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UNIVERSITY OF CINCINNATI
Date:___________________
I, _________________________________________________________,
hereby submit this work as part of the requirements for the degree of:
in:
It is entitled:
This work and its defense approved by:
Chair: _______________________________
_______________________________
_______________________________
_______________________________
_______________________________
Nov 15th/2005
Gang Xuan
Master of Science
University of Cincinnati
Performace-Based Design of a 15-story
Reinforced Concrete Coupled Core Wall Structure
Dr. Bahram M. Shahrooz
Dr. T. Michael Baseheart
Dr. Gian A. Rassati
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Abstract
The reinforced concrete coupled core wall (CCW) structures have been widely used
in the medium to high-rise buildings due to their advantages both in the architectural and
structural aspects. The structures not only accommodate the versatile architectural needs,
but provide large lateral load resistance to withstand earthquake and wind.
The design of CCWs is typically based on the traditional strength-based method,
which is the basis of current codes. However, the resulting extremely high shear stresses
in coupling beams have been a long-lasting difficulty associated with the use of
strength-based methods for seismic design of CCWs. The performance-based design
(PBD) method, as a solution to the aforementioned problem, has been recently proposed
in an attempt to capture the expected behavior of CCW buildings subjected to ground
motions, while producing safe and constructible buildings.
In this thesis, a 15-story reinforced CCW office building was initially designed by
using the strength-based design method. The resulting high shear stresses in beams
exceed the code limits, and no suitable design could be found unless unrealistic measures
such as artificial reduction of beam stiffness are used to lower the demands. Subsequently,
the PBD method was applied as an alternative to the same building. The coupling beams
and wall piers were designed with acceptable internal forces below the code limits. As
necessary, the design provisions form NEHRP 2000, ACI 318-02, and FEMA 356 were
used. An analytical model was developed to generate the force-deformation
characteristics of diagonally reinforced concrete coupling beams. This model was
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calibrated based on experimental data from previous studies on coupling beams. Using
this model and prior experience with modeling of wall piers, a detailed analytical model
of the 15-story prototype was conducted. The applicability and validity of the PBD
method used in this study were demonstrated through nonlinear static and dynamic
analyses of the prototype structure.
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Table of Contents
List of Tables.................................................................................................................v
List of Figures.............................................................................................................vii
Chapter 1 Introduction .................................................................................................1
1.1 Notations.........................................................................................................1
1.2 Reinforced Concrete Coupled Core Wall System...........................................1
1.3 Diagonally Reinforced Concrete Beam ..........................................................2
1.4 Strength-Based Design and Performance-Based Design Methodologies.......3
1.5 Scope of Thesis ............................................................................................... 5
Chapter 2 Preliminary Design......................................................................................9
2.1 Notations.........................................................................................................9
2.2 Objective........................................................................................................11
2.3 Design Preparation.........................................................................................11
2.4 Loads and Analytical Model..........................................................................12
2.4.1 Gravity Loads.......................................................................................12
2.4.2 Seismic Loads ......................................................................................13
2.4.2.1 Design Response Spectrum........................................................13
2.4.2.2 ELF Method...............................................................................13
2.4.3 Mathematical Model ............................................................................15
2.5 Comparison of Four Prototype Models..........................................................16
Chapter 3 Design of Diagonally Reinforced Concrete Coupling Beams ...................24
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3.1 Notations........................................................................................................24
3.2 Introduction....................................................................................................27
3.3 Traditional Strength-Based Design ................................................................27
3.4 Traditional Strength-Based Design Result Review........................................30
3.5 Introduction of Performance-Based Design Method .....................................33
3.5.1 Performance-Based Design Concept ...................................................33
3.5.2 Changes of Design Requirements Using PBD Method .......................35
3.5.3 Diagonally Reinforced Concrete Coupling Beam Design by PBD
Method ..................................................................................................36
Chapter 4 Design of Wall Piers...................................................................................47
4.1 Notations........................................................................................................47
4.2 Introduction....................................................................................................50
4.3 Simplified Method for Wall Pier Analyses ....................................................51
4.3.1 X Direction Analyses ...........................................................................52
4.3.2 Y Direction Analyses ...........................................................................54
4.4 Load Combinations........................................................................................55
4.5 Wall Pier Design ............................................................................................60
Chapter 5 Studies of Behaviors of Diagonally Reinforced Concrete Coupling
Beams..........................................................................................................71
5.1 Notations........................................................................................................71
5.2 Objective........................................................................................................73
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5.3 Test Data ........................................................................................................73
5.4 Evaluation of Theoretical Models..................................................................74
5.4.1 Paulays Model.....................................................................................74
5.4.2 Hindis Model ......................................................................................76
5.5 FEMA 356......................................................................................................78
5.6 Statistical Analyses and Evaluation of Methods............................................78
5.6.1 Yield Strength.......................................................................................79
5.6.2 Ultimate Strength.................................................................................79
5.6.3 Yield Chord Rotation ...........................................................................80
5.6.4 Ultimate Chord Rotation......................................................................81
5.7 Modified Model .............................................................................................81
Chapter 6 Nonlinear Static and Dynamic Analyses....................................................94
6.1 Notations........................................................................................................94
6.2 Objective........................................................................................................95
6.3 Pushover (Static Nonlinear) Analysis ............................................................95
6.3.1 Introduction..........................................................................................95
6.3.2 Computer Model ..................................................................................95
6.3.2.1 Geometry and Mass Configuration............................................95
6.3.2.2 Coupling Beam Member Properties...........................................96
6.3.2.3 Wall Member Properties ............................................................97
6.3.2.4 Applied Lateral Loads................................................................98
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6.3.3 Results and Discussions.......................................................................99
6.4 Nonlinear Dynamic Analysis .......................................................................102
6.4.1 Computer Model ............................................................................... 102
6.4.2 Results and Discussions.................................................................... 103
Chapter 7 Conclusions and Recommendations for Future Research........................132
7.1 Summary......................................................................................................132
7.2 Conclusions..................................................................................................133
7.3 Recommendations for Future Research.......................................................135
Reference ..................................................................................................................137
Appendix A Preliminary Design Calculations..........................................................A-1
Appendix B Beam Design Calculations ...................................................................B-1
Appendix C Wall Design Calculations .....................................................................C-1
Appendix D Calculated Wall Pier Parameters from XTRACT for RUAUMOKO
Modeling..............................................................................................D-1
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List of Tables
Table2.1 Design of a Typical Interior Column............................................................19
Table2.2 Design Spectrum Defined by NEHRP.........................................................19
Table2.3 Performance Comparison of Four Prototype Structures ..............................20
Table 3.1 Mass Participation of the First Two Modes in the Coupled Direction........38
Table 3.2 Base Shear Amplification Factor ................................................................38
Table 3.3.1 Beam Shears of Mode 1 after Amplifications..........................................39
Table 3.3.2 Beam Shears of Mode 2 after Amplifications..........................................39
Table 3.4 SRSS of Beam Shear Forces and Related Shear Stresses...........................40
Table 4.1.1 Lateral Load Effects and Effective Moments in the X Direction ............61
Table 4.1.2 X Direction Lateral Load Effect Distribution between Wall Piers ..........61
Table 4.2 X Direction Torsion Analysis......................................................................62
Table 4.3 Y Direction Lateral Load Effect Distribution between Wall Piers..............62
Table 4.4 Y Direction Torsion Analysis ......................................................................63
Table 4.5.1 Design Demands for Biaxial Bending Design with 1.0X+0.3Y
Combination...................................................................................................63
Table 4.5.2 Design Demands for Biaxial Bending Design with 0.3X+1.0Y
Combination......................................................................................................64
