /H = 0.10
0.0
0.00
0.02
0.04
0.06
0.08
0.10
0.12
0.14
Roof Drift Ratio, ∆r/H
Figure 6.1 Static pushover curve – IBC 2000 load pattern.
6
Design Load
Target Disp., δt
5
Max. Strength
∆r/H = 0.06
Floor #
4
∆r/H = 0.10
3
2
1
0
0.00
0.02
0.04
0.06
0.08
0.10
0.12
0.14
0.16
0.18
Interstory Drift Ratio, IDR
Figure 6.2 Distribution of interstory drift ratios up the height of the frame – pushover
results.
207
0.20
0.40
0.37
0.48
0.33
0.56
0.37
0.52
0.64
0.35
0.59
0.40
0.52
0.70
0.27
0.25
0.24
0.24
At Target Disp., δ t (∆ r/H = 0.026)
0.35
0.32
0.48
0.47
0.72
0.55
0.66
0.37
0.88
0.82
1.18
1.03
1.09
0.68
1.31
0.87
1.26
1.12
1.23
0.72
1.37
0.50
0.44
0.40
0.43
At Maximum Strength (∆ r/H = 0.039)
0.39
0.54
0.34
0.49
0.31
0.70
0.85
1.23
1.09
1.12
0.72
1.39
1.63
2.31
2.27
2.16
1.38
2.39
1.73
2.36
2.39
2.36
1.44
2.49
0.30
0.87
0.72
0.71
At ∆ r/H = 0.06
0.68
Figure 6.3 Distribution of damage indices and progression of damage – pushover
results.
208
Values of the damage index, Dθ, introduced in Chapter 4 are reported in Fig. 6.3. Note
that the cumulative damage index, Dθ, serves here as a peak ductility index with the
plastic rotation at a given section, at a member end, as its numerator and the rotation
capacity (θfailure - θyield) at that end as its denominator. Values of (θfailure - θyield) are given
in Tables 6.5 through 6.7 for all members and joints of the frame. Techniques used to
calculate these limiting values are presented and discussed in detail in Chapter 4. Values
of Dθ larger than 0.95∼1.0 mean failure of the structural component based on calibration
versus test data. Also, note that values of Dθ less than 0.3 are not drawn on the frame
elevations since it has been shown in Chapter 4 through calibration process and reported
physical damage of test specimens that such values for Dθ are associated with minor
damage. One important note is that all composite joints of the RCS frame have not
suffered any remarkable damage even up to a high roof drift ratio of 0.06 imposing
severe damage (and failure) to the base of ground floor columns and beams of the lower
three floors. At ∆r/H=0.06, average total joint distortion is about 0.8% for inner joints
with a maximum of 1.8% for one of the first floor inner joints.
Floor
1-2 Outer
1-2 Inner
3-4 Outer
3-4 Inner
5-6 Outer
5-6 Inner
Table 6.5 Limiting values of rotation capacity for RC columns.
Curvature Capacity
Plastic Hinge Length
Rotation Capacity
l p (inches)
(φ f - φ y), (rad/inches)
(θ f - θ y), (rad.)
0.0057
17.97
0.102
0.0052
19.44
0.101
0.0054
17.97
0.097
0.0058
19.44
0.113
0.0059
16.65
0.098
0.0061
17.97
0.110
209
Table 6.6 Limiting values of rotation capacity for composite and steel beams.
Floor
Positive Rotation Capacity
Negative Rotation Capacity
(θ f - θ y)+, (rad.)
(θ f - θ y)-, (rad.)
COMPOSITE BEAMS
1–4
0.052
0.037
5–6
0.054
0.058
STEEL BEAMS
1–4
0.035
0.035
5–6
0.048
0.048
Floor
1–4
5–6
Table 6.7 Limiting values for composite joints distortion.
M n,shear / M n,bearing
γ f,cycling
0.652
0.069
0.626
0.070
γ f,monotonic
0.083
0.084
6.2.1 Relating Global, IDR, and Local, θ p, Responses for Static Pushover Results
Usually throughout the design process proposed by codes, global response measures are
specified and used as acceptance criteria (e.g., drift limits in terms of inelastic interstory
drift ratios to control stiffness as per IBC 2000). However, recent seismic design codes
and guidelines such as ATC 40 and FEMA 273 specify demands in terms of a global
response measure such as the target displacement, δ t . Then, they provide response limits
in terms of peak plastic deformations to serve as acceptance criteria for the seismic
behavior of structural components (i.e., at the local level). Hence, it is very instructive
and beneficial to relate response at both global and local levels first to evaluate codes
limits and its consistency and second to thoroughly study the performance of the
structure.
In the present work, the relationship between global response, IDR, and local response,
plastic rotations in beams, θp,B, and columns, θp,C, is investigated based on results from
the static pushover analysis. As reported by Leelataviwat et al. (1998), after formation of
beam hinges in the moment frames designed by modern practice, the distribution of
moments in columns changes drastically from the elastic distribution. An abrupt increase
of moments in columns below the floor beam and corresponding decrease of moments
210
above the floor takes place. This leads to higher plastic response (and ductility demands)
at the top of columns within a story compared to the bottom, especially at high levels of
demand. Accordingly, plastic hinges in columns are usually considered to occur at the top
sections of each story, except for the first story where the bottom sections (at the
foundation) are also critical.
θ i+1
θi
i+1
Story, i
θi
θ eff = θ i+1 - θ i
= |IDR i - IDRi+1 |
i+1
θi
θ p,B = f(θ p,i+1)
Story, i
θ p,C = f(θ eff )
i -1
θi
θ i+1
θ p,B = f(θ p,i+1 )
θ p,C = f(θ eff)
i -1
Deformed configuration (I)
θ eff = |θ i+1 - θ i |
= IDR i - IDRi+1
Deformed configuration (II)
(a) Column Hinging at Top Sections
θi
θ i-1
θ i-1
θ p,B = f(θ p,i)
θ p,B = f(θ p,i )
θ i-1
θi
θ eff = θ i - θ i-1
= IDRi - IDR i-1
θ i-1
θ p,C = f(θ eff )
Deformed configuration (I)
θ eff = |θ i - θ i-1|
= |IDRi - IDRi-1|
θ p,C = f(θ eff)
Deformed configuration (II)
(b) Column Hinging at Bottom Sections
Figure 6.4 Schematic of different deformed configurations.
Based on the anticipated deformed configuration of the frame as identified above, the
local response of columns, given in terms of θp,C, is related to a corresponding global
211
response quantity. Such global response is best represented by the absolute value of the
difference between IDR at this given story and IDR at the upper story (i.e., |IDRi –
IDRi+1 |, Fig. 6.4a), referred to as ∆IDR. On the other hand, it has been found that local
response of beams, given by θp,B, is best related to the global response given by the
plastic component of IDR, referred to as IDRp . As a reasonable assumption, the elastic
component of total IDR is suggested to be a value of about 0.01; this inherently means
that the first 1% of the interstory drift ratio is due to elastic deformation with no resulting
plastic demands. Thus, IDRp is given as IDR-0.01. Again, based on the anticipated
deformed configuration of the frame mentioned previously (i.e., with columns hinging
mainly at top sections of different story columns), plastic hinges in beams of a given floor
are related to IDRp of the upper story (Fig. 6.4a). However, whenever column hinging
takes place at bottom sections of a specific story, as shown in Fig. 6.4b, θp,B should be
related instead to IDRp at the same story, and θp,C should be related to the absolute value
of the difference between IDR at this story and that at the lower one. Accordingly, the
following form for the relationship between local and global response is assumed
θp,C = f(|IDRi – IDRi+1 | or |IDRi – IDRi-1 |) = α ∆IDRβ
(6.4a)
θp,B = f(IDRp,i+1 or IDRp,i) = α IDRp β
(6.4b)
The parameters α and β are determined through a regression analysis carried out in the
log space. The advantages of the regression analysis in log-space are that we can carry
out the conventional linear regression and that the variance of the error does not depend
on the value of the independent parameter in the regression relation. The exponent β in
Equations 6.4a and 6.4b is introduced to capture any nonlinearity in the relationship. It is
also able to pick up a linear relationship if it is manifested by the data. In Figs. 6.5 and
6.6, ∆IDR versus θp,C and IDRp versus θp,B data are plotted for results from the pushover
analysis. The values shown in the figures are those corresponding to different roof drift
ratios ranging from the target displacement to an extremely large roof drift demand of
0.06 corresponding to a maximum IDR of about 0.09.
212
0.10
2
Ra = 0.9
0.08
θp,C = 1.66 ∆IDR 1.22
σlnθp,C|∆IDR = 0.277
∆IDR = 0.54 θ p,C0.77
0.06
∆ IDR
σln∆IDR|θ
= 0.220
p,C
0.04
Target Disp., δt
Max. Strength
∆r/H = 0.06
0.02
Regression given θ p,C
Regression given ∆IDR
0.00
0.00
0.02
0.04
0.06
0.08
0.10
Column Plastic Rotation, θp,C [rad.]
Figure 6.5 Global, ∆IDR, versus local, θp,C, response – pushover results.
0.10
2
Ra = 0.9
0.08
θ p,B = 1.56 IDR p1.15
σlnθp,B|IDR = 0.360
p
IDRp = 0.52 θp,B 0.80
IDRp
0.06
σlnIDR |θ
= 0.300
p p,B
0.04
Target Disp., δ t
Max. Strength
∆r /H = 0.06
0.02
Regression given θp,B
Regression given IDRp
0.00
0.00
0.02
0.04
0.06
0.08
Beam Plastic Rotation, θp,B [rad.]
Figure 6.6 Global, IDRp , versus local, θp,B, response – pushover results.
213
0.10
The least square fit has been done once conditioned on global response (i.e., given IDR)
and then conditioned on local response (i.e., given θp ). Regression lines for both cases are
also shown in Figs. 6.5 and 6.6 with the corresponding conditional dispersion as defined
in Section 6.3.2. Note the close values of the two regression relationships conditioned on
either global or local response. Moreover, one may observe that up to considerable values
of either θp,C or θp,B of about 0.06 radians, estimates of the medians of ∆IDR and IDRp
have almost the same value of 0.06 within a difference of 1% and 10%, respectively. This
clearly indicates that there is a consistent pattern of deformation associated with the
frame design which produces a proportionate increase in the member plastic rotation
demand (and consequently if needed the member rotation ductility demand) as the overall
lateral deformation increases. This proportionality not only shows the adequacy of the
design process but also facilitates linking element ductility (or plastic rotation) capacities
that are required to reach maximum drift limits associated with specific performance
level to such global drift values, or vice versa. However, this finding should be addressed
with great care until global versus local response relationships are revisited later in this
chapter and in the following chapter based on results from time history analyses of the
case study frames under general and near-fault suites of records.
6.3 Nonlinear Dynamic (Time History) Analyses
Second-order inelastic dynamic analyses are carried out using the two bins of ground
motions representing general and near-fault events under different hazard levels.
Horizontal and vertical components of each ground record are applied simultaneously
and are scaled by the same factor whenever scaling process takes place.
6.3.1 Incremental Dynamic Analysis (IDA) Concept
By performing a time history analysis of a structure, one has in mind to study its
performance at a certain hazard (or demand) level. Hence, the resulting response
parameters should be related to an appropriate intensity measure of the ground motion
214
hazard level. Drift has traditionally been considered as an efficient and simple measure to
assess global structural performance. Drift measures can include inter-story drift ratio
(i.e., IDR), drift ductility, or other drift-dependent damage index. For quantifying applied
ground motions, elastic spectral acceleration is usually considered as an “effective”
intensity measure for earthquake records (Shome et al., 1998) since it is a convenient
measure for which the record-to-record dispersion of the drift response at a given
intensity level is relatively small, and for which a hazard analysis is available. More
specifically, Shome (1999) has proven that the elastic spectral acceleration at the
fundamental period of the structure is an effective structure-specific measure of ground
motion intensity for predicting the nonlinear response of buildings. In his study, Shome
looked at two steel frame structures of 5 and 20 stories representing short and long period
Spectral Acceleration, Sa(T1,ξ)
buildings respectively.
"Hardening" behavior
Elastic Response
Multiple time history
analyses
"Softening" behavior
Drift Capacity
(Limit IDR max)
IDRmax
Figure 6.7 Schematic of typical Incremental Dynamic Analysis Curves
(IDAC).
One way to systematically relate spectral acceleration to drift is through so-called
Incremental Dynamic Analysis Curves (IDAC) that have been originally introduced by
Cornell and his coworkers (Luco and Cornell, 1998) for the SAC project. Typical IDACs
are shown in Fig. 6.7. Creation of a single incremental dynamic analysis curve entails
performing multiple nonlinear dynamic analyses for a model structure subjected to an
215
earthquake record that is incrementally scaled. The result is an IDAC which relates the
scale factor for the earthquake record (or the spectral acceleration at the fundamental
structure period) to the drift response of the structure. From the IDAC, as shown in Fig.
6.7, limit of IDRmax corresponds to the transition point at which the analytical response of
the model structure becomes “unstable” (i.e., when the dynamic drift response increases
drastically for a relatively small increase in ground motion intensity), or when the
apparent stiffness (i.e., the slope of the IDAC) decreases radically. This limiting value
may be used as a measure of IDRmax capacity for that structure, for that record. With
several estimates (from IDACs for several earthquake records) of IDRmax capacity, the
median of IDRmax for that specific structure is calculated. So far the problem that Luco
and Cornell (1998) faced was that almost all IDACs of a 3-story ductile case study steel
frame (with inherently small P-∆ effects) have showed hardening effect (refer to Fig. 6.7)
in the behavior which remained stable up to values of IDRmax > 10%, the limit
corresponding to undependable analysis results that they set for their numerical model.
Thus, they were unable to really detect a value of IDRmax corresponding to global
collapse of the system. In the following section, a technique capable of capturing global
failure state of a structure is proposed to identify IDRmax or any other response parameter
capacity (i.e., limiting response parameter value at global collapse of the structure).
6.3.2 Relationship between Spectral Acceleration and Maximum Interstory Drift
Ratio
The relationship between spectral acceleration and IDRmax is established by performing
multiple nonlinear dynamic analyses of the frame for ground motions at increasing levels
of intensity (as measured by Sa, spectral acceleration). The spectral index, Sa(T1 ,ξ=5%) is
the peak induced in a single ground motion for a SDOF elastic oscillator with period T1
and 5% viscous damping. Given a set of Sa versus IDRmax data points, a regression (or
“least squares fit”) of the form
IDRmax = α Saβ (T1 , ξ = 5% )
(6.5)
216
where IDRmax is the median maximum interstory drift response and Sa (T1 , ξ = 5% ) is the
spectral acceleration, provides an appropriate relationship between spectral acceleration
and median drift values. The exponent β in Equation 6.5 is included to capture any
“softening” or “hardening” of the nonlinear relationship between Sa and IDRmax inherent
to typical IDACs as shown in Figure 6.7. A regression of the form given in Equation 6.5
is equivalent to a linear regression of the log of drift on the log of spectral acceleration.
The advantages of the regression analysis in log-space are that we can carry out the
conventional linear regression and that the variance of the error does not depend on the
level of spectral acceleration. The advantage of this constancy of variance is clear when
using such regression functions in probabilistic seismic demand calculations (Shome and
Cornell, 1999). The dispersion of the drift response conditioned on the spectral
acceleration, σ ln IDR max |S a (T1 ,ξ ) , is calculated as the mean squared deviation of the spectral
acceleration versus drift data points from the regression fit. In other words, the dispersion
is defined herein as the standard deviation of the natural logarithms of the data.
By incrementally scaling up and/or down each record as discussed in Chapter 5, and
performing nonlinear time history analyses, different values for IDRmax are obtained for
each record at different input intensity (i.e., hazard) levels. Each set of pair of points
(IDRmax vs. Sa (T1 , ξ = 5% ) ) corresponding to a specific record throughout this scaling
procedure defines an IDAC for this structure, for that record. Another way of looking at
these data points is by considering all the points for all the eight records within each bin
as different data points corresponding to different hazard levels and then performing a
single regression fit for all points as the one defined by Equation 6.5. But the way
considered in this research is a bit different. Trying to eliminate any bias of the least
square fit due to different locations of the data points within the spectral acceleration-drift
space for the different records of each bin, first, a regression of the form given by
Equation 6.5 is performed for data points corresponding to each record alone. Thus, one
can obtain for each bin eight pair of values for α and β. Then, a relation of the same form
of Equation 6.5 but with medians of both α and β will provide the required relationship
between the drift response, IDRmax, and the spectral acceleration, Sa (T1 , ξ = 5% ) . Note
217
that this point estimate for α is calculated as the exponential of the average of the natural
logarithms of the eight values (also known as the geometric mean), while the point
estimate for β is just the regular arithmetic mean. The geometric mean is a logical
estimator of the median, especially if the data are at least approximately lognormally
distributed as for the case herein for the nonlinear response of the structure in terms of
IDRmax given Sa (T1 , ξ = 5% ) , as has been proved by Shome et al. (1997). Final values for
α and β are given in Table 6.8 for bins of general and near-fault records. For briefness, Sa
will be simply used for Sa (T1 , ξ = 5% ) in the sequel unless otherwise explicitly stated.
Figures 6.8 and 6.9 give samples of the regression fit performed on data points
corresponding to single ground records showing large record-to-record dispersion.
Collections of these spectral acceleration versus IDRmax plots for the two ground motion
bins are given in Figs. 6.10 and 6.11. Notice that β values for both bins are greater than
1.0 showing “softening” of the nonlinear relationship. Moreover, the values of both
regression parameters α and β for the near-fault records bin are larger than for the general
records bin, suggesting that the near-fault records are more damaging. For instance, given
Sa = 0.864g (a value corresponding to a 2%in50years hazard level), the expected median
values for the drift response, IDRmax, are 0.023 and 0.028 for general and near-fault
records respectively. Thus, there is about 22% difference on average between the drift
response corresponding to a 2%in50years input hazard level for general and near-fault
ground motions. This difference is even more pronounced at higher intensity (i.e.,
hazard) levels reaching on average a value of 32% at 1.5 times the hazard of
2%in50years, while it is less at lower hazard levels (8% difference at 10%in50years
hazard level).
Table 6.8 Values of α and β for the regression fit of Equation 6.5.
Parameter and Statistical
General Records
Near-Fault Records
Measure Values
0.027 (34%)
0.034 (56%)
α (C.O.V.)
1.11
(32%)
1.35 (35%)
β (C.O.V.)
218
5
5
Miyagi
Valparaiso
4
Sa (T1 ,5%)
Sa (T1,5%)
4
3
2
1
0
0.00
3
2
1
0.04
0.08 0.12
IDR max
0
0.00
0.16
0.04
0.08
5
LP89-HCA
LP89-HSP
4
Sa (T1 ,5%)
Sa (T1 ,5%)
4
3
2
0.04
0.08 0.12
IDRmax
2
0
0.00
0.16
0.04
4
Sa (T1,5%)
3
2
3
2
1
1
0
0.04
0.08
0.12
0.00
0.16
0.04
0.08
0.12
0.16
IDR max
IDRmax
5
5
LA92-YER
Mendocino
4
Sa (T1 ,5%)
4
Sa (T1,5%)
0.16
CM92-RIO
LP89-WAHO
4
3
2
3
2
1
1
0
0.00
0.08 0.12
IDRmax
5
5
Sa (T1 ,5%)
3
1
1
0
0.00
0.16
IDR max
5
0
0.00
0.12
0.04
0.08 0.12
IDR max
0
0.00
0.16
0.04
0.08 0.12
IDR max
0.16
Figure 6.8 Conditional regression relationship of IDRmax for general records.
219
5
5
LP89-LG
IV79-A6
4
Sa (T1,5%)
Sa (T1,5%)
4
3
2
1
0
0.00
3
2
1
0.04
0.08 0.12
IDR max
0
0.00
0.16
5
0.04
EZ92-EZ
4
Sa (T1,5%)
Sa (T1 ,5%)
4
3
2
1
0.04
0.08 0.12
IDR max
0
0.00
0.16
0.08 0.12
IDR max
0.16
4
Sa (T1 ,5%)
Sa (T1,5%)
0.04
NR94-RS
NR94-NH
3
2
1
3
2
1
0.04
0.08 0.12
IDR max
0
0.00
0.16
5
0.04
0.08 0.12
IDR max
0.16
5
NR94-SY
4
KB95-JM
4
Sa (T1,5%)
Sa (T1,5%)
2
5
4
3
2
1
0
0.00
3
1
5
0
0.00
0.16
5
LP89-LX
0
0.00
0.08 0.12
IDR max
3
2
1
0.04
0.08 0.12
IDR max
0
0.00
0.16
0.04
0.08 0.12
IDR max
0.16
Figure 6.9 Conditional regression relationship of IDRmax for near-fault records.
220
5
IDRmax = 0.027 Sa1.11
Sa (T1,5%)
4
3
2
1
0
0.00
Sa(2%in50years)
0.04
0.08
0.12
0.16
IDRmax
Figure 6.10 Spectral acceleration versus IDRmax for bin of general records.
5
4
Sa (T1,5%)
IDR max = 0.034 S a 1.35
3
2
1
0
0.00
Sa(2%in50years)
0.04
0.08
0.12
0.16
IDRmax
Figure 6.11 Spectral acceleration versus IDRmax for bin of near-fault records.
221
Values of median IDRmax in the previous paragraph at a given hazard, Sa with
p%innyears probability of exceedance are first order estimates of the p%innyears
response (or a mean return period T-years response). This implies that the variability in
the median response (IDRmax) obtained for a given level of Sa as shown in the regression
analysis is neglected. As mentioned by Bazzurro et al. (1998), if the variability in the
response for a given Sa is accounted for, a second order estimate of the p%innyears
IDRmax is associated with a higher spectral acceleration, Sa,H, representing a higher hazard
level, i.e., representing an event with less probability of exceedance or larger mean return
period than the mean return period of the response. In other words, the average drift
demand resulting from a 2%in50years event does not represent a 2%in50years response.
However, it should be associated with a higher probability of exceedance through a
correction factor which accounts for the dispersion in drift given spectral acceleration. As
reported by Bazzurro et al. (1998), such a correction factor in most practical cases is not
larger than 2.0. A more complete discussion is given by Luco and Cornell (1998).
Furthermore, among other benefits of such relationship between Sa(T1 ,5%) and IDRmax of
the form given by Equation 6.5 is that it can be combined with existing site hazard curves
for spectral acceleration to arrive at a drift demand hazard curve. More specifically, the
annual probability of exceeding any specified drift demand, and the drift demand
associated with a particular exceedance probability, given the intensity (i.e., hazard) level
can be computed. Moreover, with estimates of a reliable median of the drift capacity
(shown in Fig. 6.7 and discussed in the following section) and the dispersion of that drift
capacity, the annual probability of failure can also be computed. The probability of
failure may be defined as the probability that the drift demand exceeds the drift capacity
when the drift capacity is regarded as random variable. These applications related to a
probabilistic seismic hazard assessment of structures are out of the scope of this thesis.
For more details about the subject one may refer to Shome (1999), Luco and Cornell
(1998) and Bazzurro et al. (1998).
In Figures 6.12 and 6.13, IDACs are given for each story of the 6-story RCS frame for
the 16 ground records. Such figures have the merit of showing that maximum transient
222
interstory drift ratios, IDRmax, are usually much larger in the first two or three stories of
the frame compared to other stories, especially for high hazard levels. This finding is
consistent with the pushover analysis results of Section 6.2, which reveal that the frame
fails in a multiple story mechanism involving the first three stories.
Finally, it is important to note that the conventional spectral acceleration at the
fundamental period, T1 , of the structure and damping level (5%), Sa(T1 ,ξ=5%), is used as
an intensity measure of the input record since it has been proved to be as effective a
predictor of nonlinear MDOF behavior as any other measure of ground motion intensity
yet considered (see Shome et al., 1998). The damping ratio of 5% tends to smooth
variations in the acceleration response spectra values when compared to Sa values from
lower damping ratios (Shome et al., 1988). This is reflected in the decrease of the
dispersion of drift measure given intensity level as in the relationship between IDRmax
and Sa(T1 ,ξ=5%) of the form given by Equation 6.5. Another merit of the use of 5%
damping over another damping ratio is that it permits the use of widely available
attenuation laws and hazard results. Note that this is not inconsistent with the use of 2%
critical damping ratio for the modeling of viscous damping in the nonlinear time history
analyses of the frame since the relationship between SDOF elastic parameters (e.g.,
Sa(T1 ,5%)) and MDOF inelastic response (as given by IDRmax) is quite empirical and
viscous damping is an ad-hoc method that we assume to approximately model damping
within the elastic range of behavior of the structure. Hence, there is no need for having
same values in both the nonlinear dynamic analysis and the intensity measure of the
input. The adequacy of using 5% damping over the 2% damping as it decreases the
conditional dispersion of the drift given the intensity level of the input is shown in Table
6.9 for both general and near-fault records. However, one may note that the relative
change in statistical measures due to different damping levels is not significant.
Table 6.9 Conditional dispersions and coefficient of determination for IDRmax.
Statistical
Bin of General Records
Bin of Near-Fault Records
Measure
ξ = 2% ξ = 5%
ξ = 10%
ξ = 2%
ξ = 5%
ξ = 10%
σ ln IDR |S (T ,ξ )
0.448
0.416
0.379
0.468
0.449
0.426
max
R2
a
1
0.553
0.622
0.681
223
0.333
0.397
0.448
5
4
4
Sa (T1 ,5%)
Sa (T1,5%)
5
3
2
1
0
0.00
Story
Story
Story
Story
Story
Story
3
2
1
0.04
0.08
0.12
0
0.00
0.16
0.04
IDRmax
0.12
0.16
0.12
0.16
(b) Valparaiso
5
5
4
4
Sa (T1,5%)
S a (T1 ,5%)
0.08
IDR max
(a) Miyagi
3
2
1
0
0.00
1
2
3
4
5
6
3
2
1
0.04
0.08
0.12
0.16
0
0.00
0.04
0.08
IDRmax
IDRmax
(c) LP89-HCA
(d) LP89-HSP
Figure 6.12 Story IDACs for general records.
224
5
5
Story 1
Story 2
Story 3
Story 4
Story 5
Story 6
3
4
S a (T1,5%)
S a (T 1,5%)
4
2
1
2
1
0.04
0.08
0.12
0
0.00
0.16
0.08
IDRmax
(e) LP89-WAHO
(f) CM92-RIO
5
5
4
4
3
2
1
0
0.00
0.04
IDRmax
Sa (T1,5%)
S a (T 1,5%)
0
0.00
3
0.12
0.16
0.12
0.16
3
2
1
0.04
0.08
0.12
0.16
0
0.00
0.04
0.08
IDRmax
IDRmax
(g) LA92-YER
(h) Mendocino
Figure 6.12 Story IDACs for general records. (Continued)
225
5
5
Story 1
Story 2
Story 3
Story 4
Story 4
Story 6
3
4
S a (T 1,5%)
Sa (T1,5%)
4
2
1
0
0.00
3
2
1
0.04
0.08
0.12
0
0.00
0.16
0.04
IDRmax
0.16
0.12
0.16
(b) LP89-LG
5
5
4
4
Sa (T1,5%)
Sa (T1,5%)
0.12
IDR max
(a) IV79-A6
3
2
3
2
1
1
0
0.00
0.08
0.04
0.08
0.12
0.16
0
0.00
0.04
0.08
IDRmax
IDRmax
(c) LP89-LX
(d) EZ92-EZ
Figure 6.13 Story IDACs for near-fault records.
226
5
5
Story 1
Story 2
Story 3
Story 4
Story 5
Story 6
3
4
Sa (T1,5%)
S a (T 1,5%)
4
2
1
0
0.00
3
2
1
0.04
0.08
0.12
0
0.00
0.16
0.04
IDRmax
0.16
0.12
0.16
(f) NR94-RS
5
5
4
4
Sa (T1,5%)
S a (T1,5%)
0.12
IDR max
(e) NR94-NH
3
2
1
0
0.00
0.08
3
2
1
0.04
0.08
0.12
0.16
IDRmax
0
0.00
0.04
0.08
IDRmax
(h) KB95-JM
(g) NR94-SY
Figure 6.13 Story IDACs for near-fault records. (Continued)
227
Note
that
values
of
conditional
dispersions,
σlnResponse|Input , and coefficient of
determination1 , R2 , given in Table 6.9 are based on performing least square fit in the form
of Equation 6.5 for all data points of all records at different scale levels for each bin.
High values of σ ln IDR
max |S a
( T1 ,ξ )
are due to the fact that we are dealing in the regression
analysis with all data points up to very high hazard levels (corresponding to global failure
of the structure) with inherent high dispersion values due to the high non-linearity in the
response of the structure. In other words, if regression analysis is performed only for data
points at low hazard levels, σ ln IDR
max |S a
( T1 ,ξ )
will take much lower values. For instance, if the
regression fit is carried out only on the eight unscaled near-fault records (with seven out
of them representing a hazard level for the structure higher than the 2%in50 hazard
according to NEHRP 97 at that site), conditioned on Sa(T1 ,ξ=5%), σ ln IDR
max |S a
( T1 ,ξ )
= 0.259
and R2 = 0.741 as compared to values of 0.449 and 0.397 respectively, shown in Table
6.9. R2 values should be considered with care since they might represent a biased
statistical measure in the present context because of the pre-selection of the different
values of the independent parameter Sa(T1 ,ξ=5%) for each record used for the regression
analysis.
1
The coefficient of determination measures the proportionate reduction of total variation in the dependent
variable Y associated with the use of the set of X independent variables X1 ,…,Xp-1 in the regression
analysis. Also known as coefficient of multiple determination (See Neter et al., 1996). It is given as
R 2 = 1−
(∑ (Y − Y) ) / ∑(Y − Y ) )
2
i
2
i
(6.6a)
i
where Yi are the observations or the data points, Y is their mean, and Yˆi are the values of the observations
based on the fitted regression line. An adjusted coefficient of multiple determination, Ra2 , is sometimes
used that adjusts for the number of X values in the mo del. It is given as
(
)
 n −1 
R 2a = 1 − 
 ∑ Yi − Yi
n− p
(
) / ∑ (Y − Y ) )
2
2
i
(6.6b)
where n is the number of data points, and p the number of X independent variables considered in the
regression. The coefficient of determination shows how good the regression model is able to capture the
variability in the data. R2 assumes the value 1 when all observations fall directly on the fitted regression
surface, i.e., when Yi
)
= Yi for all i.
228
6.4 Identification of Collapse Limit State
The determination of the state of total collapse of a given structure is a challenging task
needed within the context of performance based design/evaluation framework. Since
collapse (or near collapse) state is defined by FEMA 273 as one of the structural
performance levels that should be investigated within the design process, accurate and
adequate quantification of collapse (i.e., global failure) of the structural system, rather
than just local failure of some of its components, should be identified. Moreover, as
mentioned in the previous section, for successful application of probabilistic seismic
hazard assessment of structures, capacity values of different response parameters at the
stage of overall collapse of the structure should be determined. One technique for the
detection of global failure of framed structures is introduced herein. The methodology is
first presented along with the necessary implementation parameters, then results from
applying the process to the dynamic analyses of the 6-story RCS frame are given.
Important relationships between the collapse criterion, λu (see Section 6.4.1 for
definition), the input intensity level of the hazard, Sa, and the response parameter IDRmax
are also provided.
6.4.1 Methodology for the Determination of the State of Global Collapse
The need for a few steps procedure for the detection of the state of global collapse of a
structure is due to the fact that available analysis programs, including DYNAMIX, are
unable to capture overall failure of the structural system under earthquake induced
dynamic effects through the implemented material models. This is also the thrust behind
the continuous need for damage indices, such as the ones proposed in Chapter 4, to detect
damage and failure of structural components. However, one should keep in mind that
most damage indices (e.g., Dθ and DE presented in Chapter 4) are local indices (i.e.,
calibrated to detect damage and failure at the level of structural components). Therefore,
their values have to be processed or combined in a certain scheme for the different
members of the structure to be able to assess the overall damage of the frame. Figure 6.14
229
shows a flow chart of the technique introduced in this research for the determination of
the global failure of a given structure.
First, second-order inelastic time history analysis of the frame is performed for a specific
record scaled incrementally at different intensity (i.e., hazard) levels. Second, the
ductility-based cumulative damage index, Dθ, proposed in Chapter 4 is calculated for the
columns, beams, and joint panels of the frame. As previously shown in Chapter 4, this
index is capable of capturing evolution of damage of a structural component. It is
preferred herein over its dual energy-based index, DE, also presented in Chapter 4, for
being less computationally demanding, yet giving comparable results. The similar results
of the two indices shown by some statistical measures in Chapter 4 reveal that, in spite of
dealing only with deformation, Dθ is still a good candidate for predicting failure and its
evolution in a computationally efficient manner. Moreover, it still captures many
cumulative effects related to energy measures.
Then, based on the values of the damage index at different critical sections (ends of
beams and columns) within the structure, new stiffness and strength properties of these
sections are reassigned to the model. This feedback step is needed to capture the updated
state of the structure after undergoing some damage at its different components due to the
applied hazard. Suggestions for values of stiffness and strength updates are discussed
below.
At this stage, residual displacements resulting from time history analysis at this specific
intensity level are also applied to the damaged structure with modified values of stiffness
and strength at different damaged critical sections of beams, columns, and joint panels.
Second-order inelastic static analysis of the damaged frame is performed under the effect
of incrementally increasing gravity loads composed of full dead load and 25% of the live
load as considered in the time history analysis. The limiting value of the load factor, λu,
which defines the maximum load – as a ratio of the applied gravity loads – that the
damaged structure can sustain is computed as a result of this static inelastic analysis.
When the value of λu reaches 1.0 or less that means theoretically global collapse of the
230
structure since the structure in its new damaged state after suffering from the earthquake
is not capable of carrying its gravity load. Note that λuo is the value of the limiting load
factor for the totally undamaged structure; it is a property of the structure under a given
gravity load and for instance it is equal to 5.5 for the 6-story RCS case study frame.
Scale up the record
under consideration.
Perform 2nd Order Inelastic
Time History Analysis
of the Undamaged
Structure.
Calculate Cumulative
Damage Index, Dθ,
at all Beams, Columns,
and Joint Panels.
Feedback Step:
1- Revise stiffness and strength values at
damaged sections
2- Apply residual displacements
Perform 2nd Order Inelastic
Static Analysis (with Gravity
Loads only) of the Revised
(i.e., Damaged) Structure to
calculate λu
NO
λu ≤ 1.0
YES
Global Failure State
Figure 6.14 Flow chart of the technique for global collapse determination.
231
6.4.1.1 New stiffness and strength values for updating the damage state of the
structure
As mentioned above, the damaged structure should be revisited again and stiffness and
strength capacities of its damaged sections should be modified before performing the
second-order inelastic static analysis to determine its ultimate load factor, λu. Updated
stiffness and strength values to model damage effects are proposed based on calibration
studies given in Chapter 4 for different structural components including steel and
composite beams and reinforced concrete columns. Figure 6.15 shows the relationship
between the proposed stiffness of the member at the critical section after damage as a
ratio of its initial undamaged stiffness and the value of the damage index Dθ. Stiffness
values at different values of Dθ correspond to secant stiffness calculated from the cyclic
test results curves at this specific damage state. These modified stiffness values due to
damage are assigned to critical sections throughout the frame modeled by short elements
at columns and beams ends with an assumed hinge formation length approximately equal
to the member cross section depth.
1.2
Damaged Stiffness / Initial Stiffness
RC Columns, Steel and Composite Beams
1.0
0.8
EI/EIo = -0.2(1-1/Dθ1.5)
0.6
0.4
0.2
0.0
0.0
0.2
0.4
0.6
0.8
1.0
1.2
Dθ
Figure 6.15 Proposed stiffness reduction as a function of the damage index Dθ.
232
Scattered values from results of different test specimens investigated in Chapter 4 are
given in Figure 6.15 along with a suggested continuous line based on an equation relating
the reduced stiffness due to damage and Dθ. For practical reasons, one might use the step
function shown in Figure 6.15 rather than the continuous curve. Thus, for a value of Dθ
below 0.3, there is no decrease of the initial stiffness (since we assume very minor
damage based on test results given in Chapter 4), while for a value of Dθ of above 0.95, a
real hinge is introduced assuming total failure of the section and total loss of strength and
stiffness. For values of Dθ between 0.3 and 0.6, a decrease of 60% from the initial
stiffness is assumed, while for a value of Dθ between 0.6 and 0.95, a total loss of 85% of
the initial stiffness is introduced. Note that a value of Dθ of about 0.6 has been suggested
in Chapter 4 as the limit for repairable damage (beyond which the structural component
should be replaced), again based on the reported physical damage in the experiments that
have been considered.
For consistency, a decrease of strength as a function of the element damage index is also
introduced. Again, for a value of Dθ between 0.3 and 0.6, the strength of the cross section
is assumed to be 90% of its undamaged capacity, and for Dθ between 0.6 and 0.95, a
decrease of 25% from the initial strength is introduced. For Dθ less than 0.3, the section
maintains its undamaged strength.
6.4.2 Relationship between Spectral Acceleration and Global Failure Criterion, λ u
Using the technique introduced in Section 6.4.1, inelastic gravity load strength ratios, λu,
of the frame are computed after the frame is subjected to each of the sixteen ground
records scaled to different intensities, Sa. In this way, the strength index of the frame λu is
related to the ground motion intensity Sa. The gravity load strength ratio of the
undamaged frame, λuo , was first determined to be λuo = 5.5, implying that initially the
frame can resist over five times the applied gravity load. As will be shown below, after
being damaged by increasingly intense ground motions, λu reduces to λu = 1.0,
corresponding to a state of incipient collapse.
233
7
Miyagi
Valparaiso
LP89-HCA
Sa (T1=1.25sec,ξ=5%)
6
Sa = 2.92 λu-0.36
σlnS |λ = 0.379
a u
5
LP89-HSP
LP89-WAHO
CM92-RIO
LA92-YER
Mendocino
Linear Regression
4
3
2
1
λu = 1.0
(collapse)
0
0
1
2
3
4
λu (based on 1.0D+0.25L)
5
λ uo
6
Figure 6.16a Spectral acceleration - λu relationship for bin of general records.
6
5
S a (T 1=1.25sec,ξ=5%)
IV79-A6
LP89-LG
LP89-LX
Sa = 2.62 λu-0.37
σlnS |λ = 0.444
a u
EZ92-EZ
NR94-NH
NR94-RS
NR94-SY
KB95-JM
Lin. Regression
4
3
2
1
λu = 1.0
(collapse)
0
0
1
2
3
4
λu (based on 1.0D+0.25L)
5
λ uo
6
Figure 6.16b Spectral acceleration - λu relationship for bin of near-fault records.
234
Figure 6.16a shows results for the bin of general records. Along with the data calculated
for each of the eight records is a regression line based on least square fit of the data.
Linear regression, conditioned on λu, is applied to all data points in the log space
excluding points with a value of λu greater than 0.95λuo , so that the fitted model can
capture the real behavior throughout the evolution of damage as reflected by the decrease
of λu from its original value, λuo , to a value of 1.0 defining the state of incipient collapse
(or stability limit point). From discrete data points at λu=1.0, the mean value of
Sa(T1 ,ξ=5%) is 3.12g with C.O.V. of 37.5%. However, an estimate of the median of this
value is 2.92g according to the regression line given in Figure 6.16a with a conditional
dispersion σ ln S
a
(T1 ,ξ =5 %)| λ u
= 0.379. This 6.4% difference in the average value of Sa at λu=1.0
is due to the fact that the regression line is minimizing the error of Sa values on average
throughout the full range of λu, conditioned on λu, and not only at the specific value of
1.0. According to these results, the 6-story frame at this specific site (with Sa(T1 ,ξ=5%) =
0.864g corresponding to a 2%in50years hazard level according to IBC 2000) is on
average at the threat of global collapse at a Sa(λu=1.0)/S a(2%in50) ratio of about 3.6
(=3.12g/0.864g). Collapse here means more than just some local collapse of several
structural components or damage beyond the repairability level or loss of functionality.
One can get similar information at any stage along the evolution of the damage status of
the structure (i.e., at any value of λu of interest) and accordingly can relate Sa(λu=λ’u) to
different spectral acceleration values associated with different hazard levels at the site
(e.g. the design level of 10%in50years). It is also very useful to note that values of
Sa(T1 ,ξ=5%) corresponding to λu=1.0 range from 1.47g for scaled Mendocino ground
motion (Scale Factor=4.3) to 4.42g for scaled LomaPrieta-WAHO record (Scale
Factor=14.6). Corresponding values for the ratio Sa(λu=1.0)/S a(2%in50) are 1.7 and 5.1,
respectively.
Similarly, Figure 6.16b shows results for the eight earthquakes forming the bin of nearfault records. Again, at λu=1.0, mean value of Sa is 2.95g with C.O.V. of 43.2%.
