Coupling Shear Wall

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: _______________________________ _______________________________ _______________________________ _______________________________ _______________________________

Performance Based Design of a 15 Story Reinforced Concrete Coupled

Core Wall Structure

By

Gang Xuan

Bachelor of Science, Tongji University, China, 1999

Master of Science, Tongji University, China, 2002

Submitted in Partial Fulfillment of the Requirements for the Degree of Master of Science

in the Department of Civil and Environmental Engineering

College of Engineering

University of Cincinnati

Nov 2005

Dr. Bahram M. Shahrooz

Department of Civil and Environmental

Engineering

Director of Thesis

Dr. T. Michael Baseheart


Engineering

Thesis Committee Member

Dr. Gian A. Rassati


Engineering

Thesis Committee Member

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

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.

Acknowledgements

First of all, I would like to show my thankfulness and appreciations to my supervisor,

Dr. Bahram M. Shahrooz, for more than two years advising and tutoring. His valuable

comments and stimulating suggestions always help me keep on the right track and

proceed to the finale of the research program. Furthermore, his devotion to research and

great responsibility for high quality work have placed deep influences on me to

understand the ethics and principles of a good engineer, which will surely benefit my

future work. Again, I would thank him for all the time and efforts he provided in advising,

discussing, and revising my research work.

Secondly, I would like to thank Dr. Baseheart and Dr. Rassati. As the thesis

committee members, they put valuable time in reviewing my thesis and providing helpful

comments.

I would also thank Dr. Patrick Fortney for his great supports to my research work.

He is a wonderful colleague and always ready to provide me a discussion whenever I met

a problem in the research work.

Finally, I would like to give my sincere and special thankfulness to my parents and

sister for their long time and constant supports, understanding, and cares.

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

i

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

ii

5.3 Test Data ........................................................................................................73

5.4 Evaluation of Theoretical Models..................................................................74

5.4.1 Paulay’s Model.....................................................................................74

5.4.2 Hindi’s 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

iii

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

iv

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

v

Table 5.2 Strengths and Deformations Calculated According to Paulay’s Model ......85

Table 5.3 Strengths and Deformations Calculated According to Hindi’s 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

vi

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

vii

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 Paulay’s Model ............................................................92

Fig.5.2 Coupling Beam Vertical Deformation of Paulay’s Model..............................92

Fig.5.3 Force Equilibrium of Hindi’s 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

viii

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

ix

Chapter 1 Introduction

1.1 Notations:

otmM --Total overturning moment caused by lateral loads

1M , --Moments resisted by the tension and compression wall piers, respectively 2M

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 piers otmM

1

( 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

2

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

3

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 owner’s 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

4

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

5

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.

6

p

TM1

L

2MT

V1 V2

C

Fig. 1.1 Lateral Load Resisting Mechanism of a Coupled Core Wall System

7

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 components

Acceptability checks of conceptual overall design

No Acceptability checks of preliminary design using static, dynamic linear

and nonlinear analysis methods

Numerical preliminary design to comply simultaneously with at least two limit states

Yes

No

Final design and detailing using available experimental data and presenting material codes and

regulations

Acceptability checks of final design using static, 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)

8

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

9

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

avgδ : Average displacement of the floor

maxδ : Maximum displacement of the floor

φ : Strength reduction factor

10

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 100×100 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

11

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.

12

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

13

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 =(avgδ

δ2.1

max ) (2.1) 2

14

maxδ is the maximum displacement occurred at the corner of the building and avgδ 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

15

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 0.35E 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

16

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 in×24 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

17

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 20in ×24in to 20in ×30in. 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.

18

Table 2.1 Design of a Typical Interior Column Dead Loads (psf)

8 in Slab 100Partitions 10Devices 5Total 115

Live Loads (psf) For Office 50

Loads Combination 1.2Dead Load+1.6Live Load (psf) 1.2x115+1.6x50=218Tributary Area (ft2) 20x20=400Total Design Load on One Story (kips) 218x400/1000=87.2Total Design Load of 15 Storys (kips) 15x87.2=1308Required Area of the Column (in2) Assuming fc'=6 ksi φ=0.7 1308/(0.7x6)=311Square Root of the Required Area (in) 3110.5=18Actual 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 NEHRPS1 0.65g Directly from maps of NEHRPFa 1 Determined by Table 4.1.2.4a of NEHRPFv 1.3 Determined by Table 4.1.2.4b of NEHRPSMs 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)

0.0000.200

0.4000.6000.800

1.0001.200

0 1 2 3 4 5

19

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 flange wall in the X direction is 9 feet long and 20 inches thick. The web wall in the Y direction is 18 feet long and 16 inches thick. The coupling beam is 6 feet long with a 20in×24in section.