Table 4.6.1 Design Demands for Shear Design with 1.0X+0.3Y Combination .........64
Table 4.6.2 Design Demands for Shear Design with 0.3X+1.0Y Combination .........65
Table 5.1 Diagonally Reinforced Concrete Beam Test Database ...............................84
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Table 5.2 Strengths and Deformations Calculated According to Paulays Model......85
Table 5.3 Strengths and Deformations Calculated According to Hindis Model........86
Table 5.4 Strengths and Deformations Calculated According to FEMA 356
Method ................................................................................................................88
Table 5.5 Evaluation of All Models ............................................................................89
Table 5.6 Strengths and Deformations Calculated According to Modified Model.....91
Table 6.1 Beam Member Properties..........................................................................106
Table 6.2 Values of Four Control Points for Quadratic Beam-Column Elements ....106
Table 6.3 Wall Member Properties............................................................................106
Table 6.4 Strength Degradation Factors....................................................................107
Table 6.5 Maximum Chord Rotations under Five Selected Ground Motions ..........107
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List of Figures
Fig.1.1 Lateral Load Resisting Mechanism of a Coupled Core Wall System .............7
Fig.1.2 Flow Chart of a Conceptual Framework for the Performance-Based
Design (Bertero, 1997).....................................................................................8
Fig.2.1 Elevation View of the 15-story Coupled Core Wall Building ........................21
Fig.2.2 Column Tributary Area and X Y Coordinate System.....................................21
Fig.2.3 Planar View of Prototype I .............................................................................22
Fig.2.4 Planar View of Prototype II ............................................................................22
Fig.2.5 Planar View of Prototype III...........................................................................23
Fig.2.6 Planar View of Prototype IV ..........................................................................23
Fig.3.1 Labels of Wall Piers Used in the Redundancy Factor Calculation.................41
Fig.3.2 Deformation Relationship between Coupling Beam and Wall Piers..............41
Fig.3.3 Tri-Stage Mechanism of CCWs in PBD.........................................................42
Fig.3.4 Comparison of Design Demands on CCW Elements between Strength-Based
Method and Performance-Based Method .......................................................42
Fig.3.5 Assignment of Coupling Beam Design Shear Stresses ..................................43
Fig.3.6.1 Section Details of Beam Group I.................................................................44
Fig.3.6.2 Section Details of Beam Group II ...............................................................45
Fig.3.6.3 Section Details of Beam Group III ..............................................................46
Fig.4.1.1 X Direction Lateral Load Analysis..............................................................66
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Fig.4.1.2 X Torsion Analysis ......................................................................................66
Fig.4.2.1 Y Direction Lateral Load Analysis ..............................................................67
Fig.4.2.2 Y Torsion Analysis.......................................................................................67
Fig.4.3.1 Section Details of Wall Group I...................................................................68
Fig.4.3.2 Section Details of Wall Group II .................................................................69
Fig.4.3.3 Section Details of Wall Group III................................................................70
Fig.5.1 Force Equilibrium of Paulays Model ............................................................92
Fig.5.2 Coupling Beam Vertical Deformation of Paulays Model..............................92
Fig.5.3 Force Equilibrium of Hindis Truss Model.....................................................93
Fig.5.4 Shear-Chord Rotation Relationship Defined by FEMA 356 ..........................93
Fig.5.5 Shear-Chord Rotation Relationship Defined by Modified Model..................93
Fig.6.1 Nonlinear Analyses Model ...........................................................................108
Fig.6.2 Axial Load-Moment Interaction Diagram for Quadratic Beam-column
Element .............................................................................................................. 109
Fig.6.3 Pushover Analysis Result .............................................................................110
Fig.6.4 Beam Vertical Deformation Caused by Rigid Link Rotations......................111
Fig.6.5 Chord Rotation Distributions at LS and CP States.......................................111
Fig.6.6 Modified Takeda Hysteresis Model..............................................................112
Fig.6.7 Strength Degradation Model Used in RUAUMOKO...................................112
Fig. 6.8 Selected Earthquake Ground Motions.........................................................113
Fig. 6.9 Acceleration Response Spectra of Earthquake Records Induced by 5
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Selected Ground Motions ...............................................................................114
Fig.6.10 Roof Displacement History ........................................................................115
Fig.6.11 Story Drift Envelope...................................................................................116
Fig.6.12 Member Responses under El Centro Ground Motion ................................117
Fig.6.13 Member Responses under Simulated LS Ground Motion..........................120
Fig.6.14 Member Responses under Simulated CP Ground Motion..........................123
Fig.6.15 Member Responses under Northridge Pacoima Ground Motion ...............126
Fig.6.16 Member Responses under Northridge Slymar Ground Motion..................129
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Chapter 1 Introduction
1.1 Notations:
otmM --Total overturning moment caused by lateral loads
1M , --Moments resisted by the tension and compression wall piers, respectively2M
T L --Moment due to the coupling effect; T is equal to the axial force at the base of
tension wall pier; L is the coupling arm, the distance between the centroids of two
wall piers.
1.2 Reinforced Concrete Coupled Core Wall System
The reinforced concrete coupled core wall (CCW) systems have been widely used in
mid to high-rise buildings due to the architectural and structural advantages. The concrete
cores in the middle of the structures accommodate elevator shafts, stairwells and service
ducts to meet versatile architectural requirements. Additionally, the use of flat slab floors
in CCW systems provides more architectural efficiency by reducing story heights. Most
of all, CCW systems are very effective in resisting lateral loads in earthquakes and
hurricanes. The effectiveness of the systems is demonstrated by the way they withstand
the lateral loads: the structural lateral load resisting capacities are not increased through
enlarging the member sizes, but through introducing the frame action. As Fig. 1.1 shows,
two cantilever wall piers are connected by the coupling beams in between. Due to the
frame action of the system, a tension force and a compression force are produced in the
left and right wall piers, respectively. The magnitudes of the tension and compression are
identical, either of which is equal to the sum of all coupling beam shear forces. The total
overturning moment from the lateral loads ( ) is resisted not only by the wall piersotm
M
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( and ), but also by the coupling effect (1M 2M T L ) due to the frame action. Hence, the
frame action greatly decreases the internal forces on wall piers and then reduces the
deformation of the building. The degree of the frame action is expressed by a term known
as the degree of coupling (DOC), which is defined as the ratio of T L to . DOC
equal to 0 means that no frame action exists and the system behaves as two isolated
cantilever walls. On the other hand, DOC equal to 1 represents that two walls act in the
way as a single solid wall. The national building code of Canada (NBCC) quantifies
DOC to indicate the effectiveness of CCW systems. The buildings with DOC less than
66% are classified as partially coupled walls and those with DOC greater than 66% are
considered as effectively coupled walls.
otmM
1.3 Diagonally Reinforced Concrete Beam
The use of diagonally reinforced concrete beams instead of conventional concrete
beam is recommended by ACI 318-02 when the ratio of the beam span to depth is less
than 4. The preference of diagonally reinforced concrete beams is based on their good
performance in terms of ductility and strength under cyclic loads.
Experiments have illustrated the following disadvantages of conventional concrete
beams with small span-to-depth ratio under seismic loads (Park and Paulay, 1975). (1)
The compression stress of concrete is not reduced by placing compression reinforcement
and correspondingly the increase of ductility of the beam should not be expected. The
reason is that the diagonal cracks of the beam under reverse loads cause a radical
redistribution of the tensile forces and tensile stress exists where conventional flexure
theory indicates that compression stresses should be present. Therefore, the compression
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reinforcement actually carries the tension forces instead of resisting the compression as
expected. (2) The insufficiency of shear capacities of the interfaces between the beams
and wall piers results in the direct sliding shear failure. Considering the flexure
reinforcement dowel action can only transmit small amount of shear forces from the
beams to wall piers, the bulk of the beam shear must be transferred across the concrete
compression zones into the wall piers. However, the compression concrete zones have
little shear-transferring ability because they have already been cracked during the
preceding load cycles. (3) The stiffness of the conventional coupling beams with
sufficient web reinforcement after the onset of diagonal cracking is reduced to 1/5 of the
stiffness before crack. For the conventional beams without sufficient web reinforcement,
the stiffness degradation is greater. The drastic loss of stiffness considerably reduces the
frame action and increases the deformation of the buildings.
In contrast to the conventionally reinforced concrete beams, diagonally reinforced
concrete beams have superior cyclic responses even under high intensity alternating loads
(Park and Paulay, 1975). Experiments show that the hysteretic loop for a diagonally
reinforced concrete beam exhibits small stiffness degradation. Also, the beam displays
little strength reduction with the cumulative ductility. Due to its good seismic
performance, the diagonally reinforced concrete beams are employed in the design of the
building presented in this research.
1.4 Strength-Based Design and Performance-Based Design Methodologies
The strength-based design method requires that each individual member in the
system has sufficient capacities to resist the forces induced by predetermined loads. The
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strength-based design method is the basis of current building codes. ASCE 7-02 and IBC-
2003 codes provide the guidelines for determining the design loads and analytical
methods. ACI 318-02 and AISC-99 codes are the design specifications for the concrete
and steel members, respectively.
The application of the strength-based design method to the design of CCW systems
causes a problem: the design shear stresses in the coupling beams exceed the code-
defined (ACI 318-02) limit (Harries et al., 2004). The high shear stresses are attributed to
the assumption that the wall piers and beams yield simultaneously at the code specified
base shear. However, the 1964 Alaska earthquake indicates that all or most coupling
beams yielded before the strength of the coupled walls was attained. Theoretical studies
also verify that the critical coupling beams yield before the required ductility of the
systems is achieved (Park and Paulay, 1975).