However, the estimate of the median of Sa is 2.62g according to the regression line given
in Figure 6.16b with a conditional dispersion σ ln S
235
a
(T1 ,ξ =5 %)| λ u
= 0.444. Thus, there is about
11.2% difference in the average value of Sa at λu=1.0 when calculated directly from data
points as opposed to the regression relationship. According to these results, one can
notice that the 6-story frame at this specific site is on average at incipient collapse at a
Sa(λu=1.0)/S a(2%in50) ratio of about 3.4 (=2.95g/0.864g). It is again useful to report that
values of Sa(T1 ,ξ=5%) corresponding to λu=1.0 range from 1.21g for scaled Imperial
Valley ground motion (Scale Factor=3.1) to 4.61g for scaled LomaPrieta-Lexington
record (Scale Factor=2.5). Corresponding values for the ratio Sa(λu=1.0)/S a(2%in50) are
1.4 and 5.3 (i.e., stability limit point of the 6-story frame is expected for values of Sa
ranging from 1.4 to 5.3 times the spectral acceleration at the fundamental period of the
frame associated with 2%in50years hazard for that site according to IBC 2000). Also,
note the smaller scale factors at which the frame reaches the state of incipient collapse
under the effect of near-fault records as compared to the general records.
The
reasons
for
the
considerably
high
values
of
about
3.5
for
the
ratio
Sa(λu=1.0)/S a(2%in50) for the RCS frame under both types of records can be explained as
follows. First is the large “actual” lateral overstrength (Ω = 6.3) of the frame as discussed
in Section 6.2 leading to high values of Sa at λu=1.0. Furthermore, the high mean values
of Sa at λu = 1.0 are reduced if averages minus one standard deviation are reported instead
to consider some confidence bands in the results. Accordingly, mean minus one standard
deviation values for Sa(λu=1.0) are 1.95g and 1.68g for general and near-fault records,
respectively, corresponding to Sa(λu=1.0)/S a(2%in50) ratios of 2.3 and 1.9. Finally,
another reason that might be contributing to high Sa(λu=1.0) values is the limitation of the
analytical and material models implemented in DYNAMIX to automatically detect
structural collapse under earthquake induced dynamic effects and hence the use of the
global collapse determination technique proposed in Section 6.4.1. The use of such
technique in a subsequent step to the time history analysis and the lack of its integrity
with the analysis process explain part of these high values of the collapse load (i.e., the
stability limit point). In other words, if the damage identification process was
implemented in DYNAMIX as one entity, with all stiffness and strength deterioration
rules based on a continuous calculation of the damage index Dθ at different time steps
236
throughout the loading history, the collapse limit state (or global failure of a given
structure) would have been automatically and faithfully captured. This would have lead
to lower values of Sa at λu = 1.0. Furthermore, collapse limit state would be always
associated with a clear softening behavior of all IDACs as shown in Fig. 6.7 and for some
of the cases in Figs 6.8 (e.g., LP89-HCA) and 6.9 (e.g., IV79-A6).
One can observe from the results of the near-fault records that the response curves show
two distinct tendencies, depending on the record. This is obvious from IDRmax versus Sa
relationship given in Fig. 6.11 and Section 6.3.2 and also from λu versus Sa relationship
presented here in Figure 6.16b. In both figures, there are two distinct (especially near the
point of incipient collapse, i.e., at λu = 1.0) batches of curves composed of four records
each. The two subsets of records are: (1) IV79-A6, LP89-LG, EZ92-EZ and NR94-SY,
and (2) KB95-JM, NR94-RS, NR94-NH, and LP89-LX. A way of differentiating
between these two subsets is by using the idea of the pulse period, Tp , proposed and
defined by Krawinkler and Alavi (1998) and Somerville (1998). The pulse period, Tp , is
determined from the peaks of the velocity spectra of the near-fault record. More details
are given in Chapter 5. Four of the eight near-fault records forming subset (1) have a
pulse period much larger than the fundamental period of the frame (Tp /T1 =1.84 to 2.72),
while the other four records (subset (2)) have a pulse period a little less or almost equal to
the period of the frame (Tp /T1 =0.72 to 1.04). Accordingly, one can suppose that the
severity of their effect on the frame might be governed by the ground motion pulse
characteristics, as reflected by the ratio of Tp /T1 . The effect of the pulse period and its
probable good correlation with the response is further studied in Section 6.6. Thus, for
the first set with Tp /T1 =1.84 to 2.72, at λu=1.0, average value of Sa(T1 ,ξ=5%) is 1.81g
with a C.O.V. of 25.4%. For the second set with Tp /T1 =0.72 to 1.04, and again at λu=1.0,
average value of Sa(T1 ,ξ=5%) is 4.09g with a C.O.V. of 8.6%. It is then quite obvious
that for near-fault records with Tp much higher than the fundamental period of the
structure, global collapse might be expected at much lower Sa values than for the case
with Tp a little less or about the same value of the fundamental period of the structure.
This finding is further investigated while looking at the performance of the 12-story RCS
and 6-story STEEL frames in Chapter 7. For the first, more critical set, the ratio
237
Sa(collapse)/S a(2%in50) ranges from 1.4 to 2.5 in a quite narrow band. The scaling
factors at which collapse is reached for this subset range from 1.48 to 3.11 with an
average value of 2.34.
Finally, according to the regression equations shown in Figures 6.16a and b, on average,
one can further say that the frame is at the state of incipient collapse at a value of spectral
acceleration which is about 1.8 (for both general and near-fault records) times the value
causing excessive yielding and irrepairable damage of some members (i.e., Sa(T1 ,ξ=5%)
corresponding to 0.95λuo ). Irrepairable damage of some structural components (i.e., the
components should be replaced) is identified at 0.95λuo through values of Dθ larger than
about 0.6 as proposed in Chapter 4 based on reported physical damage from several test
specimens.
Again, as was done for the spectral acceleration-IDRmax relationship in Section 6.3.2,
trying to eliminate any bias of the least square fit due to different locations of data points
from different scaled events within the spectral acceleration-λu space, a linear regression
is first performed for data points corresponding to each record alone. Note that data
points with a value of λu greater than 0.95λuo are excluded from the least square fit for the
same reason mentioned earlier. Thus, one can obtain for each bin eight least square fit
lines with eight pair of regression coefficients: α and β. Then, a relation of the form
Sa(T1 ,ξ=5%) = a λßu
(6.7)
where a is the geometric mean of the eight α values for that bin, while ß is the
arithmetic mean of the eight β values, can be easily obtained. Values of a and ß are
given in Table 6.10. Note that these values are very close to regression coefficients
computed by applying regression on all records of each bin at once (see Figures 6.16a
and b) resulting in almost identical curves to those shown in Figs. 6.16a and b.
238
Table 6.10 Values of a and ß for Equation 6.7.
Parameter and Statistical
General Records Bin
Near-Fault Records Bin
Measure Values
a (C.O.V.)
2.94 (40%)
2.71 (50%)
-0.37 (54%)
-0.41 (43%)
ß (C.O.V.)
Spectral Acceleration, Sa
Collapse
Stiffness and Strength
Residual
reduction
Displacement
λ u = 4.2
0.95λ uo
1.0
λ uo =5.5
λu
Figure 6.17 Schematic of the effect of residual displacements on λu.
As previously mentioned, the technique used for the identification of the collapse limit
state involves reducing element stiffnesses and strengths based on the accumulated
damage and incorporating the residual (permanent) building drift into the structural
topology. The following question has to be answered: how much is the value of the
failure criterion λu affected by the stiffness and strength reduction versus residual
displacements? To answer this question, values of residual displacements associated with
the state of incipient collapse (i.e., λu = 1.0) and representative of values already
monitored through time history analyses for both general and near-fault records at this
damaged state are assigned to the frame. A second-order inelastic analysis of the frame
with undamaged stiffness and strength properties but with these pre-specified permanent
displacements is performed under the applied gravity loads. A stability limit load of λu of
about 4.2 is then reached. Therefore, at a given Sa corresponding to collapse limit state as
239
identified by the proposed methodology, about 29% of the reduction in the failure
criterion λu from λuo = 5.5 to λu = 1.0 is due to residual displacements. The rest is due to
stiffness and strength reduction in presence of that permanent drift. Carrying out the same
process but at a damage state corresponding to λu = 0.95λuo , the reduction in λu due to
residual building drift is about 15%. Figure 6.17 provides a simple explanation of the
results.
6.4.2.1 Conditional regression of λ u
So far, the relationship between the spectral acceleration at the fundamental period of the
structure and λu has been derived through a conditional regression of Sa(T1 ,ξ=5%) values
given λu. This form of the regression equation is useful to compute the probability of
having a specific Sa corresponding to a certain hazard level (e.g., 2%in50years), or the
probability of being at an Sa value less than or equal to that corresponding to a certain
hazard level, given a specific damage state, i.e., a specific value of λu.
Another useful way to view the data is to perform a conditional regression of λu given
spectral acceleration, Sa(T1 ,ξ=5%). Such a regression relationship of λu conditioned on
spectral acceleration is used to compute, for instance, the probability of reaching collapse
of the structure at a given hazard level. It can be further extended to calculate the
probability of failure of that structure at a given site due to all hazard levels expected to
occur at that site as follows
P[λu ≤ 1.0] = ∫ P[λu ≤ 1.0 | Sa (T1 , ξ )] f S a (Sa (T1 , ξ )) dSa
(6.8)
where the first term at the right hand side, showing a conditional probability of λu, can be
calculated through the regression model mentioned above, while the second term, the
probability distribution of Sa(T1 ,ξ), can be obtained from a conventional site hazard
analysis. For more details and different techniques to carry out such probabilistic seismic
240
demand analysis, one might refer to Shome (1999), Bazzurro et al. (1998), and Luco and
Cornell (1998).
The conditional regression of λu on Sa for both bins of records is shown in Fig. 6.18. Here
we see that despite its usefulness for a probabilistic seismic demand analysis, this
regression does not accurately capture the actual average value of the spectral
acceleration Sa at collapse (i.e., at λu=1.0). For instance, for the bin of general records, at
λu=1.0, Sa=5.38g versus its actual average of 3.12g. Similarly, these values are 5.73g
versus 2.95g for the bin of near-fault records.
6.4.3 Relationship between Maximum Interstory Drift Ratio and Global Failure
Criterion, λ u
In the previous section the global failure criterion, λu, has been related to the intensity
parameter Sa. In this present section, λu is related to a global response measure, IDRmax.
Among the merits of such relationship is to get an estimate of IDRmax given a certain
level of damage. Note that IDRmax is a global measure of the MDOF inelastic response
that might then be related to an SDOF elastic response parameter such as spectral
displacement.
Linear log-space regression conditioned on λu is carried out for data points with a value
of λu < 0.95λuo . Figure 6.19a shows results for the bin of general records. Results reveal
that the correlation between IDRmax and λu is quite good as manifested by a narrow band
of curves throughout the damage evolution from λuo up to total collapse with a
conditional dispersion σ ln IDR
max |λu
= 0.229. At λu=1.0, the average IDRmax = 0.087 with a
C.O.V. of 22.5%. An estimate of the median IDRmax at λu = 1.0 is 0.085 according to the
regression line given in Figure 6.19a. One general observation is that all values of IDRmax
are clustered within a narrow band except for the Valparaiso record where the IDRmax
values are much smaller for a given λu. An explanation for this behavior is that this
record is a very long one with duration of strong motion, tSM = 38 seconds. Long records
241
of high intensity such as this accumulate more damage with smaller peak values of the
response parameters. Thus, collapse of structures subjected to such records is mainly due
to accumulation of damage over many cycles rather than a peak single pulse that is
characteristic of near-fault records.
Figure 6.19b gives similar results for the bin of near-fault records. Again, the correlation
between IDRmax and λu is good as reflected by a narrow band of curves throughout the
damage evolution from λuo up to total collapse with a conditional dispersion σ ln IDR
max |λu
=
0.191. At λu=1.0, the average IDRmax = 0.116 with a C.O.V. of 20.9%. An estimate of the
median IDRmax at λu = 1.0 is 0.111 according to the regression line given in Figure 6.19b.
Observed higher value of mean IDRmax at λu=1.0 than for the bin of general records is
due to the pulse effects of the near-fault records. One may note that at the state of overall
failure, mean value of IDRmax for near-fault records is about 30% higher than for general
records.
The distinction of response within the set of near fault records, previously reported in
Section 6.4.2, is not obvious in the IDRmax versus λu data. Moreover, the two records with
highest IDRmax (Loma Prieta at Lexington, LP89-LX, and Imperial Valley at array 06,
IV79-A6) have the highest and lowest Sa(T1 ,ξ=5%) at failure. Review of the results
shown in Figure 6.19b will explain the reasons for such behavior. The Imperial Valley
record, is the one with the highest Tp /T1 ratio (2.72) and the most severe pulse effect, as
seen in its ground velocity record (Appendix A). Consequently, such record will cause
higher IDRmax values when applied to the frame. The Lexington record with a Tp /T1 ratio
of 0.8, produces a large value of IDRmax but the smallest value of residual displacements
among all near-fault records. Thus, when applying the technique for computing λu, low
values of residual displacements will lead to less severity of P-∆ effects, and
consequently, higher values of Sa(T1 ,ξ=5%) for λu = 1.0. Again, the subset of the four
most critical records within the bin has a higher mean IDRmax (0.121) at λu=1.0 relative to
the other subset (mean IDRmax= 0.110), though the difference between the values are not
as pronounced as the differences between Sa values at λu =1.0.
242
7
6
Sa (T1=1.25sec,ξ=5%)
Miyagi
Valparaiso
LP89-HCA
λu = 4.22 Sa-0.86
σlnλ |S = 0.589
u a
LP89-HSP
LP89-WAHO
CM92-RIO
LA92-YER
Mendocino
Lin. Regr. given Sa
Sa = 2.92 λu -0.36
σlnS |λ = 0.379
a u
5
4
Lin. Regr. given λu
3
2
1
λu = 1.0
(collapse)
0
0
1
2
3
4
5
6
λu (based on 1.0D+0.25L)
Figure 6.18a Conditional regression of λu given Sa for bin of general records.
6
λu = 3.47 Sa-0.72
σlnλ |S = 0.624
u a
-0.37
Sa = 2.62 λu
σlnS |λ = 0.444
a u
5
Sa (T1=1.25sec,ξ=5%)
IV79-A6
LP89-LG
LP89-LX
4
EZ92-EZ
NR94-NH
NR94-RS
NR94-SY
KB95-JM
Lin. Regr. given Sa
3
Lin. Regr. given λu
2
1
λu = 1.0
(collapse)
0
0
1
2
3
4
5
λu (based on 1.0D+0.25L)
Figure 6.18b Conditional regression of λu given Sa for bin of near-fault records.
243
6
0.20
Miyagi
Valparaiso
LP89-HCA
λu = 1.0
(Collapse)
0.16
LP89-HSP
LP89-WAHO
CM92-RIO
LA92-YER
Mendocino
Linear Regression
IDRmax
IDRmax = 0.085 λu -0.39
0.12
σlnIDR
max|λu = 0.229
0.08
0.04
2
Ra = 0.556
0.00
0
1
2
3
4
λu (based on 1.0D+0.25L)
5
6
Figure 6.19a IDRmax - λu relationship for bin of general records.
0.20
0.16
λu = 1.0
(Collapse)
IV79-A6
LP89-LG
LP89-LX
-0.49
IDRmax = 0.111 λu
EZ92-EZ
NR94-NH
NR94-RS
NR94-SY
KB95-JM
Linear Regression
IDRmax
σlnIDR
max |λu = 0.191
0.12
0.08
0.04
2
Ra = 0.753
0.00
0
1
2
3
4
λu (based on 1.0D+0.25L)
5
Figure 6.19b IDRmax - λu relationship for bin of near-fault records.
244
6
In FEMA 273, both the peak transient and residual (i.e., permanent) interstory drifts are
used to define performance levels. Tables are provided that contain limiting drift values
for various structural systems, at various performance levels. FEMA 273 emphasizes that
these drift values should be used as general indicators of performance and not as strict
design or evaluation limits. Table 6.11 summarizes the drift values cited in FEMA 273
for concrete frames and steel moment frames. Based on the design and behavioral
considerations described in Chapter 5, the steel moment frames criteria in Table 6.11 may
be thought of as representative of RCS frames.
Table 6.11 Indicative drift values at different performance levels (FEMA 273).
Structural
Collapse Prevention
Life Safety
Immediate Occupancy
System
Drift (%)
Drift (%)
Drift (%)
Transient Residual Transient Residual Transient
Residual
Concrete
4
4
2
1
1
Negligible
Frames
Steel
5
5
2.5
1
0.7
Negligible
Moment
Frames
For the 6-story RCS frame, the average transient IDRmax at λu=1.0 (collapse) is 8.7%
(C.O.V.=22.5%), while at λu=0.95λuo (excessive yielding) is 3.2% (C.O.V.=10.1%) for
the general records. Corresponding values for the near-fault records are 11.6%
(C.O.V.=20.9%) and 3.3% (C.O.V.=23.9%), respectively. Note the similar values at the
excessive yielding stage while the considerable difference at failure due to the
pronounced effect of the pulse at such a high intensity level of the record causing global
collapse of the structure. Also note that average transient IDRmax at λu = 1.0 and λu =
0.95λuo are above the limits of Life Safety performance level transient drift proposed by
FEMA 273 but the values at λu = 0.95λuo are close to the 2.5% drift at Life Safety
performance level. Another useful observation is the point made in Section 6.4.2 that, on
average, Sa at λu = 1.0 is 1.8 times its value at λu = 0.95λuo . This shows the narrow range
(in terms of intensity of the input) that can get us from around the life safety limit to
collapse. Results presented herein also reinforce the adequacy of the indicative drift
values suggested by FEMA 273 for collapse prevention. However, they also emphasize
245
the fact that the range available between the so-called Life Safety performance level and
Collapse in terms of the driving input intensity, Sa, is small and sometimes difficult to
accurately quantify.
One final observation is that for the same 6-story RCS frame, the average residual IDRmax
at λu=1.0 is 6.8% (C.O.V.=28.0%), while at λu=0.95λuo is 1.1% (C.O.V.=48.7%) for the
bin of general records. Corresponding values for the bin of near-faults records are 7.4%
(C.O.V.=29.7%) and 1.3% (C.O.V.=40.6%), respectively. It is also worth to point that up
to the drift values proposed by FEMA 273 for the Life Safety performance level, the
global load ratio λu is almost stable and nearly equal to λuo . Beyond this, i.e., λu <
0.95λuo , λu deteriorates very quickly down to λu = 1.0.
6.4.4 Spatial Damage Distribution
After identifying the two damaged states at λu = 0.95λuo and λu = 1.0 and relating them to
specific performance levels, it is important to look at the distribution of damage in terms
of Dθ throughout the frame. Figs. 6.20 and 6.21 show values of Dθ at different sections
for two general records: Valparaiso (1985) and Mendocino (1992), respectively. At λu =
0.95λuo , the damage is much more accentuated due to the Valparaiso record. Moreover, it
is distributed throughout the whole frame (involving columns, beams, and joint panels)
with local failure of a few beam sections. Local failure is represented by a gray fill in the
figures. Values of Dθ > 0.60 take place at many locations showing severe damage.
However, at the same damage state, i.e., at λu = 0.95λuo , the Mendocino record causes
much less damage to the frame. The damage is mainly confined to the ground floor
columns bases and the beams of the first three stories. Furthermore, only very few
sections of the beams surpass the level of repairable damage (i.e., damaged parts should
be replaced) and no damage has been observed for composite joints. The difference in the
severity and spread of damage due to both records is mainly explained by the much
longer strong motion duration of the Valaparaiso record (tSM = 38 seconds) compared to
the Mendocino record (tSM = 18 seconds). This difference is even more pronounced at the
collapse limit state (i.e., λu = 1.0) as shown in Figs. 20 and 21. However, at λu = 1.0,
246
more beams suffer local failure due to the Mendocino record but with no failure or even
severe damage of any joint or any column section aside from the base of the ground floor
columns. Thus, one can conclude that the damage due to the Valparaiso record is mainly
of the cumulative type and not a peak response type of failure. Fig. 6.19a reveals same
observation showing that the collapse limit state due to the Valparaiso record is reached
at the smallest IDRmax (0.044). Figs 6.22 and 6.23 give the distribution of plastic rotations
at all columns and beams sections and joint distortions of all composite joints up the
height of the frame at λu = 1.0 for the Valparaiso and the Mendocino records,
respectively. These figures show that collapse limit state occurs at much lower plastic
rotations for beams and columns for the case of the Valparaiso record than the
Mendocino record yet with high values of Dθ.
Figures 6.20 through 6.23 also show that severe damage is mainly located at the lower
three stories of the frame as previously seen from the static pushover results (Section 6.2
and Figs. 6.2 and 6.3) and story IDACs (Fig. 6.12).
Figure 6.24 gives the spatial distribution of Dθ at λu = 0.95λuo and λu = 1.0 for Erzincan
(1992) record as an example of near-fault records with severe pulse effect (subset (1),
Section 6.4.2). Fig. 6.25 shows the values of columns and beams plastic rotations and
joint distortions throughout the frame at λu = 1.0. The two figures show the severe
damage at λu = 1.0 with high values of θp,C and θp,B corresponding to high values of Dθ,
revealing “peak response” type of damage. The damage is only confined to the first three
stories with much more severe damage (and failure) of beams than columns. No
composite joint failure takes place; an observation that holds for nearly all near-fault
records.
The less spread of damage throughout the frame due to the Erzincan record as compared
to the Valparaiso and the Mendocino records is because of the shorter strong motion
duration of this near-fault record (tSM = 7.1 seconds). Accordingly, damage due to
Erzincan record is more of the pulse type (i.e., peak response), a characteristic of nearfault events, rather than of the cumulative type.
247
0.87
0.49
0.82
0.62
0.85
0.65
0.85
0.49
0.90
0.36
0.94
0.34
0.62
0.63 0.53
0.36
0.57
0.79 0.48
0.53
0.86 0.35
0.52
0.34 0.81
0.54
0.39
0.35
0.44
0.43
0.46 0.57 0.70
0.37
0.44
0.51 0.65 0.62
0.47
0.62
0.34
0.62 0.69 0.74
0.40
0.39
0.64 0.64 0.83
0.47
0.43
0.90 0.38
0.49
0.42
0.69 0.82 0.56
0.33
0.76 0.63 0.76
0.39
0.92
0.92
0.92
0.92
0.89
0.36
0.55
0.42 0.46
0.35
0.53
0.58
(a) λ u = 0.95 λ uo
0.92
0.66
0.83
0.79
0.91
0.79
0.89
0.66
0.91 0.64
0.62
0.58
0.58
0.43 0.84
0.44
0.81
0.67 0.91
0.61
0.73
0.38 0.94 0.66
0.52
0.41
0.73
0.44
0.77
0.60
0.50
0.69
0.65
0.58 0.78 0.91
0.58
0.63
0.77 0.89
0.71
0.56
0.75 0.92 0.85
0.64
0.82
0.49
0.59
0.74
0.39
0.71
0.79
0.66
0.32
0.74
0.67
0.54
0.74
0.91
0.77
0.73 0.41
0.62
0.85
0.62
0.48
0.94
0.61
0.91
0.41
0.49
(b) λ u = 1.0
Figure 6.20 Distribution of Dθ at different λu values- Valparaiso (1985) record.
248
0.48
0.37
0.31
0.35
0.32
0.52
0.35
0.57
0.31
0.65
0.32
0.48
0.35
0.62
0.32
0.63
0.51
0.50
0.49
0.50
0.55
(a) λ u = 0.95 λ uo
0.87
0.46 0.40 0.47
0.41
0.73
0.33 0.74
0.33
0.84
0.36
0.39
0.58
0.67
0.60
0.36
0.38
0.34
0.42
0.35 0.82
0.44
0.38
0.40
0.46
(b) λ u = 1.0
Figure 6.21 Distribution of Dθ at different λu values- Mendocino (1992) record.
249
6
5
Floor #
4
3
2
1
0
0.00
0.02
0.04
0.06
Max. Transient Col. Plastic Rot., θ p,C [rad.]
0.08
0.02
0.04
0.06
Max. Transient Beam Plastic Rot., θp,B [rad.]
0.08
0.01
0.02
0.03
0.04
0.05
Max. Transient Joint Panel Distortion, γJP [rad.]
0.06
6
5
Floor #
4
3
2
1
0
0.00
6
5
Floor #
4
3
2
1
0
0.00
Figure 6.22 Plastic rotation values at λu = 1.0 – Valparaiso (1985) record.
250
6
5
Floor #
4
3
2
1
0
0.00
0.02
0.04
0.06
0.08
0.10
Max. Transient Col. Plastic Rot., θp,C [rad.]
0.12
0.02
0.04
0.06
0.08
0.10
Max. Transient Beam Plastic Rot., θ p,B [rad.]
0.12
0.01
0.02
0.03
0.04
0.05
Max. Transient Joint Panel Distortion, γJP [rad.]
0.06
6
5
Floor #
4
3
2
1
0
0.00
6
5
Floor #
4
3
2
1
0
0.00
Figure 6.23 Plastic rotation values at λu = 1.0 – Mendocino (1992) record.
251
0.40
0.66
0.73
0.32
0.38
0.52
0.34
0.53
0.33
0.59
0.44
0.59
0.41
0.38
0.42
0.41
( a) λ u = 0.95 λ uo
0.44
0.67
0.77
0.49
0.43
0.32
0.32
0.79
0.38
0.48
0.48
0.45
0.91
(b) λ u = 1.0
Figure 6.24 Distribution of Dθ at different λu values – Erzincan (1992) record.
252
6
5
Floor #
4
3
2
1
0
0.00
0.03
0.06
0.09
0.12
Max. Transient Col. Plastic Rot., θp,C [rad.]
0.15
0.03
0.06
0.09
0.12
Max. Transient Beam Plastic Rot., θ p,B [rad.]
0.15
0.006
0.012
0.018
0.024
Max. Transient Joint Panel Distortion, γJP [rad.]
0.030
6
5
Floor #
4
3
2
1
0
0.00
6
5
Floor #
4
3
2
1
0
0.000
Figure 6.25 Plastic rotation values at λu = 1.0 – Erzincan (1992) record.
253
6.5 Global versus Local Response
The relationship between global and local response previously presented in Section 6.2.1
is revisited here based on the results from the time history analyses under the two bins of
records. In this section, we deal with maximum (or peak) values of global and local
response measures that occur through each time history analysis. Accordingly, Equation
6.4 will consider maximum values and it is given below in its new form.
θp,C|max = f(|IDRi – IDRi+1 |max or |IDRi – IDRi-1 |max) = α ∆IDRmaxβ
(6.9a)
θp,B|max = f(IDRp,i+1|max or IDRp,i|max) = α IDRp,maxβ
(6.9b)
Values used for Equation 6.9 are again based on the deformed configuration of the frame,
i.e., whether columns hinging takes place at top or bottom sections of the story.
6.5.1 Relationship between ∆IDRmax and Peak θ p,C
In Figures 6.26 and 6.27, ∆IDRmax versus peak θp,C data are plotted for the general and
near-fault records at λu =1.0. Again, a power model regression (Eq. 6.9) is performed in
the log space for results corresponding to each bin. The least square fit has been done
once conditioned on global response (i.e., given ∆IDRmax) and then conditioned on local
response (i.e., given θp,C). Regression lines for both cases are also shown with the
conditional dispersions as defined in Section 6.3.2. The benefit of performing regression
conditioned on global response is that for a given maximum change in IDR that one may
anticipate (i.e., corresponding to any level of performance), an estimate of the median
peak plastic rotation in columns can be identified. Then, this estimate can be compared to
acceptance criteria and limits set within ATC 40 or FEMA 273. Thus, one can rate the
performance of the structure according to local acceptance criteria set by codes by only
processing data corresponding to global response results. This is provided that local
versus global response relationship is quite stable and not a function of the level of
damage or the type of record (e.g. general versus pulse record, etc…). On the other hand,
254
if the fit is carried out conditioned on local response, this implies that given a specific
peak plastic rotation θp,C, an estimate of median ∆IDRmax can be obtained.
To further elaborate on this, Figure 6.28 compares least square fit relationship for general
and near-fault records, again conditioned on either global or local response. It is very
obvious that one get almost identical results for general and near-fault records if the
regression analysis is performed conditioned on local response, and very close results if
conditioned on global response. Note that the regression coefficient β is close to 1.0
meaning that there is almost linear relationship between ∆IDRmax and peak θp,C. Also note
the low dispersion values and high R 2a values (close to 1.0) reported in Figs 6.26 and
6.27 showing the good correlation between the two response measures. This clearly
indicates that there is a consistent pattern of deformation associated with the frame design
which produces a proportionate increase in the column plastic rotation demand (or the
column rotation ductility demand) as the overall lateral deformation increases. This
proportionality not only shows the adequacy of the design process but also facilitates
linking element ductility (or plastic rotation) capacities that are required to reach
maximum drift limits associated with specific performance level to such global drift
values, or vice versa. One may also observe that up to considerable values of θp,C of
about 0.06 radians, ∆IDRmax has almost the same value of 0.06 within a difference
between 2 and 10%.
After showing that ∆IDRmax versus θp,C relationship is quite stable irrespective of the type
of record, it is worth showing the effect of the level of damage. Similar regression
analyses have been carried out for results associated with values of λu ≅ 0.55λuo . Again
comparable results concerning the β coefficients in the regression form close to 1.0, and
reasonable conditional dispersion values (e.g. σ ln θ
p ,C | ∆IDRmax
=0.238 and σ ln ∆IDR
max |θ p ,C
=0.241)
are obtained for both cases of near-fault and general records. Figure 6.29 gives ∆IDRmax
versus θp,C relationship at the two levels of damage (λu = 1.0 and λu ≅ 0.55λuo ). Again,
very comparable relationships are obtained up to high values of plastic rotations and
change in maximum interstory drift ratios.
255
0.12
Maximum Change in IDR
2
Ra = 0.9
0.09
θp,C = 0.69 ∆IDR max
0.06
0.91
σlnθ |∆IDR
= 0.214
p,C
max
∆IDRmax = 1.00 θp,C
0.99
σln∆IDR
= 0.224
max|θp,C
0.03
Values from Analysis
Regression given θp,C
Regression given IDR
0.00
0.00
0.03
0.06
0.09
0.12
Max. Transient Col. Plastic Rot., θ p,C [rad.]
Figure 6.26 Global versus local response (θp,C) for bin of general records at λu=1.0.
0.16
Ra2 = 0.9
Maximum Change in IDR
θp,C = 0.91 ∆IDRmax0.99
0.12
σlnθ |∆IDR
= 0.139
p,C
max
0.98
∆IDR max = 1.00 θp,C
0.08
σln∆IDR
= 0.139
max|θp,C
0.04
Values from Analysis
Regression given θp,C
Regression given IDR
0.00
0.00
0.04
0.08
0.12
0.16
Max. Transient Col. Plastic Rot., θp,C [rad.]
Figure 6.27 Global versus local response (θp,C) for bin of near-fault records at λu=1.0.
256
0.12
Maximum Change in IDR
Columns
0.09
0.06
0.03
General Records
Near-Fault Records
0.00
0.00
0.03
0.06
0.09
0.12
Max. Transient Col. Plastic Rot., θ p,C [rad.]
(a) Regression conditioned on local response, θp,C.
0.12
Maximum Change in IDR
Columns
0.09
0.06
0.03
Genral Records
Near-Fault Records
0.00
0.00
0.03
0.06
0.09
0.12
Max. Transient Col. Plastic Rot., θp,C [rad.]
(b) Regression conditioned on global response, ∆IDRmax.
Figure 6.28 ∆IDRmax - θ p,C relationship for general and near-fault records at λu=1.0.
257
0.12
Maximum Change in IDR
0.10
Columns
0.08
0.06
Regression conditioned on θp,C
0.04
λu = 0.55λuo
λu = 1.0
0.02
0.00
0.00
0.02
0.04
0.06
0.08
Max. Transient Col. Plastic Rot., θp,C [rad.]
0.10
0.12
0.12
Columns
Maximum Change in IDR
0.10
0.08
0.06
Regression conditioned on ∆IDR max
0.04
λ u = 0.55λuo
0.02
0.00
0.00
λ u = 1.0
0.02
0.04
0.06
0.08
0.10
0.12
Max. Transient Col. Plastic Rot., θp,C [rad.]
(a) Bin of general records
Figure 6.29 ∆IDR max - θp,C relationship at different levels of damage based on values of λ u.
258
0.12
Maximum Change in IDR
0.10
Columns
0.08
0.06
Regression conditioned on θp,C
0.04
λu = 0.55λuo
λu = 1.0
0.02
0.00
0.00
0.02
0.04
0.06
0.08
0.10
0.12
Max. Transient Col. Plastic Rot., θp,C [rad.]
0.12
Maximum Change in IDR
0.10
Columns
0.08
0.06
Regression conditioned on ∆IDR max
0.04
λ u = 0.55λ uo
0.02
0.00
0.00
λ u = 1.0
0.02
0.04
0.06
0.08
0.10
0.12
Max. Transient Col. Plastic Rot., θp,C [rad.]
(b) Bin of near-fault records
Figure 6.29 ∆IDRmax - θp,C relationship at different levels of damage. (Continued)
259
6.5.2 Relationship between IDRp,max and Peak θ p,B
Similar to what has been done for the relationship between ∆IDRmax and θp,C|max, one can
relate IDRp,max and θp,B|max as previously mentioned. Again, Figure 6.30 gives the
IDRp,max versus peak θp,B regression relationship for bin of general records at collapse
state (i.e., λu =1.0), while Figure 6.31 gives it for the bin of near-fault records, along with
data points from different time history analyses results. Figure 6.32 compares least square
fit relationships for general and near-fault records, again conditioned on either global or
local response. It is very obvious that one gets almost identical results for general and
near-fault records if the regression analysis is performed conditioned on global response,
and very close results if conditioned on local response. The regression coefficient β is
also close to 1.0 as for the case of columns’ response. Conditional dispersion values
shown in the figures (e.g. σ ln θ
p ,B |IDRp, max
=0.379 and σ ln IDR
| p, B
p, max θ
=0.358 for general records) are
a bit higher than for the case of ∆IDRmax versus θp,C relationship but might still be
considered reasonable. These values along with high R 2a still indicate that there is a
consistent pattern of deformation associated with the frame design which produces a
proportionate increase in the beam plastic rotation demand (and consequently its rotation
ductility demand) as the overall lateral deformation increases.
Finally, Figure 6.33 gives IDRp,max versus θp,B relationship at the two levels of damage
(λu = 1.0 and λu ≅ 0.55λuo ) as previously done for columns. Again, very comparable
relationships are obtained up to high values of plastic rotations and plastic interstory drift
ratios. A 1:1 relationship between the values of IDRp,max and θp,B|max holds on average
(e.g., for near-fault records, at both damage levels, at a given global response
IDRp,max=0.06, an estimate of the median of the local response θp,B|max is 0.057 radians).
260
0.12
2
Ra = 0.9
0.10
0.98
θp,B = 0.88 IDRp,max
IDRp,max
0.08
σlnθ |IDR
= 0.379
p,B
p,max
0.06
0.04
0.87
IDRp,max = 0.66 θp,B
σlnIDR
= 0.358
p,max|θp,B
0.02
Values from Analysis
Regression given θp,B
Regression given IDR
0.00
0.00
0.02
0.04
0.06
0.08
0.10
0.12
Max. Transient Beam Plastic Rot., θp,B [rad.]
Figure 6.30 Global versus local response (θ p,B) for bin of general records at λu=1.0.
0.18
2
Ra = 0.9
0.15
1.01
θp,B = 0.97 IDR p,max
IDRp,max
0.12
σlnθ |IDR
= 0.393
p,B
p,max
0.09
IDRp,max = 0.78 θp,B
0.92
σlnIDR
= 0.375
p,max|θp,B
0.06
Values from Analysis
Regression given θp,B
0.03
Regression given IDR
0.00
0.00
0.03
0.06
0.09
0.12
0.15
0.18
Max. Transient Beam Plastic Rot., θp,B [rad.]
Figure 6.31 Global versus local response (θ p,B) for bin of near-fault records at λu=1.0.
261
0.12
Beams
0.10
IDRp,max
0.08
0.06
0.04
0.02
0.00
0.00
General Records
Near-Fault Records
0.02
0.04
0.06
0.08
0.10
0.12
Max. Transient Beam Plastic Rot., θp,B [rad.]
(a) Regression conditioned on local response, θ p,B.
0.12
0.10
Beams
IDRp,max
0.08
0.06
0.04
0.02
0.00
0.00
General Records
Near-Fault Records
0.02
0.04
0.06
0.08
0.10
0.12
Max. Transient Beam Plastic Rot., θp,B [rad.]
(b) Regression conditioned on global response, IDR p,max .
Figure 6.32 IDRp,max - θp,B relationship for general and near-fault records at λu=1.0.
262
0.12
0.10
Beams
IDRp,max
0.08
0.06
Regression conditioned on θp,B
0.04
λ u = 0.55λuo
0.02
0.00
0.00
λ u = 1.0
0.02
0.04
0.06
0.08
0.10
0.12
Max. Transient Beam Plastic Rot., θ p,B [rad.]
0.12
0.10
Beams
IDRp,max
0.08
0.06
Regression conditioned on IDR p,max
0.04
λu = 0.55λ uo
λu = 1.0
0.02
0.00
0.00
0.02
0.04
0.06
0.08
0.10
0.12
Max. Transient Beam Plastic Rot., θp,B [rad.]
(a) Bin of general records
Figure 6.33 IDRp,max - θp,B relationship at different levels of damage based on values of λ u.
263
0.12
Beams
0.10
IDRp,max
0.08
0.06
Regression conditioned on θp,B
0.04
λ u = 0.55λ uo
0.02
0.00
0.00
λ u = 1.0
0.02
0.04
0.06
0.08
0.10
0.12
Max. Transient Beam Plastic Rot., θ p,B [rad.]
0.12
0.10
Beams
IDRp,max
0.08
0.06
Regression conditioned on IDRp,max
0.04
λu = 0.55λuo
λu = 1.0
0.02
0.00
0.00
0.02
0.04
0.06
0.08
0.10
0.12
Max. Transient Beam Plastic Rot., θp,B [rad.]
(b) Bin of near-fault records
Figure 6.33 IDRp,max - θp,B relationship at different levels of damage. (Continued)
264
6.5.3 Estimates of Local Response Given Global Response and Input Intensity Le vel
– Benefits and Implications
If a structure is subjected to different events with same Sa(T1 ,ξ), different maximum IDR
values are obtained. In other words, for different records representing the same hazard
level different values of the global response measures are obtained. The dispersion of the
values of this response parameter (which is considered as an MDOF inelastic response
measure that can be related to some SDOF elastic response parameter through some
empirical formulae) depends on the level of non-linearity the structure has undergone.
One further useful step in this whole process is to estimate an average value of a local
response measure given both: an input hazard level defined by a specific Sa(T1 ,ξ), and a
global MDOF response parameter given in terms of some maximum IDR, for instance.
This can be achieved through regression analysis of the form
2
θ p = α Saβ1 (T1 , ξ ) IDR βmax
(6.10)
where θp is an estimate of the median of the peak transient plastic rotation from a time
history analysis (i.e., a measure of local response at the element level).
As such, for any anticipated hazard level and any pre-specified global response that might
be associated with this hazard, one may estimate an average value of some local demand
parameter such as members plastic rotation. This demand can then be checked against
limiting values given by codes as acceptance criteria corresponding to that specific
hazard. This whole process serves to rate the performance of the structure at the local
level and not just globally, and to judge the adequacy of the design.
Another benefit of having such relationship as the one suggested by Equation 6.10 is its
usefulness in calculating joint probabilities of having a specific local response, θp , along
with a specific global response, IDRmax, for a given structure at a given site. This joint
probability, P(θp ,IDRmax) can be calculated by integrating the following product
265
[P(θp |IDRmax,Sa).P(IDRmax|Sa)] of conditional probabilities over all hazard levels at the
specific site known through a hazard spectrum curve of the site.
Regression analyses of the form given by Equation 6.10 are performed in the log space
and resulting relationships are given in Table 6.12 and Figures 6.34 and 6.35. Note that
local response of members is represented by element peak plastic rotation (θp,B|max for
beams and θp,C|max for columns), and global response is given as before in terms of
∆IDRmax for columns and IDRp,max for beams.
Table 6.12 Regression equations for local response given global response and input
intensity level.
Beams
Columns
General Records
General Records
θ p, B = 0.95 S0.02
IDR 1p,.00
a
max
. 93
θ p, C = 0.62 S0.19
∆IDR 0max
a
σ ln θ p,B |Sa (T1 ,ξ ), IDR p,max = 0.386
σ ln θ p,C |Sa ( T1 ,ξ ),∆ IDR max = 0.242
R a2 = 0.9
Near-Fault Records
R a2 = 0.9
Near-Fault Records
θ p, B = 1.24 S-0.07
IDR 1p,.07
a
max
σ ln θ p,B |Sa (T1 ,ξ ), IDR p,max = 0.398
.97
θ p, C = 0.76 S0.11
∆IDR 0max
a
σ ln θ p,C |Sa ( T1 ,ξ ),∆ IDR max = 0.190
R a2 = 0.9
R a2 = 0.9
Figures 6.34 and 6.35 show the global versus local response calibrated for data at certain
hazard el vels. Again as before, the linear relationship between selected local response and
global response measures for beams and columns (as expected from values of β 2 close to
1.0) is obvious. Moreover, given the same global response value, and same hazard level
(i.e., same Sa(T1 ,ξ)), local response in terms of plastic rotation in either beams or columns
is very close for bins of general records and near-fault records. Though, values are a bit
higher for the case of near-fault records; for instance, a difference of 8.6% in median θp,B
between both types of records is observed at a given IDRp,max=0.06 and Sa(T1 ,ξ=5%)
266
representing a 2%in50years hazard level, and a difference of 11.1% in median θp,C is
observed at a given ∆IDRmax=0.06 and similar hazard level.