3.93% 4.28% 79.7%

The maximum X and Y

story 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, and the web wall in the Y direction is from 18 feet to 20 feet. Each wall thickness is also increased from 18 inches to 20 inches.

2.81% 2.95% 75.5%

The maximum X and Y


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 maximum X and Y


limit.

IV

The difference of this model (see Fig. 2.6) and Prototype III is that the web wall in the Y direction is lengthened from 22 feet to 25 feet, and the beam is enlarged from 20 in × 24 in to 20in×30in.

1.97% 1.73% 79.7%

This model meets the

2% deformation

limit and 66% DOC

limit.

20

12'-2

"14

stor

ies a

t 9'-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

21

Beam section is 20in by 24 in

9' 6' 9'

18'

20"16"

X

Y

Fig 2.3 Planar View of Prototype I


10' 6' 10'

20'

20"20"

X

Y

Fig 2.4 Planar View of Prototype II

22


10' 6' 10'

22'

20"20"

X

Y

Fig 2.5 Planar View of Prototype III


10' 6' 10'

25'

20"20"

X

Y

Fig 2.6 Planar View of Prototype IV

23

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 floors xA

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 load D

: Reinforcement bar diameter bd

: Length of wall section (see Fig. 3.2) wD

E : Structural response from seismic loads

: Lateral load of Mode m in the coupled direction xmF


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 pier wl

: Accidental torsion associated with taxmM xmF

EQ : Structural response from horizontal seismic loads

s : Span of link beam


24

: Ductility factor bu

: Code-defined base shear calculated by the ELF method bV

: Beam shear due to bfV xmF

btV : Beam shear due to taxmM

: Shear at the base when the link beams yield byV

xmV : Base Shear of Mode m in the coupled direction

: SRSS of base shear forces of all modes under consideration tV

: Shear at the base when the wall piers yield wyV

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 direction xmW

α : Inclination of diagonal reinforcement

maxγ : 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 beam by∆

yε : Steel yield strain


wyϕ : Wall yield curvature

bθ : Link beam chord rotation

byθ : Link beam yield chord rotation

25

wθ : Wall pier rotation

wyθ : Wall pier yield rotation

ρ : Redundancy factor

υ : 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.

26

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

27

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 ( ρ ) 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 = ρ 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

28

due to and were magnified by the redundancy factor (xmF taxmM ρ ). For wall piers, the

factor ( ρ ) is calculated as the following.

ρ =2-Amax

20γ

(3.2)

A is the total area of the floor, which is equal to 100×100 ft2. γ is the ratio of the shear

in a single element (torsional shear included) to the story shear. The subscript of max of

γ means that the maximum γ from all the elements should be taken. Additionally, per

Section 5.2.4.2 of NEHRP, the calculated γ 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

γ values. The maxγ used in the magnification is the greatest γ 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 maxγ and ρ .

The third scaling factor for the beam shear forces is strength reduction factor (φ ).

Per Section 9.3.4 (c) of ACI, φ 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

29

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

30

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∆ = wθ × +1c wθ ×( - )=wD 2c wθ ×( + - ) (3.4) wD 1c 2c

If the distance is equal to , Equation 3.4 can be rewritten as: 1c 2c

AB∆ = wθ wD (3.5)

The vertical deformation ( AB∆ ) can also be expressed using the chord rotation of

the coupling beam ( bθ ) as the following.