Recently, researchers (Harries et al., 2004) have proposed a performance-based
design (PBD) method as an alternative of the strength-based design method in CCW
design. Concisely, the PBD method is defined as Design and Engineering of buildings
for targeted performance objectives (Bertero, 1997). The selection of the performance
objectives involves several factors as the following. Firstly, the selection is made by the
owner in consultation with the designers, based on the owners expectations, economic
analysis, and the accepted risks. Secondly, the selected performance needs to meet the
structural actual seismic behavior. Thirdly, the performance objectives need to be
determined for different earthquake hazard levels. The multi-level design methodology
has been advocated (Bertero, 1997) to replace the current code one-level design
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methodology because the multi-level method improves the design safety, reliability, and
also optimizes the design procedures to reduce the cost.
A complete set of design steps using PBD method is illustrated in Fig. 1.2.
Especially, the following steps can be specified (Harries et al. 2004) for seismic design of
CCWs: (1) Define the desired performance objectives; (2) Design coupling beams; (3)
Design wall piers; (4) Develop nonlinear force-deformation relationship for beams and
wall piers; and (5) Conduct nonlinear static and dynamic analyses to check the design
results.
1.5 Scope of Thesis
A 15-story reinforced concrete coupled core wall building was initially designed by
using the traditional strength-based method. The difficulty of the traditional method
meeting the design shear limit in current building codes was encountered. Subsequently,
the PBD method was used as an alternative to the same building. The performance of the
building, designed by following PBD method, was evaluated by nonlinear static and
dynamic analyses. Before the nonlinear analyses, an analytical model for establishing the
nonlinear behavior of diagonally reinforced concrete beams was developed and verified
through the use of experimental data available in literature.
The thesis is organized in seven chapters. Chapter 1 briefly presents the current state
of knowledge about coupled core wall systems. Chapter 2 shows the preliminary design
of the 15-story building to determine its specific structural layouts. Chapter 3 provides
the design procedures of the diagonally reinforced concrete coupling beams with the
strength-based method and performance-based method. Chapter 4 presents the
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calculations for the wall piers by using the performance-based method. Chapter 5 shows
the development of a theoretical model to characterize the nonlinear behaviors of
diagonally reinforced concrete beams. Chapter 6 presents the nonlinear analyses of the
designed coupled core wall system. Chapter 7 provides the conclusions and the
suggestions for the future research.
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p
TM1
L
2MT
V1 V2
C
Fig. 1.1 Lateral Load Resisting Mechanism of a Coupled Core Wall System
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Check Suitability of site
Yes
Discuss with client the performance levels and
select the minimum performance design objectives
Yes
No
Yes
Conduct conceptual overall design, selecting configuration,structural layout, structural system, structural material and
nonstructural com onents
Acceptability checks of
conceptual overall design
NoAcceptability checks of preliminary
design using static, dynamic linear
and nonlinear analysis methods
Numerical preliminary design to complysimultaneously with at least two limit states
Yes
No
Final design and detailing using availableexperimental data and presenting material codes and
re ulations
Acceptability checks of final design usingstatic, dynamic linear and nonlinear
analysis methods and experimental data
Yes
Quality assurance during construction
Monitoring, maintenance and function
Fig 1.2 Flow Chart of a Conceptual Framework for Performance-Based Design
(Bertero, 1997)
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Chapter 2 Preliminary Design
2.1 Notations:
xA : Torsion amplification factor
sC : Seismic response coefficient in the ELF method
E: Elastic modulus
aF : Site coefficient
'
cf : Concrete compression strength
yf : Steel yield strength
vF : Site coefficient
g: Gravity acceleration
I : Occupancy important factor
gI : Section gross moment of inertia
taM : Accidental torsion
R : Response modification factor
DsS : Design spectral response acceleration at short period
1DS : Design spectral response acceleration at 1 second period
MsS : Adjusted maximum considered earthquake spectral response acceleration at
short period
1MS : Adjusted maximum considered earthquake spectral response acceleration at
1 second period
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sS : Maximum considered earthquake spectral response acceleration at short
period
1S : Maximum considered earthquake spectral response acceleration at 1 second
period
0T : Period parameter used to determine the design response spectrum, equals to
0.2 /1DS DsS
1T : Period parameter used to determine the design response spectrum, equals to
/1DS DsS
bV : Design base shear from the ELF method
W : Building total weight
avg : Average displacement of the floor
max : Maximum displacement of the floor
: Strength reduction factor
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2.2 Objective
The detailed layouts of a 15-story reinforced concrete coupled core wall office
building are presented specified in this chapter. The layouts to be configured include the
following: (i) story and building total height; (ii) locations of the perimeter columns, wall
piers, and coupling beams; (iii) dimensions of walls, beams, columns, and floor slabs.
The initial layout was based on a previous similar research focused on a 10-story
reinforced concrete core wall structure (Harries et al., 2004). The results of the
preliminary design were evaluated by two criteria from current building codes. The first
is that the maximum story drift should not be more than 2% as required by NEHRP 2000.
The second is that the degree of coupling (DOC) should be greater than 66%, which is
the minimum value defined by NBCC 1995 for effectively coupled systems.
2.3 Design Preparation
The structure (see Fig. 2.1) is a 15-story reinforced concrete coupled core wall
office building assumed to be located in San Francisco, CA in class C site. Stories 2
through 15 each are 9 feet and 2 inches high and the ground story is 12 feet and 2 inches
high. The total building height, therefore, is 140 feet and 6 inches. Post-tensioned
reinforced concrete slabs, 8 inches thick and 100100 square feet large, are used in every
floor of the building.
The building has two load resisting systems: (a) columns uniformly distributed
around the floors (see Fig. 2.2) and (b) a coupled core wall in the middle of the building.
The core wall consists of two C shaped wall piers, which are connected by two coupling
beams located at the ends of wall flanges. Considering the lateral stiffness of the central
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core is much larger than that of the columns, it is assumed that the concrete core carries
all of the lateral loads and resists the gravity loads in conjunction with the perimeter
columns. The design of a typical interior column is shown in Table 2.1. The gravity load
within its tributary area is used. Also for simplicity, it is assumed that all other columns
in a floor have the same dimensions as interior columns.
2.4 Loads and Analytical Model
2.4.1 Gravity Loads
Section 5.3 of NEHRP states that the gravity loads in the seismic design should
cover the total dead loads and applicable portion of other loads listed in the following. (i)
25 percent of floor live load shall be applicable in areas used for storage. The selected
building is for office usage; hence, this item is not included. (ii) Partition load should not
be less than 10 psf. The minimum partition load of 10 psf is taken into account in the
calculations. (iii) Operation equipment load. A 5 psf mechanical device load is included.
(iv) Snow load. It is not included in the design because of the location of the building.
Other than these code-defined gravity loads, a cladding load of 15 psf on each side of the
building surfaces is included. The dead loads include the self-weight of the building, i.e.,
the weights of the post-tensioned floor slab, wall piers, link beams, and columns.
In the analytical model, the gravity loads from columns and walls are
concentrated at the center of mass of each floor. The floor heights above and below are
used to calculate the floor mass. Accordingly, the gravity loads assigned to the top and
ground floor will be less and more, respectively, than typical floors in the middle of the
building.
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2.4.2 Seismic Load
2.4.2.1 Design Response Spectrum
NEHRP describes the earthquake motion with the following two factors. is the
maximum ground motion at short period and is that at 1 second. In San Francisco,
and are taken as 1.5g and 0.65g, respectively. The values of and should be
modified to include the influence from specific site conditions by using factors and
. ( ) and ( ) are the results after the site effect adjustment to
represent the structural acceleration response at the short period and the period of 1
second, respectively. These values are based on the exceedance probability of 2 percent
in 50 years, which is defined as the collapse prevention (CP) level earthquake by
NEHRP. Hence, the calculated values need to be multiplied by 2/3 to generate the design
response spectrum. The design response spectrum in NEHRP is based on the exceedance
probability of 10 percent in 50 years, which is defined as life safety (LS) level
earthquake. Additionally, two period values, and , are used to separate the spectrum
into three parts, which are short period section, peak value section, and long period
section, respectively. Table 2.2 shows the shape and the calculations of the design
response spectrum.
sS
1S sS
1S sS 1S
aF
vF MsS sS aF 1MS MsS vF
0T sT
2.4.2.2 Equivalent Lateral Force (ELF) Method
The structure is classified into seismic design category D by its specific site
condition. Based on the seismic design category and structural symmetrical
configuration, the equivalent lateral force (ELF) method may be used to calculate the
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lateral seismic loads on the prototype. The basic idea of ELF is to calculate the maximum
seismic response ( ) of the building from the design response spectrum (see Table 2.2).