One other important note is that the effect of the level of the spectral acceleration (i.e.,
the input hazard level) on the relationship between local and global response values is
almost negligible. This observation shows again the stability of such relationship and its
usefulness to get reliable estimates of the median of local response measures given a
specific global response value in terms of IDR irrespective of the level of damage, type of
record, or even the intensity of the input. But definitely further study is to be made for
other types of structures (different geometries, construction materials, amounts of
overstrength, etc.) to confirm or modify this finding. Part of this study is carried out in
the following chapter for a 12-story RCS and a 6-story STEEL frames.
6.6 Global Response Dependency on Different Ground Motion Input Parameters
We have already investigated the correlation between spectral acceleration and different
global response parameters including maximum transient interstory drift ratio, IDRmax,
and global failure criterion, λu. As shown by R a2 values, Sa(T1 ,ξ=5%) explains most of
the dispersion of drift response while it is not performing that satisfactorily for λu.
Therefore, we will include other independent variables (i.e., input parameters) to check
for any additional significant reduction in the regression conditional dispersion. If
successful, an improved correlation between input variables and response would reduce
the number of nonlinear time history analyses necessary to reach a pre-specified
accuracy. These additional input parameters may be seismological parameters, e.g.,
magnitude, M, and distance, R, of the records (out of the scope of this research), and/or
record parameters, e.g., strong motion duration, tSM, higher frequency spectral
acceleration, spectral acceleration at a target longer (i.e., damaged) period, TF, of the
structure, Sa(TF,ξ=5%), or pulse period, Tp , for near-fault records.
267
Max. Transient Column Plastic Rot., θp,C [rad.]
0.10
Columns
0.08
0.06
0.04
10%in50years Sa
0.02
2%in50years Sa
1.5*2%in50years S a
0.00
0.00
0.02
0.04
0.06
0.08
0.10
Max. Transient Beam Plastic Rot., θp,B [rad.]
∆IDRmax
0.10
Beams
0.08
0.06
0.04
10%in50years S a
0.02
2%in50years Sa
1.5*2%in50years Sa
0.00
0.00
0.02
0.04
0.06
0.08
0.10
IDRp,max
Figure 6.34 Global versus local response at different hazard levels for bin of general records.
268
Max. Transient Column Plastic Rot., θp,C [rad.]
0.10
Columns
0.08
0.06
0.04
10%in50years Sa
2%in50years Sa
0.02
1.5*2%in50years S a
0.00
0.00
0.02
0.04
0.06
0.08
0.10
Max. Transient Beam Plastic Rot., θp,B [rad.]
∆IDRmax
0.10
Beams
0.08
0.06
0.04
10%in50years S a
0.02
2%in50years Sa
1.5*2%in50years Sa
0.00
0.00
0.02
0.04
0.06
0.08
0.10
IDRp,max
Figure 6.35 Global versus local response at different hazard levels for bin of near-fault records.
269
Strong Motion Duration: The strong motion duration, tSM, is defined as the difference in
times corresponding to 95% and 5% of the total input energy carried by a record
(Trifunac and Brady, 1975). The input energy up to a time, T, is calculated as the integral
of the square of acceleration time history which is given as
E (T ) =
∫
T
0
a 2 (t ) dt
(6.11)
where, a(t) is the ground acceleration at a time t. Values of tSM are given in Tables 5.7
and 5.8 in Chapter 5 for general and near-fault records, respectively.
Pulse Period: It is important to mention that duration may be misleading if there exists a
large acceleration and/or velocity pulse in the record. Therefore, for near-fault records
characterized by their pulse effects, a pulse period, Tp , is also considered. Tp is
determined as the period at the peak of the velocity spectra as suggested by Krawinkler
and Alavi (1998). Values of Tp for the eight near-fault records are given in Table 5.8,
Chapter 5.
Damaged Period: The damaged period of the structure, TF, reflects the loss of lateral
stiffness of the structure due to damage. While period lengthening apparent in TF is a
phenomenon rather than just a specific value, TF is suggested as a numerical value that
serves picking a spectral acceleration correlating with the tendency of the structure to
damage. This specific spectral acceleration value is thought of to be of great importance
in studying the nonlinear response dependency on record parameters for a specific
structure. There are several ways that might be proposed to calculate TF. Some methods
are trying first to compute a certain lateral target displacement of the structure. Then, by
carrying a static pushover analysis of the structure, the initial period of the undamaged
structure, Ti, could be related to the final period of the damaged structure, TF, through the
decrease in corresponding lateral stiffness values, Ki and Kδt as follows
TF = Ti
(6.12)
K i / K δt
270
where Kδt is easily calculated as the secant stiffness at the target displacement δ t . In the
present work, two methods of calculating δ t are suggested. The first is by calculating the
target displacement as given by Equation 5.8 in Chapter 5 (as per FEMA 273). The
second is based on the “Equal Displacement” concept. Accordingly, one can perform an
elastic pushover analysis of the structure and get the lateral displacement at a base shear
corresponding to 2%in50years hazard; this value might be considered as a reasonable
value for δ t . According to this method, TF has been determined as 2.18 seconds for the 6story RCS frame. (Note that T1 =1.25 seconds).
A linear regression analysis of the two response global damage measures (IDRmax and λu)
on different input independent variables will be carried out in the log space. The
independent parameters considered herein are the spectral acceleration at the fundamental
period of the structure and 5% critical damping, Sa(T1 ,ξ=5%), the ratio, R Sa , of the
spectral accelerations at TF and at T1 (Sa(TF,ξ=5%)/S a(T1 ,ξ=5%)), the duration of the
strong motion, tSM, and the pulse period, Tp , for near-fault records. R Sa is a way to pick
up the shape of the response spectrum at a period TF representing the damaged structure
after the partial loss of its lateral stiffness due to seismic loading. It is believed that the
response of a structure to a given ground motion is better correlated to spectral
acceleration values corresponding to different stages of its performance than solely to the
spectral acceleration at the undamaged state represented by Sa(T1 ). Values of R Sa for the
16 records are given in Table 6.13.
The functional dependency of response measures, IDRmax or λu, denoted here by Y for
compactness, on different independent input parameters is of the form
Y = α S βa 1 (T1 , ξ ) R βSa2 e β3 t SM
(6.13)
where α, β 1 , β 2 and β 3 are regression parameters. The results of the regression analysis
for both bins are given in Tables 6.14 and 6.15. Values of the adjusted coefficient of
271
determination, R a2 , indicate that most of the variability of the response is explained by
considering the effect of the spectral acceleration at the lengthened period of the
structure, TF, in addition to the spectral acceleration at the fundamental period. The effect
of considering either the strong motion duration or the pulse period (for near-fault
records), beside Sa(T1 ,ξ=5%) and R Sa , has been shown to be ineffective in further
reducing the conditional dispersion of the response, and consequently the standard error
of estimation of the median response. The latter is defined by Shome (1999) as
σ lnY| Indep.Parameters divided by the square root of the sample size, n, or in other words the
number of nonlinear time history analyses. It is also worth pointing that considering
either tSM effect or Tp effect along with Sa(T1 ,ξ=5%), but ignoring the effect of the
spectral acceleration at TF, shows some benefit over considering Sa(T1 ,ξ=5%) alone.
However, this gives higher dispersion of the response than considering instead R Sa
beside Sa(T1 ,ξ=5%). Note that Sa 1 has been used for Sa(T1 ,ξ=5%) in Tables 6.14 and 6.15
for briefness.
Table 6.13 R Sa values for different records.
General Records
Near-Fault Records
Record
Record
RSa
RSa
Miyagi
Valparaiso
LP89-HCA
LP89-HSP
LP89-WAHO
CM92-RIO
Mendocino
LA92-YER
0.22
0.23
0.59
0.50
0.15
0.26
0.96
0.28
IV79-A6
LP89-LG
LP89-LX
EZ92-EZ
NR94-NH
NR94-RS
NR94-SY
KB95-JM
1.09
0.85
0.38
0.64
0.27
0.34
0.68
0.39
Note that instead of describing the intensity (or destructiveness) of the record solely by
the spectral acceleration at the fundamental period of the structure, one can further define
a new earthquake intensity index comprising R Sa beside Sa(T1 ,ξ=5%). Such index, as
proved by results shown in Tables 6.14 and 6.15, might be an efficient index that reduces
272
the conditional dispersion of the response and accordingly the number of analyses
required for a target confidence band width of the response, or more precisely for a target
standard error of estimation of the median response. One of the drawbacks of such index
is that the exponents (β 1 and β 2 ) of each of its two terms are not fixed for different types
of records (e.g. general versus near-fault), or for different types of response (e.g. IDRmax
versus λu), or maybe even for different types of structures. Another disadvantage is that
seismic hazard information (curves and maps) is only available in terms of spectral
acceleration and not in terms of this index.
Table 6.14 Regression results for IDRmax conditioned on different input parameters.
General Records
Near-Fault Records
IDR max = 0.030S0a.79
IDR max = 0.046S0a. 64
σ ln IDR max |S a1 = 0.416
σ ln IDR max |S a1 = 0.449
R a2 = 0.622
R a2 = 0.397
1
1
IDR max = 0.051S1a.00 R 0S.59
IDR max = 0.064S1a. 26 R1S.14
σ ln IDR max |S a1 , R Sa = 0.268
σ ln IDR max |S a1 , R Sa = 0.220
R 2a = 0.843
R a2 = 0.856
IDR max = 0.061S1a.00 R S0. 59 e-0.01t SM
IDR max = 0.075S1a.25 R1S. 18e-0.02t SM
σ ln IDR max |S a1 , R Sa , t SM = 0.260
σ ln IDR max |S a1 , R Sa , t SM = 0.220
R a2 = 0.852
R 2a = 0.855
1
1
a
1
a
1
a
a
0.06T p
IDR max = 0.053S1a.26 R1S. 03e
N/A
1
a
σ ln IDR max |S a1 , R Sa , Tp = 0.221
R a2 = 0.854
As a quick example of the benefit of reducing the conditional dispersion of the response
by considering R Sa beside Sa(T1 ,ξ=5%), the uncertainty in the estimation of median
IDRmax due to limited sample size has dropped from 14.7% (=0.416/√8) to 9.5%
(=0.268/√8) for bin of general records, and from 15.9% (=0.449/√8) to 7.8% (=0.220/√8)
for bin of near-fault records. Similar drops in the uncertainty in the estimation of median
273
λu are from 20.8% to 15.5% and from 22.1% to 15.2% for general and near-fault records
bins, respectively. Note that the sample size herein is 8 records per bin. Also note that, as
mentioned by Shome (1999), we are still ignoring here the uncertainty in the response
due to uncertainty in modeling and in physical properties of the structure and of its
components.
Table 6.15 Regression results for λu conditioned on different input parameters.
General Records
Near-Fault Records
λu = 4.23S−a 0. 86
λu = 3.47Sa−0.72
σ ln λu |S a1 = 0.589
σ ln λu |S a1 = 0.624
R 2a = 0.305
R a2 = 0.264
1
1
λu = 2.98S−a 1.71RS−0.94
λu = 2.55S−a 1.62 R −S1.43
σ ln λu |S a1 , R Sa = 0.438
σ ln λu |Sa1 , R Sa = 0.431
R a2 = 0.617
R a2 = 0.649
λu = 3.35S−a 1. 72 R −S0.95 e -0.01t SM
λu = 4.81S−a 1.66 R S−1.28 e-0.07t SM
σ ln λu |Sa1 , R Sa , tSM = 0.439
σ ln λu |Sa1 , R Sa , tSM = 0.423
R a2 = 0.614
R 2a = 0.661
1
1
a
1
a
1
a
a
λu = 2.39S−a 1. 62 RS−1.47 e
0.02T p
N/A
1
a
σ ln λu |S a1 , R Sa , Tp = 0.434
R 2a = 0.643
Another form of the term related to the strong motion duration which is t βSM3 has been
tried, but it has been found that this form has lower correlation with the response
parameter and gives higher conditional dispersion of the response than the other form
adopted in Equation 6.13 for the case of IDRmax. Both forms give quite comparable
results for the case involving λu as the response parameter. This observation has also
been made by Shome (1999) for the drift response. Moreover, we should be very careful
about evaluating the response dependency on tSM in this research since all records
considered within each bin have very close strong motion duration values with a few
274
exceptions (Valparaiso and Loma Prieta at WAHO records for general bin and Loma
Prieta at Lexington for near-fault bin). The narrow spectrum of tSM values prevents us
from picking up a reliable estimate of response dependency on strong motion duration, if
any. It is also worth pointing that Sa at a shorter period than T1 , taking care of higher
mode effects, has been tried in the regression model given by Equation 6.13. Results
revealed that it is not an issue for this 6-story RCS frame.
6.7 Summary
In this chapter a detailed study of the performance of the 6-story RCS frame designed
according to current seismic provisions has been presented. The findings of the major
issues are summarized herein.
1- Both static pushover and dynamic time history analyses indicate that most of the
damage is confined to the base of the ground floor columns and the beams of the first
three stories. More specifically, the damage always starts at the column bases and
even if any damage takes place in higher stories (i.e., above the third floor) it finally
migrates to the bottom when the records are scaled to higher intensity levels. It has
been generally observed that at global collapse of the frame (as identified in this
chapter by λu=1.0), the maximum interstory drift ratio, an estimator of global damage
is almost always taking place at the first story.
2- A relationship between maximum interstory drift ratio, IDRmax, and spectral
acceleration at the fundamental period of the frame and at 5% critical damping has
been presented. It is given separately for the two bins of general versus near-fault
records. The relationship is based on first carrying a power law regression analysis of
the data points for each record scaled at different intensity levels up to global collapse
of the frame, conditioned on Sa(T1 ,ξ=5%), then taking an average of the regression
coefficients associated with every record to describe the final relationship for the
whole bin. The regression coefficient β for both general (β=1.11) and near-fault
275
(β=1.35) bins of records is greater than 1.0 showing “softening” behavior in the
IDRmax-Sa(T1 ,ξ=5%) relationship. Furthermore, at a given spectral acceleration,
median IDRmax of the frame due to near fault records is larger than for general
records, and the difference is more pronounced at higher intensity levels. Spectral
acceleration at the fundamental period of the structure with 5% critical damping has
been considered in the relationship as it is an “effective” intensity measure for
earthquake records with a relatively small record-to-record dispersion of the drift
response given the intensity level, and for which a hazard analysis is available.
3- A methodology for an identification technique of global collapse of the frame has
been introduced as one of the major thrusts of this chapter. A technique is needed
since our analytical models are not solely able to adequately capture collapse; this has
always been the reason behind the continuous need for damage indices, such as the
ones proposed in Chapter 4, to detect damage and failure of structural components.
The procedure shown in this chapter is based on performing second order inelastic
analysis of the damaged frame, after the earthquake, under just the gravity loads. The
damaged frame is a modified original frame by introducing a decrease of the stiffness
and strength of the damaged sections according to the cumulative damage index
introduced in Chapter 4, along with considering the residual (i.e., permanent)
displacements. The load factor, λu, giving the ratio of the applied gravity loads the
structure can sustain is the estimator of global collapse, since a value of less than 1.0
means the structure is not able to carry its gravity loads. Values between 1.0 and λuo
(the gravity load capacity of the undamaged structure) describe different levels of
global damage. One major benefit of such technique is that the global damage is
calculated avoiding the use of an averaging or weighting procedure of local damage
indices to integrate the effect of damage of different elements of the frame, a
procedure that is always questionable. It has been shown that averaging procedures of
local indices used in the literature to calculate a global damage estimator may give
incorrect, and sometimes physically impossible, results in some cases as proved by
Ghobarah et al. (1999). Another benefit of the proposed technique is that it is also
capable of capturing damage due to mechanisms other than flexural yielding (e.g.,
276
shear failure in existing non-ductile reinforced concrete frames) provided the models
used in the analysis include these possible failure modes.
4- A relationship between Sa(T1 ,ξ=5%) and λu is developed to describe different states
of damage in terms of the input intensity parameter Sa. It has been observed that the
ratio between the spectral acceleration at the state of incipient collapse (i.e., λu=1.0)
and the state of excessive yielding in the building assumed to be at about 0.95λuo ,
Sa(λu=1.0)/S a(λu=0.95λuo ), is 1.8. The latter state of damage may be also looked at as
a performance level corresponding to Life Safety as introduced by FEMA 273. This
implies that the hazard intensity for near collapse (λu=1.0) is about twice that
corresponding to the point when the structure begins to significantly degrade (i.e.,
λu=0.95λuo , or Life Safety as previously suggested). This margin is larger than the
ratio of 1.5 implied by modern codes between the “design level” earthquake response
(geared to life safety) and the maximum considered earthquake (geared to near
collapse). One may further use such relationship of Sa(T1 ,ξ=5%) versus λu introduced
herein to adequately calculate the probability of collapse of a given structure, at a
given site, using probabilistic seismic demand analysis techniques.
5- Another interesting value is the ratio Sa(λu=1.0)/S a(2%in50) which is about 3.5 for the
frame for both types of records. One reason for the high values of Sa at λu = 1.0
observed herein is the large “actual” lateral overstrength (Ω = 6.3) of the frame as
reported in Section 6.2 when ignoring accidental torsion and the upper cap on the
period imposed by code design procedures. Among other sources of overstrength are:
(1) expected versus minimum specified material strengths, (2) minimum stiffness
(drift) limitations, (3) structural redundancy, (4) SCWB criterion, (5) discrete member
sizing, and (6) the use of a distributed space frame with relatively shallow members.
One can show for example, that when stiffness governs the design, the shallow beams
used in space frames will lead to higher overstrength than deeper beams commonly
found in perimeter frame systems. Moreover, it is believed that the use of the collapse
limit state determination technique, presented earlier, in a subsequent step to the time
277
history analysis and the lack of its integrity with the analysis process explain part of
these high values of Sa(λu=1.0). Finally, the high mean value of Sa at λu = 1.0 might
be reduced if averages minus one standard deviation are reported instead to consider
some confidence bands in the results. Accordingly, mean minus one standard
deviation values for Sa(λu=1.0) are 1.68g and 1.95g for near-fault and general records,
respectively, corresponding to Sa(λu=1.0)/ Sa(2%in50) ratios of 1.9 and 2.3 instead of
the large value of 3.5.
6- An interesting observation from the IDAs of near-fault records (Fig. 6.10, Section
6.3.2) is that the response curves generally fall into one of two groups, where the
lower collection of curves in the figure has clearly more softening than the upper
group. A likely reason for this is that all of the more damaging records in the lower
group have a pulse period, Tp , that is larger than the natural period T1 of the structure.
For example, the four records in this lower group (subset 1) have ratios of Tp /T1 = 1.8
to 2.7, whereas the upper group records (subset 2) have ratios of Tp /T1 = 0.7 to 1.0.
Here the pulse period is defined based on the peak of the pulse observed in the
velocity spectra of the records. The reason for this behavior is that when Tp /T1 > 1.0,
the structure softens into the more damaging pulse effect of the records whereas in the
other case, i.e., Tp /T1 < 1.0, the opposite occurs. Differences of this sort indicate that
for near-fault effects, the intensity scaling technique should involve both Sa(T1 ) and a
second index that reflects the frequency content of the record, as might be reflected
by a spectral velocity measure.
Adopting this disaggregation of the results based on Tp /T1 ratio, the ratio
Sa(λu=1.0)/S a(2%in50) for subset 1 equals 2.1 which is much less than the value of
3.5 previously reported for both general and near-fault records. On the other hand, the
ratio is 4.1 for subset 2, which is even larger than that corresponding to general
records. To conclude, response due to near-fault records even with forward directivity
and severe pulse properties depends on the ratio between the pulse period and the
fundamental period of the structure. Therefore, in order to predict reliable
performance, statistics applied to results from a series of near-fault records must be
278
evaluated with great care so as to not consider an average value of a given response
parameter computed from events with totally different effects on the structure.
7- A similar relationship to the one presented between Sa(T1 ,ξ=5%) and λu is also
introduced for transient IDRmax versus λu. It has been shown that the two variables are
reasonably correlated. Moreover, values of IDRmax have been compared to proposed
values in FEMA 273 at different performance levels. One may note that values
corresponding to Life Safety as given by FEMA (0.025 for steel moment frames) are
very close to those associated with λu=0.95λuo (IDRmax = 0.03). Collapse is however
reached at higher values (IDRmax = 0.09 and 0.12 for general and near-fault records
respectively) than those cited in FEMA as for Near Collapse or Collapse Prevention
performance level (0.05). Average values of transient IDRmax associated with global
collapse for near-fault records were found to be larger than for the case of general
records due to the pulse effects characterizing the former events. Values of residual
IDRmax also showed good correlation with values proposed by FEMA for the two
performance levels mentioned above (Section 6.4.3 and Table 6.11).
8- It has been shown that local response in terms of plastic rotations of beams and
columns can be successfully related (with a very good correlation) to interstory drift.
This drift quantity has been proved to offer good correlation if it is given in terms of
maximum plastic transient interstory drift ratio, IDRp,max, for the case of beams and
maximum change in transient interstory drift ratios, ∆IDRmax, for columns based on
the anticipated deformed configuration of the frame. Whenever column hinging takes
place at bottom sections of a specific story, plastic beam rotations should be related to
IDRp,max at the same story, and plastic column rotations should be related to the
maximum absolute value of the difference between IDR at this story and that at the
lower one. On the other hand, if column hinging takes place at top sections of a given
story, plastic beam rotations should be related to IDRp,max at the upper story, and
plastic column rotations should be related to the maximum absolute value of the
difference between IDR at this story and that at the upper one. It has been also found
that for the 6-story frame under investigation, local versus global response
279
relationship in terms of the variables cited above is on average quite stable
irrespective of the type of record (i.e., general versus near-fault) and the level of the
overall damage as given in terms of λu. Moreover, applying a conditional regression
analysis of local response given both global response and the intensity of the input (in
terms of spectral acceleration), one may look at the relationship between local and
global response at different hazard levels. Such relationship still shows the stable and
almost constant (on average) relationship between local and global response in terms
of the parameters introduced herein. A finding that has to be further investigated for
other types of structures.
9- It has been shown that the variability in the response (either IDRmax or λu) is mainly
explained by the spectral acceleration at the fundamental period of the structure. It
has also been shown that the uncertainty in the response due to limited sample size
(i.e., limited number of records or number of nonlinear time history analyses) is
further reduced by considering the change in the spectral shape corresponding to a
specific record by introducing a ratio, R Sa . R Sa is the ratio between the spectral
acceleration at a long (damaged) period of the structure and the spectral acceleration
at the fundamental period. Thus, Sa(T1 ,ξ=5%). R Sa might serve as an “effective”
earthquake intensity index. Scaling records to the same Sa(T1 ,ξ=5%). R Sa may reduce
the standard error of estimation of the median response (or demand) and will
accordingly decrease the number of expensive nonlinear time history analyses
needed. The main disadvantages are that first, each of the two terms of this index is
raised to a different power which is even different according to the type of record
(general versus near-fault) and the response parameter (IDRmax versus λu), and
second, hazard information is not available in terms of such index. Strong motion
duration of records and the pulse period of near-fault records have also been
considered with the spectral acceleration. However, no definite conclusions can yet
be made concerning their effectiveness in further reducing the dispersion in the
response conditioned on the input parameters because of the narrow range of strong
motion duration and pulse period values of the records considered. Magnitude and
280
distance are out of the scope of this research, but it has been previously proved by
Shome (1999) that there is a mild dependency of the response on these parameters
which can be neglected for all practical issues.
In general, the RCS frame has performed much better than expected by codes as reflected
by its high collapse limit load (at λu = 1.0) and moderate damage at a lower level (λu =
0.95λuo ) corresponding to Life Safety performance level. Among main reasons is the high
lateral overstrength of the frame as previously discussed.
Finally, it should be remembered that all the above conclusions are strictly based on the
study of the 6-story RCS frame designed according to current seismic design practice as
given in Chapter 5. Further study is to be made for other types of structures, e.g.,
different heights, construction materials (RCS versus steel or reinforced concrete), etc., to
confirm or modify the breadth of the conclusions reported here. In the following chapter,
a parallel study is conducted for comparable 12-story RCS frame and 6-story Steel frame.
281
Chapter 7
Comparative Assessment of RCS and STEEL
Moment Frames
A detailed study of the seismic behavior and performance of a 6-story composite RCS
Special Moment Frame (SMF) has been presented in the previous chapter. The present
chapter is divided into two parts. Part one describes the seismic performance of a 12story RCS moment frame with the same structural configuration as the 6-story one
investigated in Chapter 6. Members’ cross-section dimensions and main seismic
properties are given in Chapter 5. Contrasting the behavior of the 6- and 12-story RCS
frames serves in better understanding the effects of higher modes, if any, on the
performance of composite RCS systems. Furthermore, it provides some insight into the
suitability of such systems for low- to mid-rise construction in high seismic zones leading
to their broader acceptance and utilization.
Part two presents a comparative study of the 6-story RCS frame previously investigated
and a 6-story STEEL frame with the same structural configuration. The 6-story steel
frame members’ cross sections and main seismic properties are also given in Chapter 5.
This comparative assessment will put into perspective all the issues that should be
282
addressed for evaluating composite RCS construction by comparing its seismic
performance, as a new system, with the well established system of moment resisting steel
frames.
PART I: 12-Story RCS Special Moment Frame
A 12-story RCS special moment frame in the short direction of the theme structure with
dimensions and detailing as given in Chapter 5 is studied within this chapter. The
analysis results will be evaluated both on their own merits and in comparison with the
behavior of the 6-story RCS frame investigated in Chapter 6.
7.1 Modeling of the 12-Story RCS Frame
The main issues considered for the analytical modeling of the frame are identical to these
discussed in Chapter 6 for the 6-story RCS frame. Among these issues are the frame
loading and mass characteristics, dimensions and member sizes, boundary conditions,
elements in DYNAMIX used to model the different members, discretization strategy for
the different members, and all control parameters needed to provide reasonable accuracy
for the numerical solution. Refer to Tables 5.2 and 5.4 to 5.6, and Figures 5.5 and 5.11
for all relevant details. Tables 7.1 through 7.3 give stiffness and strength properties for
columns, beams, and composite joints as modeled in DYNAMIX.
Viscous damping is again modeled for the frame through mass and stiffness proportional
(Rayleigh) damping where 2% of critical damping in the first and fourth modes is
assumed. Applying this value of damping at the first and fourth modes is based on the
study of modal properties of the frame. The cumulative effective modal masses of the
first four modes of the frame constitute about 94.7% of the total mass suggesting that
assigning the critical damping to the first and fourth modes is a reasonable assumption.
Applying Equation 6.2 to calculate percentages of critical damping associated with
283
different modes reveals a smallest critical damping value of 1.3% for the second mode
and a largest critical damping value of 8.2% for the twelfth mode. The critical damping
values range from 1.3% to 3.5% for the first six modes. The values are believed to be
reasonable, encompassing adequate range of damping for the type of frame under
investigation.
Story #
1-3
Outer
1-3
Inner
4-6
Outer
4-6
Inner
7-9
Outer
7-9
Inner
10-12
Outer
10-12
Inner
Floor #
1-6
7-9
10-12
1-6
7-9
10-12
Table 7.1 Stiffness and strength values of RC columns.
Axial Properties
Bending Properties
Squash Balance Tensile
EA
Strength
EI
Load
Load
Strength
(kips)
at P=Pbal (kips.in2 )
(kips)
(kips)
(kips)
(kips.in)
6
8824
2841
1147
5.24x10
34620
2.38x108
Shear
GA
(kips)
4.78x105
8824
2841
1147
5.24x106
34620
2.46x108
5.11x105
7950
2512
1105
4.68x106
29490
1.87x108
4.13x105
7950
2512
1105
4.68x106
29490
1.91x108
4.32x105
6968
1986
899
4.08x106
23370
1.41x108
3.55x105
6968
1986
899
4.08x106
23370
1.44x108
3.65x105
5446
1258
839
3.12x106
16180
8.09x107
2.58x105
5446
1258
839
3.12x106
16180
8.13x107
2.61x105
Table 7.2 Stiffness and strength values of composite and steel beams.
Flexural Strength
Flexural Stiffness, EI
Shear
2
(kips.in)
(kips.in )
Stiffness, GA
(kips)
Positive
Negative
Positive
Negative
COMPOSITE BEAMS
22860
16010
2.08x108
9.48x107
1.47x105
18420
12900
1.58x108
6.87x107
1.26x105
8
7
14920
10200
1.31x10
5.31x10
1.10x105
STEEL BEAMS
16010
16010
9.48x107
9.48x107
1.47x105
7
7
12900
12900
6.87x10
6.87x10
1.26x105
10200
10200
5.31x107
5.31x107
1.10x105
284
Floor #
1-3
Outer
1-3
Inner
4-6
Outer
4-6
Inner
7-9
Outer
7-9
Inner
10-11
Outer
10-11
Inner
12
O&I
Table 7.3 Properties of composite joint panels.
Dimensions
Strength, M joint
Stiffness
(inches)
(kips.in)
(kips.in)
Horizontal
Vertical
Shear
Bearing
Shear
Bearing
6
30.1
33.7
37380
56770
9.00x10
1.42x107
30.1
33.7
37380
56770
9.42x106
1.51x107
28.4
33.7
35180
50290
8.28x106
1.23x107
28.4
33.7
35180
50290
8.61x106
1.29x107
26.6
30.1
27020
39910
6.16x106
9.34x106
26.6
30.1
27020
39910
6.36x106
9.72x106
23.0
29.7
22330
29800
4.89x106
6.64x106
23.0
29.7
22330
29800
5.01x106
6.85x106
23.0
23.7
22330
29800
4.47x106
5.96x106
7.2 Static Push-Over Analysis
A static inelastic push-over analysis is performed for the 12-story RCS frame using the
IBC 2000 equivalent lateral force distribution. Geometric nonlinearity (P-∆) effects are
considered. The full dead load and 25% of the live load were applied first prior to
ramping up the lateral loading. Base shear/weight ratio versus roof drift ratio is shown in
Figure 7.1. The figure reveals that the static lateral overstrength of the frame is about Ω =
Vu/Vd = 4.4. The frame has been designed for a base shear ratio (including accidental
torsion effect as imposed by codes and based on a period of 1.2Ta = 1.6 seconds) of Vu/W
= 0.069. However, ignoring accidental torsion effects and considering the calculated
period, T1 = 2.07 seconds, the “actual” lateral overstrength of the frame is in the order of
Ω * = Vu/Vd* = 6.9 (=4.4x(0.069/0.044)), refer to Table 5.6 (Chapter 5) for more details
about these values.
285
0.35
Base Shear-Weight Ratio, V/W
0.30
0.25
0.20
Static POC
Design Load Level
Target Disp., FEMA-273
Max. Lateral Strength
∆r/H = 0.04
0.15
0.10
∆r/H = 0.05
0.05
0.00
0.00
∆r/H = 0.06
0.01
0.02
0.03
0.04
0.05
0.06
0.07
Roof Drift Ratio, ∆r /H
Figure 7.1 Static pushover curve - IBC 2000 lateral load pattern.
12
11
Design Load Level
Target Disp., FEMA-273
Max. Lateral Strength
∆r /H = 0.04
10
9
∆r /H = 0.05
∆r /H = 0.06
Floor #
8
7
6
5
4
3
2
1
0
0.00
0.02
0.04
0.06
0.08
0.10
0.12
Interstory Drift Ratio, IDR
Figure 7.2 Distribution of interstory drift ratios up the height of the frame - static
pushover results.
286
This large overstrength is due to: (1) expected versus minimum specified material
strengths, (2) minimum stiffness (drift) limitations, (3) structural redundancy, (4) SCWB
criterion, (5) discrete member sizing, and (6) the use of a distributed space frame with
relatively shallow members instead of a perimeter frame system with deeper beams such
as has been commonly applied in recent U.S. practice.
The target displacement, δ t , for the frame calculated according to Equation 5.8 and a
2%in50years hazard level (reflected in the value of Sa(Te,ξ)) is 41 inches, corresponding
to a roof drift ratio, ∆r/H, of about 0.022. At this pre-specified target displacement the
structure has not yet reached its maximum lateral capacity of Vu = 0.30W which is
reached at a roof drift ratio, ∆r/H ≅ 0.030. This ultimate roof drift ratio is less than for the
6-story RCS frame where the maximum lateral strength is reached at ∆r/H ≅ 0.039.
Distributions of interstory drift ratios, IDR, up the height of the frame are given in Figure
7.2 at δ t and at different other deformation levels corresponding to different total roof
drift ratios. At the target displacement, interstory drift ratios range between IDR = 0.02 to
0.028 for the first 9 stories, with a very uniform distribution between the second and
eighth floors. These values are not very different from the IBC code limit of 0.02.
Figure 7.2 shows that most of the inelastic behavior, as reflected by the high interstory
drift ratios at large roof drifts is constrained to the first five stories, with the maximum
occurring at the second through fourth floors. Note that this observation is for the
assumed distribution of the lateral loading (IBC 2000), and it is known that the push-over
results may be sensitive to the applied lateral load pattern. It is also important to report
that the sections with highest demands are located at the base of the ground floor level
columns, reflected by high values of plastic rotation demands at high values of IDR.
These results so far identify the critical regions of the frame and probable overall
behavior under a real earthquake record provided the structure responds in its first mode.
But again we should keep in mind that this behavior is also ground record dependent
since a certain record with a specific frequency and energy content might trigger higher
modes of the structure. Accordingly, the pushover results presented above should be
287
considered in light of the inelastic time history analysis results under several types of
ground motions as will be presented later.
7.3 Incremental Dynamic Analyses
The Incremental Dynamic Analysis (IDA) concept as introduced by Cornell and his coworkers (1998) and explained in Chapter 6 is applied herein for the 12-story RCS frame.
Second-order inelastic time history analyses of the frame are performed for every record
in the two bins of general and near-fault records presented in Chapter 5. For each record,
the analysis is performed at different hazard levels as defined by the spectral acceleration
at the fundamental period T1 = 2.07 seconds and 5% damping, Sa(T1 ,ξ=5%). Thus, each
IDA curve associated with each record entails several inelastic second-order time history
analyses. Then, as done for the 6-story RCS frame in Chapter 6, a regression power
model, as given by Equation 6.5 and herein by Equation 7.1 for completeness, is applied
in the log space to the response data points defining the IDA curve for each record,
conditioned on the input parameter, Sa(T1 ,ξ=5%)
IDRmax = α Saβ (T1 , ξ = 5% )
(7.1)
where IDRmax is the median maximum interstory drift response. Eight pair of values for α
and β are thus obtained for each bin of records. Medians of the regression parameters, α
and β, are given in Table 7.4 for the two bins.
Table 7.4 Values of α and β for the regression fit of Equation 7.1.
Parameter and Statistical
General Records
Near-Fault Records
Measure Values
0.058 (23%)
0.053 (56%)
α (C.O.V.)
1.03 (15%)
1.16 (28%)
β (C.O.V.)
Figures 7.3 and 7.4 show IDA curves for all records of the two bins along with the final
spectral acceleration versus IDRmax relationships based on Eq. 7.1 with the values given
288
in Table 7.4. Note that given a specific hazard level as defined by a value of Sa(T1 ,ξ), the
median response IDRmax is less for the case of near-fault records than for the case of
general records. This conclusion is valid for all hazard levels up to high values of
Sa(T1 ,ξ), but the difference in response is very small and it is decreasing with the increase
of Sa as shown in Figure 7.5. This observation is different from what was found in
Chapter 6 concerning the performance of the 6-story RCS frame where the median
IDRmax response is always larger for near-fault records than for general records; a result
that we have attributed to the pulse effect characterizing the near-fault ground motions.
Values of β given in Table 7.4, when compared to those reported for the 6-story RCS
frame in Table 6.8, show that the behavior of the 12-story frame reveals less softening in
the nonlinear relationship between Sa and median IDRmax. However, larger values of the
median response IDRmax at a given input Sa(T1 ,ξ) are observed for the 12-story RCS
frame. Furthermore, the relationship is almost linear (β=1.03) for the bin of general
records for the 12-story frame. A possible explanation for the higher IDRmax values for
the 12-story frame, compared to the 6-story frame, is the greater flexibility of the 12-story
structure and chances of triggering higher mode effects as well.
Another useful observation to report is that given Sa(T1 ,ξ=5%) = 0.522g (the value
corresponding to a 2%in50years hazard level for the 12-story building), the estimated
median values for the drift response, IDRmax , are 0.030 and 0.025 for general and nearfault records, respectively. On the other hand, given Sa(T1 ,ξ=5%) = 0.864g (a value
corresponding to a 2%in50years hazard level for the 6-story building), the estimated
median values for the drift response, IDRmax , are 0.023 and 0.028 for general and nearfault records, respectively. These values show that given a specific hazard level in terms
of Sa(T1 ) associated with a certain probability of occurrence might result in comparable
response in terms of median IDRmax for both frames (6- and 12-story) under both types of
records (general and near-fault). The median response estimates are not that close if they
are calculated at a same given value of Sa(T1 ,ξ=5%) for the two frames. Note that same
value of Sa(T1 ,ξ=5%) for the two frames means different hazard level for each frame.
This finding further proves the suitability of the spectral acceleration at the fundamental
289
period of the structure as an effective intensity measure for earthquake records that is
reliably correlated with the hazard level.
In Chapter 6, looking at the results and the relationship between the response and the
input parameters for the bin of near-fault records for the 6-story frame, we found that the
response can be distinguished into one of two trends. These two trends are defined based
on the ratio between the pulse period of a near-fault record and the fundamental period of
the structure, Tp /T1 . The conclusion that we then made was that for records with Tp /T1 >>
1.0, global collapse and performance deterioration are expected at lower Sa(T1 ,ξ=5%)
values than for records with Tp /T1 in the vicinity of 1.0. This trend seems to also hold for
the 12-story RCS frame. For records with Tp /T1 ≈ 1.0 (as for the case of the Erzincan
(1992) record, Tp /T1 =1.11, and the Northridge (1994) record at Sylmar, Tp /T1 =1.16),
global collapse and response deterioration are observed at higher Sa(T1 ,ξ=5%) values
than for other records with Tp /T1 << 1.0 or Tp /T1 >> 1.0. Thus, the modification to the
observation made in last chapter is that having the structure with a fundamental period
value far away on either sides of the pulse period (and not only smaller than Tp ) will
increase the vulnerability of that structure under this near-fault record.
It has also been observed that the dispersion in the response given by IDRmax conditioned
on the input intensity, Sa(T1 ,ξ=5%), is smaller for the 12-story frame when compared to
the 6-story frame results. For instance, for general records, σ ln IDR
max
|Sa (T1 ,ξ )
= 0.240 and
0.416 for the 12- and 6-story frames, respectively, and for near-fault records,
σ ln IDR
max
|Sa (T1 ,ξ )
= 0.258 and 0.449 for the 12- and 6-story frames. This considerable
decrease in the dispersion of the response conditioned on the input is automatically
reflected in the decrease of the uncertainty in the estimation of median IDRmax due to
limited sample size. Or in other words, it leads to the decrease of the number of nonlinear
time history analyses required for demand hazard calculations to meet a certain prespecified accuracy or standard error of estimation of the median response.
290
1.5
IDRmax = 0.058 S a
1.03
Sa(T1=2.07sec,5%)
1.2
0.9
0.6
S a (2%in50years)
0.3
0.0
0.00
0.02
0.04
0.06
Miyagi
Valparaiso
LP89-HCA
LP89-HSP
LP89-WAHO
CM92-RIO
LA92-YER
Mendocino
Regress. Line
0.08
0.10
IDRmax
Figure 7.3 Spectral acceleration versus IDRmax for bin of general records.
2.5
1.16
IDR max = 0.053 Sa
S a(T1=2.07sec,5%)
2.0
1.5
1.0
0.5
0.0
0.00
Sa (2%in50years)
0.03
0.06
0.09
IV79-A6
LP89-LG
LP89-LX
EZ92-EZ
NR94-NH
NR94-RS
NR94-SY
KB95-JM
Regress. Line
0.12
0.15
IDR max
Figure 7.4 Spectral acceleration versus IDR max for bin of near-fault records.
291
2.0
Sa(T1,5%)
1.6
1.2
0.8
0.4
General Records
Near-Fault Records
0.0
0.00
0.02
0.04
0.06
0.08
0.10
0.12
IDR max
Figure 7.5 Comparison of regression results of spectral acceleration versus
IDRmax relationship for general and near-fault records.