AB∆ = bθ s (3.6)

31

The results from Equations 3.5 and 3.6 should be equal. Hence, the following

equation is obtained.

bθ / wθ = / (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

bθ =10/6 wθ =1.67 wθ (3.8)

Paulay suggested the following equation for calculating the yield rotation of wall

pier ( wyθ ) (Paulay, 2002).

wyθ = wyϕ eh /2 (3.9)

In the prototype structure, is 108 feet provided by ETABS analyses. eh wyϕ is assumed

to be 1.55 yε / (Paulay, 2002). The steel yield strain (wD yε ) is approximately 0.002. By

substituting all these parameters into Equation 3.9, the following result is calculated.

wyθ =1.55×0.002/10×108/2=0.0167 rad (3.10)

At the time when the wall pier yields, the corresponding coupling beam chord

rotation can be computed by substituting wyθ into Equation 3.8.

bθ =1.67×0.0167=0.0280 rad (3.11)

Paulay also recommended the following equation for computing the yield chord

rotation of coupling beam ( byθ ) (Paulay, 2002).

byθ = by∆ / =1.3( /coss s α +16 )bd yε / (3.12) s

32

bd is 1.41 inches assuming that No. 11 bars are used, and the inclination of the diagonal

bars (α ) is roughly taken as tan (beam height/its length)=tan (30/72)=22.6 . After

substituting these values into Equation 3.12,

1− 1−

byθ is calculated from Equation 3.13.

byθ =1.3× (72/cos22.6+16×1.41) ×0.002/72=0.0036 rad (3.13)

By comparing the results of Equations 3.13 and 3.11, the ductility factor ( bµ ), is

calculated with Equation 3.14.

bµ = bθ / byθ =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.

33

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 yield

but 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 are

permitted 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.

34

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

35

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

36

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.

37

Table 3.1 Mass Participation of the First Two Modes in the Coupled Direction Mode 1 Mode 2 Total

Mode Mass (kips) xmW 17039 3869Building Actual Mass W (kips) 22987 22987Mass Participation = /W xmW 74% 17%

91%

Table 3.2 Base Shear Amplification Factor bC

Mode 1 Mode 2 Vxm (kips) 1110 645Vt SRSS of both Vxm (kips) 12840.85Vb from ELF (kips) 2227Cb =0.85Vb/Vt 1.73

38

Table 3.3.1 Beam Shears of Mode 1 after Amplifications Story Vbf

(kips) Vbt

(kips) (Vbf+AxavgVbt)

(kips) (Vbf+AxavgVbt)ρ

(kips) (Vbf+AxavgVbt)ρ/φ

(kips) Cb(Vbf+AxavgVbt)ρ/φ

(kips) 15 53.6 13.3 67.1 99.0 116.4 202.014 63.0 14.8 78.1 115.2 135.6 235.113 76.7 16.2 93.3 137.6 161.9 280.712 92.9 17.8 111.1 163.8 192.8 334.311 109.9 19.5 129.8 191.4 225.2 390.510 126.6 21.1 148.1 218.4 256.9 445.6

9 142.1 22.5 165.0 243.4 286.3 496.68 155.8 23.7 180.0 265.4 312.3 541.67 167.2 24.6 192.2 283.5 333.6 578.56 175.8 25.0 201.2 296.7 349.1 605.55 180.8 24.8 206.0 303.9 357.5 620.04 181.2 24.0 205.7 303.3 356.8 618.93 175.6 22.4 198.4 292.7 344.3 597.22 161.7 19.8 182.0 268.3 315.7 547.51 135.8 16.1 152.2 224.5 264.1 458.1

Notation: (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 ρ. (5) φ 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)ρ

(kips) (Vbf+AxavgVbt)ρ/φ

(kips) Cb(Vbf+AxavgVbt)ρ/φ

(kips) 15 -38.54 -5.98 -47.0 -66.1 -77.8 -134.914 -44.23 -6.65 -53.7 -75.5 -88.8 -154.013 -50.01 -7.00 -60.0 -84.3 -99.2 -172.012 -53.40 -7.03 -63.4 -89.1 -104.8 -181.811 -52.64 -6.65 -62.1 -87.3 -102.7 -178.110 -46.96 -5.79 -55.2 -77.6 -91.3 -158.3

9 -36.44 -4.49 -42.8 -60.2 -70.8 -122.88 -21.74 -2.81 -25.7 -36.2 -42.6 -73.87 -4.08 -0.89 -5.3 -7.5 -8.8 -15.36 14.98 1.11 16.6 23.3 27.4 47.55 33.58 3.00 37.8 53.2 62.6 108.54 49.70 4.58 56.2 79.0 93.0 161.23 61.18 5.66 69.2 97.3 114.5 198.52 65.67 6.06 74.3 104.4 122.8 213.11 60.42 -1.88 57.7 81.2 95.5 165.6

Notation: (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 ρ. (5) φ is 0.85, defined by ACI 318-02. (6) Refer to Table 3.2 for Cb.