The code defined base shear ( ) is determined as the product of with the building
total weight (W). The base shear ( ) is distributed to various floors based on the weight
and height of each floor.
sC
bV sC
bV
The following parameters are required for the ELF method. The response
modification factor (R ) was selected as 6 in accordance with the structure type specified
in NEHRP Table 5.2.2. The occupancy important factor (I ) was taken as 1 (see NEHRP
Table 1.4) considering the structure is an ordinary office building.
The accidental torsion ( ) corresponding to the lateral loads in each main
direction should be included in the calculations, as the required by Section 5.4.4.2 of
NEHRP. The inclusion of the accidental torsion for a symmetric building is to account
for some factors that have not been explicitly considered in NEHRP, such as the
rotational component of ground motion, unforeseeable differences between computed and
actual values of stiffness, etc. The magnitude of the accidental torsion at one level is
equal to the lateral force at that level multiplied by 5 percent of the building dimension
perpendicular to the direction of the applied lateral load. Furthermore, Section 5.4.4.3 of
NEHRP states that for structures in the seismic design category D, the accidental torsion
at each level needs to be scaled up by a torsion amplification factor ( ), defined as the
following.
taM
xA
xA =(avg
2.1
max ) (2.1)2
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max is the maximum displacement occurred at the corner of the building and avg is the
average displacement at the center of building. The average value of in all levels
representing the average torsion influence was used in the calculation (Brienen, 2002).
xA
2.4.3 Mathematical Model
ETABS (CSI Berkeley, 1997) was employed to conduct the elastic analyses. The
following types of elements were used to represent the different structural members of
the building.
(a) The columns were modeled by column elements. The elements have been
formulated to include the effect of axial, shear, bending, and torsional deformations.
Considering that the columns in the building are assumed to carry the vertical loads only
without any lateral resistance, the column elements in the model are pinned both at the
top and bottom. (b) The post-tensioned concrete slabs in the building are modeled as rigid
diaphragms, which have infinite in-plane stiffness. (c) The flanges and webs of the C
shaped walls are represented by ETABS panel elements. Each panel element has been
formulated as a membrane member with iso-parametric properties. The panels are
continuous from level to level and fixed at the base of the building. ETABS automatically
assembles three adjacent panels together to form the C shaped wall, which is considered
as one unit in the analyses. (d) The coupling beams are represented by the beam
elements, which have been formulated to include the effect of axial, shear, bending, and
torsional deformations. The beam elements are rigidly connected to the wall panels.
ACI 318-02 was used to determine the stiffness of various components. Per
Section 10.11.1 of ACI, the member stiffness should account for the presence of axial
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loads, cracks along the length of the member, and duration of the loads. Also, the
following values are suggested by ACI for typical reinforced concrete structural
members. For a cracked wall, the stiffness is taken as 0.35 E gI ; and for an un-cracked
wall, it is taken as 0.70E gI . Usually, the wall piers in the ground story suffer more
damage, and as a result the stiffness is less than that in other stories. Hence, in the
analyses, the stiffness for the ground story wall piers was taken as 0.35E gI , and the
stiffness for the walls in other stories was assumed to be 0.70E gI . Moreover, per ACI,
0.35E gI was used as the effective stiffness for coupling beams. Note that other
equations are available to establish stiffness of diagonally reinforced coupling beams
(Paulay, 1992). For consistency, ACI recommendations were used both for the walls and
coupling beams. The distribution of mass is described in Section 2.4.1.
This ETABS model also includes the P- effect in the force and deformation
analyses. The concrete used is normal weight concrete with compression strength ( ) of
6 ksi, and the reinforcement is Grade 60 with yield strength ( ) of 60 ksi.
'
cf
yf
2.5 Comparison of Four Prototype Models
The computer model described in the previous section was used to evaluate four
structures shown in Figures 2.3 to 2.6. These analyses were conducted to finalize the
layouts of the prototype structure. The accepted prototype must be proportioned such that
two criteria are satisfied. One is the maximum story drift of the building should be within
the 2% limit defined by NEHRP. The other is that the degree of coupling (DOC) should
be greater than 66 percent, as NBCC states. Table 2.3 provides a brief review of the
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configurations and performance of these 4 models. The evolution of these 4 models is
detailed in the following.
Prototype I (see Fig. 2.3) was directly extracted from a 10-story building
investigated in a previous study (Harries et al., 2004). The flange wall is 9 feet long and
20 inches thick, and the web wall is 18 feet long and 16 inches thick. The coupling beams
connecting the two wall piers are 6 feet long with a section of 20 in24 in. The building
is symmetrical about the X and Y axes. For simplicity, it is assumed that the wall
dimensions remain the same over the total height of the building.
The calculations of loads, internal forces and deformations of this prototype are
listed in Tables A.1.1 to A.1.5 in Appendix A. The results show that the maximum story
drift in the X direction is 3.93% and 4.28% in the Y direction, which exceed the 2% limit.
Hence, the prototype is unacceptable. The DOC of the building is 79.7%, which satisfies
the 66% minimum DOC requirement.
The flange walls in Prototype II (see Fig. 2.4) were changed from 9 feet to 10
feet, and the web walls were changed from 18 feet to 20 feet. The thickness of the flanges
and webs was changed from 16 inches to 20 inches. The beam dimensions remain the
same as Prototype I. The purpose of the changes is to increase the structural stiffness and
correspondingly reduce the maximum story drift to meet the 2% limit. The calculations
shown in Tables A.2.1 to A.2.5 indicate that the maximum story drift in the X direction is
2.81% and 2.95% in the Y direction. The results also show that the DOC is 75.5%. Hence,
Prototype II also does not meet the 2% story drift limit.
The difference between Prototype III (see Fig. 2.5) and II is that the web walls
were changed from 20 feet to 22 feet long. All other dimensions were kept the same. The
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maximum X story drift is 2.62% and the Y story drift is 2.41% (see Tables A.3.1 to
A.3.5). The structure has a DOC of 75.5%. Prototype III still does not meet the 2%
deformation limit.
The differences between Prototype IV (see Fig. 2.6) and Prototype III are that the
length of the web walls was extended from 22 feet to 25 feet, and the dimensions of
coupling beams were enlarged from 20in24in to 20in30in. The enlargement of the
beam sections can keep the relative stiffness between the wall and the beam in order to
maintain the degree of coupling, and provide more construction space to avoid
congestion problems. The calculations of the maximum displacements and degree of
coupling shown in Tables A.4.1 to A.4.5 (see Appendix A) indicate that Prototype IV
meets both design criteria. This structure has a maximum story drift of 1.97% and 1.73%
in the X and Y direction, respectively. The DOC of the structure is 79.7%. Prototype IV
is selected for all the subsequent analyses and discussions.
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Table 2.1 Design of a Typical Interior Column
Dead Loads (psf)
8 in Slab 100
Partitions 10
Devices 5
Total 115Live Loads (psf)
For Office 50
Loads Combination
1.2Dead Load+1.6Live Load (psf) 1.2x115+1.6x50=218
Tributary Area (ft2) 20x20=400
Total Design Load on One Story (kips) 218x400/1000=87.2
Total Design Load of 15 Storys (kips) 15x87.2=1308
Required Area of the Column (in2) Assuming fc'=6 ksi =0.7 1308/(0.7x6)=311
Square Root of the Required Area (in) 3110.5=18
Actual Size of the Square Column (in) 20
Table 2.2 Design Spectrum Defined by NEHRP
Item Value Comments
Ss 1.5g Directly from maps of NEHRP
S1 0.65g Directly from maps of NEHRP
Fa 1 Determined by Table 4.1.2.4a of NEHRP
Fv 1.3 Determined by Table 4.1.2.4b of NEHRP
SMs 1.5g SMs=SsxFa
SM1 0.845g SM1=S1xFv
SDs
1.0g SDs
=2/3xSMs
SD1 0.563g SD1=2/3xSM1
T0 0.113 T0=0.2SD1/SDS
Ts 0.563 Ts=SD1/SDS
Sa=SD1/T
TsT0 T (s)
Sa(g)
0.000
0.200
0.400
0.600
0.800
1.000
1.200
0 1 2 3 4 5
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Table 2.3 Performance Comparison of Four Prototype Structures
Prototype Description
Max X
Story
Drift
Max Y
Story
Drift
DOC Comments
I
The layouts of this model (see
Fig.2.3) are from a previous 10-story
CCW building design. The flangewall in the X direction is 9 feet long
and 20 inches thick. The web wall in
the Y direction is 18 feet long and 16inches thick. The coupling beam is 6
feet long with a 20in24in section.