7.3.1 Story Incremental Dynamic Analysis Curves
In Figures 7.6 and 7.7, IDA curves are given for each story of the 12-story RCS frame for
the two records: Cape Mendocino at Rio Del Overpass station, and Imperial Valley at
Array 06, representing bins of general and near-fault records, respectively. Story IDA
curves for all other records of the two bins are given in Appendix B. Such figures have
the merit of showing that, at a very wide range of hazard levels, maximum transient
interstory drift ratios, IDRmax, are much larger at the lower stories (stories 1-4) of the
frame than at other higher stories for almost all records of the bin of near-fault ground
motions. However, for the Kobe record at JMA station, IDRmax values are almost equally
large for higher stories (floors 7-10). On the other hand, for general records, IDRmax
values at higher stories (stories above the sixth) are larger than (Valparaiso and Cape
Mendocino at Rio Del Overpass) or almost as equally large as (Loma Prieta WAHO and
292
Landers at Yermo Fire Station) values at lower stories. This observation shows that the
behavior of the 12-story RCS frame for different hazard levels under general records is
more affected by higher modes, affecting higher stories, while under near-fault records,
the pulse effect working simultaneously with P-∆ effects seriously attacks the lower
stories. The higher mode effects reflected in the response due to general records may be
justified by their larger Sa(T2 )/Sa(T1 ) ratio when compared to near-fault records. Mean
value of Sa(T2 )/Sa(T1 ) for the eight general records is 5.5 (with a median of 4.1) while it is
3.0 for the eight near-fault records (with a median of 2.7).
Note that the static pushover analysis results, based on the equivalent lateral load pattern
of IBC 2000 and shown in Section 7.2, are able to estimate the general performance and
the vulnerable areas of the frame under the near-fault type of records. The lateral load
pattern, on the other hand, does not mimic the effects of general records as well and does
not succeed in giving suitable ideas of the seismic demands for the frame when higher
modes are triggered by a given ground motion. If the static pushover is to be used for
seismic assessment of such buildings with probable high mode effects, suitable lateral
load patterns should be used.
7.4 Global Failure Analysis of the 12-Story RCS Frame
Owing to the limitations of the analysis to fully capture strength/stiffness degradation, the
Incremental Dynamic Analysis curves of Figures 7.3 and 7.4 in themselves do not reveal
a clear stability limit (or global collapse limit). This is evident from the fact that many of
the response curves in Figs. 7.3 and 7.4 continue to have a positive slope at very large Sa
and IDRmax.
293
1.50
1.25
Sa(T1,5%)
1.00
0.75
Story
Story
Story
Story
Story
Story
0.50
0.25
0.00
0.00
0.02
0.04
1
2
3
4
5
6
0.06
0.08
IDR max
1.50
1.25
Sa(T 1,5%)
1.00
0.75
Story 7
Story 8
Story 9
Story 10
Story 11
Story 12
0.50
0.25
0.00
0.00
0.02
0.04
0.06
0.08
IDR max
Figure 7.6 Story IDACs for the 12-story RCS frame under the general record,
Cape Mendocino (1992) at Rio Del Overpass station.
294
1.00
Sa(T1,5%)
0.75
0.50
Story
Story
Story
Story
Story
Story
0.25
0.00
0.00
0.02
0.04
0.06
1
2
3
4
5
6
0.08
0.10
IDR max
1.00
Sa(T1,5%)
0.75
0.50
Story 7
Story 8
Story 9
Story 10
Story 11
Story 12
0.25
0.00
0.00
0.02
0.04
0.06
0.08
0.10
IDRmax
Figure 7.7 Story IDACs for the 12-story RCS frame under the near-fault record,
Imperial Valley (1979) at Array 06
295
As previously described in Chapter 6, damage indices calculated from each time history
analysis provide the basis for modifying the structural analysis model to approximate the
damaged condition after an earthquake. Primarily, these modifications involve reducing
element stiffnesses and strengths based on the accumulated damage and incorporating the
residual (permanent) building drift into the structural topology. This modified model is
then reanalyzed by a second-order inelastic analysis under gravity loads to determine the
resulting gravity load stability index, λu, defined as the normalized gravity load capacity
(refer to Section 6.4 for more details). This index provides a failure criterion for each
earthquake record and intensity, Sa(T1 ,ξ), which can then be related to a specific hazard
level. Values of λu range from λuo = 6.8 for the undamaged 12-story RCS frame to λu =
1.0 for conditions at incipient collapse. The large initial stability index, λuo = 6.8, is a
function of the structure being designed for high seismic forces with stringent drift
limitations. λu thus serves as a global failure criterion that integrates the effect of local
damage on reducing the system stability.
7.4.1 Relationship between Spectral Acceleration and Global Failure Criterion, λ u
Adopting the global collapse identification technique, Figures 7.8 and 7.9 show the
evolution of damage from λuo to λu=1.0 for the general and near-fault records. In addition
to plots formed by the data points calculated from incremental scaling of each record, a
regression line based on least square fit of all data points is given in these figures. As was
done previously in Chapter 6, linear regression, conditioned on λu, is applied to all data
points in the log space excluding points with a value of λu > 0.95λuo . Alternatively, using
pairs of regression coefficients (α and β) for curves fit to each record alone, the average
curve for each bin of records is given as,
Sa(T1 ,ξ=5%) = a λßu
(7.2)
where a is the geometric mean of the eight α values for each bin, and ß is the arithmetic
mean of the eight β values. Values of a and ß are given in Table 7.5. Note that these
296
values are very close to regression coefficients computed by applying regression on all
records of each bin at once (see Figures 7.8 and 7.9).
Table 7.5 Values of a and ß for Equation 7.2.
Parameter and Statistical
General Records
Near-Fault Records
Measure Values
a (C.O.V.)
1.21 (19%)
1.47 (34%)
-0.26
(43%)
-0.26
(47%)
ß (C.O.V.)
One way of looking at these results is to calculate the ratio Sa(λu=1.0)/S a(2%in50), which
on average is equal to 2.3 and 2.9 for general and near-fault records, respectively. That
means the 12-story frame is at incipient collapse at Sa(T1 ) of about 2.3 and 2.9 times the
spectral acceleration associated with 2%in50years hazard defined by the IBC 2000. These
two ratios are 3.6 and 3.4, respectively, for the 6-story frame.
The large difference in the ratio Sa(λu=1.0)/S a(2%in50) between general and near-fault
records (2.3 and 2.9) for the 12-story frame and the fact that the ratio for near-fault
records is larger than for general records are counter-intuitive and different than results
for the 6-story frame. One way to explain these results is to look at the performance of
the structure under near-fault records based on the ratio between the pulse period of the
record and the fundamental period of the frame, Tp /T1 . Excluding the two records
(Erzincan and Northridge at Sylmar station) with values of Tp /T1 ≈ 1.0 (and accordingly
less damaging effects), the ratio Sa(λu=1.0)/S a(2%in50) for the remaining six near-fault
records is about 2.5 which is closer to the value of 2.3 for the bin of general records. One
may further point out that Sa(λu=1.0)/S a(2%in50) ratio ranges from 1.7 to 2.9 and from
1.6 to 3.1 for bins of general and near-fault (with 6 records only) records, respectively.
To conclude, near-fault records even with forward directivity and severe pulse properties
might produce less damage to structures than general records, when both are scaled up to
same high hazard levels. Accordingly, great care should be advised when dealing with
results from a series of near-fault records statistically so as to not consider an average
value of a given response parameter computed from events with totally different effects
on the structure.
297
2.0
Miyagi
Valparaiso
LP89-HCA
LP89-HSP
LP89-WAHO
CM92-RIO
LA92-YER
Mendocino
Linear Regres.
S a = 1.18 λu-0.24
σlnS |λ = 0.271
a u
S a(T 1=2.07sec, ξ =5%)
1.6
1.2
0.8
0.4
λu = 1.0
(collapse)
0.0
0
1
2
3
4
5
6
λu (based on 1.0D+0.25L)
7
λuo
8
Figure 7.8 Spectral acceleration-λu relationship for bin of general records.
3.0
2.5
S a(T 1=2.07sec,ξ=5%)
IV79-A6
LP89-LG
LP89-LX
EZ92-EZ
NR94-NH
NR94-RS
NR94-SY
KB95-JM
Linear Regres.
-0.25
Sa = 1.47 λ u
σlnS |λ = 0.277
a u
2.0
1.5
1.0
λu = 1.0
(collapse)
0.5
0.0
0
1
2
3
4
5
λu (based on 1.0D+0.25L)
6
7
λuo
8
Figure 7.9 Spectral acceleration-λu relationship for bin of near-fault records.
298
One reason for the high values of Sa at λu = 1.0 due to both types of records is the large
lateral overstrength (Ω = 4.4, Ω * = 6.9) of the frame. Moreover, these high Sa(λu=1.0)
values are not as large as suggested by looking at only mean values. If averages minus
one standard deviation are reported instead to consider some confidence bands in the
results, values of about 1.01g and 1.03g for general and near-fault records, respectively,
are
observed
corresponding
to
Sa(λu=1.0)/S a(2%in50)
ratios
of
1.9
and
2.0.
Corresponding values for the 6-story frame are 2.3 and 1.9. Finally, it is believed that the
use of the collapse limit state determination technique, presented earlier, in a subsequent
step to the time history analysis and the lack of its integrity with the analysis process
explain part of these high values of Sa(λu=1.0).
Looking from another perspective, and according to the regression parameters given in
Table 7.5, on average, the frame is near collapse at a value of spectral acceleration,
Sa(λu=1.0) = 1.64 (for both general and near-fault records) times the value causing
excessive yielding and severe damage of a few members (i.e., Sa(T1 ,5%) corresponding
to 0.95λuo ). Note that this ratio is a little less than for the case of the 6-story RCS frame
(1.8 for general and near-fault records). This finding implies that for the 12-story frame
the threat of collapse is closer to the state of excessive yielding of the structure than for
the 6-story frame. This further indicates that the hazard intensity for near collapse
(λu=1.0) for the 12-story frame is about 1.6 times that corresponding to the point when
the structure begins to significantly degrade (i.e., λu=0.95λuo , or Life Safety as previously
suggested in Chapter 6). This margin is very close to the ratio of 1.5 implied by modern
codes between the “design level” earthquake response (geared to life safety) and the
maximum considered earthquake (geared to near collapse).
7.4.2 Relationship between Maximum Interstory Drift Ratio and Global Failure
Criterion, λ u
Figures 7.10 and 7.11 show IDRmax-λu relationship for each of the two bins, including a
regression fit using a power law format conditioned on λu. The regression analysis was
299
done in the log space using all data points within each bin excluding points with
λu>0.95λuo . Results reveal that the correlation between IDRmax and λu is quite good as
manifested by a narrow band of curves throughout the damage evolution from λuo up to
collapse with a conditional dispersion σ ln IDR
max |λ u
= 0.190 and 0.177 for general and near-
fault records, respectively.
At λu=1.0, the average value of IDRmax is 0.071 with C.O.V. of 16% for the general
records and 0.088 with C.O.V. of 18% for the near-fault records. The differences between
IDRmax values at λu=1.0 for the near-fault and general records are due to the pulse effects
of the former events and the longer duration characterizing the latter events. The long
strong motion duration leads to more degradation – hence, the stability limit load is
reached at smaller deformations. Also note the decrease of the average value, as well as
C.O.V., of IDRmax at λu=1.0 for the 12-story frame compared to the 6-story frame, where
average values of IDRmax are 0.087 and 0.116 for general and near-fault records,
respectively. Another observation is that the location of IDRmax at failure is almost
always within the first couple of stories for the 6-story frame for nearly all records. On
the other hand, for the 12-story frame, IDRmax at λu=1.0 occurs at different locations
between the first to the ninth story. It migrates towards higher stories for general records
much more than for near-fault records. Such information can be seen from the story IDA
curves given in Section 7.3.1 and Appendix B.
At λu=0.95λuo (i.e., excessive yielding), the average transient IDRmax is 0.033
(C.O.V.=19%) and 0.034 (C.O.V.=20%) for general and near-fault records, respectively.
Note the similar values at the excessive yielding stage for both types of records while the
larger difference at failure as shown in the previous paragraph due to the pronounced
effect of the pulse at such a high intensity level of the record causing global collapse of
the structure.
300
0.10
0.08
IDR max
Miyagi
Valparaiso
LP89-HCA
LP89-HSP
LP89-WAHO
CM92-RIO
LA92-YER
Mendocino
Linear Regres.
-0.29
IDRmax = 0.072 λ u
σlnIDR
= 0.190
max|λu
0.06
0.04
0.02
λu = 1.0
(collapse)
2
Ra = 0.608
0.00
0
1
2
3
4
5
6
λ u (based on 1.0D+0.25L)
7
8
Figure 7.10 IDRmax-λu relationship for bin of general records.
0.15
IV79-A6
LP89-LG
LP89-LX
EZ92-EZ
NR94-NH
NR94-RS
NR94-SY
KB95-JM
Linear Regres.
-0.38
IDRmax = 0.083 λ u
0.12
IDR max
σlnIDR
= 0.177
max |λu
0.09
0.06
0.03
λ u = 1.0
Ra2 = 0.756
(collapse)
0.00
0
1
2
3
4
5
6
7
λu (based on 1.0D+0.25L)
Figure 7.11 IDR max -λu relationship for bin of near-fault records.
301
8
Overall, IDRmax values for the two frames subjected to various ground motions are
remarkably consistent. At λu = 0.95λuo , the average IDRs range between 3.2% to 3.4%,
and there are no perceptible differences between drifts for the different ground motion
bins. The range of 3.2% to 3.4% is slightly larger than the value of 2.5% suggested by
FEMA 273 for Life Safety (Table 6.11, Chapter 6) and, referring to the static pushover
results in Figs. 7.1 and 7.2, corresponds to the point where the 12-story frame reaches its
maximum lateral capacity. At λu = 1.0, there are consistent differences between response
for the general and near-fault records. For the 12-story frame, IDRmax = 7.1% and 8.8%
for general and near-fault records, respectively, while for the 6-story frame, the
corresponding values are 8.7% and 11.6%. The smaller IDRmax for the general records is
probably due to their longer strong motion duration that leads to larger cumulative
damage and stiffness/strength degradation, which in turn causes the stability limit (λu =
1.0) to be reached at smaller drift ratios.
7.4.3 Spatial Distribution of Damage
The damage distribution in terms of the cumulative damage index, Dθ, is presented herein
for selected records from the two bins at the two previously identified performance
levels: λu = 0.95λuo and λu = 1.0 (near collapse).
As previously discussed in Chapter 4, Dθ ≤ 0.3 means “minor” damage, Dθ between 0.3
and 0.6 defines “moderate” damage, between 0.6 and 0.95 “severe” damage, and finally,
Dθ > 0.95 indicates “collapse” (or failure). Figure 7.12 shows values of Dθ at the different
critical sections of the 12-story RCS frame due to the Cape Mendocino record as an
example of general records. Figs. 7.12a and 7.12b give the damage distribution at λu =
0.95λuo and λu=1.0, respectively. At λu = 0.95λuo (Fig. 7.12a), moderate damage is spread
throughout the frame that is most severe (Dθ > 0.60) at a few beams sections from the
seventh to the ninth floor. Moderate damage (0.3 < Dθ < 0.6) is observed at the base of
the ground floor columns and at very few sections at the top and bottom of the inner
columns of the ninth to the eleventh stories. No (or minor) damage (Dθ < 0.30) is
302
observed at any composite joint. In Figure 7.12b (λu = 1.0), severe damage is observed all
over the frame with collapse (Dθ > 0.95, shown in gray fill on the frame elevation) at
various critical sections of the beams at the first three floors and floors seven to ten.
Failure (i.e., full damage) also takes place at the ground floor columns bases. Moderate
damage of some inner composite joints (floors 7 to 9) has also been observed.
Concentration of damage at upper and lower stories reveals some higher mode effects
that have also been identified in the IDA curves of Figure 7.6 (see Section 7.3.1).
Figure 7.13 gives the damage distribution under the Loma Prieta Lexington station record
as an example of near-fault records with significant pulse effects. Fig. 7.13a shows that at
λu = 0.95λuo , almost all the damage occurs at the beams of the first four stories and the
base of the ground floor columns. No damage (Dθ < 0.3) is observed in the composite
joints or other columns sections.
At incipient collapse, i.e., λu = 1.0, Fig. 7.13b shows that most of the damage due to the
Lexington record is still mainly confined to the lower stories. There is collapse of all end
sections of the beams of the first three floors and severe damage of the ground floor
columns bases. These results correspond to observations from story IDA curves under the
Lexington record given in Appendix B showing higher IDRmax values associated with the
first four stories.
An interesting observation is that there is less spread of damage throughout the frame due
to the Lexington record as compared to the Cape Mendocino record. This is primarily due
to the shorter strong motion duration (tSM = 3.3 seconds) of the former near-fault record
as opposed to the latter general record (tSM = 15.4 seconds). Accordingly, damage due to
the Lexington record is more of the peak response type (pulse effect), while damage due
to the Cape Mendocino record is more of the cumulative type. Peak plastic rotations for
the former (near-fault) event are θp,C = 0.068 rad and θp,B = 0.076 rad at λu=1.0 while
corresponding values are 0.058 rad and 0.056 rad, respectively, for the latter (general)
event.
303
0.32
0.32
0.36
0.45
0.34
0.34
0.34
0.32
0.53
0.52
0.37
0.35
0.69
0.48
0.46
0.52
0.46
0.63
0.62
0.41
0.45
0.53
0.47
0.62
0.34
0.40
0.33
0.38
0.32
0.37
0.52
0.36
0.38
0.31
0.37
0.38
0.32
0.34
0.35
0.32
0.33
0.34
0.40
0.38
0.33
0.46
0.47
0.50
Figure 7.12a Distribution of Dθ at λu = 0.95λuo – Cape Mendocino (1992) record.
304
0.47
0.36
0.57
0.69
0.69
0.56
0.45
0.58 0.37
0.71
0.49
0.32
0.81
0.39 0.34 0.31
0.85
0.79
0.31 0.33
0.86
0.31
0.55
0.82 0.41 0.85
0.59
0.56
0.87 0.40 0.64
0.60
0.64
0.85 0.37 0.90
0.41
0.51
0.57
0.39 0.79
0.40
0.77
0.65
0.42
0.90
0.62
0.61
0.82
0.31
0.53
0.52
0.50
0.48
0.85
0.92
0.38
0.71
0.31
0.82
0.62
0.56
0.44
0.93
0.39
0.65
0.32
0.33
0.76
0.59
0.35
0.63
0.32
0.34
0.94
0.88
0.84
0.89
0.83
0.36
0.35
0.31
0.39
0.41
0.32
0.38
0.45
0.46
0.94
Figure 7.12b Distribution of Dθ at λu = 1.0 – Cape Mendocino (1992) record.
305
0.36
0.31
0.32
0.41
0.36
0.36
0.32
0.31
0.38
0.44
0.45
0.42
0.39
0.39
0.50
0.46
0.49
0.45
0.43
0.42
0.54
0.43
0.44
0.41
0.39
0.41
0.52
0.44
0.44
0.44
Figure 7.13a Distribution of Dθ at λu = 0.95λuo – Loma Prieta (1989) record at Lexington.
306
0.34
0.32
0.46
0.39
0.44
0.42
0.38
0.49
0.32
0.33
0.32
0.33
0.45
0.36
0.40
0.34
0.76
0.72
0.69
0.67
0.31
0.35
0.34
0.86
0.90
0.92
0.34
0.48
0.62
0.88
0.34
Figure 7.13b Distribution of Dθ at λu = 1.0 – Loma Prieta (1989) record at Lexington.
307
7.5 Global versus Local Response
In Chapter 6, it has been shown for the 6-story RCS frame that local response in terms of
peak plastic rotations of beams and columns can be successfully related (with good
correlation) to global drift response. This drift quantity provides good correlation if it is
given in terms of maximum plastic transient interstory drift ratio, IDRp,max, for the beams
and maximum change in transient interstory drift ratios, ∆IDRmax, for the columns (see
Section 6.2.1). Whenever column hinging takes place at bottom sections of a specific
story, plastic beam rotations should be related to IDRp,max at the same story, and plastic
column rotations should be related to the maximum absolute value of the difference
between IDR at this story and that at the lower one. On the other hand, if column hinging
takes place at top sections of a specific story, plastic beam rotations should be related to
IDRp,max at the upper story, and plastic column rotations should be related to the
maximum absolute value of the difference between IDR at this story and the upper one.
7.5.1 Relationship between ∆IDRmax and Peak θ p,C
Figures 7.14 and 7.15 present ∆IDRmax versus θp,C|max data, measured at the collapse state
(i.e., λu =1.0), for general records and near-fault records, respectively. Again, a power
form regression fit is performed in the log space for results corresponding to each bin.
The least square fit has been done once conditioned on global response (i.e., given
∆IDRmax) and then conditioned on local response (i.e., given θp,C). Regression lines for
both cases are also shown in the same figures with values of conditional dispersion, σ,
and coefficient of determination, Ra2 . These regression models are compared in Figure
7.16. It is clear that one gets on average close results for general and near-fault records
especially al lower values of the response parameters.
Studying the effect of the level of damage, similar regression analyses have been carried
out for results associated with values of λu ≈ 0.55λuo , i.e., about midway between
λu=0.95λuo and λu=1.0. Figure 7.17 gives ∆IDRmax versus θp,C relationship from
regression analysis at the two levels of damage (λu = 1.0 and λu = 0.55λuo ) for the 6- and
308
12-story RCS frames, for general and near-fault records. Two main observations can be
made. First, very comparable relationships are obtained for each frame up to high values
of plastic rotations and change in maximum interstory drift ratios showing that the
∆IDRmax versus θp,C relationship is quite stable irrespective of the level of damage.
Second, comparing results for the 6- and 12-story frames, on average the ∆IDRmax versus
θp,C relationship is close for both frames especially when regression is conditioned on
global response, ∆IDRmax. This result reinforces the finding that ∆IDRmax versus θp,C
relationship is still on average stable to a good extent even for structures with different
heights. The reason for the lower curves for the 12-story RCS frame in Figure 7.17 when
regression is performed conditioned on local response is the larger number of data points
with lower rotations (due to the larger number of stories) compared to the 6-story frame.
Hence, the difference in the curves is more a function of the effect of these data points on
the regression rather than a difference in the underlying behavior of the frames.
7.5.2 Relationship between IDRp,max and Peak θ p,B
Figures 7.18 and 7.19 give IDRp,max versus θp,B regression relationship at collapse state
(i.e., λu =1.0) for bins of general and near-fault records, respectively, along with data
points from different time history analyses results. Figure 7.20 compares least square fit
relationships for general and near-fault records, again conditioned on either global or
local response. It is obvious that one gets almost identical results for general and nearfault records if the regression analysis is performed conditioned on local response, and
very close results if conditioned on global response. Conditional dispersions are in the
order of σ ln θ
p ,B
| IDRp,max
= 0.479 and 0.473 and σ ln IDR
and near-fault records, respectively.
309
p, max
|θ p, B =
0.469 and 0.413 for general
0.12
2
Ra =0.6
1:1
Maximum Change in IDR
0.10
0.78
θp,C =0.43 ∆IDRmax
σ lnθ |∆IDR
=0.419
p,C
max
0.08
0.06
0.75
∆IDRmax=0.34 θp,C
σln∆IDR
=0.413
max|θp,C
0.04
Values from Analysis
Regression given θp,C
0.02
Regression given IDR
0.00
0.00
0.02
0.04
0.06
0.08
0.10
0.12
Max. Transient Col. Plastic Rot., θp,C [rad.]
Figure 7.14 Global versus local response (θp,C ) for bin of general records at λu=1.0.
0.12
2
R a =0.8
Maximum Change in IDR
0.10
0.08
1:1
θ p,C =0.79 ∆IDRmax0.95
σ lnθ |∆IDR
=0.321
p,C
max
0.06
0.87
∆IDRmax =0.59 θp,C
σ ln∆IDR
=0.306
max|θ p,C
0.04
Values from Analysis
Regression given θp,C
0.02
Regression given IDR
0.00
0.00
0.02
0.04
0.06
0.08
0.10
0.12
Max. Transient Col. Plastic Rot., θp,C [rad.]
Figure 7.15 Global versus local response (θp,C) for bin of near-fault records at λu=1.0.
310
0.10
Columns
Maximum Change in IDR
0.08
0.06
0.04
0.02
General Records
Near-Fault Records
0.00
0.00
0.02
0.04
0.06
0.08
0.10
Max. Transient Col. Plastic Rot., θp,C [rad.]
(a) Regression conditioned on local response, θp,C
0.10
Columns
Maximum Change in IDR
0.08
0.06
0.04
0.02
General Records
Near-Fault Records
0.00
0.00
0.02
0.04
0.06
0.08
0.10
Max. Transient Col. Plastic Rot., θp,C [rad.]
(b) Regression conditioned on global response, ∆IDR max
Figure 7.16 ∆IDRmax-θp,C relationship for general and near-fault records at λ u=1.0.
311
0.10
Maximum Change in IDR
Columns
0.08
Regression conditioned on θp,C
1:1
0.06
0.04
λu=0.55λuo (RCS6)
λu=1.0 (RCS6)
λu=0.55λuo (RCS12)
λu=1.0 (RCS12)
0.02
0.00
0.00
0.02
0.04
0.06
0.08
0.10
Max. Transient Col. Plastic Rot., θp,C [rad.]
0.10
Maximum Change in IDR
Columns
0.08
Regression conditioned on ∆IDR max
1:1
0.06
0.04
λu=0.55λuo (RCS6)
λu=1.0 (RCS6)
λu=0.55λuo (RCS12)
λu=1.0 (RCS12)
0.02
0.00
0.00
0.02
0.04
0.06
0.08
0.10
Max. Transient Col. Plastic Rot., θ p,C [rad.]
(a) Bin of general records
Figure 7.17 ∆IDRmax - θp,C relationship at different levels of damage based on values of λu.
312
0.10
Columns
Maximum Change in IDR
0.08
Regression conditioned on θp,C
1:1
0.06
0.04
λu=0.55λuo (RCS6)
λu=1.0 (RCS6)
λu=0.55λuo (RCS12)
λu=1.0 (RCS12)
0.02
0.00
0.00
0.02
0.04
0.06
0.08
0.10
Max. Transient Col. Plastic Rot., θp,C [rad.]
0.10
Maximum Change in IDR
Columns
0.08
Regression conditioned on ∆IDR max
1:1
0.06
0.04
λu=0.55λuo (RCS6)
λu=1.0 (RCS6)
λu=0.55λuo (RCS12)
λu=1.0 (RCS12)
0.02
0.00
0.00
0.02
0.04
0.06
0.08
0.10
Max. Transient Col. Plastic Rot., θ p,C [rad.]
(b) Bin of near-fault records
Figure 7.17 ∆IDRmax - θp,C relationship at different levels of damage. (Continued)
313
Figure 7.21 shows IDRp,max versus θp,B relationship at the two levels of damage (λu = 1.0
and λu = 0.55λuo ) as previously done for columns, for the 6- and 12-story frames, for
general and near-fault bins of records. Again, very comparable relationships are obtained
for each frame independently up to high values of plastic rotations and plastic maximum
interstory drift ratios. Furthermore, comparing results for the 6- and 12-story frames
simultaneously, on average, IDRp,max versus θp,B relationship is close for both frames
especially for near-fault records. This result still validates that global (IDRp,max) versus
local (θp,B|max) response relationship is on average stable irrespective of the type of
records (general versus near-fault), the level of damage ((λu = 1.0 versus λu = 0.55λuo ),
and the height of the structure for the RCS frames studied herein.
7.5.3 Estimates of Local Response Given Global Response and Input Intensity Level
As previously done for the 6-story RCS frame, applying a conditional regression analysis
of local response in terms of members plastic rotations given both global response (IDR)
and the ground motion intensity Sa, one may look at the relationship between local and
global response at different hazard levels. As such, for any anticipated hazard level and
any pre-specified global response that might be associated with this hazard, one may
estimate the median of a local demand measure such as members’ plastic rotation. This
demand can then be checked against limiting values given by codes as acceptance criteria
corresponding to that specific hazard.
Regression analyses of the form given by Equation 6.10 (Chapter 6) are performed in the
log space, and the resulting relationships are given in Table 7.6 and Figures 7.22 and
7.23. Note that local response of members is represented by element peak plastic rotation
(θp,B for beams and θp,C for columns), and global response is given as before in terms of
∆IDRmax for columns and IDRp,max for beams.
314
0.12
θp,B=0.56 IDRp,max 0.91
0.10
σlnθ |IDR
=0.479
p,B
p,max
1:1
IDR p,max
0.08
0.06
IDRp,max=0.76 θp,B0.87
σlnIDR
=0.469
p,max |θp,B
0.04
Values from Analysis
Regression given θp,B
0.02
2
R a =0.8
0.00
0.00
0.02
0.04
0.06
Regression given IDR
0.08
0.10
0.12
Max. Transient Beam Plastic Rot., θp,B [rad.]
Figure 7.18 Global versus local response (θp,B) for bin of general records at λu=1.0.
0.12
1:1
1.08
θp,B=1.03 IDRp,max
0.10
σlnθ |IDR
=0.473
p,B
p,max
IDRp,max
0.08
0.06
IDRp,max=0.64 θp,B0.82
σlnIDR
=0.413
p,max|θp,B
0.04
Values from Analysis
Regression given θp,B
0.02
2
R a =0.9
0.00
0.00
0.02
0.04
0.06
Regression given IDR
0.08
0.10
0.12
Max. Transient Beam Plastic Rot., θp,B [rad.]
Figure 7.19 Global versus local response (θp,B) for bin of near-fault records at λ u=1.0.
315
0.10
Beams
IDRp,max
0.08
0.06
0.04
0.02
General Records
Near-Fault Records
0.00
0.00
0.02
0.04
0.06
0.08
0.10
Max. Transient Beam Plastic Rot., θp,B [rad.]
(a) Regression conditioned on local response, θp,B
0.10
Beams
IDR p,max
0.08
0.06
0.04
0.02
General Records
Near-Fault Records
0.00
0.00
0.02
0.04
0.06
0.08
0.10
Max. Transient Beam Plastic Rot., θp,B [rad.]
(b) Regression conditioned on global response, IDR p,max
Figure 7.20 IDRp,max-θp,B relationship for general and near-fault records at λ u=1.0.
316
0.10
1:1
Beams
0.08
IDRp,max
Regression conditioned on θp,B
0.06
0.04
λu=0.55λuo (RCS6)
λu=1.0 (RCS6)
λu=0.55λuo (RCS12)
λu=1.0 (RCS12)
0.02
0.00
0.00
0.02
0.04
0.06
0.08
0.10
Max. Transient Beam Plastic Rot., θp,B [rad.]
0.10
Beams
0.08
1:1
IDRp,max
Regression conditioned on IDR p,max
0.06
0.04
λu=0.55λuo (RCS6)
λu=1.0 (RCS6)
λu=0.55λuo (RCS12)
λu=1.0 (RCS12)
0.02
0.00
0.00
0.02
0.04
0.06
0.08
0.10
Max. Transient Beam Plastic Rot., θp,B [rad.]
(a) Bin of general records
Figure 7.21 IDR p,max - θ p,B relationship at different levels of damage based on values of λu.
317
0.10
1:1
Beams
IDRp,max
0.08
Regression conditioned on θp,B
0.06
0.04
λu=0.55λuo (RCS6)
λu=1.0 (RCS6)
λu=0.55λuo (RCS12)
λu=1.0 (RCS12)
0.02
0.00
0.00
0.02
0.04
0.06
0.08
0.10
Max. Transient Beam Plastic Rot., θp,B [rad.]
0.10
Beams
0.08
IDRp,max
Regression conditioned on IDR p,max
1:1
0.06
0.04
λu=0.55λuo (RCS6)
λu=1.0 (RCS6)
λu=0.55λuo (RCS12)
λu=1.0 (RCS12)
0.02
0.00
0.00
0.02
0.04
0.06
0.08
0.10
Max. Transient Beam Plastic Rot., θ p,B [rad.]
(b) Bin of near-fault records
Figure 7.21 IDRp,max - θ p,B relationship at different levels of damage. (Continued)
318
Table 7.6 Regression equations for local response given global response and input
intensity level for the 12-story RCS frame.
Beams
Columns
General Records
General Records
θ p, B = 0.75 S -0.02
IDR 1p,.00
a
max
σ ln θ p, B |Sa ( T1 ,ξ ), IDR p,max = 0.461
.81
θ p, C = 0.46 S0.31
∆IDR 0max
a
σ ln θ p, C |Sa ( T1 ,ξ ), ∆IDR max = 0.404
R 2a = 0.796
Near-Fault Records
R 2a = 0.632
Near-Fault Records
θ p, B = 1.24 S-0.004
IDR 1p,.13
a
max
σ ln θ p,B |Sa (T1 ,ξ ),IDR p, max = 0.484
. 93
θ p, C = 0.69 S0.16
∆IDR 0max
a
σ lnθ p, C |Sa (T1 ,ξ ), ∆IDR max = 0.331
R 2a = 0.881
R 2a = 0.816
For comparison purposes, Figures 7.22 and 7.23 show the global versus local response
for both the 6- and 12-story RCS frames at various relevant hazard levels. As shown from
the figures, given the same global response value, and same Sa(T1 ,ξ), local response in
terms of plastic rotation of the beams or columns at different hazard levels is very close
for the general and near-fault records for each of the 6- and the 12-story frames.
Furthermore, it is clear from Figures 7.22 and 7.23 that the effect of Sa(T1 ,ξ) on the local
response in the presence of the global response is almost negligible for beams, especially
for the 12-story frame. This is obvious from the small values of the exponent of Sa(T1 ,ξ)
in Table 7.6.
The comparisons made in Figures 7.22 and 7.23 extend previous findings in this section
and Chapter 6. It shows again the stability of the relationship between local and global
response values and its usefulness to get reliable “mean estimates” of local response
measures (in terms of members’ plastic rotations) given a specific global response value
in terms of some suitable IDR parameter irrespective of 1) the level of damage, 2) type of
record, 3) intensity of the input, or even 4) height of the structure. Before generalizing
these findings, further study should be made for other RCS frames with different
geometries, periods, amounts of overstrength, etc.
319
Max. Transient Column Plastic Rot., θp,C [rad.]
0.10
1:1
0.08
0.06
10%in50years Sa (RCS6)
2%in50years Sa (RCS6)
1.5*2%in50years Sa (RCS6)
10%in50years Sa (RCS12)
2%in50years Sa (RCS12)
1.5*2%in50years Sa (RCS12)
0.04
0.02
Columns
0.00
0.00
0.02
0.04
0.06
0.08
0.10
Max. Transient Beam Plastic Rot., θp,B [rad.]
∆IDRmax
0.10
0.08
0.06
1:1
10%in50years Sa (RCS6)
2%in50years Sa (RCS6)
1.5*2%in50years Sa (RCS6)
10%in50years Sa (RCS12)
2%in50years Sa (RCS12)
1.5*2%in50years Sa (RCS12)
0.04
0.02
Beams
0.00
0.00
0.02
0.04
0.06
0.08
0.10
IDRp,max
Figure 7.22 Global versus local response at different hazard levels for bin of general records.
320
Max. Transient Column Plastic Rot., θp,C [rad.]
0.10
0.08
0.06
1:1
10%in50years Sa (RCS6)
2%in50years Sa (RCS6)
1.5*2%in50years Sa (RCS6)
10%in50years Sa (RCS12)
2%in50years Sa (RCS12)
1.5*2%in50years Sa (RCS12)
0.04
0.02
Columns
0.00
0.00
0.02
0.04
0.06
0.08
0.10
Max. Transient Beam Plastic Rot., θp,B [rad.]
∆IDRmax
0.10
0.08
0.06
1:1
10%in50years Sa (RCS6)
2%in50years Sa (RCS6)
1.5*2%in50years Sa (RCS6)
10%in50years Sa (RCS12)
2%in50years Sa (RCS12)
1.5*2%in50years Sa (RCS12)
0.04
0.02
Beams
0.00
0.00
0.02
0.04
0.06
0.08
0.10
IDR p,max
Figure 7.23 Global versus local response at different hazard levels for bin of near-fault records.
321
7.6 Global Response Dependency on Different Ground Motion Input Parameters
As presented in Sections 7.3 and 7.4, we have already investigated the correlation
between spectral acceleration at first mode and different global response parameters
including maximum transient interstory drift ratio, IDRmax, and the global failure
criterion, λu. Sa(T1 ,ξ=5%) explains most of the dispersion of drift response while ti is not
performing that satisfactorily for λu as shown by R a2 values (refer to Tables 7.7 and 7.8
for more details). Therefore, as done for the 6-story RCS frame, we will next include
other independent variables (i.e., input parameters) to check for any additional significant
reduction in the regression conditional dispersion. Such reduction, if any, will reduce the
uncertainty in the estimation of the median response, conditioned on the input earthquake
intensity parameters, due to limited sample size. Such input parameters (considered either
individually or through a combination) might then serve as potential “effective”
earthquake intensity indices.
Additional input parameters considered include: 1) R S a - T = Sa(TF,ξ=5%)/S a(T1 ,ξ=5%) as
F
defined in Chapter 6; 2) R S a-1,2 = Sa(T1 ,ξ=5%)/ Sa(T2 ,ξ=5%), a way to pick up any higher
mode effects; 3) strong motion duration, tSM; and 4) pulse period, Tp , for near-fault
records. Other forms and parameters have also been tried within this research to
incorporate higher mode effects, such as through ratios of spectral accelerations at other
higher modes rather than just the second, or using a combination of spectral accelerations
at different modes weighted by mass modal participation factors as suggested by Shome
(1999). However, of these, R S a-1,2 , representing the second mode effect, has been proved
to be the one giving the best correlation.
A linear regression analysis of the two global response measures (IDRmax and λu) on the
different input independent variables mentioned above is carried out in the log space. The
functional dependency of these two response measures, denoted here by Y, on different
independent input parameters is of the form
322
β
β
β
Y = α S a 1 (T1, ξ ) R S 2 R S 3 e β 4 t SM
a - TF
a-1,2
(7.3)
where α, β 1 , β 2 , β 3 and β 4 are regression parameters. Note that another form of the term
β4
related to the strong motion duration, equal to t SM
, has been tried for the 6-story RCS
frame in Chapter 6. It has been found that this form has lower correlation with the
response parameter and gives higher conditional dispersion of the response for that
β4
frame. Both forms, e β 4 tSM and t SM
, are also tried herein for the 12-story frame and the
one that gives the lower conditional dispersion of the median response for the frame is
the one to be reported in Tables 7.7 and 7.8. Also note that for the bin of near-fault
records considered in this research, a term involving the pulse period (given by e
β 4 Tp
or
Tpβ 4 ) has been also tried in the regression model given by Equation 7.3. As mentioned in
Chapter 6, two methods for calculating δ t and consequently TF needed for R Sa -T =
F
Sa(TF,ξ=5%)/S a(T1 ,ξ=5%) calculation are suggested. One method is based on the target
displacement (or coefficient method as per FEMA 273) and the second is based on the
equal displacement rule. Both values of TF are tried for the 12-story frame and the one
with the best correlation (i.e., lower conditional dispersion) is the one to be reported in
Tables 7.7 and 7.8. It is useful to report that Sa(TF,ξ=5%) based on TF calculated
according to the equal displacement rule is the one that correlated best with the response
for the 6-story frame. TF has been determined for the RCS 12-story frame under
consideration as 2.62 seconds and 3.81 seconds according to the target displacement and
equal displacement rules, respectively. (Note that T1 =2.07 seconds). Also note that Sa 1 is
used for Sa(T1 ,ξ=5%) in Tables 7.7 and 7.8 for briefness.
323
Table 7.7 Regression results for IDRmax conditioned on different input parameters.
Bin of General Records
EQ
Bin of Near-Fault Records
Param.