39

Table 3.4 SRSS of Beam Shear Forces and Related Shear Stresses Story Shear from Mode 1

(kips) Shear from Mode 2

(kips) Shear by SRSS

(kips) Shear Stress

υ (psi) υ over root fc'

15 202.0 -134.9 242.9 404.8 5.214 235.1 -154.0 281.1 468.4 6.013 280.7 -172.0 329.2 548.7 7.112 334.3 -181.8 380.6 634.3 8.211 390.5 -178.1 429.2 715.3 9.210 445.6 -158.3 472.9 788.1 10.2

9 496.6 -122.8 511.6 852.7 11.08 541.6 -73.8 546.6 911.0 11.87 578.5 -15.3 578.7 964.5 12.56 605.5 47.5 607.3 1012.2 13.15 620.0 108.5 629.5 1049.1 13.54 618.9 161.2 639.5 1065.9 13.83 597.2 198.5 629.3 1048.8 13.52 547.5 213.1 587.5 979.2 12.61 458.1 165.6 487.1 811.8 10.5

40

Y

P202

P203

P102

P103 X

P101 P201

Fig. 3.1 Labels of Wall Piers Used in the Redundancy Factor Calculation

θbθw θw

c1 c2

Dw DwD

A

B

Lines through the N.A.

s

Fig. 3.2 Deformation Relationship between Coupling Beam and Wall Piers

41

(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

42

To Meet ACI Minimum

Reinforcement Requirement 7Story

2

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

0.0 2.0

' 'f

Fig. 3.5 Ass

.1

Group II shifting

Group III shifting

10

13

3

3

4.0 6.

cf10 cf

c

'cf'

cf

'cf'

'cf

ignment of Cou

6

Group I shifting

cf

.8
10.9
0 8.0 10.0

pling Beam Design Shea

43

.9

.2

12.0 1

r Stresse

.1

4 'f

s

.0

.8

ACI Limit'

Shear Stresses from ElasticAnalysis of Strength-Based

Design

Group Average ShearStresses Based on Elastic Analytical

Results

Group Shear Stresses Used in PBD

c

44

#4 distributed bars @5c-c

#4 distributed bars @5.2c-c

#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

6#10

Group III Beams

Group II Beams

Group I Beams

Group II Beams

diagonal box: 11"wide, 6"high (out to out)

Fig. 3.6.1 Section Details of Group I Coupling Beam

C-C3.874.87 4.64


#11 wall longitudinal bras

#4 ties@6c-c

#4 ties@4c-c

#4 distributed bars @5.2c-c

C D

C D

6#9

6#9


Fig. 3.6.2 Section Details of Group II g Beam

45

D-D

Couplin


4.87E-E

4.64

F

3.87

#11 wall longitudinal bras#4 distributed bars @5.2c-c

#4 ties@4c-c

#4 ties@6c-c

6#7

E F

E F

6#7


Fig. 3.6.3 Section Details of Group III Coupling eam

46

-F

B

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 levels xA

yA : Torsion amplification factor of each level in the Y direction

yavgA : Average of of all levels yA

: Base shear amplification factor bC

: Dead Load D

E : Elastic modulus

: Lateral loads in the X direction xF

: Lateral loads in the Y direction yF

: Gross moment of inertia gI

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 direction taxM

: Accidental torsion associated with lateral loads in the Y direction tayM

xM1 : Moment in the X direction on P100 due to lateral loads in the Y direction

xM 2 : Moment in the X direction on P200 due to lateral loads in the Y direction

yM 1 : Moment in the Y direction on P100 due to lateral loads in the X direction

47

yM 2 : 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


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

48

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


ρ : Redundancy factor

xρ : Redundancy factor in the X direction

yρ : Redundancy factor in the Y direction

: Coupling moment ∑ LVby

49

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 assuming

that 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 response

method is also used.