3.93% 4.28% 79.7%
The
maximum
X and Ystory drift
are both
over 2%limit.
II
The difference between this model
(see Fig. 2.4) and Prototype I is that
the flange wall in the X direction is
increased from 9 feet to 10 feet, andthe web wall in the Y direction is
from 18 feet to 20 feet. Each wallthickness is also increased from 18
inches to 20 inches.
2.81% 2.95% 75.5%
The
maximum
X and Ystory drift
are both
over 2%limit.
III
The difference between this model(see Fig. 2.5) and Prototype II is the
web wall in the Y direction is
increased from 20 feet to 22 feet.
2.62% 2.41% 75.5%
The
maximumX and Y
story drift
are bothover 2%
limit.
IV
The difference of this model (see Fig.
2.6) and Prototype III is that the webwall in the Y direction is lengthened
from 22 feet to 25 feet, and the beam
is enlarged from 20 in 24 in to
20in30in.
1.97% 1.73% 79.7%
This modelmeets the
2%
deformation
limit and66% DOC
limit.
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12'-2"
14
stories
at9
'-2"
Level 1
Level 2
Level 3
Level 4
Level 5
Level 6
Level 7
Level 8
Level 9
Level 10
Level 11
Level 12
Level 13
Level 14
Level 15
Fig. 2.1 Elevation View of the 15-story Coupled Core Wall Building
tributary area:20X20 ft2
20' 20' 20' 20' 20'
20'
20'
20'
20'
20'
of a typical interior column
X
Y
Fig 2.2 Column Tributary Area and X Y Coordinate System
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Beam section is 20in by 24 in
9' 6' 9'
18'
20"
16"
X
Y
Fig 2.3 Planar View of Prototype I
Beam section is 20in by 24 in
10' 6' 10'
20
'
20"
20"
X
Y
Fig 2.4 Planar View of Prototype II
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Beam section is 20in by 24 in
10' 6' 10'
22'
20"
20"
X
Y
Fig 2.5 Planar View of Prototype III
Beam section is 20in by 30 in
10' 6' 10'
25
'
20"
20"
X
Y
Fig 2.6 Planar View of Prototype IV
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Chapter 3 Design of Diagonally Reinforced Concrete Coupling Beams
3.1 Notations
A : Floor area
xA : Torsion amplification factor in the coupled direction
xavgA : Average of of all floorsxA
bC : Base shear amplification factor
, : Distances from the wall neutral axis to the edge of tension wall pier or
compression wall pier, respectively (see Fig. 3.2)
1c 2c
: Dead loadD
: Reinforcement bar diameterbd
: Length of wall section (see Fig. 3.2)wD
E: Structural response from seismic loads
: Lateral load of Mode m in the coupled directionxmF
'
cf : Concrete compression strength
eh : Effective building height, measured from the building base to the resultant
force position of the first mode in the coupled direction
: Length of a rectangular wall pierwl
: Accidental torsion associated withtaxmM xmF
EQ : Structural response from horizontal seismic loads
s : Span of link beam
DsS : Design spectral response acceleration at short period
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: Ductility factorbu
: Code-defined base shear calculated by the ELF methodbV
: Beam shear due tobfV xmF
btV : Beam shear due to taxmM
: Shear at the base when the link beams yieldbyV
xmV : Base Shear of Mode m in the coupled direction
: SRSS of base shear forces of all modes under considerationtV
: Shear at the base when the wall piers yieldwyV
uV : Ultimate base shear corresponding to structural ultimate displacement or
ultimate limit state
W : Building total weight
: Effective weight of Mode m in the coupled directionxmW
: Inclination of diagonal reinforcement
max : Maximum ratio of the shear on one single element to the story shear
: Vertical displacement different between point A and B (see Fig. 3.2)AB
: Vertical displacement between two ends of a link beamby
y : Steel yield strain
: Strength reduction factor
wy : Wall yield curvature
b : Link beam chord rotation
by : Link beam yield chord rotation
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w : Wall pier rotation
wy : Wall pier yield rotation
: Redundancy factor
: Beam shear stress
LS (life safety) and CP (collapse prevention) level seismic loads: the LS level
earthquake loads represent the seismic loads with 10 percent exceedance in 50
years, and NEHRP design spectrum is generated correspondingly to the LS level
ground motion. The CP level earthquake loads represent the loads with 2 percent
of exceedance in 50 years. The CP level seismic loads are much more intensive
than the LS level loads. The acceleration spectrum of CP level in NEHRP is 1.5
times that of LS level.
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3.2 Introduction
At the beginning of this chapter, the traditional strength-based design was carried
out by following NEHRP provisions. However, it is concluded that diagonally reinforced
concrete coupling beams cannot be designed because the shear stresses in coupling beams
exceed the ACI defined limit. After investigating plausible reasons for the large shear
stresses, the performance-based design (PBD) methodology is introduced. The PBD
method recognizes the expected seismic behavior of a CCW building by proposing a tri-
stage failure mechanism. As a result, the shear forces in beams were regenerated to an
accepted level. Finally, the coupling beams were detailed by following the requirements
in Chapter 21 of ACI 318-02.
3.3 Traditional Strength-Based Design
The modal response spectrum analysis (MRSA) method was selected to replace
the equivalent lateral force (ELF) method to calculate the lateral seismic loads and related
structural responses. The MRSA method allows the inclusion of higher modes of
structures in addition to the fundamental mode. Therefore more precise results are
possible. Per Section 5.5.2 of NEHRP, the MRSA method should include sufficient
modes to obtain the total modal mass participation of at least 90 percent. According to the
results listed in Table 3.1, the first two modes in the coupled direction, which
respectively correspond to the first and fifth mode of the structure, have provided 91
percent of mass participation, and should be sufficient for the required analyses.
Two types of seismic loads, the lateral loads ( ) and the accidental torsion
( ), are included in the modal analysis. The inclusion of is required by
xmF
taxmM taxmM
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NEHRP 5.4.4.2 to cover unforeseeable issues, which are not explicitly defined in the
code. Calculations of and are summarized in Tables B.1.1 and B.1.2 in
Appendix B. A 3-dimensional ETABS computer model, which includes two transverse
and one torsional degrees of freedom, was developed to calculate structural elastic
seismic responses. Per NEHRP, the results from ETABS elastic analyses still need to be
magnified by four different factors to obtain the design shear demands for coupling
beams.
xmF taxmM
The first magnification factor is the torsion amplification factor ( ). The
equation defining is provided in Section 2.4.2.2. The factor has been introduced by
NEHRP as an attempt to account for the structural torsional dynamic instability. The
shear forces from ETABS due to the accidental torsion ( ) were magnified by
before being combined with the shear forces induced by the lateral loads ( ). The
calculations of for the first two modes in the coupled direction are provided in
Tables B.2.1 and B.2.2, respectively.
xA
xA
taxmM xavgA
xmF
xavgA
The second factor to be considered is the redundancy factor () which is defined by
NEHRP as an index to increase the design reliability. Per Section 5.2.7 of NEHRP, the
response of the structure due to seismic loads (E) is defined as the following.
E= EQ 0.2 (3.1)DsS D
EQ is the responses due to horizontal seismic loads, which includes the effects from
horizontal lateral forces ( ) and associated torsion ( ). The item of 0.2
represents the effect of the vertical ground motion component, which is not considered in
the beam shear analyses. Hence, following Equation 3.1 the sum of beam shear forces
xmF taxmM DsS D
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due to and were magnified by the redundancy factor (xmF taxmM ). For wall piers, the
factor () is calculated as the following.
=2- Amax
20
(3.2)
A is the total area of the floor, which is equal to 100100 ft2. is the ratio of the shear
in a single element (torsional shear included) to the story shear. The subscript ofmax of
means that the maximum from all the elements should be taken. Additionally, per
Section 5.2.4.2 of NEHRP, the calculated needs to be multiplied by 10/ . Note that the
value of 10/ should not be greater than 1.0 per NEHRP. Walls in the C shaped section
are classified into two groups (see Fig. 3.1). The walls in the X direction are labeled as
P101, P102, P201, and P202 in Group I. The walls in the Y direction are labeled as P103
and P203 in Group II. Due to the symmetry of the building, the wall piers in the same
group resist the shear forces equally. Therefore, the elements in the same group produce
identical
wl
wl
values. The max used in the magnification is the greatest from these two
groups among all stories in the building. Table B.3.1 and B.3.2 illustrate the details of the
calculations of max and .