IDR max = 0.056S0a.93
IDR max = 0.049S0a. 96
Sa 1
1
1
σ ln IDR max |S a = 0.240
σ ln IDR max |S a = 0.258
R a2 = 0.866
IDR max = 0.071S0a.96 RS0. 62
1
a- TF
R a2 = 0.816
IDR max = 0.069S1a.03R 0S.38
1
a -TF
1
1
σ ln IDR max |S a , R S −T = 0.187
1
a F
Sa 1
&
R Sa -T
F
R a2 = 0.919
(TD)
σ ln IDR max |S a , R S −T = 0.218
1
a F
R a2 = 0.869
(ED)
IDR max = 0.037S0a.98 RS−0.27
Sa 1
σ ln IDR max |S a , R S −1 ,2 = 0.197
1
a
R Sa -1,2
σ ln IDR max |S a , R S −1 ,2 = 0.259
1
a
Sa 1
IDR max = 0.040S0a.96e 0.03t SM
1
a-1,2
R a2 = 0.910
−0.51
IDR max = 0.243S0a.96 t SM
1
σ ln IDR max |S a , t SM = 0.175
1
IDR max = 0.047S0a.96 RS−0.03
1
&
R a2 = 0.815
1
&
tSM
σ ln IDR max |S a , t SM = 0.255
1
R a2 = 0.929
R a2 = 0.820
IDR max = 0.057S1a.01R 0S.44 RS−0.34
Sa 1 ,
σ ln IDR max |Sa , R S −T , R S = 0.144
1
a F
a -1,2
R Sa -T
1
a-1,2
a -TF
R a2 = 0.952
a-1,2
F
&
IDR max = 0.066S1a.03R 0S.39 RS−0.06
1
a -TF
a-1,2
σ ln IDR max |S a , R S −T , R S = 0.211
1
a F
a-1,2
R Sa -1,2
R a2 = 0.871
−0.37
IDR max = 0.178S0a.96 RS0. 23t SM
Sa 1 ,
IDR max = 0.073S1a. 03RS0.39e -0.01t SM
σ ln IDR max |S a , R S , tSM = 0.172
1
a -TF
R Sa -T
1
(ED)
a- TF
R a2 = 0.931
(TD)
F
&
tSM
Sa 1 ,
σ ln IDR max |Sa , R S , t SM = 0.164
1
a -1,2
R Sa -1,2
R a2 = 0.937
N/A
a-1,2
a -TF
σ ln IDR max |S a , R S , tSM = 0.219
1
a -TF
R a2 = 0.868
0. 40
IDR max = 0.141S0a. 96R S−0. 14 t −SM
1
1
(ED)
&
tSM
(ED)
IDR max = 0.038S0a. 96R S−0.04 e0.03 tSM
1
a-1,2
σ ln IDR max |Sa , R S , t SM = 0.256
1
a -1,2
R a2 = 0.819
Sa 1 ,
R Sa -T
F
&
Tp
TD = TF based on Target Displacement.
IDR max = 0.096S1a.05R 0S.53e
1
σ ln IDR max |S a , R S , Tp = 0.207
1
a -TF
R a2 = 0.882
ED = TF based on Equal Displacement
324
-0.10T p
a-TF
(ED)
Table 7.8 Regression results for λu conditioned on different input parameters.
Bin of General Records
EQ
Bin of Near-Fault Records
Param.
λu = 2.49S−a 1.43
λu = 3.37Sa−1. 42
Sa 1
1
1
σ ln λu |Sa = 0.660
σ ln λu |Sa = 0.664
R 2a = 0.345
R a2 = 0.349
1
1
λu = 1.84Sa−1.59 RS−0.76
Sa 1
σ ln λu |Sa , R S = 0.639
1
a -TF
R Sa -T
1
a- TF
1
&
F
R a2 = 0.385
(TD)
λu = 5.38S−a 1. 81R 0S.52
1
λu = 1.50Sa−1.95R S−0.95
Sa 1
&
R Sa -1,2
a-1,2
σ ln λu |S a , R S = 0.615
1
a-1,2
R a2 = 0.431
a-TF
σ ln λu |Sa , R S = 0.587
1
a -TF
R a2 = 0.492
(ED)
λu = 3.42Sa−1.42 RS0. 02
1
a-1,2
σ ln λu |S a , R S = 0.669
1
a-1,2
R a2 = 0.340
.62
λu = 0.41S−a 1. 61t 0SM
λu = 4.04S−a 1. 42e
Sa 1
1
&
tSM
σ ln λu |Sa , t SM = 0.634
1
−0.03 tSM
1
σ ln λu |Sa , t SM = 0.668
1
R a2 = 0.395
R a2 = 0.343
λu = 2.54S−a 2.25 R −S0.95 R 0S.75
Sa 1 ,
λu = 1.60S−a 1. 95R S−0.96 RS0. 08
σ ln λu |Sa , R S , R S = 0.559
1
a -TF
a -1,2
R Sa -T
σ ln λu |Sa , R S , R S = 0.590
1
a -TF
a -1,2
1
a -TF
a -1,2
R 2a = 0.530
(ED)
F
&
R Sa -1,2
λu = 0.58S−a 1.62 RS−0.27 t 0.47
SM
Sa 1 ,
σ ln λu |Sa , R S , t SM = 0 .638
1
a -TF
R Sa -T
1
a -TF
R a2 = 0.389
F
&
(TD)
Sa 1 ,
σ ln λu |Sa , R S , t SM = 0.614
1
a -1,2
R Sa -1,2
a -1,2
R a2 = 0.434
N/A
&
Sa 1 ,
R Sa -T
F
325
R a2 = 0.488
a-1,2
(ED)
λu = 0.84S−a 2.01R −S 1. 08e0.07t SM
1
a -TF
σ ln λu |Sa , R S −T , t SM = 0.583
1
a F
(ED)
λu = 4.17S−a 1.42 RS0. 03e0.03 tSM
1
a-1,2
σ ln λu |S a , R S , t SM = 0.673
1
a -1,2
R a2 = 0.334
tSM
&
Tp
a-TF
R a2 = 0.500
tSM
0. 31
λu = 1.87Sa−1.83R 0S.42 t SM
1
1
λu = 0.89S−a 2.10 RS−1.30 Tp0.41
1
a- TF
σ ln λu |Sa , R S −T , Tp = 0.567
1
a F
R a2 = 0.527
(ED)
Referring to results given in Tables 7.7 and 7.8, considering R Sa -1,2 (representing the
effect of second mode on the response) beside Sa(T1 ) does not offer good correlation with
the median response, either IDRmax or λu, for near-fault records. This is easily observed
by the lack of reduction of the conditional dispersion of the median response (or the nonincrease of the adjusted coefficient of determination, R a2 ). On the other hand, R Sa -1,2
works well if considered along with Sa(T1 ) for general records. This finding reveals that
higher mode effects represent an important factor affecting the response due to general
records. This can be further understood by carefully inspecting the story IDA curves of
general records (Fig. 7.6 and Appendix B) with more than half of the records showing
larger deformations within the upper stories (above the sixth), especially at high hazard
levels. For near-fault ground motions, all records (see Fig. 7.7 and Apendix B) except for
Kobe cause the maximum deformation to occur within the lower stories (below the
fourth). Similar information can be observed from the distribution of damage at various
performance levels (i.e., at λu = 0.95λuo , life safety, and λu = 1.0, near collapse) as shown
in Figs. 7.12 and 7.13.
Furthermore, as for R Sa -1,2 , considering strong motion duration, tSM, as an extra input
parameter for decreasing the conditional dispersion of the median response shows almost
same trends. It works well in reducing the uncertainty in the estimation of the median
response for general records while it fails to behave that successfully for near-fault
records. Interpretation of such an observation is easy since it is well appreciated that
structures subjected to general records might accumulate more damage before reaching
global collapse (as can be easily reflected on their response) as function of the strong
motion duration. On the other hand, damage due to near-fault records is more of the peak
type due to the inherent impulsive aspect of the record. However, this effect of tSM should
be considered herein with great care, in part because of the narrow range of strong motion
duration values in the bins of ground records used for the time history analyses.
Finally, a general conclusion that we can draw is that considering R Sa -T (representing the
F
effect of period lengthening on the spectral acceleration values due to structural damage)
326
beside Sa(T1 ,ξ=5%) correlates well with the response of both general and near-fault
records as was the case for the 6-story RCS frame. However, considering R Sa -1,2 along
with Sa(T1 ,ξ=5%) and R Sa -T
will further reduce the conditional dispersion of the
F
response for general records for the 12-story frame. While considering the pulse period,
Tp , along with Sa(T1 ,ξ=5%) and R Sa -T
is best for reducing the uncertainty in the
F
estimation of the median response due to limited sample size for near-fault records. But
we should keep in mind that we are still ignoring here the uncertainty in the response due
to uncertainty in modeling and in physical properties of the structure and of its
components as previously mentioned in Chapter 6.
327
PART II: 6-Story STEEL Special Moment Frame
A 6-story STEEL special moment frame in the short direction of the theme structure with
dimensions and detailing as given in Chapter 5 is studied in this section. Similarly to
what has been done for the 12-story RCS frame in Part I of this chapter, the analysis
results of the 6-story steel frame will be evaluated both on their own merits and in
comparison with the behavior of the 6-story RCS frame.
An important aspect of the 6-story steel frame under investigation is its significant lateral
overstrength (Ω=6.0 and Ω * =8.4) above the design force level. This can be attributed to
the fact that member sizes in most of the frames designed according to current seismic
provisions and code procedures are governed by drift requirements. Accordingly,
member sizes are relatively large regardless of the base shear and the strength demand.
Moreover, steel members (i.e., available standard steel sections) are usually manufactured
in such a way that their strength and stiffness are in proportion. Therefore, increasing
member sizes to satisfy imposed drift (i.e., stiffness) requirements will proportionally add
unneeded lateral strength to the system leading to higher overstrength. This situation has
also been identified by Leelataviwat et al. (1998) when they found that a 5-story 3-bay
steel frame designed according to UBC-94 for seismic zone 4 possesses an overstrength
of six times the UBC design base shear.
7.7 Modeling of the 6-Story STEEL Frame
The same procedure adopted in modeling the RCS frames has been followed in modeling
the steel frame. Tables 5.3 through 5.6 and Figures 5.5 and 5.12 in Chapter 5 provide all
relevant
details
of
the
steel
frame
including
structural
configuration,
members’
dimensions, boundary conditions, design gravity and lateral loading, seismic mass
characteristics, etc. Tables 7.9 to 7.11 give stiffness and strength properties for different
columns, beams, and joints as modeled in DYNAMIX.
328
Floor #
1-4
5-6
Floor #
1-4
5-6
1-4
5-6
Floor #
1-4
5-6
Table 7.9 Stiffness and strength values of steel columns.
Axial Properties
Bending Properties
Shear Stiffness
Squash
Tensile
EA
Flexural
EI
GA
2
Load
Strength
(kips)
Strength
(kips.in )
(kips)
(kips)
(kips)
(kips.in)
3571
3571
1.80x106
22460
7.71x107
1.72x105
6
7
2984
2984
1.50x10
18430
6.21x10
1.41x105
Table 7.10 Stiffness and strength values of composite and steel beams.
Flexural Strength
Flexural Stiffness, EI
Shear
2
(kips.in)
(kips.in )
Stiffness, GA
(kips)
Positive
Negative
Positive
Negative
COMPOSITE BEAMS
14920
10200
1.31x108
5.31x107
1.10x105
12230
8294
9.98x107
3.86x107
9.37x104
STEEL BEAMS
10200
10200
5.31x107
5.31x107
1.10x105
8294
8294
3.86x107
3.86x107
9.37x104
Table 7.11 Properties of joint panels.
Dimensions
Strength
(inches)
M joint
(kips.in)
Horizontal
Vertical
15.7
23.7
15780
15.2
21.0
11340
Stiffness
(kips.in)
3.87x106
2.81x106
The mathematical model of the joint panel proposed by Krawinkler and Popov (1982) is
employed to calculate joint panels stiffness and strength values for the steel frame. Based
on basic mechanics supported by experimental data, the yield moment strength and the
yield strain of steel joint panels are assumed as
My = 0.55 Fy dc tcw db
(7.4)
γ y = Fy /
(7.5)
3G
329
where Fy is the yield strength of the column, dc is the depth of the column, db is the depth
of the beam, tcw is the thickness of the column web, and G is the shear modulus. Using
these values, the elastic stiffness of the joint panels reported in Table 7.11 can be
calculated as
Kjoint = My / γy = 0.95 tcw db dc G
(7.6)
The ultimate moment strength of the joint panel, Mjoint, reported in Table 7.11 is given by

3.45 b c t 2cf

M joint = 0.55 Fy d c d b t cw 1 +

d b d c t cw





(7.7)
in which tcf is the thickness of the column flange, and bc is the width of the column
flange.
Viscous damping is again modeled for the frame through mass and stiffness proportional
(Rayleigh) damping. 2% of critical damping in the first and third modes as for the 6-story
RCS frame are assumed based on the study of modal properties of the frame. The
cumulative effective modal masses of the first three modes of the frame constitute about
96.3% of the total mass suggesting that assigning the critical damping to the first and
third modes is a reasonable assumption. Applying Equation 6.2 to calculate percentages
of critical damping associated with different modes reveals the smallest critical damping
value of 1.5% for the second mode and the largest critical damping value of 5.2% for the
sixth mode.
7.8 Static Push-Over Analysis
A static inelastic push-over analysis similar to what has been previously done for the case
study RCS frames (Section 6.2, Chapter 6, and Section 7.2, Chapter 7) is performed for
330
the 6-story steel frame. Base shear/weight ratio versus roof drift ratio is shown in Figure
7.24. The figure reveals that the static lateral overstrength, Ω, of the frame is about 6.0,
i.e., Ω = Vu/Vd ≅ 0.60/0.099 ≅ 6.0. The frame has been designed for a base shear
(including accidental torsion effect and based on a period of 1.2Ta = 1.11 seconds) to
weight ratio of 0.099. However, when ignoring torsion effects and considering the
calculated period of 1.26 seconds the lateral overstrength is in the order of Ω * = 8.4
(=6.0x(0.099/0.071)); refer to Table 5.6, Chapter 5, for more details. This particularly
large overstrength in the steel frame is attributed as previously mentioned to the
minimum stiffness (drift) requirements imposed by codes and the use of a distributed
space frame with relatively shallow beams (W24 and W21). One can show, for example,
that for stiffness controlled designs, the shallower beams in the space frames will result in
higher seismic overstrength than with deeper beams (e.g., W36) commonly found in
perimeter frame systems. Moreover, the remarkably large column base moment strength
in the steel frame as compared to the case study 6-story RCS frame is another important
reason contributing to the larger lateral overstrength of the steel frame. The moment
strength at zero axial load of the column cross-section of the steel frame is about twice
that of the RCS frame, and about 1.5 times at a level of axial force corresponding to
gravity load considered for time history analyses (i.e., 1.0 Dead Load + 0.25 Live Load).
The target displacement, δ t , for the frame calculated according to Equation 5.8 and a
2%in50years hazard level is 23.6 inches, corresponding to a total roof drift ratio, ∆r/H, of
about 0.025. At this pre-specified target displacement the structure has not yet reached its
maximum lateral capacity of Vu = 0.60W with a corresponding roof drift ratio, ∆r/H, of
about 0.095. Both values, lateral capacity and associated roof drift, are much larger than
for the 6-story RCS frame that reaches its maximum lateral strength of 0.46W at ∆r/H ≅
0.039.
331
0.7
Base Shear-Weight Ratio, V/W
0.6
0.5
0.4
Static POC
Design Load
∆r/H = 0.02
0.3
Target Disp., δt
0.2
∆r/H = 0.04
0.1
∆r/H = 0.06
∆r/H = 0.08
Max. Strength
0.0
0.00
0.02
0.04
0.06
0.08
0.10
0.12
0.14
Roof Drift Ratio, ∆r/H
Figure 7.24 Static pushover curve - 6-story STEEL frame, IBC 2000 load pattern.
6
Design Load
∆r/H = 0.02
5
Target Disp., δt
∆r/H = 0.04
∆r/H = 0.06
Floor #
4
∆r/H = 0.08
Max. Strength
3
2
1
0
0.00
0.02
0.04
0.06
0.08
0.10
0.12
0.14
0.16
Interstory Drift Ratio, IDR
Figure 7.25 Distribution of IDR up the height of the frame - static pushover
results.
332
Distributions of interstory drift ratios, IDR, up the height of the frame are given in Figure
7.25 at various roof drift ratios. Figure 7.25 shows that most of the inelastic behavior, as
reflected by the high interstory drift ratios, takes place between the first and fourth
stories, with the maximum occurring at the second and third stories. Comparing general
trends of the results of the 6-story steel frame from Fig. 7.25 with Fig. 6.2 for the 6-story
RCS frame, there is slightly more uniformity in the IDR distribution up the height of the
steel frame with lower maximum values even at high demands. Figure 7.26 compares
distribution of interstory drift ratios for both frames up the height. This loosely identified
“uniformity” in IDRs is mainly attributed to the larger overstrength of the steel frame
with stronger but more flexible columns.
6
∆r /H = 0.04 (RCS-6)
∆r /H = 0.04 (STEEL-6)
5
∆r /H = 0.06 (RCS-6)
∆r /H = 0.06 (STEEL-6)
∆r /H = 0.10 (RCS-6)
Floor #
4
∆r /H = 0.10 (STEEL-6)
3
2
1
0
0.00
0.02
0.04
0.06
0.08
0.10
0.12
0.14
0.16
0.18
0.20
Interstory Drift Ratio, IDR
Figure 7.26 Comparison of IDR values for 6-story RCS and STEEL frames static pushover results.
333
7.9 Incremental Dynamic Analyses
Figures 7.27 and 7.28 show IDA curves for the two bins of records along with the
spectral acceleration versus IDRmax relationships defined by Equation 7.1 and the values
given in Table 7.12. Comparing Figs 7.27 and 7.28, for a specific hazard level as defined
by a value of Sa(T1 ,ξ), the median response IDRmax is less for the general records than for
the near-fault records. This agrees with results of the 6-story RCS frame and is valid for
all hazard levels up to high values of Sa(T1 ) defining collapse limit state of the frame.
Table 7.12 Regression parameters α and β for the 6-story steel frame.
Parameter and Statistical
General Records
Near-Fault Records
Measure Values
0.029 (16%)
0.034 (29%)
α (C.O.V)
0.85 (25%)
0.99 (35%)
β (C.O.V.)
For comparison purposes, Fig. 7.29 shows Sa-IDRmax regression fits for both frames for
both bins of records. Comparing values of median response IDRmax for the RCS
(T1 =1.25sec.) and steel (T1 =1.26sec.) frames, at Sa(T1 ,ξ=5%) below a hazard level of
about 2%in50years, median IDRmax is slightly less for the RCS frame than for the steel
frame. For instance, at 2%in50years hazard level (i.e., Sa(T1 ,5%) = 0.864g for RCS frame
and 0.857g for steel frame), the ratio of median IDRmax values for the RCS and steel
frames are 0.91 and 0.95 for bins of general and near-fault records, respectively. These
close values reflect the fact that both frames have almost identical fundamental periods
and are still fairly elastic at Sa(2%in50). However, at higher hazard levels up to the
collapse limit, median IDRmax values for the RCS frame are larger than those for the steel
frame for both types of records. The difference in median values of the response of the
two frames is also revealed through the values of the regression parameter β. Average
values of β of 1.11 and 1.35 are given in Table 6.8 for the RCS frame for general and
near-fault records, respectively. These values clearly show a “softening” behavior in the
nonlinear relationship between the input in terms of Sa(T1 ) and the response in terms of
median IDRmax. On the other hand, average β values are 0.85 and 0.99 for the steel frame
334
for general and near-fault records, respectively. These β values even show on average
some “hardening” behavior for general records and linear behavior for near-fault ones.
As reported in the previous paragraph, identical fundamental period, T1 , for both frames
(RCS versus STEEL) is the result of the steel frame being less stiff but with a smaller
seismic mass than the RCS frame. Hence, the elastic behavior of both frames is very
close, as previously reported, for 2%in50year hazards and lower. However, at higher
hazard levels accompanied by moderate to severe global damage, the RCS frame is
seeing more stiffness degradation of its RC columns under cyclic earthquake loading
leading to higher inelastic deformations (see Fig. 7.29) than for the steel frame. This
effect is more accentuated by the early yielding of the RC columns cross-sections as
opposed to steel columns with inherent higher strength as designed.
Investigation of the relationship between IDRmax and Sa(T1 ) for the steel frame for the bin
of near-fault records leads to the same observation made in Chapter 6 for the RCS frame.
The response may be classified into two categories each involving four of the eight nearfault records, as characterized by the ratio Tp /T1 . Reapplying regression analysis but for
each subset of four records alone, average values of the regression parameters α and β are
given in Table 7.13. Subset 1 which is more damaging to the structure due to the severe
pulse effect shows “softening” of the IDRmax-Sa(T1 ) relationship, while subset 2 with less
damaging effect (according to Tp /T1 ) shows considerable “hardening” in the behavior.
Moreover, comparing local regression fit for subset 2 with that for general records, one
may note lower values of median IDRmax for this subset than for general records at any
given input intensity Sa(T1 ). Accordingly, as pointed out earlier, some near-fault records
with forward directivity might be less damaging to a given structure than general records.
This must be recognized when evaluating response results for near-fault records.
Parameters such as Tp /T1 ratio that may serve as good indicator for predicting potential
damage effect of near-fault records should be further investigated.
335
6
Miyagi
Valparaiso
LP89-HCA
Sa (T1=1.26sec,ξ=5%)
5
0.85
IDRmax = 0.029 Sa
LP89-HSP
LP89-WAHO
CM92-RIO
LA92-YER
Mendocino
Regression Line
4
3
2
1
Sa (2%in50years)
0
0.00
0.02
0.04
0.06
0.08
0.10
IDRmax
Figure 7.27 Sa -IDR max relationship for bin of general records.
6
5
Sa(T1=1.26sec,ξ=5%)
IDRmax = 0.034 S a0.99
IV79-A6
LP89-LG
LP89-LX
EZ92-EZ
NR94-NH
NR94-RS
NR94-SY
KB95-JM
Regression Line
4
3
2
1
Sa (2%in50years)
0
0.00
0.02
0.04
0.06
0.08
0.10
0.12
IDRmax
Figure 7.28 Sa-IDRmax relationship for bin of near-fault records.
336
0.14
5
General (RCS6)
Near-Fault (RCS6)
General (STEEL6)
Near-Fault (STEEL6)
Sa(T1,5%)
4
3
2
1
Sa (2%in50years)
0
0.00
0.02
0.04
0.06
0.08
0.10
0.12
IDR max
Figure 7.29 Comparison of regression results of Sa-IDR max relationship for 6-story
RCS and STEEL frames.
Table 7.13 Average regression parameters α and β for near-fault records.
Parameter and Statistical
Near-Fault Records
Near-Fault Records
Measure Values
Subset 1
Subset 2
0.041
(24%)
0.028
(7%)
α (C.O.V.)
1.20 (33%)
0.78 (13%)
β (C.O.V.)
Dispersion in the response given by IDRmax conditioned on the input intensity, Sa(T1 ), is
smaller for the 6-story steel frame when compared to the 6-story RCS frame. For general
records, σ ln IDR
max
|Sa (T1 ,ξ )
= 0.416 and 0.233 for RCS and steel frames, respectively, while
for near-fault records, σ ln IDR
max
|Sa (T1 ,ξ )
= 0.449 and 0.302 for RCS and steel frames,
respectively. This considerable decrease in the dispersion of the response conditioned on
the input is automatically reflected on the decrease of the uncertainty in the estimation of
median IDRmax, for the steel frame than for the RCS frame, due to limited sample size.
337
7.9.1 Story Incremental Dynamic Analysis Curves
Figures 7.30 and 7.31 show IDA curves for each story of the 6-story steel frame for the
Cape Mendocino at Rio Del Overpass station (general) and Erzincan (near-fault) records.
Story IDA curves for all other records are given in Appendix B. Such figures reveal that
maximum transient interstory drift ratios, IDRmax, are usually larger at the first three or
four stories of the frame than at the upper two stories, with the maximum IDR usually at
either the second or third story, for almost all records. This finding is consistent with the
pushover analysis results given in Figure 7.25, which show that the frame inelastic
behavior is mainly confined to the first four stories with maximum effect at the second
and third stories. Therefore, a conclusion can be made that IBC 2000 equivalent lateral
loading pattern is successful in estimating the location of probable damage in the 6-story
steel frame under different earthquakes with various types and intensity levels.
A general observation of the behavior of the 6-story RCS and steel frames inspired by
story IDA curves is that the estimated damage, and location of expected high inelastic
behavior, occurs at lower stories for the RCS frame as opposed to the steel frame, with
damage confined to the lower two thirds of the structure for both frames. One may refer
to story IDA curves in Figs. 7.30 and 7.31 and Appendix B for the steel frame and to
Figs. 6.12 and 6.13 (Chapter 6) for the RCS frame. This is also clear from the results of
the static pushover analysis presented in Figure 7.26.
7.10 Global Failure Analysis of the 6-Story STEEL Frame
The gravity load stability index, λu, defining a failure criterion for each earthquake record
is calculated with values ranging from λuo = 5.76 for the undamaged 6-story steel frame
to λu = 1.0 for conditions at incipient collapse.
338
5
Sa(T 1,5%)
4
3
2
Story 1
Story 2
Story 3
Story 4
Story 5
Story 6
1
0
0.00
0.02
0.04
0.06
0.08
0.10
0.12
IDRmax
Figure 7.30 Story IDACs for the 6-story steel frame under the Cape Mendocino (1992)
record at Rio Del Overpass station - general record.
3.0
2.5
Sa(T1 ,5%)
2.0
1.5
1.0
Story
Story
Story
Story
Story
Story
0.5
0.0
0.00
0.02
0.04
0.06
0.08
0.10
1
2
3
4
5
6
0.12
IDRmax
Figure 7.31 Story IDACs for the 6-story steel frame under the Erzincan (1992) record
in Turkey - near-fault record.
339
7.10.1 Relationship between Spectral Acceleration and Global Failure Criterion, λ u
Figures 7.32 and 7.33 show the evolution of damage from λuo to λu=1.0 for the general
and near-fault records, respectively. Also shown in figures are regression lines from least
square fit performed in log space on all data points with λu < 0.95λuo for all eight records
of each bin. However, as previously done with other case study frames, if the regression
is instead performed for data points corresponding to each record alone (Equation 7.2),
one can obtain for each bin of records eight least square fit lines with eight pairs of
regression coefficients. Average values of these regression coefficients, a and ß , are
given in Table 7.14. Note the close values of these parameters compared to the ones
shown in Figures 7.32 and 7.33.
Table 7.14 Values of a and ß for the 6-story steel frame.
Parameter and Statistical
General Records
Near-Fault Records
Measure Values
a (C.O.V.)
3.61 (43%)
3.36 (44%)
-0.44 (35%)
-0.53 (25%)
ß (C.O.V.)
According to the regression parameters given in Table 7.14 and Equation 7.2, one can
relate the two performance levels identified by λu = 0.95λuo and λu =1.0. On average, the
6-story steel frame is at incipient collapse (λu =1.0) at a value of spectral acceleration
which is about 2.1 and 2.5 for general and near-fault records, respectively, times the
value causing excessive yielding and severe damage of a few critical members (i.e.,
Sa(T1 ,5%) corresponding to 0.95λuo ). Note that these ratios are much higher than for the
case of the 6-story RCS frame (about 1.8 for both general and near-fault records). This
finding can be explained by the fact that the 6-story steel frame has much higher lateral
overstrength as designed.
340
6
5
Sa (T 1=1.26sec, ξ=5%)
Miyagi
Valparaiso
LP89-HCA
-0.38
Sa = 3.63 λu
σlnS |λ = 0.373
a u
LP89-HSP
LP89-WAHO
CM92-RIO
LA92-YER
Mendocino
Lin. Regression
4
3
2
1
λu = 1.0
(collapse)
0
0
1
2
3
4
5
λu (based on 1.0D+0.25L)
6
λuo
Figure 7.32 Spectral acceleration-λu relationship for bin of general records.
6
5
Sa(T 1=1.26sec,ξ=5%)
IV79-A6
LP89-LG
LP89-LX
-0.46
Sa = 3.43 λ u
σlnS |λ = 0.408
a u
EZ92-EZ
NR94-NH
NR94-RS
NR94-SY
KB95-JM
Lin. Regression
4
3
2
1
λ u = 1.0
(collapse)
0
0
1
2
3
4
λu (based on 1.0D+0.25L)
5
6
λuo
Figure 7.33 Spectral acceleration-λu relationship for bin of near-fault records.
341
On the other hand, relating performance to hazard, the ratio Sa(λu=1.0)/S a(2%in50) for
that frame is on average 4.5 (C.O.V. = 36%) and 4.1 (C.O.V. = 37%) for general and
near-fault records, respectively. That means the 6-story steel frame is at incipient collapse
at a spectral acceleration at its fundamental period of about 4.5 and 4.1 times the spectral
acceleration associated with 2%in50years hazard for general and near-fault records,
respectively. These two ratios are 3.6 and 3.4 respectively for the 6-story RCS frame
which reinforces what has been mentioned earlier about relative lateral overstrength of
the two frames. These considerably high values can be also justified as before (see
Section 6.4.2 for the 6-story RCS frame and Section 7.4.1 for the 12-story RCS frame).
Note the difference in the ratio Sa(λu=1.0)/S a(2%in50) between general and near-fault
records (4.5 and 4.1) for the steel frame which is consistent with results of the RCS
frame. The ratio takes a smaller value for near-fault records due to their impulsive effect
and consequently their expected more severe damaging potential. The pulse effect on the
behavior is more pronounced and further clarified by looking at the performance of the
structure under the effect of near-fault records based on the ratio between the pulse period
of the record and the fundamental period of the frame, Tp /T1 , as mentioned before. The
eight records of the near-fault bin can thus be divided into two subsets as previously done
for the 6-story RCS frame in Chapter 6. Subset 1 includes four records (IV79-A6, LP89LG, EZ92-EZ, and NR94-SY) with Tp /T1 >> 1.0 with more damaging effect. Subset 2
includes the remaining four records with milder and less damaging effect on the structure
due to values of Tp /T1 in the vicinity of 1.0. Adopting this classification, regression
analysis is applied again but for each subset alone, and average values of the regression
parameters
a
and
ß
are
given
in
Table
7.15.
Accordingly,
the
ratio
Sa(λu=1.0)/S a(2%in50) for subset 1 equals 2.8 which is much less than the value of 4.5
for the bin of general records. On the other hand, Sa(λu=1.0)/S a(2%in50) = 5.4 for subset
2 which is even much larger than that corresponding to general records.
342
Table 7.15 Average a and ß values for near-fault records.
Parameter and Statistical
Near-Fault Records
Near-Fault Records
Measure Values
Subset 1
Subset 2
a (C.O.V.)
2.37 (31%)
4.78 (16%)
-0.49 (22%)
-0.57 (28%)
ß (C.O.V.)
Given the rather large variability in spectral values at the λu=1.0 performance level, as
reflected by the C.O.V. of Sa(λu=1.0) values, one may consider relating performance
level to seismic hazard by comparing mean minus standard deviation values – rather than
mean values. Accordingly, for the most critical case of the STEEL frame under near-fault
motions (i.e., subset 1), performance/hazard ratio reduces to Sa(λu=1.0)/S a(2%in50) = 2.0,
and ratio for the general records reduces to 2.9. These ratios still exceed unity, indicating
that the frame would exceed the desired performance at the 2%in50year hazard level
expected by codes.
7.10.2 Relationship between IDRmax and Global Failure Criterion, λ u
Figures 7.34 and 7.35 show IDRmax-λu data for general and near-fault records,
respectively. The figures also present linear regression fits performed in log space using a
power law format conditioned on λu and excluding points with a value of λu > 0.95λuo .
Results reveal that the correlation between IDRmax and λu is quite good as manifested by
a narrow band of curves throughout the damage evolution from λuo up to collapse limit
state with a conditional dispersion σ ln IDR
max |λ u
= 0.193 and 0.170 for general and near-fault
records, respectively.
At λu=1.0, average value of IDRmax is 0.082 with C.O.V. of 12.4% for bin of general
records, while it is 0.103 with C.O.V. of 15.0% for bin of near-fault records. Note that the
observed higher value of mean IDRmax at λu=1.0 for near-fault records than for general
records is due to the pulse effects characterizing the former events along with the longer
duration of the general records causing accumulation of damage and collapse at lower
levels of inelastic deformation. Moreover, comparing these IDRmax values to the ones
343
associated with the 6-story RCS frame (average values of IDRmax are 0.087 and 0.116 for
general and near-fault records, respectively) reveals fairly close values.
Furthermore, a similar observation to that made for the 6-story RCS frame is obvious for
the steel frame under general records. All values of IDRmax are clustered within a narrow
band except for values associated with one record, Valparaiso, with much less IDRmax at
λu values associated with high hazard levels. This record with the longest duration of
strong motion among all records considered, tSM = 38 seconds, will accumulate more
damage with less peak values of the response parameters. Thus, collapse is mainly due to
accumulation of damage more than to peak single response type of behavior.
On the other hand, at the performance level defined at λu = 0.95λuo (i.e., excessive
yielding), average transient IDRmax is 0.035 (C.O.V.=8.4%) and 0.033 (C.O.V.=10.9%)
for general and near-fault records, respectively. Note the nearly similar values at this
damage stage associated with both types of records while the larger difference at failure
as shown in the previous paragraph due to the pronounced effect of the pulse at such a
high intensity level causing global collapse of the structure.
Another useful observation is that average values of IDRmax corresponding to λu =
0.95λuo reported herein for the 6-story steel frame are very close to those for the 6-story
RCS frame (0.032 and 0.033 for general and near-fault records, respectively) and those
for the 12-story RCS frame (0.033 and 0.034 for general and near-fault records,
respectively). As previously mentioned, these average values are fairly close to the 0.025
indicative drift value proposed by FEMA 273 for steel moment frames at the Life Safety
performance level. Therefore, a performance level defined by λu = 0.95λuo may be chosen
as a reasonable and stable performance level that might be further related to the Life
Safety performance level proposed by FEMA 273.
344
0.10
IDRmax = 0.079 λu -0.34
σlnIDR
IDRmax
0.08
= 0.193
max|λu
0.06
Miyagi
Valparaiso
LP89-HCA
LP89-HSP
LP89-WAHO
CM92-RIO
LA92-YER
Mendocino
Lin. Regress.
0.04
0.02
λu = 1.0
(collapse)
2
Ra = 0.68
0.00
0
1
2
3
4
5
6
λu (based on 1.0D+0.25L)
Figure 7.34 IDR max-λu relationship for bin of general records.
0.14
IDR max = 0.102 λu-0.52
0.12
IDR max
EZ92-EZ
NR94-NH
NR94-RS
NR94-SY
KB95-JM
Lin. Regress.
σlnIDR
= 0.170
max |λ u
0.10
IV79-A6
LP89-LG
LP89-LX
0.08
0.06
0.04
λu = 1.0
(collapse)
0.02
Ra 2 = 0.84
0.00
0
1
2
3
4
5
λu (based on 1.0D+0.25L)
Figure 7.35 IDRmax-λu relationship for bin of near-fault records.
345
6
7.10.3 Spatial Distribution of Damage
The cumulative damage index, Dθ, distribution throughout the frame is presented in this
section for the Mendocino (general) and Erzincan (near-fault) ground motions. The
distribution is given at the two limit states, λu = 0.95λuo and λu = 1.0.
Figure 7.36 shows values of Dθ at various sections of the 6-story steel frame due to the
Mendocino record. At λu = 0.95λuo , severe damage (and failure) of various end sections
of beams takes place throughout the first four stories. Failure at a given section is shown
in the figure by a gray fill. It is noticeable that at this performance level, no (or minor,
i.e., Dθ<0.3) damage has been observed at any column section including the ground floor
columns bases. However, moderate (0.3<Dθ<0.6) damage is seen at almost all inner joint
panels. On the other hand, at collapse limit state, i.e., λu = 1.0, failure of more than half of
the end sections of all beams takes place along with severe damage and failure of many
inner joint panels. Failure of the ground floor columns bases is observed with no damage
at any other columns sections. Comparing Fig. 7.36 (STEEL frame) to Fig. 6.21 (RCS
frame) reveals more spread of damage for the steel frame than for the RCS one,
especially at the performance level defined by λu = 0.95λuo . Furthermore, damage of joint
panel regions is a characteristic of the behavior of the 6-story steel frame even at lower
overall damage levels.
Figure 7.37 shows the damage distribution due to the Erzincan record, as an example of a
damaging near-fault ground motion (subset 1, Section 7.10.1). Again, comparing damage
pattern shown herein with that of the 6-story RCS frame given in Figure 6.24 under the
same record reveals that the severity and spread of the damage are more accentuated for
the steel frame at both performance levels (λu = 0.95λuo and λu = 1.0). A key aspect of the
damage seen by the steel frame is the noticeably severe damage of its inner joint zones
and the nearly no (or minor) damage of its columns except for the ground floor columns
bases.
346
0.38
0.73
0.40
0.32 0.45
0.48 0.33
0.45 0.48
0.44 0.56 0.43
0.51 0.49
0.50 0.54 0.45
0.36 0.47
0.43 0.42 0.32
0.33
0.71
0.74
0.33
(a) λ u = 0.95 λ uo
0.61
0.82
0.82
0.63
0.44 0.91 0.40
0.53 0.93 0.35
0.92 0.86 0.83
0.91 0.93
0.81
0.77
0.78
0.69
0.85
0.53
(b) λ u = 1.0
Figure 7.36 Distribution of Dθ at different λu values – Mendocino (1992) record.
347
0.37
0.32
0.32
0.63
0.44 0.32
0.44 0.32
0.37
0.83
0.48 0.36
0.32 0.47 0.39
0.47
0.55
0.40 0.35
0.33 0.39 0.34
0.39
(a) λ u = 0.95 λ uo
0.37
0.37
0.62
0.55
0.41 0.79 0.62
0.39 0.76 0.60
0.55
0.58
0.93
0.79
0.85
0.80
0.91
0.88
(b) λ u = 1.0
Figure 7.37 Distribution of Dθ at different λu values – Erzincan (1992) record.
348
7.11 Global versus Local Response
As previously applied to studied RCS frames, local response in terms of maximum plastic
rotations of beams and columns are related within this section to global response in terms
of IDRp,max and ∆IDRmax, respectively, for the 6-story steel frame. The plastic rotation of
different structural components at every floor level is again associated with the
corresponding relevant drift quantity based on the deformed configuration of the frame as
presented in Section 6.2.1, Chapter 6.
The value of IDRmax has an elastic component and a plastic component. One way of
determining the elastic component of IDR associated with any target demand level is by
scaling up IDR values corresponding to the elastic level of behavior by the ratio of the
base shear at this target demand level and the base shear corresponding to that elastic
level. For levels of demand higher than 0.95λuo , all case study frames in this thesis are
above their ultimate lateral capacity (i.e., ultimate base shear). Accordingly, at these
levels of demand, the elastic component of IDR can be easily calculated as
IDRelastic = IDRd (Vu/Vd)
(7. 8)
where Vu is the ultimate base shear, Vd is the design base shear or any lateral load level
associated with fully elastic behavior of the frame, and IDRd is the interstory drift
corresponding to Vd. IDRelastic values are therefore structure-dependent. The values are
also different from story to story but with no dramatic changes between floors for frames
with no stiffness irregularities. It has been found that on average, IDRelastic up the height
of the RCS frames (corresponding to high level of demands above ultimate base shear
level) is about 0.01. This value has been used to calculate IDRplastic, which is correlated to
beam plastic rotations in Sections 6.2.1, 6.5.2 and 7.5.2. On the other hand, for the 6story steel frame investigated in this section, an average value of IDRelastic up the height
of the frame is about 0.02, twice that of the RCS frames.
349
Similarly, one can calculate ∆IDRelastic up the height of the frame using the distribution of
IDRelastic determined as previously mentioned. ∆IDRelastic values have been found
negligible for the case study RCS frames. However, for the 6-story steel frame, the
average value is about 0.01 for the first couple of stories where there is the highest
inelastic demand in the steel columns. Accordingly, ∆IDRmax should be modified by
subtracting the elastic component (0.01) before studying its possible correlation with the
columns peak transient plastic rotation, θp,C. The stronger (reflected by high Vu/Vd ratio)
and less stiff steel columns of the steel frame are the reason why ∆IDRelastic is not
negligible in the steel frame, whereas it was negligible for the RCS frames.
In Sections 6.5.2 and 7.5.2, a least square fit of the local versus global response data has
been performed once conditioned on global response and then conditioned on local
response for the RCS case study frames. In this section, regression analysis will be
carried out only conditioned on global response (either ∆IDRp,max or IDRp,max). The
benefit of performing regression conditioned on global response is that for a given
maximum response quantity (IDR) corresponding to any level of performance, an
estimate of the median peak plastic rotation in columns and beams can be identified.
Then, this estimate can be compared to acceptance criteria and limits set within FEMA
273, or other performance standards. Thus, one can rate the performance of the structure
according to local acceptance criteria set by codes by only processing data corresponding
to global response results.
7.11.1 Relationship between ∆IDRp,max and Peak θ p,C
Figures 7.38 and 7.39 show ∆IDRp,max versus θp,C|max data for the general and near-fault
records, respectively, at λu = 1.0. From these figures, the maximum change in the plastic
IDR is not proportional to the maximum plastic rotation, θp,C, of the columns for low
values of θp,C, i.e., θp,C < 0.02 radians. However, above this minimum plastic rotation
threshold, θp,C ≈ 0.02 radians, large ∆IDRp,max values are associated with proportionally
large θp,C|max values. This non-proportionality for low θp,C values is due to the large
350
moment capacity of the columns cross-sections compared to beams, forcing all the
nonlinearity to occur in beams while the columns remain nearly elastic. But once the
demand on the columns is large enough to cause remarkable inelastic behavior, any
increase in the ∆IDRp,max value will cause a proportional increase in the columns plastic
rotation.