(4) In addition to the lateral loads in the X and Y directions ( and ), the

accidental 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. The

resulting 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

50

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

51

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

yM 1 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

52

direction of lateral loads. Similarly, is divided equally between wall piers P201 and

P202. Once again, wall pier P203 carries no shear.

xV2

The modal response spectrum analysis method is employed to calculate the wall

pier design forces. The overturning moment and story shear are the SRSS results from the

first two modes in the X direction. The lateral load and the effects of each mode are

provided in Table C.1.1 and Table C.1.2. The base shear amplification factor ( listed

in Table 3.2) is also included to increase the values of the overturning moment and story

shear. The resulting X direction lateral loads along with the effective overturning

moments are listed in Table 4.1.1. The axial load, moment, and shear in the wall piers

making up the tension and compression wall piers are summarized in Table 4.1.2.

bC

Figure 4.1.2 illustrates the effects of the accidental torsion in the X direction. The

shear force in each wall pier is calculated by using the following equations, which ignore

warping of the core wall system.

txV =∑ + )( 22

yx

ytax

IyIxyIM

(4.1)

txV =∑ + )( 22

yx

xtax

IyIxIxM

(4.2)

Equation 4.1 is used for the wall piers in the X direction, i.e., P101, P102, P201, and

P202. Equation 4.2 is used for the wall piers in the Y direction, i.e., P103 and P203. The

term symbolizes the shears in wall piers caused by the X direction torsion. and

are the distances measured from the geometry center of the wall pier to the story

center of rigidity. and are the moments of inertia about the local axes parallel to the

global X and Y axes through the centroid of each wall pier, respectively. To account for

txV x

y

xI yI

53

the effects of axial load, the stiffness or for piers P101, P102, and P103 is taken as

0.3 . or for pier P201, P202, and P203 is taken as 0.7 . Table 4.2 lists the

results of the X direction torsion analysis. Combination of the lateral load and torsional

effects is discussed in Section 4.4.

xI yI

gI xI yI gI

4.3.2 Y Direction Analyses

Figure 4.2.1 shows the free-body diagram of the coupled wall under the action of

the Y direction lateral loads. The applied lateral forces from the first two modes of the Y

direction, as calculated from the design spectrum are used. (See Tables C.3.1 and C.3.2.)

Because there is no coupling effect in the Y direction, the two wall piers act as two

cantilever members with the same stiffness and equally resist the lateral loads. Hence, the

moments and shear forces in the wall piers ( and in the left wall pier, and

and in the right wall pier, see Fig. 4.2.1) are equal to one half of the total overturning

moment (OTM in Table C.3.3) and the total story shear ( in Table C.3.3),

respectively. The values of the total overturning moment and story shear were calculated

by using modal response spectrum analysis. As NEHRP requires, the first two modes in

the Y direction were included. These two modes contribute to 92 percent of mass

participation (see Table C.2.1). The lateral loads and their load effects in each mode are

listed in Table C.3.1 and Table C.3.2. The SRSS results of load effects scaled up by the

base shear amplification factor ( in Table C.2.2) are listed in Table C.3.3. The

resulting forces after the load distribution between two wall piers are summarized in

Table 4.3. Note that the walls in the perpendicular direction do not carry any shear unless

xM1 yV1 xM 2

yV2

strV

bC

54

the effects of torsion are taken into account. Wall pier P103 in the Y direction carries the

entire story shear in the left wall pier ( in Fig. 4.2.1) and wall piers P101 and P102

perpendicular to the direction of later loads carry no shear. Similarly, wall pier P203

carries the entire story shear in the right wall pier ( in Fig. 4.2.1) and wall pier P201

and P202 carries no shear.

yV1

yV2

The additional shear forces due to torsion, which in this case is accidental torsion,

are shown in Fig. 4.2.2. The analysis procedure is the same as that descried for

computing the effects of accidental torsion in the X direction. The only difference is that

the wall piers have the same stiffness values, and as a result shear forces are distributed

based on their gross stiffness. The results are listed in Table 4.4. The combinations of

additional shear forces due to torsion with shear forces from lateral loads are presented in

Section 4.4.