The third scaling factor for the beam shear forces is strength reduction factor ( ).
Per Section 9.3.4 (c) of ACI, is taken as 0.85 for the design of coupling beams.
The last magnification factor is the base shear amplification factor ( ). Section
5.5.7 of NEHRP states if the SRSS of the base shear forces of all the modes considered
( ) is less than 85% of the base shear from the ELF method ( ), all the seismic
bC
tV bV
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responses of the structure should be scaled up by multiplying with the factor of . is
defined by Equation 3.3 and its value is listed in Table 3.2.
bC bC
bC =0.85 / (3.3)bV tV
The applications of the aforementioned factors for the first two modes in the
coupled direction are listed respectively in Tables 3.3.1 and 3.3.2. Subsequently, the
SRSS of beam shears in these two modes were generated as the design demands. Table
3.4 lists the resulting shear and shear stresses along the building stories (in psi and in
terms of'
cf ).
3.4 Traditional Strength-Based Design Result Review
Section 21.7.7.4 of ACI 318-02 specifies 10 'cf as the beam maximum nominal
shear stress. By referring to Table 3.4, the maximum coupling beam shear stress is
13.8'
cf occurring in level 4. Furthermore, the shear stresses from level 1 to 10 all
exceed the ACI defined maximum shear limit. Based on the code design requirement,
these coupling beams can not be designed due to the large shear stresses.
The practical construction conditions place another limit on the shear stress in
coupling beams. The shear stress equal to 6'
cf has been recommended as the upper
limit in design in order to avoid congestion problems in diagonally reinforced concrete
coupling beams (Harries, 2003). The congestion likely happens at two locations. The first
location is the middle span, where the reinforcement in two diagonal directions meets
together. The second location is the intersections between the coupling beams and wall
piers, where the beam reinforcing bars interface with the wall reinforcement. A series of
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coupling beam design studies have been conducted (Fortney, 2005) to investigate the
congestion problem. These design cases have proved that a coupling beam with a shear
stress close to 6'
cf is designable, but a coupling beam with a shear stress close to
10'
cf is very difficult or impractical to be designed. Hence, the value of 6'
cf is taken
as the maximum shear stress in this study. The shear stresses of the beams except that in
the top level exceed 6 'cf (Table 3.4). From the constructability point of view, these
coupling beams can not be designed in view of the high shear stresses.
The large shear stresses in coupling beams are due to an implausible assumption
used in the traditional strength-based design. It has been assumed that the wall piers and
coupling beams yield simultaneously at the code-defined base shear level. However, the
deformation relationship between the wall piers and coupling beams (Paulay, 2002)
proves that this assumption is not correct.
As Figure 3.2 shows, the vertical difference between points A and B ( ) due to
the wall rotation (It is assumed that the two wall piers have the same rotations.) can be
calculated from the following equation.
AB
AB = w +1c w ( - )=wD 2c w ( + - ) (3.4)wD 1c 2c
If the distance is equal to , Equation 3.4 can be rewritten as:1c 2c
AB = w wD (3.5)
The vertical deformation ( AB ) can also be expressed using the chord rotation of
the coupling beam ( b ) as the following.
AB = b s (3.6)
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The results from Equations 3.5 and 3.6 should be equal. Hence, the following
equation is obtained.
b / w = / (3.7)wD s
Equation 3.7 indicates that the ratio of beam chord rotation to wall pier rotation is always
equal to the ratio of the wall length to beam span. In the selected prototype, is equal
to 10 feet and is taken as 6 feet. Substituting these values into Equation 3.7, the
following result is obtained.
wD
s
b =10/6 w =1.67 w (3.8)
Paulay suggested the following equation for calculating the yield rotation of wall
pier ( wy ) (Paulay, 2002).
wy = wy eh /2 (3.9)
In the prototype structure, is 108 feet provided by ETABS analyses.eh wy is assumed
to be 1.55 y / (Paulay, 2002). The steel yield strain (wD y ) is approximately 0.002. By
substituting all these parameters into Equation 3.9, the following result is calculated.
wy =1.550.002/10108/2=0.0167 rad (3.10)
At the time when the wall pier yields, the corresponding coupling beam chord
rotation can be computed by substituting wy into Equation 3.8.
b =1.670.0167=0.0280 rad (3.11)
Paulay also recommended the following equation for computing the yield chord
rotation of coupling beam ( by ) (Paulay, 2002).
by = by / =1.3( /coss s +16 )bd y / (3.12)s
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bd is 1.41 inches assuming that No. 11 bars are used, and the inclination of the diagonal
bars ( ) is roughly taken as tan (beam height/its length)=tan (30/72)=22.6. After
substituting these values into Equation 3.12,
1 1
by is calculated from Equation 3.13.
by =1.3 (72/cos22.6+161.41) 0.002/72=0.0036 rad (3.13)
By comparing the results of Equations 3.13 and 3.11, the ductility factor ( b ), is
calculated with Equation 3.14.
b = b / by =0.028/0.0036=7.8 (3.14)
The ductility factor indicates that the beam chord rotation when the wall yields is 7.8
times its yield chord rotation. It is impossible for the coupling beams to remain elastic
until the wall piers yield. The traditional strength-based design assumption of enforcing
elastic behavior of coupling beams prior to yielding of the wall piers generates
unrealistically high shear stresses in the coupling beams. As a matter of fact, the coupling
beams in CCW systems yield much earlier than wall piers do. The early yielding of the
beams helps transfer more loads to the wall piers which in turn reduces the beam shear
stress dramatically.
3.5 Introduction of Performance-Based Design Method
3.5.1 Performance-Based Design Concept
The traditional strength-based design method does not accurately address the real
seismic performance of CCW systems. As an alternative approach, a performance-based
design (PBD) method has been proposed (Harries et al., 2004) in an attempt to capture
the expected seismic behavior of CCW buildings.
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The PBD method divides the seismic behavior of a CCW system into three stages
in terms of yielding sequence of the members. Figure 3.3 provides a schematic view of
this tri-stage yielding mechanism. The first stage is the elastic stage, in which all the
structural members (beams and wall piers) are elastic. The second stage is the transition
stage, in which the beams begin to yield and the wall piers still stay elastic. The final
stage is the yield stage, in which wall piers yield and beams may reach their ultimate
deformation capacities. Note that at this stage the wall piers have not reached their
ultimate capacity and can continue to provide resistance. The structure reaches the
ultimate displacement after the plastic hinges are formed at the base of the building, and a
collapse mechanism is developed. The following performance requirements for CCW
systems under seismic loads are proposed to meet the tri-stage mechanism. These
requirements are for the structural behaviors at the life safety (LS) and collapse
prevention (CP) limit states (Refer to Section 3.1 for explanations of LS and CP limit
states.).
(1)Under the life safety (LS) level earthquake loads, the beams are allowed to yieldbut the wall piers are required to remain elastic. The maximum building story drift
should be less than NEHRP-defined 2% limit.
(2)Under the collapse prevention (CP) level earthquake loads, the wall piers arepermitted to yield, and the beams may reach their ultimate deformation capacities.
The aforementioned performance criteria coincide with the definitions of
structural performance at the LS and CP levels in FEMA 356. Section 1.5.1.3 of FEMA
356 states that at the LS level earthquake, the structural components can be damaged but
the structure shall still maintain a margin against onset of partial or total collapse.
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Correspondingly, in the proposed LS level performance, the beams are damaged but the
wall piers still remain essentially elastic to prevent the total collapse of the building.
Additionally, according to Section 1.5.1.5 of FEMA 356 the structure under the CP level
earthquake loads needs to continue to support gravity loads but retains no margin against
collapse. In the proposed CP level performance, the beams and walls are allowed to yield
or enter into the ultimate limit state, and the collapse mechanism is allowed to occur
when plastic hinges formed at the building base.