A power form regression fit is performed in the log space for data shown in Figures 7.38
and 7.39. It is obvious that the regression does not accurately capture the behavior due to
the non-proportionality between the values of the two response measures at low θp,C
values. This non-proportionality will be even more exaggerated at lower demand levels
(i.e., at λu > 1.0). If one is interested in reliably estimating medians of the large values of
the local response θp,C and capturing the real behavior at this high demand level of λu =
1.0 (i.e., near collapse), regression should be carried out only on the values in the range
of interest. Accordingly, the same power model is applied again but only on data points
with θp,C > 0.02 radians, mainly corresponding to the first story where serious inelastic
behavior in the bases of the columns is observed. Resulting regression lines are shown in
Figs. 7.38 and 7.39.
Shown in Figures 7.38 and 7.39 are also 1:1 lines based on the anticipated behavior and
mechanistic models shown in Figure 6.4, Chapter 6. The 1:1 relationship between local
and global response measures has also been observed, on average, for the RCS case study
frames. It is clear that the regression line fitted only to the data associated with the
ground floor columns bases, with the highest local demands, agrees to a good extent with
the 1:1 line. Values of conditional dispersion, σ, and coefficient of determination, R2 , are
also given in Figs. 7.38 and 7.39 for the two regression fits showing a good correlation
for the latter performed only on a small subset of the data. For completeness, shown in
Figs 7.38 and 7.39 are σ and R2 values computed, involving all data points, for the
assumed 1:1 “theoretical” line.
351
0.08
1.27
θ p,C = 1.29 ∆IDR p,max
σ lnθ |∆IDR
= 0.94
p,C
p,max
0.06
σ = 1.36
∆IDRp,max
2
R = 0.63
R2 = 0.23
0.87
θp,C = 0.66 ∆IDR p,max
σlnθ |∆IDR
= 0.14
p,C
p,max
0.04
R2 = 0.83
0.02
Values from Analysis
Regression on all data points
Regression - Ground Floor Cols
0.00
0.00
0.02
0.04
0.06
0.08
Max. Transient Col. Plastic Rot., θp,C [rad.]
Figure 7.38 ∆IDRp,max - θp,C relationship for general records at λ u = 1.0.
0.12
0.10
1.06
θp,C = 0.56 ∆IDRp,max
σlnθ |∆IDR
= 0.86
p,C
p,max
R2 = 0.65
σ = 1.19
R2 = 0.34
∆IDRp,max
0.08
1.10
θp,C = 1.15 ∆IDR p,max
0.06
σlnθ |∆IDR
= 0.11
p,C
p,max
2
R = 0.96
0.04
Values from Analysis
Regression on all data points
Regression - Ground Floor Cols.
0.02
0.00
0.00
0.02
0.04
0.06
0.08
0.10
0.12
Max. Transient Col. Plastic Rot., θp,C [rad.]
Figure 7.39 ∆IDRp,max - θp,C relationship for near-fault records at λu = 1.0.
352
7.11.2 Relationship between IDRp,max and Peak θ p,B
Data points showing the relationship between IDRp,max and θp,B|max are plotted in Figures
7.40 and 7.41 for general and near-fault records, respectively, at collapse state (λu = 1.0).
Least square fit relationships using a power model and performed in log space
conditioned on IDRp,max are also given in the figures. The large scatter of data points
shown in Figs. 7.40 and 7.41 along with the considerably large conditional dispersion, σ,
and low R2 values reported there reveal that there is no reliable correlation between the
two response parameters. For instance, in Figure 7.40 for general records, at a given
IDRp,max of about 0.03, corresponding maximum beam plastic rotations at a given floor
under a given record range from about 0.04 to 0.11 radians. Before making any
conclusion, one should first investigate the reason behind that large scatter which takes
place for both types of records but is worse under general records. Some scatter was also
observed in the IDRp,max versus θp,B|max relationship for the RCS case study frames (Figs.
6.30 and 6.31 for 6-story RCS and Figs. 7.18 and 7.19 for 12-story RCS) but it is not as
accentuated as that associated with the 6-story steel frame shown in Figs. 7.40 and 7.41.
7.11.3 Explanation of Large Dispersion in Beams Plastic Rotation θ p,B Values
Unlike a steel beam, response of the composite beam is different under negative and
positive bending. As a result, under inelastic cyclic loading, the composite beam
deformation keeps always shifting towards its less stiff and weaker negative direction
leading to an unsymmetrical load-deformation curve. This drifting of the deformation is
especially accentuated if the first loading cycle has already pushed the composite beam
far into negative deformation. Then, under the reverse cycle with the same load
magnitude in a load-controlled test, the beam will reach a smaller deformation in the
opposite (i.e., positive) direction than the negative deformation attained in its first cycle.
This is not the case for a conventional structural steel beam with same flexural stiffness
and strength properties in both directions. Under a cyclic loading with same force
magnitude, the load-deformation curve for this regular steel beam will be always
symmetric.
353
0.12
0.95
θp,B = 0.83 IDRp,max
σlnθ |IDR
= 0.578
p,B
p,max
2
R = 0.259
0.10
IDRp,max
0.08
0.06
0.04
0.02
Values from Analysis
Regression given IDR
0.00
0.00
0.02
0.04
0.06
0.08
0.10
0.12
Max. Transient Beam Plastic Rot., θp,B [rad.]
Figure 7.40 IDRp,max - θp,B relationship for general records at λu = 1.0.
0.14
θp,B = 0.87 IDRp,max
0.12
σlnθ |IDR
= 0.537
p,B
p,max
R2 = 0.600
0.10
IDRp,max
0.99
0.08
0.06
0.04
Values from Analysis
Regression given IDR
0.02
0.00
0.00
0.02
0.04
0.06
0.08
0.10
0.12
0.14
Max. Transient Beam Plastic Rot., θp,B [rad.]
Figure 7.41 IDRp,max - θp,B relationship for near-fault records at λu = 1.0.
354
A set of data points from Figure 7.40 with a large dispersion of θp,B values at a given
IDRp,max has been identified. This set of points corresponds to θp,B values at the ends of
the first floor beams ranging from 0.03 to 0.11 radians resulting from the time history
analysis of the steel frame under the LP89-WAHO record at λu = 1.0. The corresponding
IDRp,max is around 0.028 as may be observed in Figure 7.40. This value converts to a total
IDR of 0.048 for the first story. Detailed results from the time history analysis are shown
in Figure 7.42 for the first floor beams. Among these results are: IDR time history for the
first story, plastic rotation time history for all end sections of the first floor beams, and
moment-plastic rotation curves at a few end sections pertaining to outer (composite) and
inner (steel) beams.
The large dispersion of θp,B|max values is obvious from Figure 7.42 with all composite
beams end sections (CB1 to CB4) drifting towards negative deformation causing
permanent (i.e., residual) negative plastic rotation. On the other hand, mid span steel
beam end sections did not show that shift in the deformation. Accordingly, maximum
values of θp,B are much different for composite and steel beams. Furthermore, as
previously explained, even composite beams will reach considerably different maximum
plastic rotation values depending on the direction of their first few loading cycles in the
inelastic range.
Modeling the joint panel flexibility is an equally important factor as the composite beam
action contributing to the large scatter in the maximum beam plastic rotations at a given
floor level of the frame. Among other factors causing the large dispersion of θp,B|max
values are (a) the unequal spans of the frame with longer outer spans imposing more
demands, in terms of plastic rotation, at the outer ends of these beams with no continuity,
and (b) the fact that we are investigating in Figure 7.40 the plastic rotation demands at
considerably large drifts of the building (at incipient collapse, i.e., at λu = 1.0). The
dispersion of θp,B|max values is less accentuated if one looks at the performance at lower
damage levels (i.e., at smaller drift demands) corresponding to higher values of λu.
355
0.06
0.04
IDR
0.02
0.00
-0.02
CB1
CB2
SB1 SB2
CB3
CB4
Story 1
-0.04
Composite
Beam
Steel
Beam
Composite
Beam
-0.06
0
Frame Elevation
5
10
15
20
25
30
35
40
Time [sec.]
15000
0.02
SB2
0.00
SB1
Moment [kips.in]
Plastic Rotation [rads.]
0.04
-0.02
CB2
CB3
-0.04
-0.06
CB4
-0.08
-0.10
-0.12
5
10
15
20
25
30
5000
0
-5000
-10000
CB1
0
35
-15000
-0.12
40
15000
15000
SB1
10000
10000
5000
0
-5000
-10000
-0.08
-0.04
-0.08
-0.04
0.00
0.04
Plastic Rotation [rads]
Moment [kips.in]
Moment [kips.in]
Time [sec.]
-15000
-0.12
CB1
10000
0.00
0.04
Plastic Rotation [rads]
CB3
5000
0
-5000
-10000
-15000
-0.12
-0.08
-0.04
0.00
0.04
Plastic Rotation [rads]
Figure 7.42 Results from time history analysis under LP89-WAHO at λu =1.0.
Finally, as previously mentioned, some scatter has been observed in the IDRp,max versus
θp,B|max relationship for the RCS case study frames but it is much less than for the 6-story
steel frame. This may be directly attributed to the stronger joint panels and weaker
356
columns in the composite RCS frames when compared to the steel frame. The much less
damage observed in the columns of the steel frame along with the more extensive joint
damage than for the RCS frame further reinforces the argument stated above. For more
details, refer to Section 6.4.4 and Figs. 6.21 and 6.24 for the 6-story RCS frame and
Section 7.10.3 and Figs. 7.36 and 7.37 for the steel frame.
7.12 Response Dependency on Ground Motion Parameters
Data given in Sections 7.9 and 7.10 relate global response measures (IDRmax and λu) to
the single intensity parameter, Sa(T1 ), for the 6-story steel frame. Shown in Tables 7.16
and 7.17 are results of a regression analysis of IDRmax and λu conditioned on Sa(T1 ) for
general and near-fault records using a power law model. The statistical indices
(conditional dispersion, σ, and adjusted coefficient of determination, R 2a ) reported in
Table 7.16 for the regression of IDRmax conditioned on Sa(T1 ) show good correlation,
where Sa(T1 ) is able to capture most of the variability in the response. However, values of
σ of about 0.6 and R 2a of about 0.4 given in Table 7.17 for the regression of λu
conditioned on Sa(T1 ) suggest that there is room for improvement of the correlation.
Accordingly, additional input parameters will be tried, as has been done in Section 6.6
(for the 6-story RCS frame) and Section 7.6 (for the 12-story RCS frame), to check for
any additional reduction in the dispersion of the global response measure λu. The effect
of these input parameters will be also checked for Sa-IDRmax relationship looking for any
extra reduction in the dispersion of IDRmax conditioned on the input.
Same candidate input parameters used for other frames will be tried herein for the steel
frame. These parameters include (1) the ratio R S a- T which reflects the shape of the
F
ground acceleration response spectrum in the vicinity of T1 due to earthquake induced
damage; (2) the strong motion duration, tSM; and (3) the pulse period, Tp , for near-fault
records. The same functional form given by Equations 6.13 and 7.3 relating IDRmax or λu
to the various input parameters is used herein. The period TF used in the calculation of
357
R S a- T
has been determined as 1.55 seconds and 1.89 seconds according to target
F
displacement and equal displacement methods, respectively. The value of TF providing a
ratio R S a- T
that shows the best correlation (i.e., lower conditional dispersion of the
F
response) is the one used and reported in Tables 7.16 and 7.17.
Table 7.16 Regression results for IDRmax conditioned on various input parameters.
General Records
Near-Fault Records
IDR max = 0.029S 0a.69
IDR max = 0.034S0a.73
σ ln IDR max |Sa = 0.233
σ ln IDR max |Sa = 0.302
R a2 = 0.835
IDR max = 0.038S0a. 76R 0S.36
1
a -TF
R a2 = 0.746
IDR max = 0.047S0a.88 R0S.83
1
a-TF
σ ln IDR max |S a , R S = 0.160
1
a-TF
σ ln IDR max |S a , R S = 0.182
1
a-TF
1
1
1
1
R 2a = 0.922
R a2 = 0.908
(TD)
(ED)
−0.13
IDR max = 0.058S0a. 78R 0S.84 t SM
IDR max = 0.045S0a.88R 0S.81e 0.01t SM
σ ln IDR max |Sa , R S , tSM = 0.155
1
a-TF
σ ln IDR max |Sa , R S , tSM = 0.183
1
a-TF
1
R 2a = 0.927
a -TF
1
a-TF
R a2 = 0.907
(TD)
(ED)
0.06T p
IDR max = 0.039S0a.88R 0S.69 e
N/A
1
a-TF
σ ln IDR max |S a , R S ,Tp = 0.181
1
a-TF
R a2 = 0.909
TD = TF based on Target Displacement.
(ED)
ED = TF based on Equal Displacement.
Tables 7.16 and 7.17 show that relating R S a - T
and Sa(T1 ) to the response quantities
F
causes the decrease in conditional dispersion, σ, and increase in R 2a indicating an
improved correlation with the data. The net effect is that uncertainty in the estimation of
median IDRmax with a limited sample size of eight records is cut down from 8%
(=0.233/√8) to 6% (=0.160/√8) and from 11% (=0.302/√8) to 6% (=0.182/√8) for general
and near-fault records, respectively. Similarly, uncertainty in the estimation of median λu
358
reduces from 23% to 18% and from 20% to 13% for general and near-fault records,
respectively.
Table 7.17 Regression results for λu conditioned on various input parameters.
General Records
Near-Fault Records
λu = 6.81Sa−1.09
λu = 4.94S−a 0.88
σ ln λ u |Sa = 0.636
σ ln λ u |Sa = 0.566
1
1
1
1
R a2 = 0.413
4.81S−a 1.60 R −S 1. 94
1
a -TF
R a2 = 0.406
λu = 3.39Sa−1.43R −S 1. 67
1
a -TF
σ ln λu |Sa , R S = 0.510
1
a-TF
σ ln λ u |Sa , R S = 0.366
1
a-TF
λu =
R 2a = 0.622
R a2 = 0.752
(TD)
(ED)
0. 16
λu = 2.94S−a 1.60 R −S2.05 t SM
λu = 5.13S−a 1. 45R −S 1. 52e -0.05t SM
σ ln λ u |Sa , R S , tSM = 0.512
1
a-TF
σ ln λ u |Sa , R S , t SM = 0.361
1
a- TF
1
a -TF
R 2a = 0.620
1
a -TF
R a2 = 0.758
(TD)
λu = 4.18S−a1.44 R −S1. 51e
N/A
1
(ED)
-0.07T p
a -TF
σ ln λ u |Sa , R S ,Tp = 0.368
1
a-TF
R a2 = 0.750
(ED)
Strong motion duration, tSM, and pulse period, Tp , show no or very tiny benefit in further
reducing the conditional dispersion if considered along with R S a - T and Sa(T1 ). However,
F
they show some benefit if considered with Sa(T1 ) alone, but the resulting decrease in the
conditional dispersion of the response is less than when using R S a - T and Sa(T1 ) together.
F
Care is advised when evaluating the efficiency of considering either tSM or Tp in reducing
the conditional dispersion of the response measures in this research, in part because of the
narrow range of strong motion durations and pulse period values in the bins of ground
records used for the time history analyses.
359
7.13 Summary
This chapter presents a comparative assessment study of a 12-story RCS frame and a 6story STEEL frame contrasting their seismic performance to the 6-story RCS frame with
same structural configuration investigated in Chapter 6. The main findings summarized at
the end of Chapter 6 are either confirmed or modified based on the performance of the
case study frames in this chapter. Main issues are discussed herein.
1. At a 2%in50years hazard level characterized by Sa(T1 ,5%) = 0.864g and 0.522g for
the 6- and 12-story RCS frames, respectively, the estimated median values for the
drift response, IDRmax, are quite close for both frames. For example, this median of
IDRmax is 0.025 and 0.028 for the 12- and 6-story RCS frames, respectively, under
near-fault records. Conversely, these median response estimates are not close if
calculated instead at a same Sa(T1 ), implying different hazard levels for each frame.
For instance, at a fixed Sa(T1 ) = 0.864g, median IDRmax is 0.045 and 0.028 for the 12and 6-story frames, respectively. This finding further proves the suitability of the
spectral acceleration at the fundamental period of the structure as an effective
intensity measure for earthquake records that is reliably correlated to the hazard level.
2. Identical fundamental period, T1 , for the 6-story RCS and STEEL frames has been
observed as the result of having a less stiff steel frame (same beam sizes but more
flexible columns) but accompanied with smaller seismic mass. Hence, the response of
both frames is very close when responding primarily in the elastic range (or at a mild
stage of damage), showing nearly equal median transient IDRmax values at a given
Sa(T1 ) corresponding to a 2%in50years hazard level and lower. A more or less mild
overall damage and elastic response are observed up to the 2%in50years hazard level
for both frames due to their inherent large lateral overstrength as previously reported.
However, at a higher hazard level accompanied by moderate to severe global damage,
the RCS frame, undergoing more stiffness degradation of its weaker RC columns
under cyclic earthquake loading, shows higher inelastic deformations (see Fig. 7.29)
than the steel frame, with stronger steel columns, which does not soften as much.
360
3. Ratios of average values of Sa at λu = 1.0 (near collapse) versus λu = 0.95λuo (life
safety) range from Sa(λu=1.0)/S a(λu=0.95λuo ) = 2.1 to 2.8 for the 6-story RCS frame,
from 2.0 to 2.3 for the 12-story RCS frame, and from 2.8 to 3.4 for the steel frame.
The ratios are slightly larger for the steel frame, perhaps because the steel damage
indices do not degrade as rapidly under cyclic loading as those for reinforced
concrete. These sets of ratios indicate that the hazard intensity for near collapse
(λu=1.0) is over twice that corresponding to the point when the structure begins to
significantly degrade (λu=0.95λuo ). This margin is larger than the ratio of 1.5 implied
by modern codes between the “design level” earthquake response (geared to life
safety) and the maximum considered earthquake (geared to near collapse).
4. IDRmax values for all case study frames subjected to various ground motions are
remarkably consistent. At λu = 0.95λuo average IDRmax ranges between 3.2% to 3.5%,
and there are no perceptible differences between drifts for the different ground motion
bins. This range of 3.2% to 3.5% is slightly larger than the value of 2.5% suggested
by FEMA 273 for life safety for steel moment frames. At λu = 1.0, there are
consistent differences between response for the general and near-fault records, where
IDRmax ranges between 10.3% to 11.6% for the near-fault records and 8.2% to 8.7%
for the general records, for the 6-story RCS and steel frames, respectively. The
smaller IDR for the general records is probably due to their longer strong motion
duration that leads to larger cumulative damage and stiffness/strength degradation,
which in turn causes the stability limit to be reached at smaller drift ratios. For the 12story RCS frame at λu = 1.0 the same trend for both types of records is still observed
but with smaller values; average IDRmax is 7.1% and 8.8% for general and near-fault
records, respectively. This consistent decrease of the IDR value corresponding to
collapse may be attributed to lower lateral stiffness to gravity load ratio compared to
the 6-story structures and to higher mode effects that may trigger failure at slightly
lower global deformation levels.
361
5. Dis-aggregation of the results based on Tp /T1 ratio is greatly advised in order to
predict reliable performance statistics under near-fault ground motions. For the 6story RCS and steel frames, it has been observed that the more damaging near-fault
records have a pulse period, Tp , that is larger than the natural period T1 of the
structure. The reason for this behavior is that when Tp /T1 > 1.0, the structure softens
into the more damaging pulse effect of the records. Other near-fault ground motions
with less damaging effect have a ratio Tp /T1 in the vicinity of 1.0. For the 12-story
RCS frame with a longer fundamental period T1 , it has been observed that near-fault
records with Tp /T1 ratio in the vicinity of 1.0 are also less damaging to the structure.
However, having the structure with a fundamental period far away on either side of
the pulse period of the record (and not only smaller than Tp ) will increase its
vulnerability under this near-fault record.
6. Large lateral overstrength values and high Sa(λu=1.0) values are observed for the 12story RCS frame and the 6-story steel frame. Lateral overstrength is about Ω = 4.4
and Ω = 6.0 for the RCS and steel frames, respectively. Large Sa(λu=1.0)/S a(2%in50)
ratios are 2.3 and 3.1 for the 12-story RCS frame and 4.5 and 4.1 for the 6-story steel
frame under general and near-fault records, respectively. Justification for these
considerably high values is as given in details in the summary of Chapter 6.
7. The spatial distribution of damage throughout the frames reveals that the damage is of
the peak response type (due to the pulse effect) under near-fault records, whereas it is
of the cumulative type under general records with consistently longer strong motion
durations and consequently more accumulation of damage. Furthermore, it has been
observed that the damage of the 12-story RCS frame under general records is
distributed among upper and lower stories, thus suggesting higher mode behavior. On
the other hand, the damage due to near-fault ground motions is almost always
confined to lower stories. Another interesting observation is the less spread of
damage in the 6-story RCS frame as compared to the 6-story STEEL frame,
especially at λu = 0.95λuo performance level. A key aspect of the damage undergone
by the steel frame is the noticeably severe damage of its inner joint zones and the
362
nearly no (or minor) damage of its columns, except for the ground floor columns
bases. The RCS frame has much less damage of its composite joints but larger
damage at columns sections up the height of the frame as presented in Chapter 6.
8. Comparing the performance of the 6- and 12-story RCS frames under different types
of ground motions and at various levels of damage, it has been found that local (θp,B
for beams and θp,C for columns) versus global (IDRp,max for beams and ∆IDRmax for
columns) response relationship is on average quite stable irrespective of the type of
record (i.e., general versus near-fault) and the level of the overall damage as given in
terms of λu. This relationship can be satisfactorily approximated by a 1:1 line for
practical purposes. Moreover, applying a conditional regression analysis of local
response given both global response and the intensity of the ni put (in terms of spectral
acceleration), one may look at the relationship between local and global response at
different hazard levels. Such relationship still shows the practically stable (on
average) correlation between local and global response in terms of the parameters
introduced herein for the two case study RCS frames. A finding that has to be further
investigated for other RCS frames with different geometry, amount of overstrength,
etc.
9. For the 6-story steel frame with relatively strong columns, low values of the plastic
rotation in columns, θp,C, observed throughout the frame are associated with high
values of ∆IDRp,max. However, at columns locations where considerable inelasticity
takes place (such as at the ground floor columns bases), large θp,C values are
associated with proportionally large ∆IDRp,max values. From analysis data under both
types of records (general versus near-fault), θp,C ≈ 0.02 radians can be considered as
the minimum plastic rotation threshold beyond which the local-global correlation is
good. The predominant non-proportionality between θp,C|max and ∆IDRp,max, especially
at the upper stories for all hazard levels up to collapse limit state, is due to the large
moment capacity of the columns cross-sections compared to beams, forcing all the
nonlinearity to occur in beams while the columns remain nearly elastic even at high
∆IDRp,max values. But once the demand on the columns is large enough to cause
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inelastic behavior, any increase in the ∆IDRp,max value affecting a given story will
cause a proportional increase in θp,C of the columns of that story.
10. Another aspect of the response of the case study steel frame is the large dispersion of
the beams plastic rotations, θp,B, at a given interstory drift ratio, IDRp,max. This large
dispersion renders any relationship relating such local and global measures
ineffective. It is mainly attributed to the effect of using composite beams with
different flexural stiffness and strength properties in positive versus negative bending
directions along with considering the joint panels flexibility in the analysis. The
composite beam action causes drifting of the deformation towards negative side with
considerably different amounts depending on the magnitude and direction of the first
few inelastic loading cycles. Among other factors of this large dispersion of the
beams plastic rotations are (1) the use of regular steel beams along with the composite
beams at every floor level, (2) the unequal spans of the frame, and (3) the high levels
of demand we are investigating (i.e., low values of λu). This large scatter is more
pronounced for the steel frame than for the RCS frames because of its weaker joints
and stronger columns.
11. The results presented in this chapter confirms the finding in Chapter 6 concerning the
efficiency of considering R Sa -T
(representing the effect of period lengthening on the
F
spectral acceleration values due to structural damage) beside Sa(T1 ,ξ=5%) in further
reducing the record-to-record dispersion of the response due to both general and nearfault records, for all case study frames. Moreover, considering R Sa -1,2 (representing
higher mode effects) along with the former two input parameters will further reduce
the conditional dispersion of the response for general records for the 12-story RCS
frame. While considering the pulse period, Tp , along with Sa(T1 ,ξ=5%) and R Sa -T is
F
best for reducing the uncertainty in the estimation of the median response due to
limited sample size for near-fault records.
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Chapter 8
Conclusions and Recommendations
The main objective of this research is to achieve broader acceptance of composite RCS
moment frames in high seismic regions by demonstrating their reliability through a
modern performance-based methodology. The aim is to help improve our current
understanding of the seismic performance of structures under multi-level earthquake
hazards and our ability to perform accurate and reliable advanced inelastic static and
dynamic analyses to predict such response. This is an important step towards improving
current earthquake engineering design practice in light of performance-based design and
evaluation framework envisioned in new seismic codes and standards.
In this chapter, a summary of the work done throughout this research highlighting the
main contributions is presented, followed by conclusions and recommendations.
Suggestions for future research are also included.
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8.1 Summary
Analytical Models: A detailed review of the computer program DYNAMIX for
DYNamic
Analysis
of
MIXed
(steel-concrete)
structures,
its
nonlinear
analysis
capabilities, and the theory behind it are presented in Chapter 2. Improvements to
DYNAMIX made as part of this research are summarized in Chapters 2 and 3. Briefly,
these improvements include:
•
Development and implementation of a new composite beam model following a
flexibility formulation that tracks inelastic moment-curvature cross section response
along the member as a function of spread-of-plasticity using a one-dimensional
idealization of the bounding surface model in force space. The model includes
kinematic hardening and stiffness degradation under cyclic loading as a function of
the accumulated plastic energy in the member. The element aims to capture the
overall behavior of a composite beam, particularly differences in the member’s
stiffness and strength under positive versus negative bending, while maintaining
computational efficiency.
•
Implementation of a routine to permit monitoring plastic rotations for beam-columns
by integrating generalized strains (e.g., curvatures) along the element length. A great
benefit is that plastic rotations, as opposed to curvatures, are more commonly cited as
a basic behavioral index in experimental tests and in seismic design/evaluation
standards (e.g., FEMA 273). They are also less sensitive to numerical analysis
parameters and convergence criteria and tolerances.
•
Development of a formula to determine effective initial flexure stiffness of reinforced
concrete beam-columns, taking into account modest degrees of cracking, amount of
reinforcement, and level of axial load in the member.
Damage Indices: Aside from considering fairly standard indices such as interstory drift
and maximum hinge/joint rotations, two local damage indices are proposed in Chapter 4
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to track both peak and cumulative effects in nonlinear time history analyses. The first is
based on a damage index suggested by Kratzig et al. (1989) and is described in terms of
accumulated dissipated energy. The second index is similar except it deals only with
inelastic deformations and tracks peak and cumulative ductility demand. Among the
advantages of these indices are that they (1) reflect the “temporal” sequence of loading
cycles, (2) account for cumulative type of damage, and (3) can easily deal with structural
components exhibiting un-symmetrical behavior (e.g., composite beams). The two
indices are calibrated and tested using several sets of experimental test results including
five reinforced concrete columns specimens, five steel and composite beams, and twelve
composite RCS joint sub-assemblages tested under quasi-static cyclic loading schemes.
Results obtained and statistical measures show that the proposed indices are promising
measures of damage and failure under seismic type of loading. An attempt is made to
further correlate the observable damage to the value of damage index as much as possible
through the available information reported from experiments. This correlation is useful in
terms of the index’s usefulness for the performance-based design/evaluation procedures
adopted in new seismic codes which classify the status of the structure according to the
consequences of its level of damage, e.g., the immediate occupancy, life safety, near
collapse levels specified in FEMA 273.
Case Study Buildings: Chapter 5 explains the design procedure for the case study
buildings investigated in this research. A brief outline of seismic design methods and
criteria proposed by recent seismic codes is first presented. Descriptions of the 6- and 12story RCS-framed buildings and 6-story steel-framed building are given including the
controlling design criteria and member sizes. These case study buildings follow the
general layout of the theme structure proposed as part of the US-Japan program on hybrid
structures, Phase 5. Finally, selection of records for the time history analyses of the
proposed designs is discussed. The selected ground motions fall into two bins,
distinguished between general records and near-fault records with forward directivity.
Each bin consists of eight records adjusted by Somerville (1997) to reflect conditions for
stiff soil sites (site class D as per IBC 2000).
367
Seismic performance of the case study buildings is assessed through nonlinear static and
time history analyses using the two sets of acceleration records and detailed results are
provided in Chapters 6 and 7. For multi-hazard analyses, it is assumed that the
acceleration component of the records can be linearly scaled based on the spectral
acceleration computed at the fundamental period of the structure, Sa(T1 ). Shome and
Cornell (1999) have demonstrated that, compared to other approaches, scaling based on
Sa(T1 ) will reduce the record-to-record dispersion in the response data and will not bias
the results. The spectral accelerations of the scaled earthquake records Sa(T1 ) are then
related to the maximum interstory drift ratio, IDRmax, from corresponding time-history
analyses creating for each record what is referred to as Incremental Dynamic Analysis
(IDA) curve.
Collapse Analyses: Owing to the limitations of the time history analyses to fully capture
the actual strength and stiffness degradation in the structure, the Incremental Dynamic
Analysis (IDA) results do not in themselves provide a definitive means of establishing a
stability (or near-collapse) limit of the structure. This is evident from the fact that some
of the response (IDA) curves shown in Chapters 6 and 7 continue to have a positive slope
at very large Sa and IDRmax. While the ductility damage index introduced earlier can
characterize localized conditions, one still needs a means of integrating the local damage
indices to understand their effect on overall structural stability. To address the question of
global stability, a multi-step procedure is developed to post process the time history
analysis with a gravity load stability analysis that accounts for the distribution of damage
that develops during each time history analysis. This procedure entails the following
steps:
(1) Perform a nonlinear time-history analysis and calculate the cumulative damage
indices. This provides the basis to quantify the localized (distributed) damage
following an earthquake.
(2) Modify the analysis model based on the damage incurred during the time-history
analysis. Specifically, this involves reducing element stiffnesses and strengths as a
368
function of the cumulative damage indices and incorporating the residual (permanent)
building drift into the structural topology.
(3) Reanalyze the modified structural model through a second-order inelastic static
analysis under gravity loads up to the point of reaching an inelastic stability limit. The
resulting gravity load stability index, λu, is defined as the ratio of the vertical gravity
load capacity to the unfactored gravity loads.
The index, λu, thus serves as a global criterion that rationally integrates the destabilizing
effects of local damage and residual drifts, thus avoiding the need for more ad-hoc
averaging techniques sometimes employed to relate local indices to global response. λu
values describe the evolution of damage of a given structure from its original undamaged
state, λu=λuo , to incipient collapse at λu=1.0 under each ground record. A large initial
stability index, λuo , is expected as a function of the structure being designed for high
seismic forces. These data can then be used, for example, to relate stability performance
limits to seismic hazard (intensity) levels.
By definition, the stability criterion of λu ≤ 1.0 describes a state of collapse (or near
collapse) when the structure can no longer support its gravity load. Another limit state
that can be identified is the point at which the lateral stability begins to significantly
degrade. Here we have defined this limit at λu = 0.95λuo , which occurs where there is a
sharp transition in the stability curve relating Sa(T1 ) to λu. One might consider these two
limit states, λu = 0.95λuo and λu = 1.0, as corresponding to the “life safety” and “nearcollapse” performance levels envisioned in such documents as FEMA 273.
Correlation of Local and Global Response: Relationships between local response (in
terms of beams and columns transient maximum plastic rotations) and global response (in
terms of some maximum transient interstory drift quantity) are investigated. Whenever
column hinging takes place at bottom sections of a specific story, plastic beam rotations
should be related to IDRp,max at the same story, and plastic column rotations should be
related to the maximum absolute value of the difference between IDR at this story and
369
that at the lower one. On the other hand, if column hinging takes place at top sections of a
specific story, plastic beam rotations should be related to IDRp,max at the upper story, and
plastic column rotations should be related to the maximum absolute value of the
difference between IDR at this story and that at the upper one. Such relationships, if
effective and stable, would be viable to get reliable estimates of the median local
response measure given a specific global response in terms of IDR irrespective of the
level of damage (in terms of λu), type of record (general versus near-fault), or even the
intensity of the input (in terms of Sa(T1 )). These estimates could then be compared to
acceptance criteria and limits set within ATC 40 or FEMA 273. Thus, one could rate the
performance of the structure according to local acceptance criteria set by codes by only
processing data corresponding to global response results.
Correlation of Performance Indices and Ground Motions Parameters: Response
dependency on ground motion parameters is studied by relating global response
measures, such as interstory drift ratios IDR and the failure index λu, to various
earthquake intensity parameters such as Sa(T1 ), strong motion duration, tSM, pulse period,
Tp , for near-fault records, or the ratio RSa which reflects the shape of the ground
acceleration response spectrum in the vicinity of T1 . RSa is defined as the ratio of the
spectral acceleration at a longer period, TF, representing a decrease of lateral stiffness
due to earthquake induced damage, to the spectral acceleration at the fundamental elastic
period, T1 , i.e., RSa = Sa(TF)/Sa(T1 ). Of all the parameters investigated, RSa (in addition to
Sa(T1 )) was found to most improve the correlation between the building response and the
input ground motion. Introducing such earthquake input parameters can reduce the
standard error of estimation of the median response and, thereby, decrease the number of
nonlinear time history analyses needed to achieve a certain confidence level in estimating
such response.
8.2 Main Findings and Conclusions
The main findings and general conclusions from this work can be summarized as follows:
370
8.2.1 Large Static Lateral Overstrength
All case study frames designed according to current seismic codes possess large static
lateral overstrength, Ω. Note that Ω is defined as Vu/Vd where Vu is the ultimate base
shear under the code lateral load pattern, and Vd is the design lateral load considering
accidental torsion and based on the upper cap (1.2Ta) on the period imposed by IBC 2000
for design base shear calculation. Values of Ω are 3.9, 4.4, and 6.0 for the 6-story RCS,
12-story RCS and 6-story STEEL frames, respectively. However, as described in
Chapters 6 and 7, these overstrength values are a lower bound on the “actual” lateral
overstrength of the frames if one re-computes the design base shear without the effect of
accidental torsion or the cap on the fundamental period imposed by the code.
The main reasons for this high overstrength observed herein as compared to the assumed
upper bound value of Ω = 3 in the AISC Seismic Provisions (1997) may be attributed to:
(1) expected versus minimum specified material strengths, (2) minimum stiffness (drift)
limitations, (3) structural redundancy, (4) SCWB criterion, (5) discrete member sizing,
and (6) the use of a distributed space frame with relatively shallow members. One can
show for example, that when stiffness governs the design, the shallow beams used in
space frames will lead to higher overstrength than deeper beams commonly found in
perimeter frame systems.
8.2.2 Disaggregation of Response under Near-Fault Ground Records
Distinction of the near-fault records based on Tp /T1 ratio is greatly advised in order to
more reliably predict the performance statistics. Tp is defined as the pulse period of a
given near-fault record and it is the period corresponding to the peaks in the velocity
response spectrum, whereas T1 is the fundamental period of the structure. For the 6-story
RCS and STEEL frames, it is observed that the more damaging near-fault records have a
pulse period, Tp , that is larger than the natural period T1 = 1.25seconds of the structure.
The reason for this behavior is that when Tp /T1 > 1.0, the structure softens into the more
371
damaging pulse effect of the records. Other near-fault ground motions with less
damaging effect have a ratio Tp /T1 in the vicinity of 1.0. For the 12-story RCS frame with
a longer fundamental period T1 = 2.07seconds, it has been observed that near-fault
records with Tp /T1 ratio in the vicinity of 1.0 are still less damaging to the structure than
Tp /T1 > 1.0. However, with the structure fundamental period far away on either side of
the pulse period of a given near-fault record (and not only smaller than Tp ), the
vulnerability of the 12-story RCS frame under such record is still accentuated.
Accordingly, to minimize dispersion in the response due to systematic effects associated
with pulse periods in the near-fault records, and in order to predict reliable performance,
statistics applied to results from a series of near-fault records must be evaluated based on
Tp /T1 ratio.
8.2.3 High Collapse Limit Hazard, Sa (λ u =1.0)
By definition, λu = 1.0 defines a collapse (or near collapse) limit state. High values of
Sa(λu=1.0) at the fundamental period T1 of the structure have been observed for all case
study frames when compared to the 2% in 50 year hazard spectra geared to near collapse.
For the case study buildings, the 2% in 50 year hazard taking the soil effect at the site
into consideration is characterized by spectral acceleration of Sa(2%in50) = 0.86g and
0.52g for the 6- and 12-story frames, respectively. Table 8.1 gives a summary of the
mean and coefficient of variation of Sa(λu=1.0) for all frames under the two bins of
records investigated in this research. Distinction between results due to near-fault records
based on Tp /T1 ratio is highlighted; values are given for the two subsets of the bin of
near-fault records with subset (1) referring to the more damaging events whereas subset
(2) refers to the less damaging events. Table 8.1 also provides Sa values at λu=0.95λuo
identifying the point where there is a sharp transition in the stability curves Sa versus λu
marking the start of a quick stability deterioration up to collapse at λu=1.0. One might
consider these two limit states, λu=0.95λuo and λu=1.0, as corresponding to the “life
safety” and “near collapse” performance levels envisioned in such documents as FEMA
273.
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Table 8.1 Summary of Sa statistical values at various performance levels.
Frame
Records
λ u = 0.95λ uo
λ u = 1.0
Sa (mean)
C.O.V.
Sa (mean)
C.O.V.
General
1.45g
32%
3.09g
37%
RCS-6
NF – Subset 1
0.82g
19%
1.83g
26%
NF – Subset 2
1.44g
44%
4.05g
9%
General
1.30g
18%
3.82g
36%
STEEL-6 NF – Subset 1
0.85g
16%
2.39g
28%
NF – Subset 2
1.30g
24%
4.59g
9%
General
0.61g
27%
1.20g
16%
RCS-12
NF – Subset 1
0.75g
29%
1.29g
20%
NF – Subset 2
0.72g
2%
2.23g
1%
The main reasons for the large values of Sa(λu=1.0) for all case study frames are the
inherent large overstrength values of the frames as previously reported, and the fact of
reporting mean values ignoring the rather large variability in Sa values as shown in Table
8.1. Reporting mean minus standard deviation rather than mean values to recognize the
variability in spectral performance levels and to consider some confidence levels in the
results will lower Sa values at λu=1.0. The application of the collapse analysis in a
subsequent step to the time history analysis and the lack of its integrity with the analysis
process to automatically update step by step the effect of local damage on the overall
structural performance is another factor causing high collapse values, Sa(λu=1.0). It is
believed that integration of the collapse identification technique with the time history
analysis process through DYNAMIX would result in lower values of Sa(λu=1.0).
8.2.4 Relating λ u =0.95λ uo to λ u =1.0 Performance Levels
Ratios of average values of Sa at λu = 1.0 versus λu = 0.95λuo range from
Sa(λu=1.0)/S a(λu=0.95λuo ) = 2.1 to 2.8 for the 6-story RCS frame, from 2.0 to 2.3 for the
12-story RCS frame, and from 2.8 to 3.4 for the steel frame under the two types of
ground records. The ratios are slightly larger for the steel frame, perhaps because the steel
damage indices do not degrade as rapidly under cyclic loading as those for reinforced
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concrete. These sets of ratios indicate that the hazard intensity for near collapse (λu=1.0)
is over twice that corresponding to the point when the structure begins to significantly
degrade (λu=0.95λuo ). Presuming that the limits λu=0.95λuo and λu=1.0 do correspond to
performance levels of “life safety” and “near collapse”, this margin is larger than the ratio
of 1.5 implied by modern codes between the “design level” earthquake response (geared
to life safety) and the maximum considered earthquake (geared to near collapse).
This result shows that current seismic codes design procedures provide safe structures,
and, in the case study structures, even a bit too conservative. Accordingly, there may be
room for relaxing some of the design requirements imposed by codes such as the stiffness
(i.e., drift) criteria and furnishing some consistency between various design parameters
(e.g.,
stiffness/strength
requirements).
Furthermore,
re-evaluating
code-specified
structural response modification factors, R and Cd, is another important issue for
providing a safe but more economic structure with reliable prediction of its performance
under various hazard levels.