4.4 Load Combinations

The lateral loads in the X and Y directions ( and ) and their associated

tonsions ( and ) result in four sets of forces that need to be combined in order to

obtain the design demands for wall piers. The load combinations are carried out based on

the following NEHRP provisions. (1) Section 5.2.5.2.2 states that the most critical load

effect is assumed to be 100 percent of the forces for one direction plus 30 percent of the

forces for the perpendicular direction. (2) Section 5.4.4.2 states that torsion shall be

considered simultaneously with lateral loads in the same direction. (3) Section 5.4.4.3

states that torsional effects shall be enlarged using the torsion amplification factor ( ).

(4) Section 5.2.7 states that effects from horizontal seismic loads shall be scaled up by

xF yF

taxM tayM

xA

55

multiplying them by the redundancy factor ( ρ ). (5) Section 5.5 states that the vertical

seismic loads (taken as 0.2 ) and gravity loads (summarized in Table C.4) shall be

taken into account for load combinations in the gravity direction.

DsS D

With the aforementioned summary of NEHRP specifications, axial force, moment in

X, moment in Y, and shear force are completed from 1.0X+0.3Y and 0.3X+1.0. In each

load condition, the tension wall pier (P100 in Fig. 4.1.1) and the compression wall pier

(P200 in Fig. 4.1.1) are distinguished from each other. Note that the focus is on the wall

piers in the coupled direction.

(a) Axial Force

(a.1) 1.0X+0.3Y (100 Percent of the X Direction Loads Plus 30 Percent of the Y

Direction Loads)

For tension Wall Pier (P100) 0.9 -0.2 -D DsS D xρ T (4.3)

For Compression Wall Pier (P200) xρ P +0.2 +1.2 +0.5DsS D D L (4.4)

The redundancy factor in the X direction ( xρ in Table C.5.1) is used because for

this load condition it is the major direction (the direction with 100 percent lateral loads).

The term 0.2 represents the effects of vertical component of ground motion per

NEHRP. The terms and

DsS D

D L are the gravity loads on the wall section due to the dead

and live load, respectively (given in Table C.4). The signs of vertical earthquake load

effects in Equations 4.3 and 4.4 were selected to produce the largest axial loads in the

tension and compression wall piers.

(a.2) 0.3X+1.0Y (30 Percent of the X Direction Loads Plus 100 Percent of the Y

Direction Loads)

56

For tension Wall Pier (P100) 0.9 -0.2 -0.3D DsS D yρ T (4.5)

For Compression Wall Pier (P200) 0.3 yρ P +0.2 +1.2 +0.5DsS D D L (4.6)

For this case, 30 percent of the X direction loads are included, and therefore the

axial loads (T and P ) from the X direction loads are reduced by multiplying them by

0.3. The use of the redundancy factor yρ implies the major direction changes to the Y

direction in this condition. The resulting design axial forces, which account for biaxial

loading, are summarized in Tables 4.5.1 and 4.5.2.

(b) Moment in the X direction

(b.1) 1.0X+0.3Y (100 Percent of the X Direction Loads Plus 30 Percent of the Y

Direction Loads)

For Tension Wall Pier (P100) 0.3 xρ xM1 (4.7)

For Compression Wall Pier (P200) 0.3 xρ xM 2 (4.8)

(b.2) 0.3X+1.0Y (30 Percent of the X Direction Loads Plus 100 Percent of the Y

Direction Loads)

For Tension Wall Pier (P100) yρ xM1 (4.9)

For Compression Wall Pier (P200) yρ xM 2 (4.10)

Note that only the loads in the Y direction produce moments in the X direction.

Therefore, only and are used for case (b.1) or (b.2). However, the redundancy

factors are different because they are a function of the major direction of lateral loads.

xM1 xM 2

(c) Moment in the Y Direction

(c.1) 1.0X+0.3Y (100 Percent of the X Direction Loads Plus 30 Percent of the Y

Direction Loads)

57

For Tension Wall Pier (P100) xρ yM 1 (4.11)

For Compression Wall Pier (P200) xρ yM 2 (4.12)

(c.2) 0.3X+10Y (30 Percent of the X Direction Loads Plus 100 Percent of the Y

Direction loads)

For Tension Wall Pier (P100) 0.3 yρ yM 1 (4.13)

For Compression Wall Pier (P200) 0.3 yρ yM 2 (4.14)

The moments as computed from aforementioned cases are summarized in Table

4.5.1 and 4.5.2.