3.5.2 Changes of Design Requirements Using PBD Method
The aforementioned expected seismic response of CCW systems is different from
that based on the strength-based design method. The PBD method changes the design
demands for the coupling beams and wall piers. Figure 3.4 compares the design demands
between the strength-based method and the PBD method. The strength-based design
method requires the beams and walls yield at the code-defined base shear level. Therefore,
and are rather close to the value of , as illustrated in Figure 3.4.a. Note that
is not required to be checked because the ductility requirements and detailing
measurements for structural members in the current building codes are assumed to
guarantee to be developed.
byV wyV bV
uV
uV
In PBD method, it is acceptable that beams yield before the code-defined base
shear ( ) is reached. The value of in the figure is below the value of . This means
that the design forces in the beams are reduced because of the early yielding of the
coupling beams. On the other hand, more loads are transferred from the beams to wall
piers due to the beam yielding and therefore the PBD method increases the design forces
bV byV bV
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of wall piers. In Figure 3.4, the value of is above the value of . The value of is
related to the onset of collapse mechanism due to plastic hinges at the building base or
when the inter-story drift for any floor reaches 2.5% of the story height, which ever
occurs first.
wyV bV uV
3.5.3 Diagonally Reinforced Concrete Coupling Beam Design by PBD Method
This section presents a group of steps to calculate the design shear stresses of
beams. Two criteria are adopted in these steps. The first criterion is that the maximum
shear stress shall not exceed 6 'cf based on the constructability issues. The second
criterion is that the parabolic distribution of coupling beam shear stresses from the
strength-based analysis shall still be reasonably retained, and different shear stresses are
assigned to the beams in different groups. The objective of allocating different shear
stresses is to make the beams yield approximately at the same time. The specific
descriptions of these steps are as follow. The beams are classified into three groups based
on their shear stresses from strength-based analysis as discussed in Section 3.4. In this
project, beams from level 2 to level 7 are classified as Group I. Beams in level 1 and from
level 8 to 10 are grouped together as Group II. The remaining beams from level 11 to
level 15 are grouped as Group III. After grouping the beams, the average shear stress in
each group is calculated. Groups I, II, and III have an average shear stress of 13.1'
cf ,
10.9'
cf , and 7.2'
cf , respectively. The average shear stress of Group I is decreased
from 13.1 'cf to 6'
cf . The required reduction is 7.1'
cf . Similarly, the other two
groups are shifted back by 7.1'
cf . Finally, the minimum coupling beam steel ratio is
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reviewed. ACI 21.4.3.1 defines the minimum steel ratio to be 1 percent, which results in a
shear stress of 2.1 'cf . With the exception of Group III, for which the reduced shear
stress drops below ACI minimum requirement, the reduced shear stresses for Group I and
II are acceptable. As shown in Fig. 3.5, the final shear stresses for Group I, II, and III are
6 'cf , 3.8'
cf , and 2.1'
cf , respectively.
The design of the diagonally reinforced concrete beam is carried out by following
the requirements in Chapter 21 of ACI 318-02. The details of the resulting coupling
beams are shown in Figs. 3.6.1, 3.6.2 and 3.6.3. These coupling beams have the same
configurations with slight difference in the amount of provided diagonal reinforcement.
The beams in Group I have 12 No. 10 bars in the diagonal cores. The beams in Group II
have 12 No. 9 bars, and beams in Group III have 12 No. 7 bars. Tables B.4.1, B.4.2, and
B.4.3 in Appendix B provide design details for the coupling beams in Groups I, II, and
III, respectively.
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Table 3.1 Mass Participation of the First Two Modes in the Coupled Direction Mode 1 Mode 2 Total
Mode Mass (kips)xm
W 17039 3869
Building Actual Mass W (kips) 22987 22987
Mass Participation = /W xm
W 74% 17%
91%
Table 3.2 Base Shear Amplification Factorb
C
Mode 1 Mode 2Vxm (kips) 1110 645
Vt SRSS of both Vxm (kips) 1284
0.85Vb from ELF (kips) 2227
Cb =0.85Vb/Vt 1.73
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Table 3.3.1 Beam Shears of Mode 1 after Amplifications
StoryVbf
(kips)
Vbt
(kips)
(Vbf+AxavgVbt)
(kips)
(Vbf+AxavgVbt)
(kips) (Vbf+AxavgVbt)/(kips) Cb(Vbf+AxavgVbt)/(kips)15 53.6 13.3 67.1 99.0 116.4 202.0
14 63.0 14.8 78.1 115.2 135.6 235.1
13 76.7 16.2 93.3 137.6 161.9 280.7
12 92.9 17.8 111.1 163.8 192.8 334.3
11 109.9 19.5 129.8 191.4 225.2 390.5
10 126.6 21.1 148.1 218.4 256.9 445.6
9 142.1 22.5 165.0 243.4 286.3 496.6
8 155.8 23.7 180.0 265.4 312.3 541.6
7 167.2 24.6 192.2 283.5 333.6 578.5
6 175.8 25.0 201.2 296.7 349.1 605.5
5 180.8 24.8 206.0 303.9 357.5 620.0
4 181.2 24.0 205.7 303.3 356.8 618.9
3 175.6 22.4 198.4 292.7 344.3 597.2
2 161.7 19.8 182.0 268.3 315.7 547.5
1 135.8 16.1 152.2 224.5 264.1 458.1Notation: (1) Vbf is calculated by ETABS. (2) Vbt is calculated by ETABS. (3) Refer to Table B.2.1 for
Axavg. (4) Refer to Table B.3.1 for. (5) is 0.85, defined by ACI 318-02. (6) Refer to Table 3.2 for Cb.
Table 3.3.2 Beam Shears of Mode 2 after Amplifications
Story Vbf(kips) Vbt (kips)(Vbf+AxavgVbt)
(kips)
(Vbf+AxavgVbt)
(kips) (Vbf+AxavgVbt)/(kips) Cb(Vbf+AxavgVbt)/(kips)15 -38.54 -5.98 -47.0 -66.1 -77.8 -134.9
14 -44.23 -6.65 -53.7 -75.5 -88.8 -154.0
13 -50.01 -7.00 -60.0 -84.3 -99.2 -172.0
12 -53.40 -7.03 -63.4 -89.1 -104.8 -181.8
11 -52.64 -6.65 -62.1 -87.3 -102.7 -178.1
10 -46.96 -5.79 -55.2 -77.6 -91.3 -158.3
9 -36.44 -4.49 -42.8 -60.2 -70.8 -122.8
8 -21.74 -2.81 -25.7 -36.2 -42.6 -73.8
7 -4.08 -0.89 -5.3 -7.5 -8.8 -15.3
6 14.98 1.11 16.6 23.3 27.4 47.5
5 33.58 3.00 37.8 53.2 62.6 108.5
4 49.70 4.58 56.2 79.0 93.0 161.2
3 61.18 5.66 69.2 97.3 114.5 198.5
2 65.67 6.06 74.3 104.4 122.8 213.1
1 60.42 -1.88 57.7 81.2 95.5 165.6Notation: (1) Vbf is calculated by ETABS. (2) Vbt is calculated by ETABS. (3) Refer to Table B.2.2 for
Axavg. (4) Refer to Table B.3.2 for. (5) is 0.85, defined by ACI 318-02. (6) Refer to Table 3.2 for Cb.
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Table 3.4 SRSS of Beam Shear Forces and Related Shear Stresses
StoryShear from Mode 1
(kips)
Shear from Mode 2
(kips)
Shear by SRSS
(kips)
Shear Stress
(psi) over root fc'15 202.0 -134.9 242.9 404.8 5.2
14 235.1 -154.0 281.1 468.4 6.0
13 280.7 -172.0 329.2 548.7 7.1
12 334.3 -181.8 380.6 634.3 8.2
11 390.5 -178.1 429.2 715.3 9.2
10 445.6 -158.3 472.9 788.1 10.2
9 496.6 -122.8 511.6 852.7 11.0
8 541.6 -73.8 546.6 911.0 11.8
7 578.5 -15.3 578.7 964.5 12.5
6 605.5 47.5 607.3 1012.2 13.1
5 620.0 108.5 629.5 1049.1 13.5
4 618.9 161.2 639.5 1065.9 13.8
3 597.2 198.5 629.3 1048.8 13.5
2 547.5 213.1 587.5 979.2 12.6
1 458.1 165.6 487.1 811.8 10.5
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Y
P202
P203
P102
P103
X
P101 P201
Fig. 3.1 Labels of Wall Piers Used in the Redundancy Factor Calculation
bw w
c1 c2
Dw DwD
A
B
Lines through the N.A.