8.2.5 Relating Performance to Hazard Levels
While a clear consensus has yet to emerge on linking structural performance to seismic
hazard levels, documents such as the IBC 2000 and FEMA 273 suggest that buildings
should exceed near-collapse performance for a 2% in 50 year hazard and life safety
performance for the design earthquake, nominally a 10% in 50 year hazard. The 10 % in
50 year hazard may be assumed as about 2/3 of the 2% in 50 year hazard resulting in
Sa(10%in50) = 0.57g and 0.35g for the 6- and 12-story frames, respectively, at the
buildings site. Comparing Sa(2%in50) and Sa(10%in50) to the mean values of Sa(λu=1.0)
and Sa(0.95λuo ) in Table 8.1, both the RCS and STEEL buildings exceed the minimum
requirements by a considerable margin – due in large part to their high overstrength. In
the most severe case - the 6-story RCS frame subjected to damaging near-field pulse
motions with Tp /T1 > 1.0 - the performance/hazard ratios are Sa(λu=1.0)/S a(2%in50)=2.1
and Sa(λu=0.95λuo )/Sa(10%in50) = 1.4. For the RCS frame subjected to the general
records, the ratios are Sa(λu=1.0)/S a(2%in50) = 3.6 and Sa(λu=0.95λuo )/Sa(10%in50) =
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2.5. Critical spectral acceleration values for the 6-story STEEL frame are about the same
as those in the 6-story RCS frame at the λu=0.95λuo level and about 25% larger at the
λu=1.0
level.
For
the
12-story
RCS
frame,
Sa(λu=1.0)/S a(2%in50)
and
Sa(λu=0.95λuo )/Sa(10%in50) ratios are about 2.4 and 2.0, respectively, for both general
records and severe near-fault records showing that general and near-fault ground motions
are nearly equally severe for that frame.
Given the rather large variability in spectral performance levels, as reflected by the
coefficient of variation of mean Sa reported in Table 8.1, one may again consider relating
performance levels to seismic hazards by comparing mean minus standard deviation
levels – rather than mean levels. Accordingly, for instance, for the most critical case of
the 6-story RCS frame under near-fault motions, performance/hazard ratios reduce to
Sa(λu=1.0)/S a(2%in50) = 1.6 and Sa(λu=0.95λuo )/Sa(10%in50) = 1.2, and ratios for the
general records reduce to 2.3 and 1.7, respectively. These ratios still exceed unity in all
cases, indicating that the frame would exceed the desired performance at the 10% and 2%
in 50 year hazard levels envisioned in current seismic codes.
8.2.6 Consistency of Drift versus Stability Criterion
Overall, average IDRmax values for all case study frames subjected to various ground
motions are remarkably consistent. At λu = 0.95λuo average IDRmax ranges between 3.2%
to 3.5%, and there are no perceptible differences between drifts for the different ground
motion bins. The range of 3.2% to 3.5% is slightly larger than the value of 2.5%
suggested by FEMA 273 for life safety for steel moment frames. At λu = 1.0, there are
consistent differences between response for the general and near-fault records, where
average IDRmax ranges between 10% to 12% for the near-fault records and 8% to 9% for
the general records, for the 6-story RCS and STEEL frames, respectively. The smaller
IDR for the general records is probably due to their longer strong motion duration that
leads to larger cumulative damage and stiffness/strength degradation, which in turn
causes the stability limit to be reached at smaller drift ratios. For the 12-story RCS frame
at λu = 1.0 the same trend for both types of records is still observed but with slightly
375
smaller values; average IDRmax is 7% and 9% for general and near-fault records,
respectively. This consistent decrease of the IDRmax value corresponding to collapse for
the 12-story frame can be attributed to: (1) lower lateral stiffness to gravity load ratio
compared to the 6-story structures, and (2) higher mode effects. These two factors trigger
failure at slightly lower global deformation levels than for the 6-story case study frames.
8.2.7 Spatial Distribution of Damage
The damage indices reveal that damage is governed by peak response (due to the pulse
effect) under near-fault records, whereas cumulative effects are more apparent under
general records with consistently longer strong motion durations and consequently more
accumulation of damage.
Furthermore, it has been observed that damage of the 12-story RCS frame under general
records is distributed among upper and lower stories, at different levels of demand, as a
result of higher mode effects attacking both upper and lower floors. On the other hand,
damage due to near-fault ground motions is almost always confined to lower stories since
such records put high demand on the lower floors of a building increasing their
vulnerability to P-∆
effects (Anderson and Bertero, 1987). Another interesting
observation is the less spread of damage in the 6-story RCS frame as compared to the 6story STEEL frame, especially at the life safety performance level (i.e., at λu = 0.95λuo )
showing a better performance of the RCS frame from an economic point of view
concerning cost of repair. Beside the considerable damage of the beams in all case study
frames, a key aspect of the damage undergone by the STEEL frame is the noticeably
severe damage of its inner joint zones and nearly no (or minor) damage of its relatively
strong columns, except for the ground floor columns bases that undergo large plastic
rotations at high hazard levels. The RCS frame has much less damage of its composite
joints but larger damage at various sections of its reinforced concrete columns up the
height of the frame. For more details about typical damage patterns in each frame one
should refer to Sections 6.4.4 (Chapter 6) and 7.10.3 (Chapter 7).
376
8.2.8 Local versus Global Response Relationships
For the 6- and 12-story RCS frames local response in terms of peak plastic rotations of
beams and columns can be successfully related (with a very good correlation) to
interstory drift ratios. Correlations to interstory drift are best when the drift is expressed
in terms of the maximum plastic transient interstory drift ratio, IDRp,max, for the case of
beams and maximum change in transient interstory drift ratios, ∆IDRmax, for columns
based on the deformed configuration of the frame as previously mentioned in Section 8.1.
The relationship between such local and global responses is stable and can be
satisfactorily approximated by a 1:1 line for all practical purposes. Such result is useful
for estimating median local response parameters in terms of plastic rotations (or rotation
ductility demands) at a given IDR value and then comparing them to available acceptance
criteria and limits set within ATC 40 and FEMA 273. However, before generalizing these
findings, further study is to be made for other RCS frames with different geometries,
periods, amounts of overstrength, etc.
For the 6-story STEEL frame with relatively strong columns, low values of the plastic
rotation in columns, θp,C, observed throughout the frame are associated with high values
of ∆IDRp,max. However, at columns locations where considerable inelasticity takes place
(such as at the ground floor columns bases), large θp,C values (greater than about 0.02
radians) are associated with proportionally large ∆IDRp,max values offering good
correlation of the two response quantities. The predominant non-proportionality between
θp,C|max and ∆IDRp,max, especially at the upper stories for all hazard levels up to collapse
limit state, is due to the large moment capacity of the columns cross-sections compared to
beams, forcing all the nonlinearity to occur in beams while the columns remain nearly
elastic even at high ∆IDRp,max values. But once the demand on the columns is large
enough to cause inelastic behavior (e.g., at the columns bases of the ground floor), any
increase in the ∆IDRp,max value affecting a given story will cause a proportional increase
in θp,C of the columns of that story.
377
Another aspect of the response of the case study STEEL frame is the large dispersion of
the beams plastic rotations, θp,B, at a given interstory drift ratio, IDRp,max. This large
dispersion invalidates any relationship relating local to global deformations. It is mainly
attributed to the effect of modeling composite beams with different flexural stiffness and
strength properties in positive versus negative bending directions, along with considering
joint panel flexibility in the analysis. This causes drifting of the deformation towards
negative side with considerably different amounts depending on the magnitude and
direction of the deformation of the first few inelastic loading cycles. Among other factors
for this large dispersion are the use of regular steel beams along with the composite
beams at every floor level, and the unequal spans of the frame. The dispersion of θp,B|max
values is less accentuated if one looks at the performance at low damage levels (i.e., at
small drift demands) corresponding to high values of λu. This large scatter is more
pronounced for the STEEL frame than for the RCS frames because of its weaker joints
and stronger columns.
8.2.9 Reducing the Variability in the Response through a Dual Earthquake Intensity
Index
Based on the results from the present research for all case study frames, we can suggest
that a dual earthquake intensity index of Sa(T1 ,ξ=5%) and RSa = Sa(TF)/Sa(T1 ) would be
effective for reducing the record-to-record dispersion of the response (be it the maximum
transient interstory drift ratio, IDRmax, or the global stability index, λu). TF is a longer
period than the fundamental period T1 of the structure in consideration representing a
decrease of lateral stiffness due to earthquake induced damage. TF is calculated as the
period associated with a secant lateral stiffness derived from the pushover analysis using
a target displacement beyond yield. This dual index would in turn reduce the standard
error of estimation of the median response and decrease the number of nonlinear time
history analyses needed to achieve a certain confidence level. As an example, for the 6story RCS frame under considered near-fault ground records, the net effect is that
uncertainty in the estimation of median IDRmax with a limited sample size of eight
records is cut in half from 16% (=0.45/√8) to 8% (=0.22/√8), and uncertainty in the
378
estimation of λu reduces from 22% to 15%. For general records, corresponding drops are
from 15% to 10% and from 21% to 16% for IDRmax and λu, respectively.
The main disadvantage of this dual index is that current hazard maps only report
Sa(T1 ,ξ=5%) and do not distinguish hazards on the basis of RSa. It would be worthwhile,
to confirm whether the promising results shown here using the two terms Sa(T1 ,ξ=5%)
and RSa apply for other types of records and other structures. If so, then this would
suggest a direction for improving seismic hazard maps by adding this sort of information
for engineers to include in seismic hazard analyses.
Other potential candidates for an earthquake intensity index are the strong motion
duration, tSM, and the pulse period, Tp , for near-fault ground records. These input
parameters did not offer extra benefit beyond Sa(T1 ,ξ=5%) and RSa in reducing the
dispersion in the response for the 6-story case study frames, whereas they offered some
benefit for the 12-story frame. However, no definite conclusions can be drawn
concerning these parameters, in part because of the narrow range of strong motion
durations and pulse period values in the bins of ground records considered in this
research.
Furthermore, systematic differences in the response due to near-fault records associated
with Tp /T1 ratios as reported in Section 8.2.2 indicate that an improved intensity scaling
technique for near-fault ground records should involve both Sa(T1 ) and a second index
that reflects the frequency content of the record, as might be reflected by the spectral
velocity measure. This might be a very interesting subject for future work.
8.3 Suggestions for Future Work
Three potential areas for future research may be identified as (1) enhancing and
modifying analytical models for the nonlinear analysis and collapse detection of
composite
RCS
moment
frames,
(2)
evaluating
379
current
seismic
codes
design
requirements and procedures and improving them wherever it is needed based on the
results reported herein, and (3) enlarging the scope of the current study to include more
cases and more design and modeling strategies to confirm or modify the breadth of the
conclusions reported in this work.
The collapse analysis implemented in this research is performed in a subsequent step to
the time history analysis. Incorporating such technique within the analysis program to
automatically update the localized status of damage step by step during the time history
analysis will offer a better estimate of the collapse limit state as well as of all other
intermediate damage (or performance) levels. This is equivalent to the idea of adding
stiffness/strength degradation to the material models as a function of the evolution of
damage (be it of the cumulative type or peak response type). Inclusion of such localized
damage effects with stepwise automatic update in the global structure model would make
a valuable addition to DYNAMIX and would be viable for predicting reliable estimates
of different performance limit states.
Related to the present research is work currently underway to evaluate seismic codes
design requirements, particularly the structural response modification factor, R, and the
displacement amplification factor, Cd. Estimates of R and Cd may be obtained from the
detailed static (pushover) and time history (IDA) analyses performed throughout this
research. For instance, an estimate for R may be obtained as
R=
Elastic base shear correspond ing to collapse limit state, i.e., λu = 1.0
Design base shear, Vdesign
(8.1)
and adopting a SDOF approximation, this relation might be simplified as
S ( λ = 1 .0 )
S (λ = 1.0)
R= a u
= a u
Vdesign / W
Sa, design
(8.2)
380
where Sa is the spectral acceleration at the fundamental period of the structure, and W is
the seismic weight of the building. Sa(λu=1.0) can be easily determined for a given record
from the IDA and collapse analysis curves presented in this work. Furthermore, Cd/R can
be determined as
Cd IDR inelastic correspond ing to λ u = 1.0
=
R
IDR elastic correspond ing to λ u = 1.0
(8.3)
IDRinelastic due to a given record can be easily extracted from the IDA and collapse
analysis results, while IDRelastic can be calculated by performing an elastic time history
analysis of the structure under the same record scaled to Sa(T1 ) corresponding to collapse
limit state, i.e., λu=1.0. Estimates of R and Cd might then be compared to their
corresponding values specified in codes such as IBC 2000. The ultimate goal is to
improve the reliability of constructed facilities designed using R and Cd factors and the
largely empirical foundation for the current code specified factors.
To broaden the scope of findings and conclusions of the current study, work may be done
along two fronts: (a) investigate other design and modeling strategies, and (b) study
larger number of cases to cover a wider spectrum of buildings properties and
characteristics.
Concerning design strategies, trying a perimeter frame design instead of the space frame
configuration used in designing the case study buildings is a very important issue. It is
expected that such design will lower the observed large lateral overstrength of the
moment frames and accordingly will effect all performance levels especially the collapse
limit state. Performing the design ignoring drift (stiffness) requirements might also be
another interesting way to lower the lateral overstrength and study its effect at the high
hazard levels. The ultimate goal is to come up with consistent design criteria (in terms of
stiffness and strength requirements) that produce buildings with reliable predicted
performance under multi-hazard levels.
381
Various modeling strategies should also be investigated. These might include performing
three-dimensional time history analyses of the structures under the three components of
ground records simultaneously and study torsion demands and their effect on the severity
and distribution of damage. Moreover, instead of assuming base fixity for the structures,
the effect of soil-structure interaction has to be considered in the analytical models.
Finally, a larger number of frames, with other geometric configurations, amount of
overstrength, a greater spread of natural periods should be investigated. Furthermore, in
this research we have only considered accelerograms representing general and near-fault
records with forward directivity recorded at stiff soil or rock then modified to stiff soil.
Other types of ground motions might be considered such as those recorded at soft soil
sites to study their effect on the seismic performance of the case study frames.
382
Appendix A
Selected Ground Records
Ground records selected for time history analyses are divided into two bins: general and
near-fault ground motions. Each bin is composed of eight records. In this appendix,
Figures A.1 to A.16 give acceleration, velocity, and displacement time histories and
response spectra for the eight general records, whereas A.17 to A.32 give the same
information for the eight near-fault records with forward directivity. The response spectra
are determined using a SDOF elastic oscillator and assuming 5% viscous damping. The
actual, Sv , and pseudo, PSv , velocity response spectra are superimposed in the figures.
Beside the traditional response spectra format of plotting spectral absolute acceleration,
relative velocity, or relative displacement versus period, the ADRS (AccelerationDisplacement Response Spectrum) format is also shown. In the ADRS format absolute
spectral accelerations are plotted against relative spectral displacements and the periods,
T, of the SDOF oscillator are represented by radial lines.
383
Ground Acceleration [g]
0.50
0.25
0.00
-0.25
-0.50
0
20
40
60
80
60
80
60
80
Ground Velocity [inches/sec]
Time [sec.]
(a) Ground Acceleration Record
30
15
0
-15
-30
0
20
40
Ground Displacement [inches]
Time [sec.]
(b) Ground Velocity Record
8
4
0
-4
-8
0
20
40
Time [sec.]
(c) Ground Displacement Record
Figure A.1 Miyagi-oki 1978 Ground Record - Ofuna Station
384
1.6
100
75
Sv [inches/sec]
1.2
Sa [g's]
Sv
PSv
0.8
0.4
50
25
0.0
0
0
1
2
3
4
0
1
Period, T [sec]
(a) Spectral Acceleration Response Spectrum
3
4
(b) Spectral Velocity Response Spectrum
20
1.6
15
1.2
Sa [g's]
Sd [inches]
2
Period, T [sec]
10
5
0.8
0.4
0
0
1
2
3
0.0
4
0
Period, T [sec]
5
10
15
20
Sd [inches]
(c) Spectral Displacement Response Spectrum
(d) Response Spectrum - ADRS Format
Figure A.2 Response Spectra (5% Damping) for Miyagi-oki (1978) record – Ofuna.
385
Ground Acceleration [g]
0.6
0.3
0.0
-0.3
-0.6
0
25
50
75
100
75
100
75
100
Ground Velocity [inches/sec]
Time [sec.]
(a) Ground Acceleration Record
15.0
7.5
0.0
-7.5
-15.0
0
25
50
Ground Displacement [inches]
Time [sec.]
(b) Ground Velocity Record
5.0
2.5
0.0
-2.5
-5.0
0
25
50
Time [sec.]
(c) Ground Displacement Record
Figure A.3 Valparaiso 1985 Ground Record - Llol Station
386
1.6
40
Sv
PSv
30
Sv [inches/sec]
Sa [g's]
1.2
0.8
20
10
0.4
0.0
0
0
1
2
3
4
0
1
2
3
4
Period, T [sec]
(a) Spectral Acceleration Response Spectrum
(b) Spectral Velocity Response Spectrum
8
1.6
6
1.2
Sa [g's]
Sd [inches]
Period, T [sec]
4
2
0.8
0.4
0
0
1
2
3
0.0
4
0
Period, T [sec]
2
4
6
8
Sd [inches]
(c) Spectral Displacement Response Spectrum
(d) Response Spectrum - ADRS Format
Figure A.4 Response Spectra (5% Damping) for Valparaiso (1985) record – Llol station.
387
Ground Acceleration [g]
0.30
0.15
0.00
-0.15
-0.30
0
10
20
30
40
30
40
30
40
Ground Velocity [inches/sec]
Time [sec.]
(a) Ground Acceleration Record
16
8
0
-8
-16
0
10
20
Ground Displacement [inches]
Time [sec.]
(b) Ground Velocity Record
8
4
0
-4
-8
0
10
20
Time [sec.]
(c) Ground Displacement Record
Figure A.5 Loma Prieta 1989 Ground Record - Hollister City Hall
388
40
0.6
30
Sv [inches/sec]
Sa [g's]
0.8
0.4
20
0.2
10
Sv
PSv
0.0
0
0
1
2
3
4
0
1
2
3
4
Period, T [sec]
(a) Spectral Acceleration Response Spectrum
(b) Spectral Velocity Response Spectrum
16
0.8
12
0.6
Sa [g's]
Sd [inches]
Period, T [sec]
8
4
0.4
0.2
0
0
1
2
3
0.0
4
0
Period, T [sec]
4
8
12
16
Sd [inches]
(c) Spectral Displacement Response Spectrum
(d) Response Spectrum - ADRS Format
Figure A.6 Response Spectra (5% Damping) for Loma Prieta (1989) record – Hollister
City Hall.
389
Ground Acceleration [g]
0.4
0.2
0.0
-0.2
-0.4
0
15
30
45
60
45
60
45
60
Ground Velocity [inches/sec]
Time [sec.]
(a) Ground Acceleration Record
30
15
0
-15
-30
0
15
30
Ground Displacement [inches]
Time [sec.]
(b) Ground Velocity Record
16
8
0
-8
-16
0
15
30
Time [sec.]
(c) Ground Displacement Record
Figure A.7 Loma Prieta 1989 Ground Record - Hollister South & Pine
390
64
1.2
48
Sv [inches/sec]
Sa [g's]
1.6
0.8
32
16
0.4
Sv
PSv
0.0
0
0
1
2
3
4
0
1
2
3
4
Period, T [sec]
(a) Spectral Acceleration Response Spectrum
(b) Spectral Velocity Response Spectrum
16
1.6
12
1.2
Sa [g's]
Sd [inches]
Period, T [sec]
8
4
0.8
0.4
0
0
1
2
3
0.0
4
0
Period, T [sec]
4
8
12
16
Sd [inches]
(c) Spectral Displacement Response Spectrum
(d) Response Spectrum - ADRS Format
Figure A.8 Response Spectra (5% Damping) for Loma Prieta (1989) record – Hollister
South & Pine.
391
Ground Acceleration [g]
0.4
0.2
0.0
-0.2
-0.4
0
6
12
18
24
18
24
18
24
Ground Velocity [inches/sec]
Time [sec.]
(a) Ground Acceleration Record
12
6
0
-6
-12
0
6
12
Ground Displacement [inches]
Time [sec.]
(b) Ground Velocity Record
2
1
0
-1
-2
0
6
12
Time [sec.]
(c) Ground Displacement Record
Figure A.9 Loma Prieta 1989 Ground Record - WAHO
392
1.6
52
Sv
0.8
26
13
0.4
0.0
0
0
1
2
3
4
0
1
2
3
4
Period, T [sec]
Period, T [sec]
(a) Spectral Acceleration Response Spectrum
(b) Spectral Velocity Response Spectrum
6.0
1.6
4.5
1.2
Sa [g's]
Sd [inches]
PSv
39
Sv [inches/sec]
Sa [g's]
1.2
3.0
1.5
0.8
0.4
0.0
0
1
2
3
0.0
0.0
4
Period, T [sec]
1.5
3.0
4.5
6.0
Sd [inches]
(c) Spectral Displacement Response Spectrum
(d) Response Spectrum - ADRS Format
Figure A.10 Response Spectra (5% Damping) for Loma Prieta (1989) record –
WAHO.
393
Ground Acceleration [g]
0.4
0.2
0.0
-0.2
-0.4
0
9
18
27
36
27
36
27
36
Ground Velocity [inches/sec]
Time [sec.]
(a) Ground Acceleration Record
20
10
0
-10
-20
0
9
18
Ground Displacement [inches]
Time [sec.]
(b) Ground Velocity Record
10
5
0
-5
-10
0
9
18
Time [sec.]
(c) Ground Displacement Record
Figure A.11 Cape Mendocino 1992 Ground Record - Rio Del Overpass
394
40
1.2
Sv
PSv
30
Sv [inches/sec]
Sa [g's]
0.9
0.6
20
0.3
10
0.0
0
0
1
2
3
4
0
1
2
3
4
Period, T [sec]
(a) Spectral Acceleration Response Spectrum
(b) Spectral Velocity Response Spectrum
8
1.2
6
0.9
Sa [g's]
Sd [inches]
Period, T [sec]
4
2
0.6
0.3
0
0
1
2
3
0.0
4
0
Period, T [sec]
2
4
6
8
Sd [inches]
(c) Spectral Displacement Response Spectrum
(d) Response Spectrum - ADRS Format
Figure A.12 Response Spectra (5% Damping) for Cape Mendocino (1992) record –
Rio Del Overpass.
395
Ground Acceleration [g]
0.30
0.15
0.00
-0.15
-0.30
0
11
22
33
44
33
44
33
44
Ground Velocity [inches/sec]
Time [sec.]
(a) Ground Acceleration Record
24
12
0
-12
-24
0
11
22
Ground Displacement [inches]
Time [sec.]
(b) Ground Velocity Record
20
10
0
-10
-20
0
11
22
Time [sec.]
(c) Ground Displacement Record
Figure A.13 Landers 1992 Ground Record - Yermo Fire Station
396
48
0.64
36
Sv [inches/sec]
Sa [g's]
0.48
Sv
PSv
0.32
24
0.16
12
0.00
0
0
1
2
3
4
0
1
2
3
4
Period, T [sec]
(a) Spectral Acceleration Response Spectrum
(b) Spectral Velocity Response Spectrum
16
0.64
12
0.48
Sa [g's]
Sd [inches]
Period, T [sec]
8
4
0.32
0.16
0
0
1
2
3
0.00
4
0
Period, T [sec]
4
8
12
16
Sd [inches]
(c) Spectral Displacement Response Spectrum
(d) Response Spectrum - ADRS Format
Figure A.14 Response Spectra (5% Damping) for Landers (1992) record – Yermo Fire
station.
397
Ground Acceleration [g]
0.6
0.3
0.0
-0.3
-0.6
0
15
30
45
60
45
60
45
60
Ground Velocity [inches/sec]
Time [sec.]
(a) Ground Acceleration Record
30
15
0
-15
-30
0
15
30
Ground Displacement [inches]
Time [sec.]
(b) Ground Velocity Record
8
4
0
-4
-8
0
15
30
Time [sec.]
(c) Ground Displacement Record
Figure A.15 Mendocino 1992 Ground Record - Petrolia Station
398
80
2.0
Sv
PSv
60
S v [inches/sec]
Sa [g's]
1.5
1.0
40
0.5
20
0.0
0
0
1
2
3
0
4
1
2
3
4
Period, T [sec]
(a) Spectral Acceleration Response Spectrum
(b) Spectral Velocity Response Spectrum
24
2.0
18
1.5
Sa [g's]
Sd [inches]
Period, T [sec]
12
6
1.0
0.5
0
0
1
2
3
0.0
4
0
6
12
18
24
Sd [inches]
Period, T [sec]
(c) Spectral Displacement Response Spectrum
(d) Response Spectrum - ADRS Format
Figure A.16 Response Spectra (5% Damping) for Mendocino (1992) record –
Petrolia station.
399
Ground Acceleration [g]
0.50
0.25
0.00
-0.25
-0.50
0
10
20
30
40
30
40
30
40
Ground Velocity [inches/sec]
Time [sec.]
(a) Ground Acceleration Record
50
25
0
-25
-50
0
10
20
Ground Displacement [inches]
Time [sec.]
(b) Ground Velocity Record
30
15
0
-15
-30
0
10
20
Time [sec.]
(c) Ground Displacement Record
Figure A.17 Imperial Valley 1979 Ground Record - Array 06
400
120
0.9
90
Sv [inches/sec]
Sa [g's]
1.2
0.6
60
30
0.3
Sv
PSv
0.0
0
0
1
2
3
4
0
1
2
3
4
Period, T [sec]
(a) Spectral Acceleration Response Spectrum
(b) Spectral Velocity Response Spectrum
64
1.2
48
0.9
Sa [g's]
Sd [inches]
Period, T [sec]
32
16
0.6
0.3
0
0
1
2
3
0.0
4
0
Period, T [sec]
16
32
48
64
Sd [inches]
(c) Spectral Displacement Response Spectrum
(d) Response Spectrum - ADRS Format
Figure A.18 Response Spectra (5% Damping) for Imperial Valley (1979) record –
Array 06.
401
Ground Acceleration [g]
0.8
0.4
0.0
-0.4
-0.8
0
6
12
18
24
18
24
18
24
Ground Velocity [inches/sec]
Time [sec.]
(a) Ground Acceleration Record
80
40
0
-40
-80
0
6
12
Ground Displacement [inches]
Time [sec.]
(b) Ground Velocity Record
30
15
0
-15
-30
0
6
12
Time [sec.]
(c) Ground Displacement Record
Figure A.19 Loma Prieta 1989 Ground Record - Los Gatos Station
402
180
2.4
135
Sv [inches/sec]
Sa [g's]
3.2
1.6
0.8
90
45
Sv
PSv
0.0
0
0
1
2
3
4
0
1
2
3
4
Period, T [sec]
(a) Spectral Acceleration Response Spectrum
(b) Spectral Velocity Response Spectrum
84
3.2
63
2.4
Sa [g's]
Sd [inches]
Period, T [sec]
42
21
1.6
0.8
0
0
1
2
3
0.0
4
0
Period, T [sec]
21
42
63
84
Sd [inches]
(c) Spectral Displacement Response Spectrum
(d) Response Spectrum - ADRS Format
Figure A.20 Response Spectra (5% Damping) for Loma Prieta (1989) record – Los
Gatos station.
403
Ground Acceleration [g]
0.8
0.4
0.0
-0.4
-0.8
0
10
20
30
40
30
40
30
40
Ground Velocity [inches/sec]
Time [sec.]
(a) Ground Acceleration Record
80
40
0
-40
-80
0
10
20
Ground Displacement [inches]
Time [sec.]
(b) Ground Velocity Record
30
15
0
-15
-30
0
10
20
Time [sec.]
(c) Ground Displacement Record
Figure A.21 Loma Prieta 1989 Ground Record - Lexington Station
404
2.8
180
Sv
PSv
135
Sv [inches/sec]
Sa [g's]
2.1
1.4
90
45
0.7
0.0
0
0
1
2
3
4
0
1
2
3
4
Period, T [sec]
(a) Spectral Acceleration Response Spectrum
(b) Spectral Velocity Response Spectrum
40
2.8
30
2.1
Sa [g's]
Sd [inches]
Period, T [sec]
20
10
1.4
0.7
0
0
1
2
3
0.0
4
0
Period, T [sec]
10
20
30
40
Sd [inches]
(c) Spectral Displacement Response Spectrum
(d) Response Spectrum - ADRS Format
Figure A.22 Response Spectra (5% Damping) for Loma Prieta (1989) record –
Lexington station.
405
Ground Acceleration [g]
0.50
0.25
0.00
-0.25
-0.50
0
5
10
15
20
15
20
15
20
Ground Velocity [inches/sec]
Time [sec.]
(a) Ground Acceleration Record
50
25
0
-25
-50
0
5
10
Ground Displacement [inches]
Time [sec.]
(b) Ground Velocity Record
20
10
0
-10
-20
0
5
10
Time [sec.]
(c) Ground Displacement Record
Figure A.23 Erzincan 1992 Ground Record - Erzincan Station
406
80
0.9
60
Sv [inches/sec]
Sa [g's]
1.2
0.6
40
0.3
20
Sv
PSv
0.0
0
0
1
2
3
4
0
1
2
3
4
Period, T [sec]
(a) Spectral Acceleration Response Spectrum
(b) Spectral Velocity Response Spectrum
32
1.2
24
0.9
Sa [g's]
Sd [inches]
Period, T [sec]
16
8
0.6
0.3
0
0
1
2
3
0.0
4
0
Period, T [sec]
8
16
24
32
Sd [inches]
(c) Spectral Displacement Response Spectrum
(d) Response Spectrum - ADRS Format
Figure A.24 Response Spectra (5% Damping) for Erzincan (1992) record – at
Erzincan station.
407
Ground Acceleration [g]
0.8
0.4
0.0
-0.4
-0.8
0
15
30
45
60
Ground Velocity [inches/sec]
Time [sec.]
(a) Ground Acceleration Record
60
30
0
-30
-60
0
15
30
45
60
Ground Displacement [inches]
Time [sec.]
(b) Ground Velocity Record
16
8
0
-8
-16
0
15
30
45
Time [sec.]
(c) Ground Displacement Record
Figure A.25 Northridge 1994 Ground Record - Newhall Station
408
60
2.4
120
90
Sv [inches/sec]
Sa [g's]
1.8
Sv
PSv
1.2
60
30
0.6
0.0
0
0
1
2
3
4
0
1
2
3
4
Period, T [sec]
(a) Spectral Acceleration Response Spectrum
(b) Spectral Velocity Response Spectrum
28
2.4
21
1.8
Sa [g's]
Sd [inches]
Period, T [sec]
14
7
1.2
0.6
0
0
1
2
3
0.0
4
0
Period, T [sec]
7
14
21
28
Sd [inches]
(c) Spectral Displacement Response Spectrum
(d) Response Spectrum - ADRS Format
Figure A.26 Response Spectra (5% Damping) for Northridge (1994) record –
Newhall station.
409
Ground Acceleration [g]
1.0
0.5
0.0
-0.5
-1.0
0
4
8
12
16
Ground Velocity [inches/sec]
Time [sec.]
(a) Ground Acceleration Record
80
40
0
-40
-80
0
4
8
12
16
12
16
Ground Displacement [inches]
Time [sec.]
(b) Ground Velocity Record
20
10
0
-10
-20
0
4
8
Time [sec.]
(c) Ground Displacement Record
Figure A.27 Northridge 1994 Ground Record - Rinaldi Station
410
2.4
140
105
Sv [inches/sec]
Sa [g's]
1.8
Sv
PSv
1.2
70
35
0.6
0.0
0
0
1
2
3
4
0
1
2
3
4
Period, T [sec]
(a) Spectral Acceleration Response Spectrum
(b) Spectral Velocity Response Spectrum
28
2.4
21
1.8
Sa [g's]
Sd [inches]
Period, T [sec]
14
7
1.2
0.6
0
0
1
2
3
0.0
4
0
Period, T [sec]
7
14
21
28
Sd [inches]
(c) Spectral Displacement Response Spectrum
(d) Response Spectrum - ADRS Format
Figure A.28 Response Spectra (5% Damping) for Northridge (1994) record –
Rinaldi station.
411
Ground Acceleration [g]
0.8
0.4
0.0
-0.4
-0.8
0
15
30
45
60
Ground Velocity [inches/sec]
Time [sec.]
(a) Ground Acceleration Record
60
30
0
-30
-60
0
15
30
45
60
45
60
Ground Displacement [inches]
Time [sec.]
(b) Ground Velocity Record
16
8
0
-8
-16
0
15
30
Time [sec.]
(c) Ground Displacement Record
Figure A.29 Northridge 1994 Ground Record - Sylmar Station
412
120
1.8
90
Sv [inches/sec]
Sa [g's]
2.4
1.2
60
30
0.6
Sv
PSv
0.0
0
0
1
2
3
4
0
1
2
3
4
Period, T [sec]
(a) Spectral Acceleration Response Spectrum
(b) Spectral Velocity Response Spectrum
40
2.4
30
1.8
Sa [g's]
Sd [inches]
Period, T [sec]
20
10
1.2
0.6
0
0
1
2
3
0.0
4
0
Period, T [sec]
10
20
30
40
Sd [inches]
(c) Spectral Displacement Response Spectrum
(d) Response Spectrum - ADRS Format
Figure A.30 Response Spectra (5% Damping) for Northridge (1994) record –
Sylmar station.
413
Ground Acceleration [g]
1.2
0.6
0.0
-0.6
-1.2
0
15
30
45
60
45
60
45
60
Ground Velocity [inches/sec]
Time [sec.]
(a) Ground Acceleration Record
80
40
0
-40
-80
0
15
30
Ground Displacement [inches]
Time [sec.]
(b) Ground Velocity Record
20
10
0
-10
-20
0
15
30
Time [sec.]
(c) Ground Displacement Record
Figure A.31 Kobe 1995 Ground Record - JMA Station
414
4
240
180
Sv [inches/sec]
Sa [g's]
3
Sv
PSv
2
120
60
1
0
0
0
1
2
3
4
0
1
Period, T [sec]
(a) Spectral Acceleration Response Spectrum
3
4
(b) Spectral Velocity Response Spectrum
40
4
30
3
Sa [g's]
Sd [inches]
2
Period, T [sec]
20
10
2
1
0
0
1
2
3
0
4
0
Period, T [sec]
10
20
30
40
Sd [inches]
(c) Spectral Displacement Response Spectrum
(d) Response Spectrum - ADRS Format
Figure A.32 Response Spectra (5% Damping) for Kobe (1995) record – JMA
station.
415
Appendix B
Story IDA Curves
Story Incremental Dynamic Analysis (IDA) curves are given in this appendix for the 12story RCS frame (Figures B.1 to B.16) and the 6-story STEEL frame (Figures B.17 to
B.32). For the 12-story RCS frame, Figures B.1 to B.8 show story IDA curves under the
eight selected general records, while Figures B.9 to B.16 give story IDA curves for the
eight near-fault records with forward directivity. Similarly, Figures B.17 to B.24 and
Figures B.25 to B.32 show story IDA curves for the 6-story STEEL frame under the
general and near-fault ground records, respectively.
416
Sa(T1=2.07sec,ξ =5%)
1.50
1.50
1.25
1.25
1.00
1.00
0.75
0.75
Story 1
Story 2
Story 3
Story 4
Story 5
Story 6
0.50
0.25
0.00
0.00
0.02
0.04
0.06
Story 7
Story 8
Story 9
Story 10
Story 11
Story 12
0.50
0.25
0.08
0.00
0.00
IDR max
0.02
0.04
0.06
0.08
IDR max
Sa(T 1=2.07sec,ξ=5%)
Figure B.1 Story IDA curves for Miyagi-oki (1978) record - 12-story RCS frame.
1.50
1.50
1.25
1.25
1.00
1.00
Story 1
Story 2
Story 3
Story 4
Story 5
Story 6
0.75
0.50
0.25
0.75
Story 7
Story 8
Story 9
Story 10
Story 11
Story 12
0.50
0.25
0.00
0.00 0.01 0.02 0.03 0.04 0.05 0.06
0.00
0.00 0.01 0.02 0.03 0.04 0.05 0.06
IDR max
IDR max
Figure B.2 Story IDA curves for Valparaiso (1985) record - 12-story RCS frame.
417
Sa(T1=2.07sec,ξ=5%)
1.25
1.25
1.00
1.00
0.75
0.75
Story 1
Story 2
Story 3
Story 4
Story 5
Story 6
0.50
0.25
0.00
0.00
0.02
0.04
0.06
0.08
Story 7
Story 8
Story 9
Story 10
Story 11
Story 12
0.50
0.25
0.10
0.00
0.00
0.02
IDR max
0.04
0.06
0.08
0.10
IDRmax
Sa(T1=2.07sec, ξ=5%)
Figure B.3 Story IDA curves for LP89-HCA record - 12-story RCS frame.
1.50
1.50
1.25
1.25
1.00
1.00
Story 1
Story 2
Story 3
Story 4
Story 5
Story 6
0.75
0.50
0.25
0.00
0.00
0.02
0.04
0.06
0.08
Story 7
Story 8
Story 9
Stoy 10
Story 11
Story 12
0.75
0.50
0.25
0.10
IDRmax
0.00
0.00
0.02
0.04
0.06
0.08
IDR max
Figure B.4 Story IDA curves for LP89-HSP record - 12-story RCS frame.
418
0.10
Sa(T1=2.07sec, ξ=5%)
1.00
1.00
0.75
0.75
0.50
0.25
0.00
0.00
0.50
Story 1
Story 2
Story 3
Story 4
Story 5
Story 6
0.02
0.04
0.06
Story 7
Story 8
Story 9
Story 10
Story 11
Story 12
0.25
0.00
0.00
0.08
0.02
IDR max
0.04
0.06
0.08
IDR max
Sa(T1=2.07sec, ξ=5%)
Figure B.5 Story IDA curves for LP89-WAHO record - 12-story RCS frame.
1.50
1.50
1.25
1.25
1.00
1.00
0.75
Story 1
Story 2
Story 3
Story 4
Story 5
Story 6
0.50
0.25
0.00
0.00
0.02
0.04
0.06
0.08
Story 7
Story 8
Story 9
Story 10
Story 11
Story 12
0.75
0.50
0.25
0.00
0.00
0.10
IDR max
0.02
0.04
0.06
0.08
IDR max
Figure B.6 Story IDA curves for CM92-RIO record - 12-story RCS frame.
419
0.10
Sa (T1=2.07sec,ξ=5%)
1.50
1.50
1.25
1.25
1.00
1.00
0.75
0.75
Story 1
Story 2
Story 3
Story 4
Story 5
Story 6
0.50
0.25
0.00
0.00
0.02
0.04
0.06
0.08
Story 7
Story 8
Story 9
Story 10
Story 11
Story 12
0.50
0.25
0.10
0.00
0.00
0.02
IDRmax
0.04
0.06
0.08
0.10
IDRmax
Sa(T1=2.07sec,ξ=5%)
Figure B.7 Story IDA curves for LA92-YER record - 12-story RCS frame.
1.50
1.50
1.25
1.25
1.00
1.00
Story 1
Story 2
Story 3
Story 4
Story 5
Story 6
0.75
0.50
0.25
0.00
0.00
0.02
0.04
0.06
Story 7
Story 8
Story 9
Story 10
Story 11
Story 12
0.75
0.50
0.25
0.00
0.00
0.08
0.02
0.04
0.06
IDR max
IDR max
Figure B.8 Story IDA curves for Mendocino (1992) record - 12-story RCS frame.
420
0.08
Sa (T1=2.07sec,ξ=5%)
1.00
1.00
0.75
0.75
Story 1
Story 2
Story 3
Story 4
Story 5
Story 6
0.50
0.25
0.00
0.00 0.02
0.04
0.06
0.08
0.10
Story 7
Story 8
Story 9
Story 10
Story 11
Story 12
0.50
0.25
0.00
0.00
0.12
0.02 0.04 0.06
IDRmax
0.08 0.10
0.12
IDR max
Sa(T1=2.07sec, ξ=5%)
Figure B.9 Story IDA curves for IV79-A6 record - 12-story RCS frame.
1.50
1.50
1.25
1.25
1.00
1.00
0.75
Story 1
Story 2
Story 3
Story 4
Story 5
Story 6
0.50
0.25
0.00
0.00
0.02
0.04
0.06
Story 7
Story 8
Story 9
Story 10
Story 11
Story 12
0.75
0.50
0.25
0.00
0.00
0.08
IDR max
0.02
0.04
0.06
IDR max
Figure B.10 Story IDA curves for LP89-LG record - 12-story RCS frame.
421
0.08
Sa (T1=2.07sec,ξ=5%)
1.50
1.50
1.25
1.25
1.00
1.00
0.75
0.50
0.25
0.00
0.00
0.75
Story 1
Story 2
Story 3
Story 4
Story 5
Story 6
0.02
0.04
0.06
Story 7
Story 8
Story 9
Story 10
Story 11
Story 12
0.50
0.25
0.00
0.00
0.08
0.02
IDRmax
0.04
0.06
0.08
IDR max
Sa (T1=2.07sec,ξ=5%)
Figure B.11 Story IDA curves for LP89-LX record - 12-story RCS frame.
2.5
2.5
2.0
2.0
1.5
1.5
Story 1
Story 2
Story 3
Story 4
Story 5
Story 6
1.0
0.5
0.0
0.00
0.02
0.04
0.06
0.08
0.10
Story 7
Story 8
Story 9
Story 10
Story 11
Story 12
1.0
0.5
0.0
0.00
0.12
IDRmax
0.02
0.04 0.06
0.08
0.10
IDR max
Figure B.12 Story IDA curves for EZ92-EZ record - 12-story RCS frame.
