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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