s
Fig. 3.2 Deformation Relationship between Coupling Beam and Wall Piers
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(1) Elastic Stage (2) Transition Stage (3)Yield Stage
Fig. 3.3 Tri-Stage Failure Mechanism of CCWs in PBD
VbwyV
Vby
uV Vu
Vb
byV
wyV
(a) Strength-Based Design Method (b) Performance-Based Design Method
Fig. 3.4 Comparison of Design Demands on CCW Elements between Strength-Based
Method and Performance-Based Method
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To Meet ACI Minimum
Reinforcement Requirement
10.9
Group I shifting
Group II shifting
Group III shifting
7.2
10.9
13.1
Story
3.8
2.1
3.8
6
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
0.0 2.0 4.0 6.0 8.0 10.0 12.0 14.0
'
cf ACI Limit
10 'cf
'
cf
Shear Stresses from Elastic
Analysis of Strength-Based
Design
'
cf'
cf
Group Shear Stresses
Used in PBD
'
cf'cf
Group Average Shear
Stresses Based on
Elastic AnalyticalResults
'
cf
'
cf
Fig. 3.5 Assignment of Coupling Beam Design Shear Stresses
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44
#4 distributed ba
#4 distributed bars
#4 ties@6c-c
#4 ties@4c-c
#11 wall longitudinal bras
4.643.874.87
A
A
B
B
A-AB-B
6#10
Group III Beams
Group II Beams
Group I Beams
Group II Beams
diagonal box: 11"w
Fig. 3.6.1 Section Details of Group I Coupling Beam
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C-CD-D
3.874.87 4.64
#4 distributed bars
#11 wall longitudinal bras
#4 ties@6c-c
#4 ties@4c-c
#4 distributed bars @
C D
C D
6#9
diagonal box: 11"wi
Fig. 3.6.2 Section Details of Group II Coupling Beam
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#4 distrib
4.87
E-E4.64
F-F
3.87
#11 wall longitudinal bras
#4 distribut
#4 ties@4c
#4 ties@6c
6#7
E F
E F
diagonal bo
Fig. 3.6.3 Section Details of Group III Coupling Beam
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Chapter 4 Design of Wall Piers
4.1 Notations
xA : Torsion amplification factor of each level in the X direction
xavgA : Average of of all levelsxA
yA : Torsion amplification factor of each level in the Y direction
yavgA : Average of of all levelsyA
: Base shear amplification factorbC
: Dead LoadD
E: Elastic modulus
: Lateral loads in the X directionxF
: Lateral loads in the Y directionyF
: Gross moment of inertiagI
xI : Moment of inertia of wall pier about its local axis parallel to the global X axis
yI : Moment of inertia of wall pier about its local axis parallel to the global Y axis
L : Live load
L : Coupling arm
: Accidental torsion associated with lateral loads in the X directiontaxM
: Accidental torsion associated with lateral loads in the Y directiontay
M
xM1 : Moment in the X direction on P100 due to lateral loads in the Y direction
xM2 : Moment in the X direction on P200 due to lateral loads in the Y direction
yM1 : Moment in the Y direction on P100 due to lateral loads in the X direction
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yM2 : Moment in the Y direction on P200 due to lateral loads in the X direction
OTM: Overturning moment
P: Compression force in wall pier section
DsS : Design spectral response acceleration at short period
T : Tension force in wall pier section
byV : Beam yield shear capacity
1fxV : Shear in wall pier P101, P102 or P103 caused by lateral loads in the X
direction
2fxV : Shear in wall pier P201, P202 or P203 caused by lateral loads in the X
direction
1fyV : Shear in wall pier P101, P102 or P103 caused by lateral loads in the Y
direction
2fyV : Shear in wall pier P201, P202 or P203 caused by lateral loads in the Y
direction
strV : Story Shear
1txV : Shear in wall pier P101, P102 or P103 caused by accidental torsion in the X
direction
2txV : Shear in wall pier P201, P202 or P203 caused by accidental torsion in the X
direction
1tyV : Shear in wall pier P101, P102 or P103 caused by accidental torsion in the Y
direction
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2tyV : Shear in wall pier P201, P202 or P203 caused by accidental torsion in the Y
direction
xV1 : Shear in the X direction on P100 due to lateral loads in the X direction
yV1 : Shear in the Y direction on P100 due to lateral loads in the Y direction
xV2 : Shear in the X direction on P200 due to lateral loads in the X direction
yV2 : Shear in the Y direction on P200 due to lateral loads in the Y direction
x : Abscissa of center of the wall pier
y : Ordinate of center of the wall pier
1.0X+0.3Y: Load combination with 100 percent of the X direction loads plus 30
percent of the Y direction loads
0.3X+1.0Y: Load combination with 30 percent of the X direction loads plus 100
percent of the Y direction loads
: Strength reduction factor
: Redundancy factor
x : Redundancy factor in the X direction
y : Redundancy factor in the Y direction
: Coupling moment LVby
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4.2 Introduction
The wall piers were designed by using performance-based design (PBD) method.
A simplified method, which covers the following characteristics of the PBD design
methodology, was proposed to facilitate the application of PBD method in the practical
CCW system designs.
(1)In the simplified method, internal forces on wall sections are calculated assumingthat all the coupling beams have yielded. Due to this early yielding, the forces on
wall sections are increased
(2)The tension wall and compression wall exhibit different stiffness characteristics
because of the axial load effect. Hence, they resist different percentages of the
total seismic loads. In this method, the relative stiffness ratio between the tension
wall and the compression wall is taken as 0.3/0.7 (Paulay, 2002). As a result, the
tension and compression wall piers carry 30 percent and 70 percent of the total
seismic forces, respectively.
(3)For consistency between the beam and wall analyses, modal spectrum responsemethod is also used.
(4)In addition to the lateral loads in the X and Y directions ( and ), theaccidental torsion in these two directions ( and ) associated with and
are also included.
xF yF
taxM tayM xF
yF
(5)The effects from , , , and are combined by following NEHRP. Theresulting axial forces and moments in two orthogonal directions are grouped
together as the demands for biaxial bending design. The shear forces are
considered separately as the requirement for shear design.
xF yF taxM tayM
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4.3 Simplified Method for Wall Pier Analyses
A former method involving beam modified stiffness was suggested to account for
the effect of the early yielding of coupling beams (McNeice, 2004). The purpose of
manually iterating the modification of beam stiffness is to keep all beam shear forces
between the beam shear capacity ( ) and 1.25 times the capacity (1.25 ). The range
between and 1.25 is the expected beam yielding extent after considering the
reinforcement strength hardening effect. Once all beams yield simultaneously in a
particular iteration, the internal wall forces calculated by ETABS are taken as wall design
demands. Obviously, this iterative method is time consuming. Every round of iteration
requires a complete modal response spectrum analysis. Furthermore, no methodology for
the magnitude and sequence of the needed stiffness modifications has been provided. As
a result, this method is cumbersome and time-cost.
byV byV
byV byV
The simplified method proposed in this chapter does not require iteration because all
member stiffness is determined. The following requirements need to be satisfied in the
implementation of this method. Per Section 5.2.5.2.2 of NEHRP, modal response
spectrum analysis is required independently in two orthogonal directions for buildings in
seismic design category D. The most critical load effect is from the combination of 100
percent of the forces in one direction plus 30 percent of the forces in the perpendicular
direction. Therefore, the simplified method requires two independent 2-D models
respectively in the X and Y direction. In each direction, modal response spectrum
analysis is carried out accounting the lateral forces and the associated 5 percent
accidental torsion in that direction. The wall design demands are the results from these
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two independent analyses after the combination, which is described in details in the
Section of 4.4.
4.3.1 X Direction Analyses
Figure 4.1.1 displays the free-body diagram of the coupled walls with the X
direction lateral forces as established from the design response spectrum. See Tables
C.1.1 and C.1.2 for the details of how these forces were calculated. Because the beams
are assumed to have yielded, the value of shear force at each level is equal to the beam
yield capacity ( ) at that level. As discussed previously, the axial forces (tensile on the
left walls and compressive on the right walls for the case shown in Fig. 4.1.1) change the
distribution of the lateral loads between the tension and compression walls. The tensile
wall piers (P101, P102, and P103 in Fig. 4.1.1) are assumed to resist 30% of the total
lateral loads, and the remaining 70% of the lateral loads is resisted by the compression
walls (P201, P202, and P203 in Fig. 4.1.1). The moment in the tension walls at each story
( ) is taken as 30% of the effective moment ( in Table 4.1.1), which is equal to
the overturning moment (OTM in Table 4.1.1) minus the coupling effect moment
(
byV
yM1 EM
byV L in Table 4.1.1). The moment in the compression walls at each story is 70%
taken as of the effective moment. The story shears for the tension walls ( ) and
compression walls ( ) are assumed to 30% and 70%, respectively, of the total story
shear ( ). Subsequently, is distributed equally to P101 and P102, which are in the
coupled direction. The wall pier P103 carries no shear because it is perpendicular to the
xV1
xV2
strV xV1
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direction of lateral loads. Similarly, is divided equally between wall piers P201 and
P202. Once again, wall pier P203 carries no shear.
xV2
T