422
0.12
Sa (T1=2.07sec,ξ=5%)
1.75
1.75
1.50
1.50
1.25
1.25
1.00
1.00
Story 1
Story 2
Story 3
Story 4
Story 5
Story 6
0.75
0.50
0.25
0.00
0.00
0.02
0.04
0.06
0.08
Story 7
Story 8
Story 9
Story 10
Story 11
Story 12
0.75
0.50
0.25
0.10
0.00
0.00
0.02
IDRmax
0.04
0.06
0.08
0.10
IDRmax
Sa(T1=2.07sec, ξ=5%)
Figure B.13 Story IDA curves for NR94-NH record - 12-story RCS frame.
1.75
1.75
1.50
1.50
1.25
1.25
1.00
1.00
Story 1
Story 2
Story 3
Story 4
Story 5
Story 6
0.75
0.50
0.25
0.00
0.00
0.02
0.04
0.06
0.08
Story 7
Story 8
Story 9
Story 10
Story 11
Story 12
0.75
0.50
0.25
0.00
0.00
0.10
IDRmax
0.02
0.04
0.06
0.08
IDRmax
Figure B.14 Story IDA curves for NR94-RS record - 12-story RCS frame.
423
0.10
Sa(T1=2.07sec,ξ=5%)
2.5
2.5
2.0
2.0
1.5
1.5
Story 1
Story 2
Story 3
Story 4
Story 5
Story 6
1.0
0.5
Story 7
Story 8
Story 9
Story 10
Story 11
Story 12
1.0
0.5
0.0
0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14
0.0
0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14
IDR max
IDR max
Sa(T1 =2.07sec,ξ=5%)
Figure B.15 Story IDA curves for NR94-SY record - 12-story RCS frame.
1.75
1.75
1.50
1.50
1.25
1.25
1.00
1.00
Story 1
Story 2
Story 3
Story 4
Story 5
Story 6
0.75
0.50
0.25
0.00
0.00
0.02
0.04
0.06
0.08
Story 7
Story 8
Story 9
Story 10
Story 11
Story 12
0.75
0.50
0.25
0.00
0.00
0.10
IDR max
0.02
0.04
0.06
0.08
IDR max
Figure B.16 Story IDA curves for KB95-JM record - 12-story RCS frame.
424
0.10
4
5
S a(T1 =1.26sec,ξ=5%)
Sa(T1=1.26sec, ξ=5%)
6
4
3
Story 1
Story 2
Story 3
Story 4
Story 5
Story 6
2
1
0
0.00
0.02
0.04
0.06
0.08
3
2
1
0
0.00
0.10
Story 1
Story 2
Story 3
Story 4
Story 5
Story 6
0.02
3.0
3.0
2.5
2.5
2.0
1.5
0.0
0.00
Story 1
Story 2
Story 3
Story 4
Story 5
Story 6
0.02
0.04
0.06
0.08
Figure B.18 Story IDA curves for Valparaiso
(1985) - 6-story STEEL frame.
Sa(T1=1.26sec,ξ=5%)
Sa (T1=1.26sec,ξ =5%)
Figure B.17 Story IDA curves for Miyagi (1978)
record - 6-story STEEL frame.
0.5
0.06
IDR max
IDRmax
1.0
0.04
0.08
0.10
IDRmax
2.0
1.5
Story 1
Story 2
Story 3
Story 4
Story 5
Story 6
1.0
0.5
0.0
0.00
0.02
0.04
0.06
0.08
0.10
IDR max
Figure B.19 Story IDA curves for LP89-HCA
record - 6-story STEEL frame.
Figure B.20 Story IDA curves for LP89-HSP
record - 6-story STEEL frame.
425
5
5
4
3
Story 1
Story 2
Story 3
Story 4
Story 5
Story 6
2
1
0
0.00
0.02
0.04
0.06
Sa(T1=1.26sec,ξ=5%)
Sa(T1=1.26sec,ξ=5%)
6
4
3
1
0
0.00
0.08
Story 1
Story 2
Story 3
Story 4
Story 5
Story 6
2
0.02
IDR max
0.08
0.10
Figure B.22 Story IDA curves for CM92-RIO
record - 6-story STEEL frame.
2.5
3
2
Story 1
Story 2
Story 3
Story 4
Story 5
Story 6
1
0.02
0.04
0.06
0.08
Sa(T1 =1.26sec, ξ=5%)
4
Sa(T1=1.26sec, ξ=5%)
0.06
IDR max
Figure B.21 Story IDA curves for LP89-WAHO
record - 6-story STEEL frame.
0
0.00
0.04
2.0
1.5
0.5
0.0
0.00
0.10
IDR max
Story 1
Story 2
Story 3
Story 4
Story 5
Story 6
1.0
0.02
0.04
0.06
0.08
0.10
IDR max
Figure B.24 Story IDA curves for Mendocino
(1992) - 6-story STEEL frame.
Figure B.23 Story IDA curves for LA92-YER
record - 6-story STEEL frame.
426
3.0
1.5
1.0
Story 1
Story 2
Story 3
Story 4
Story 5
Story 6
0.5
2.5
Sa(T1=1.26sec,ξ=5%)
Sa (T1=1.26sec,ξ=5%)
2.0
2.0
1.5
Story 1
Story 2
Story 3
Story 4
Story 5
Story 6
1.0
0.5
0.0
0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14
0.0
0.00
0.02
IDR max
0.06
0.08
0.10
0.12
IDR max
Figure B.25 Story IDA curves for IV79-A6
record - 6-story STEEL frame.
Figure B.26 Story IDA curves for LP89-LG
record - 6-story STEEL frame.
3.0
5
4
Sa (T1=1.26sec,ξ=5%)
Sa(T1=1.26sec,ξ=5%)
0.04
3
Story 1
Story 2
Story 3
Story 4
Story 5
Story 6
2
1
0
0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14
2.5
2.0
1.5
1.0
0.5
0.0
0.00
IDR max
Story 1
Story 2
Story 3
Story 4
Story 5
Story 6
0.02
0.04
0.06
0.08
0.10
0.12
IDR max
Figure B.27 Story IDA curves for LP89-LX
record - 6-story STEEL frame.
Figure B.28 Story IDA curves for EZ92-EZ
record - 6-story STEEL frame.
427
5
4
4
Sa(T1=1.26sec,ξ=5%)
Sa (T1=1.26sec,ξ =5%)
5
3
Story 1
Story 2
Story 3
Story 4
Story 5
Story 6
2
1
0
0.00
0.02
0.04
0.06
0.08
3
1
0
0.00
0.10
Story 1
Story 2
Story 3
Story 4
Story 5
Story 6
2
0.02
IDRmax
0.08
0.10
Figure B.30 Story IDA curves for NR94-RS
record - 6-story STEEL frame.
3.0
5
2.5
Sa(T1 =1.26sec,ξ =5%)
Sa(T1=1.26sec, ξ=5%)
0.06
IDR max
Figure B.29 Story IDA curves for NR94-NH
record - 6-story STEEL frame.
2.0
1.5
Story 1
Story 2
Story 3
Story 4
Story 5
Story 6
1.0
0.5
0.0
0.00
0.04
0.02
0.04
0.06
0.08
4
3
1
0
0.00
0.10
IDRmax
Story 1
Story 2
Story 3
Story 4
Story 5
Story 6
2
0.02
0.04
0.06
0.08
0.10
0.12
IDRmax
Figure B.32 Story IDA curves for KB95-JM
record - 6-story STEEL frame.
Figure B.31 Story IDA curves for NR94-SY
record - 6-story STEEL frame.
428
Bibliography
Ansourian, P. (1982), “Plastic Rotation of Composite Beams,” Journal of the Structural
Division, ASCE, Vol. 108, No. ST3, March, 1982, pp. 643-659.
ACI Committee 318 (1995), “Building Code Requirements for Structural Concrete (ACI
318-95) and Commentary (ACI 318R-95),” American Concrete Institute,
Farmington Hills, MI.
AIJ - Architecture Institute of Japan (1991), “AIJ Standard for Structural Calculation of
Reinforced Concrete Structures,” 654 pp., Tokyo, Japan.
AISC (1993), Load and Resistance Design Specification for Structural Steel Buildings,
2nd Ed., American Institute of Steel Construction, Chicago, IL.
AISC (1997) - American Institute of Steel Construction, Inc., “Seismic Provisions for
Structural Steel Buildings, Load and Resistance Factor Design,” 2nd Edition,
Chicago, Illinois, 1997.
Anderson, and Bertero, V.V. (1987), “Uncertainties in Establishing Design Earthquakes,”
Journal of Structural Engineering, ASCE, August, 1987, pp. 1709
ASCE 7-95 (1995), “Minimum Design Loads for Buildings and Other Structures,”
Revision of ANSI/ASCE 7-93, American Society of Civil Engineers, NY 10017.
ASCE Guidelines (1994), “Guidelines for Design of Joints Between Steel Beams and
Reinforced Concrete Columns,” Jl. Struct. Div., ASCE, Vol. 120(8), pp. 2330-2357.
Attalla, M.R., Deierlein, G.G., and McGuire, W. (1994), “Spread of Plasticity: QuasiPlastic-Hinge Approach,” Journal of Structural Engineering, ASCE, Vol. 120, No.
8, pp. 2451-2473.
ATC-40 (1996), “Seismic Evaluation and Retrofit of Concrete Buildings,” Report No.
SSC 96-01, Applied Technology Council, Redwood City, CA.
429
Azizinamini, A., Corley, W.G., and Paul Johal, L.S. (1992), “Effects of Transverse
Reinforcement on Seismic Performance of Columns,” ACI Structural Journal, Vol.
89, No. 4, July-August 1992, pp. 442-450.
Banon, H., Biggs, J.M., and Irvine, H.M. (1981), “Seismic Damage in Reinforced
Concrete Frames,” Journal of Structural Engineering, ASCE, Vol. 107, No. 9, pp.
1713-1729.
Banon, H., and Veneziano, D. (1982), “Seismic Safety of Reinforced Concrete Members
and Structures,” Earthquake Engineering and Structural Dynamics, Vol. 10, pp.
179-193.
Bathe, K.J., and Bolourchi, S. (1979), “Large Displacement Analysis of ThreeDimensional Beam Structures,” IJNME, Vol. 14, pp. 961-968.
Bazzurro, P., Cornell, C.A., Shome, N., and Carballo, J.E. (1998), “A Comparison of
Three Proposals for Characterization of MDOF Non-Linear Seismic Response,”
Journal of Structural Engineering, ASCE, November, 1998
Bertero, V.V. (1986), “Evaluation of Response Reduction Factors Recommended by
ATC and SEAOC,” Third US National Conference, Charleston, North Carolina, pp.
1667.
Bertero, V.V. (1996), “The Need for Multi-Level Seismic Design Criteria,” Proceedings
of the 11th World Conference on Earthquake Engineering, Paper No. 2120.
Bracci, J.M., Reinhorn, A.M., Mander, J.B., and Kunnath, S.K. (1989), “Deterministic
Model for Seismic Damage Evaluation of RC Structures,” Technical Report
NCEER-89-0033, National Center for Earthquake Engineering Research, State
University of New York, Buffalo, NY.
Broderick, B.M., and Elnashai, A.S. (1996a), “Seismic Response of Composite Frames –
I. Response Criteria and Input Motion,” Engineering Structures, Vol. 18, No. 9, pp.
696-706.
Broderick, B.M., and Elnashai, A.S. (1996b), “Seismic Response of Composite Frames –
II. Calculation of Behavior Factors,” Engineering Structures, Vol. 18, No. 9, pp.
707-723.
Browning, JA., Li, Y.R., Lynn, A., and Moehle, J.P. (1997), “Performance Assessment
for a Reinforced Concrete Frame Building,” Proceedings of the International
Workshop on Seismic Design Methodologies for the Next Generation of Codes,
BLED/Slovenia, 24-27 June, 1997.
Bugeja, M., Bracci, J.M., and Moore, W.P. (1999), “Seismic Behavior of Composite
Moment Resisting Frame Systems,” Tech. Report CBDC-99-01, Dept. of Civil
Engrg., Texas A & M University.
430
Bursi, O.S., and Ballerini, M. (1996), “Behavior of Steel-Concrete Composite
Substructure with Full and Partial Shear Connection,” Proceedings of 11th World
Conference on Earthquake Engineering, Paper No. 771, Acapulco, Mexico, 1996.
Capechi, D., and Vestroni, F. (1986), “Aspetti della Modellazione del Dannegiamento
nelle Struture di Cemento Armato,” Ingegneria Sismica, III, 3. (in Italian)
Carvalho, E.C. (1991), “European Seismic Regulation. Contribution to the Improvement
of Eurocode 8,” National Laboratory for Civil Engineerin, Lisbon, Res. Rep. (in
Portuguese).
Castiglioni, C.A., and Calado, L. (1996), “Comparison of Two Cumulative Damage
Approaches for the Assessment of Behavior Factors for Low-rise Steel Buildings,”
J. Construct. Steel Res., Vol. 40, No. 1, pp. 39-61.
CEB State-of-the-art Report (1996), “RC Frames Under Earthquake Loading,” Thomas
Telford Services Ltd.
Chen, C.S. (1994), “Geometric Nonlinear Analysis of Three-Dimensional Structures,”
MS Thesis, School of Civil and Environmental Engineering, Cornell University,
Ithaca, NY 14853.
Chung, Y.S., Meyer, C., and Shinozuka, M. (1987), “Seismic Damage Assessment of RC
Members,” Technical Report NCEER-87-0022, National Center for Earthquake
Engineering Research, State University of New York, Buffalo, NY.
Chung, Y.S., Meyer, C., and Shinozuka, M. (1989), “Modeling of Concrete Damage,”
Structural Journal, ACI, Vol. 86, No. 3, pp. 259-271.
Ciampoli, M., Giannini, R., Nuti, C., and Pinto, P.E. (1989), “Seismic Reliability of NonLinear Structures with Stochastic Parameters by Directional Simulation,”
Proceedings 5th International Conference on Structural Safety and Reliability
(ICOSSAR 89), San Francisco, California, Vol. II New York: ASCE, pp. 11211128.
Climenhaga, J.J., and Johnson, R.P. (1972), “Local Buckling in Continuous Composite
Beams,” The Structural Engineer, Vol. 50, No. 9, September, 1972, pp. 367-374.
Cornell, C.A. (1996), “Calculating Building Seismic Performance Reliability; A Basis for
Multi-level Design Norms,” Proceedings of the 11th Conference on Earthquake
Engineering, Paper No. 2122.
Daali, M.L., and Korol, R.M. (1996), “Adequate Ductility in Steel Beams under
Earthquake-type Loading,” Engineering Structures, Vol. 18, No. 2, pp. 179-189.
431
Dafalias, Y.H. (1975), “On Cyclic and Anisotropic Plasticity: i) A General Model
Including Material Behavior Under Stress Reversals, ii) Anisotropic Hardening for
Initially Orthotropic Materials,"” PhD Thesis, University of California, Berkeley.
Dafalias, Y.H., and Popov, E.P. (1977), “Cyclic Loading for Materials with Vanishing
Elastic Region,” Nuclear Engineering and Design, Vol. 41, pp.293-302.
Deierlein, G.G., Sheikh, T.M., Yura, J.A., and Jirsa, J.O. (1989), “Beam-Column
Moment Connections for Composite Frames: Part 2,” Journal of Struct. Engrg.,
ASCE, Vol. 115, No. 11, Nov. 1989, pp. 2877-2896.
DiPasquale, E., and Cakmak, A.S. (1989), “On the Relation Between Local and Global
Damage Indices,” Technical Report NCEER-89-0034, National Center for
Earthquake Engineering Research, State University of New York, Buffalo, NY.
Dolce, M., Kappos, A., Zuccaro G. et al. (1994), “Report of the EAEEWG:3
Vulnerability and Risk Analysis,” 10th European Conference on Earthquake
Engineering, Vienna, Austria, Aug-Sep 1994, Vol. 4, Rotterdam, Balkema, 1995,
pp. 3049-3077.
Drucker, D.C. (1951), “A More Fundamental Approach to Plastic Stress-Strain
Relations,” Proceedings of the First US National Congress of Applied Mechanics,
ASME, New York, pp.487-491.
duPlessis, D.P., and Daniels, J.H. (1972), “Strength of Composite Beam-to-Column
Connections,” Fritz Engineering Laboratory, Report No. 374.3, Lehigh University,
1973.
EERI (1994), “Expected Seismic Performance of Buildings,” Publication Number SP-10,
Earthquake Engineering Research Institute, Oakland, CA.
Ellingwood, B., Galambos, T.V., MacGregor, J.G., and Cornell, C.A., (1980),
“Development of a Probability Based Load Criteria for American National Standard
Committee A58,” Special Publication No. 577, National Bureau of Standards,
Washington, D.C.
Elnashai, A.S., and Elghazouli, A.Y. (1993), “Performance of Composite Steel/Concrete
Members Under Earthquake Loading. Part I: Analytical Model,” Earthquake
Engineering and Structural Dynamics, Vol. 22, pp.315-345.
El-Tawil, S., and Deierlein, G.G. (1996), “Inelastic Dynamic Analysis of Mixed SteelConcrete Space Frames,” Report No. 96-5, School of Civil and Environmental
Engineering, Cornell University, May 1996.
El-Tawil, S., Kanno, R., and Deierlein, G.G. (1996), “Constitutive Model for BeamColumn Joints in the Analysis of Mixed Steel-Concrete Frames,” Proc. of Comp.
Constr. III, Irsee, Germany, June 9th to 14th , 1996.
432
Engelhardt, M.D., and Kim, K. (1995), ”Development of Analytical Models for
Earthquake Analysis of Steel Moment Frames,” Report No. PMFSEL 95-2, Phil M.
Ferguson Structural Engineering Laboratory, University of Texas at Austin, April
1995.
Fajfar, P. (1992), “Equivalent Ductility Factors, Taking into Account Low-Cycle
Fatigue,” Earthquake Engineering and Structural Dynamics, Vol. 21, pp. 837-848.
Fajfar, P. (1998), “Capacity Spectrum Method Based on Inelastic Demand Spectra,”
Proceedings of the 11th European Conference on Earthquake Engineering, Paris,
France, September, 1998.
Fajfar, P., and Fischinger, M. (1988), “N2 – A Method for Non-Linear Seismic Analysis
of Regular Structures,” Proceedings of the 9th World Conference on Earthquake
Engineering, Tokyo-Kyoto, Japan, pp. V-111-116.
FEMA-273 (1997), “NEHRP Guidelines for the Seismic Rehabilitation of Buildings,”
Applied Technology Council, Redwood City, CA. (funded by Federal Emergency
Management Agency, FEM A273, Washington, D.C.), October 1997.
Freeman, S.A. (1998), “Development and Use of Capacity Spectrum Method,” Proc. 6th
US National Conference on Earthquake Engineering, Seattle, CD-ROM, EERI,
Oakland.
Freeman, S.A., Nicoletti, J.P., and Tyrell, J.V. (1975), “Evaluations of Existing Buildings
for Seismic Risk – A Case Study of Puget Sound Naval Shipyard, Bremerton,
Washington” Proc. 1st US National Conference on Earthquake Engineering, pp. 13122, EERI, Berkeley.
Gattass, M., and Abel, J.F. (1987), “Equilibrium Considerations of the Updated
Lagrangian Formulation of Beam-Columns With Natural Concepts, IJNME, Vol.
24, pp. 2119-2141.
Ghobarah, A, Abou-Elfath, H, and Biddah, A. (1999), “Response-Based Damage
Assessment of Structures,” Earthquake Engineering and Structural Dynamics, Vol.
28, pp. 79-104.
Goel, S.C. and Yamanouchi, H., Editors (1992), “Recommendations for US-Japan
Cooperative Research Program – Phase 5, Composite and Hybrid Structures,”
Research Report UMCE 92-29, The University of Michigan, November.
Gosain, N.K., Brown, R.H., and Jirsa, J.O. (1977), “Shear Requirements for Load
Reversals on RC Members,” Journal of Structural Engineering, ASCE, Vol. 103,
No. 7, pp. 1461-1476.
433
Gourley, B.C., and Hajjar, J.F. (1994), “Cyclic Nonlinear Analysis of Three-Dimensional
Concrete-Filled Steel Tube Beam-Columns and Composite Frames,” Struct. Engrg.
Report No. ST-94-3, University of Minnesota, Minneapolis, MN 55455, 1994.
Griffis, L.G. (1987), “Motivation for and Fundamentals of Composite Action in
Structures,” Proceedings of the Boston Section of ASCE on Composite
Construction, Boston.
Griffis, L.G. (1992), “Composite Frame Construction,” Constructional Steel Design, Ed.
P.J. Dowling, J.E. Harding, R. Bjorhovde, Elsevier Applied Sciences, NY, pp. 523554.
Griffis, L.G. (1995), “Future Directions and Needs for Composite Construction,” ASCE
Conference Proceedings, Boston, May 1995, pp. 1309-1312.
Gunturi, S.K.V., and Shah, H.C. (1992), “Building Specific Damage Estimation,”
Proceedings 10th World Conference on Earthquake Engineering, Madrid, Spain,
July 1992, Vol. 10, Rotterdam, Balkema, 1992, pp. 6001-6006.
Hage, S.E., and MacGregor, J.G. (1974), “Second Order Analysis of Reinforced Concrete
Frames,” Structural Engineering Report No. 9, Department of Civil Engineering,
University of Alberta, Edmonton.
Hamburger, R.O., Foutch, D.A., and Cornell, C.A. (2000), “Performance Basis of
Guidelines for Evaluation, Upgrade and Design of Moment-Resisting Steel
Frames,” Proceedings of the 12th World Conference on Earthquake Engineering,
Auckland, New Zealand, Jan-Feb, 2000.
Han, S.W., and Wen, Y.K. (1994), “Method of Reliability-Based Calibration of Seismic
Structural Parameters,” Report UILU-ENG-94-2016, Structural Research Series No.
595, University of Illinois at Urbana-Champaign, IL.
Hart, G., Skokan, M., Martin, H. (1998), “Respond Evaluation of a 20-Story Concrete
Frame Building to the Northridge and Other Earthquakes,” Presented at
California Strong Motion Instrumentation Program Seminar, Oakland, CA.
Heinge, R.A. (1992), Presentation on the design of the 73 story Interfirst Plaza Building
in Dallas, TX., Engineering Foundation Conference: Composite Construction II,
Potosi, MO.
Hilmy, S.I. (1984), “Adaptive Nonlinear Dynamic Analysis of 3-D Steel Framed
Structures with Interactive Computer Graphics,” PhD Thesis, School of Civil and
Environmental Engineering, Cornell University, Ithaca, NY 14853.
IBC 2000 (1998), “International Building Code,” International Code Council, Falls
Church, Virginia, Final Draft, July 1998.
434
Joyner, W.B., and Boore, D.M. (1988), “Measurement, Characterization, and Prediction
of Strong Ground Motion,” in Earthquake Engineering and Soil Dynamics II –
Recent Advances in Ground Motion Evaluation, Geotechnical Special Publication
20, ASCE, New York, pp. 43-102.
Kanno, R. (1993), “Strength, Deformation, and Seismic Resistance of Joints Between
Steel Beams and Reinforced Concrete Columns,” Ph.D. Thesis, Cornell University,
Ithaca, NY.
Kanno, R., and Deierlein, G.G. (1994), “Cyclic Behavior of Joints Between Steel Beams
and Reinforced Concrete Columns,” ASCE Structures Congress Proceedings.
Kanno, R., and Deierlein, G.G. (1996), “Seismic Behavior of Composite (RCS) BeamColumn Joint Subassemblies,” Proc. of Comp. Constr. III, Irsee, Germany, June 9th
to 14th , 1996.
Kappos, A.J. (1997), “Seismic Damage Indices for RC Buildings: Evaluation of Concepts
and Procedures,” Progress in Structural Engineering and Materials, Vol. 1, No. 1,
September, 1997, pp. 78-87.
Kappos, A.J., Stylianidis, K.C., and Michailidis, C.N. (1996), “A Methodology for
Developing Loss Scenarios, with an Application to the City of Thessaloniki,”
Proceedings 11th World Conference on Earthquake Engineering, Acapulco,
Mexico, June 1996. Oxford: Pergamon. 1996, Paper No. 2057.
Kemp, A.R., and Dekker, N.W. (1991), “Available Rotation Capacity in Steel and
Composite Beams,” The Structural Engineer, Vol. 69, No. 5, March, 1991, pp. 8897.
Kordina, K. (1972), “Discussion No.3 – Cracking and Crack Control,” Proceedings,
International Conference on Planning and Design of Tall Buildings, Vol. III, Lehigh
University, Bethlehem, PA, Aug., 1972, pp. 721-722.
Kratzig, W.B., Meyer, I.F., and Meskouris, K. (1989), “Damage Evolution in Reinforced
Concrete Members Under Cyclic Loading,” Proceedings 5th International
Conference on Structural Safety and Reliability (ICOSSAR 89), San Francisco, CA,
Vol. II, pp. 795-802.
Krawinkler, H. (1995), “New Trends in Seismic Design Methodology,” Proc. 10th
European Conference on Earthquake Engineering, Vienna, 1994, Vol. 2, pp. 821830, Balkema, Rotterdam.
Krawinkler, H. (1996), “A Few Basic Concepts for Performance-Based Seismic Design,”
Proc. of the 11th World Conference on Earthquake Engineering, Elsevier, Paper No.
2121.
435
Krawinkler, H., and Alavi, B. (1998), “Development of Improved Design Procedures for
Near Fault Ground Motions,” Presented at California Strong Motion
Instrumentation Program Seminar, Oakland, CA.
Krawinkler, H., and Popov, E. (1982), “Seismic Behavior of Moment Connections and
Joints,” Journal of the Structural Division, ASCE, Vol. 108, No. ST2, February
1982, pp.373-391.
Krawinkler, H., and Seneviratna, G.D.P.K. (1998), “Pros and Cons of a Pushover
Analysis for Seismic Performance Evaluation,” Engineering Structures, Vol. 20,
pp. 452-464.
Kunnath, S.K., Reinhorn, A.M., Lobo, R.F. (1992), “IDARC version 3.0: A Program for
the Inelastic Damage Analysis of RC Structures,” Technical Report NCEER-920022, National Center for Earthquake Engineering Research, State University of
New York, Buffalo, NY.
Kuramoto, H. (1994), “Test Carried in Building Research Institute (BRI), Japan. (not
published).
Lawson, R.S., Vance, V., and Krawinkler, H. (1994), “Nonlinear Static Push-Over
Analysis – Why, When, And How?,” Proceedings 5th US National Conference on
Earthquake Engineering, pp. 283-292, Chicago, IL.
Lee, S.J. (1987), “Seismic Behavior of Steel Building Structures with Composite Slabs,”
Ph.D. Thesis, Department of Civil Engineering, Lehigh University, Bethlehem,
1987.
Lee, S.J., and Lu, L.W. (1989), “Cyclic Tests of Full-Scale Composite Joint
Subassemblages,” Journal of Structural Division, ASCE, Vol. 115, No. 8, Aug.
1989, pp. 1977-1998.
Leelataviwat, S., Goel, S.C., and Stojadinovic, B. (1998), “Drift and Yield Mechanism
Based Seismic Design and Upgrading of Steel Moment Frames,” Research Report
UMCEE 98-29, Department of Civil and Environmental Engineering, University of
Michigan, Ann Arbor, MI.
Leon, R., and Bawa, S. (1990), “Performance of Large Composite Columns,”
Proceedings of IBABSE Symposium on Mixed Structures, Including New Materials,
IABSE, Zurich, Switzerland, pp. 179-184.
Leon, R., and Deierlein, G.G. (1995), “An Overview of Codes, Standards, and Guidelines
for Composite Construction,” ASCE Conference Proceedings, Boston, May 1995,
pp. 1297-1300.
436
Luco, N., and Cornell, C.A. (1998), “Effects of Random Connection Fractures on the
Demands and Reliability for a 3-Story Pre-Northridge SMRF Structures,”
Proceedings of the 6th US National Conference on Earthquake Engineering, Seattle,
WA.
MacGregor, J.G. (1993), “Design of Slender Concrete Columns - Revisited,” ACI Struct.
Jl., May-June 1993, pp. 302-309.
Mahasuverachai, M., and Powell, G.H. (1982), “Inelastic Analysis of Piping and Tubular
Structures,” EERC Report 82-27, Earthquake Engineering Research Center,
University of California, Berkeley.
Mahin, S., and Bertero, V.V. (1978), “Reliability of Inelastic Design Methods for
Seismic-Resistant Structures,” Proceedings of the 6th European Conference on
Earthquake Engineering, 2-03-2-05.
Mander, J.B., Priestly, M.N.J., and Park, R. (1988a), “Theoretical Stress-Strain Model
Confined Concrete,” Journal of Structural Engineering, ASCE, Vol. 114, No. 8,
August 1988, pp. 1804-1826.
Mander, J.B., Priestly, M.N.J., and Park, R. (1988b), “Observed Stress-Strain Behavior of
Confined Concrete,” Journal of Structural Engineering, ASCE, Vol. 114, No. 8,
August 1988, pp. 1827-1849.
McCabe, S.L., and Hall, W.J. (1989), “Assessment of Seismic Structural Damage,”
Journal of Structural Engineering, ASCE, Vol. 115, No. 9, September 1989, pp.
2166-2183.
Mehanny, S.S.F., El-Tawil, S., and Deierlein, G.G. (1999), “DYNAMIX – Software for
DYNamic Analysis of MIXed Structures, Technical Manual,” Structural
Engineering Report, Dept. of Civil Engineering, Stanford University.
Miranda, E., and Bertero, V.V. (1994), “Evaluation of Strength Reduction Factors for
Earthquake-Resistant Design,” Earthquake Spectra, Vol. 10, No. 2, EERI, Oakland,
CA.
Mork, K.J. (1992), “Stochastic Analysis of Reinforced Concrete Frames Under Seismic
Excitation,” Soil Dynamics and Earthquake Engineering, Vol. 11, No. 3, pp. 145161.
Muller, F.P., Keintzel, E., and Charlier, H. (1983), “Dynamische Probleme im
Stahlbetonbau,” Deutsche Ausschuβ fur Stahlbeton, Heft 342.
Nassar, A.A., and Krawinkler, H. (1991), “Seismic Demands for SDOF and MDOF
Systems,” Report No. 95, John A. Blume Earthquake Engineering Center, Dept. of
Civil and Environmental Engineering, Stanford University, CA 94305.
437
NEHRP (1997), “Recommended Provisions for Seismic Regulations for New Buildings
and Other Structures,” BSSC Program on Improved Seismic Safety Provisions,
FEMA 303, Washington, D.C.
Neter, J., Kutner, H.K., Nachtsheim, C.J., and Wasserman, W. (1996), “Applied Linear
Statistical Models,” Richard D. Irwin, Inc., Chicago, IL.
Newmark, N.M., and Hall, W.J. (1982), “ Earthquake Spectra and Design,” Earthquake
Engineering Research Institute, El Cerrito, California.
Newmark, N.M., Siess, C. P., and Viest, I.M. (1951), “Tests and Analysis of Composite
Beams with Incomplete Interaction,” Proceedings of the Society for Experimental
Stress Analysis, Vol. 9, No. 1, 1951.
Nielsen, S.R.K., Koyluoglu, H.U., and Cakmak, A.S. (1992), “One and TwoDimensional Maximum Softening Damage Indicators for Reinforced Concrete
Structures Under Seismic Excitation,” Soil Dynamics and Earthquake Engineering,
Vol. 11, No. 8, pp. 435-443.
NZS 3101 - New Zealand Standard (1995), “Concrete Structures Standard: Part 2 Commentary on the Design of Concrete Structures,” New Zealand.
Orbison, J. (1982), “Nonlinear Static Analysis of 3-D Steel Frames,” PhD Thesis, School
of Civil and Environmental Engineering, Cornell University, Ithaca, NY 14853.
Otes, A. (1985), “Zur werkstoffgerechten Berechnung der Erdbebenbeanspruchung in
Stahlbetontragwerken,” Mitteilungen aus dem Institut fur Massivbau der TH
Darmstadt, Heft 25.
Ozcebe, G., and Saatcioglu, M. (1987), “Confinement of Concrete Columns for Seismic
Loading,” ACI Structural Journal, July-August 1987, pp. 308-315.
Park, Y.J., and Ang, A.H-S. (1985), “Mechanistic Seismic Damage Model for Reinforced
Concrete,” Journal of Structural Engineering, ASCE, Vol. 111, No. ST4, pp. 722739.
Park, Y.J., Ang, A.H-S., and Wen, Y.K. (1985), “Seismic Damage Analysis of
Reinforced Concrete Buildings,” Journal of Structural Engineering, ASCE, Vol.
111, No. ST4, pp. 740-757.
Park, R., and Paulay, T. (1975), “Reinforced Concrete Structures,” A Wiley-Interscience
Publication, pp. 315-319.
Paulay, T., and Priestly, M.J.N. (1992), “Seismic Design of Reinforced Concrete and
Masonry Buildings,” John Wiley & Sons.
438
Powell, G.H. (1969), “Theory of Nonlinear Elastic Structures,” Journal of Structural
Engineering, ASCE, Vol. 95, No. ST12, pp. 2687-2701.
Powell, G.H., and Chen, P.F. (1986), “3D Beam-Column Element with Generalized
Plastic Hinges,” Journal of Engineering Mechanics, ASCE, Vol. 112, No. 7, pp.
627-641.
Qi, X., and Moehle, J.P. (1991), “Displacement Design Approach for Reinforced
Concrete Structures Subjected to Earthquakes,” UCB/EERC-91/02, Earthquake
Engineering Research Center, University of California, Berkeley.
Raghavendrachar, M., and Aktan, A.E. (1992), “Flexibility by Multireference Impact
Testing for Bridge Diagnostics,” Journal of Structural Engineering, ASCE, Vol.
118, No. 8, pp. 2186-2203.
Reinhorn, A.M., Kunnath, S.K., and Mander, J.B. (1992), “Seismic Design of Structures
for Damage Control,” Nonlinear Seismic Analysis and Design of Reinforced
Concrete Buildings, ed. Fajfar, P. and Krawinkler, H., Elsevier Applied Science, pp.
63-76.
Robinson, H. (1969), “Composite Beam Incorporating Cellular Steel Decking,” Journal
of Structural Division, ASCE, Vol. 95, No. ST3, March 1969.
Rodriguez-Gomez, S., and Cakmak, A.S. (1990), “Evaluation of Seismic Damage Indices
for RC Structures,” Technical Report NCEER-90-0022, National Center for
Earthquake Engineering Research, State University of New York, Buffalo, NY.
Roufaiel, M.S.L., and Meyer, C. (1987a), “Analytical Modeling of Hysteretic Behavior
of R/C Frames,” Journal of Structural Engineering, ASCE, Vol. 113, No. 3, pp.
429-444.
Roufaiel, M.S.L., and Meyer, C. (1987b), “Reliability of Concrete Frames Damaged by
Earthquakes,” Journal of Structural Engineering, ASCE, Vol. 113, No. 3, pp. 445457.
Saiidi, M., and Sozen, M.A. (1981), “Simple Nonlinear Seismic Analysis of R/C
Structures,” Journal of the Structural Division, ASCE, Vol. 107, No. ST5, May
1981, pp. 937-951.
Salari, M.R., Spacone, E., Shing, P.B., and Frangopol, D.M. (1996), “Behavior of
Composite Structures Under Cyclic Loading,” US-Japan Cooperative Research 3rd
JTCC, December 12-14, 1996, Hong Kong.
SEAOC (1995), “Performance-Based Seismic Engineering of Buildings – Vision 2000,”
SEAOC, Sacramento, CA.
439
Searer, G.R. (1994), “Inelastic Behavior and Seismic Design of Steel Moment Frames,”
MS Thesis, School of Civil and Environmental Engineering, Cornell University,
Ithaca, NY 14853.
Sheikh, T.M. (1995), “Experience with Composite Moment Frames,” ASCE Conference
Proceedings, Boston, May 1995, pp. 1301-1305.
Sheikh, T.M., Deierlein, G.G., Yura, J.A., and Jirsa, J.O. (1989), “Beam-Column
Moment Connections for Composite Frames: Part 1,” Journal of the Structural
Division, ASCE, Vol. 115, No. 11, Nov. 1989, pp. 2858-2876.
Shome, N. (1999), “Probabilistic Seismic Demand Analysis of Nonlinear Structures,”
PhD Thesis, Department of Civil and Environmental Engineering, Stanford
University, Stanford, CA 94305.
Shome, N., Cornell, C.A., Bazzurro, P., and Carballo, J.E. (1997), “Earthquake, Records
and Nonlinear MDOF Responses,” Report No. RMS-29, Reliability of Marine
Structures, Dept. of Civil and Environmental Engineering, Stanford University,
Stanford, CA 94305.
Shome, N., Cornell, C.A., Bazzurro, P., and Carballo, J.E. (1998), “Earthquake, Records
and Nonlinear MDOF Responses,” Earthquake Spectra, EERI, Vol. 14, No. 3, pp.
469-500.
Somerville, P.G. (1998), “Development of an Improved Ground Motion Representation
for Near Fault Ground Motions,” SMIP98 Seminar on Utilization of Strong-Motion
Data, Oakland, CA, Sept. 1998.
Somerville, P.G., and Graves, R.W. (1993), “Conditions that Give Rise to Unusually
Large Long Period Ground Motions,” The Structural Design of Tall Buildings 2, pp.
211-232.
Somerville, P.G., Smith, N.F., Graves, R.W., and Abrahamson, N.A. (1997),
“Modification of Empirical Strong Ground Motion Attenuation Relations to Include
the Amplitude and Duration Effects of Rupture Directivity,” Seismological
Research Letters, Vol. 68, No. 1, pp. 199-222.
Somerville, P., Smith, N., Punyamurthula, S., and Sun, J. (1997), “Development of
Ground Motion Time Histories for Phase 2 of the FEMA/SAC Steel Project, Report
No. SAC/BD-97-04.
Stephens, J.E., and Yao, J.T.P. (1987), “Damage Assessment Using Response
Measurement,” Journal of Structural Engineering, ASCE, Vol. 113, No. 4, pp. 787801.
440
Stone, W.C., and Taylor, A.W. (1993), “Seismic Performance of Circular Bridge
Columns Designed in Accordance with AASHTO/CALTRANS Standards,” NIST
Building Science Series 170, National Institute of Standards and Technology,
Gaithersburg, MD.
Sugano, T. (1968), “Experimental Study of Hysteretic Characteristics of RC Members,”
Ph.D. Thesis, University of Tokyo, Japan.
Tagawa, Y., Kato, B., and Aoki, H. (1989), “Behavior of Composite Beams in Steel
Frame Under Hysteretic Loading,” Journal of Structural Division, ASCE, Vol. 115,
No. 8, Aug. 1989, pp. 2029-2045.
Trifunac, M.D., and Brady, A.G. (1975), “A Study of the Duration of Strong Earthquake
Ground Motion,” Bulletin of the Seismological Society of America, Vol. 65, pp.
581-626.
Uang, C.M. (1985), “Experimental and Analytical Study of the Hysteretic Behavior of
Steel Composite Girders,” CE 299 Report, Department of Civil Engineering,
University of California, Berkeley, April 1985.
Uang, C.M. (1991), “Establishing R (or Rw) and Cd Factors for Building Seismic
Provisions,” Journal of Structural Engineering, Vol. 117, No. 1, pp. 21.
Uang, C.M., and Maarouf, A. (1993), “Seismic Displacement Amplification Factor in
Uniform Building Code,” Structural Engineers Association of Northern California
Research Committee, San Francisco, California, June, pp. B-1.
UBC (1997), “Uniform Building Code,” International Conference on Building Officials,
Whittier, CA.
Vecchio, F.J., and Emara, M.B. (1992), “Shear Deformations in Reinforced Concrete
Frames,” ACI Structural Journal, Vol. 89, No. 1, Jan.-Feb. 1992, pp. 46-56.
Watson, S., and Park, R. (1994), “Simulated Seismic Load Tests on Reinforced Concrete
Columns,” Journal of Structural Engineering, ASCE, Vol. 120, No. 6, pp. 18251849.
Williams, M.S., and Sexsmith, R.G. (1995), “Seismic Damage Indices for Concrete
Structures: A State-of-the-Art Review,” Earthquake Spectra, Vol. 11, No. 2, May
1995, pp. 319-349.
Wight, J. (1997-98), “Seismic Resistance of Exterior RCS Connections,” Presentation at
4th and 5th JTCC Meetings of US-Japan Program.
Yang, Y.B., and Kuo, S.R. (1994), “Theory and Analysis of Nonlinear Framed
Structures,” Prentice Hall, Simon and Schuster (Asia) Pte Ltd, 1st Edition.
441
Zhao, Y. (1993), “Modeling of Inelastic Cyclic Behavior of Members, Connections, and
Joint Panels of Steel Frames,” PhD Thesis, School of Civil and Environmental
Engineering, Cornell University, Ithaca, NY 14853.
442
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