(d) Shear Force

(d.1) 1.0X+0.3Y (100 Percent of the X Direction Loads Plus 30 Percent of the Y

Direction Loads)

For Tension Wall Piers Parallel to the X Direction (P101 and P102):

xρ ( + +0.3 ) (4.15) 1fxV xavgA 1txV yavgA 1tyV

For Tension Wall Piers Parallel to the Y Direction (P103):

xρ ( +0.3 +0.3 ) (4.16) xavgA 1txV 1fyV yavgA 1tyV

For Compression Wall Piers Parallel to the X Direction (P201 and P202):

xρ ( + +0.3 ) (4.17) 2fxV xavgA 2txV yavgA 2tyV

For Compression Wall Piers Parallel to the Y Direction (P203):

xρ ( +0.3 +0.3 ) (4.18) xavgA 2txV 2fyV yavgA 2tyV

The Y direction loads have no effect in the wall piers parallel to the X direction, and

correspondingly the components or do not appear in the Equations 4.15 and

4.17. Similarly, Equations 4.16 and 4.18 do not include the effects from the X direction

1fyV 2fyV

58

lateral loads ( or ) in the wall piers parallel to the Y direction (P103 and P203).

As discussed previously (Section 4.3.1), the compression and tension wall piers resist

70% and 30% of the total X direction lateral loads, respectively. Therefore, and

are 30% and 70% of the total lateral loads in the X direction. Since the walls are not

coupled in the Y direction, the tension and compression wall piers resist 50% of the total

lateral loads in the Y direction, i.e., and are equal to 50% of the total Y

direction lateral loads. For simplicity, the average torsion amplification factors ( in

Table C.5.2 and in Table C.6.2) are used to scale up the responses due to torsion

irregularity per NEHRP 5.4.4.3. In contrast to other studies (Brienen, 2002) that have

ignored the minor direction torsion, bidirectional accidental torsion is included in this

research.

1fxV 2fxV

1fxV 2fxV

1fyV 2fyV

xavgA

yavgA

(d.2) 0.3X+1.0Y (30 Percent of the X Direction Loads Plus 100 Percent of the Y

Direction Loads)

For Tension Wall Piers Parallel to the X Direction (P101 and P102):

yρ (0.3 +0.3 + ) (4.19) 1fxV xavgA 1txV yavgA 1tyV

For Tension Wall Piers Parallel to the Y Direction (P103):

yρ (0.3 + + ) (4.20) xavgA 1txV 1fyV yavgA 1tyV

For Compression Wall Piers Parallel to the X Direction (P201 and P202):

yρ (0.3 +0.3 + ) (4.21) 2fxV xavgA 2txV yavgA 2tyV

For Compression Wall Piers Parallel to the Y Direction (P203):

yρ (0.3 + + ) (4.22) xavgA 2txV 2fyV yavgA 2tyV

59

Based on the discussion for the previous case, and are equal to 30% and

70% of the total X direction lateral loads, respectively. and are equal to 50% of

the total Y direction lateral loads.

1fxV 2fxV

1fyV 2fyV

The combined shear forces are listed in Tables 4.6.1 and 4.6.2. The shaded cells

represent the maximum shear demands for wall piers, which are used for wall design

discussed in the next section.

4.5 Wall Pier Design

The design moments (see Table 4.5.1 and 4.5.2) decrease along the building height.

The building floors are grouped into three types. Levels 1 to 3 are classified as Group I,

levels 4 to 7 as Group II , and levels 8 to 15 as Group III. Each group was designed based

on the requirements in Chapter 21 of ACI 318-02.

As Section 21.7.6 of ACI requires, boundary elements are provided at the ends of

wall flanges and the intersections between flanges and webs. The boundary element

design calculations are presented in Tables C.7.1 through C.7.3 in Appendix C. The

biaxial bending moment demands (see Tables 4.5.1 and 4.5.2) establish the longitudinal

reinforcement in the wall sections. The effects of confinement due to boundary elements

were taken into account for computing the bending moment and axial capacities of the

wall piers. A program called XTRACT (Imbsen, 2002) was used to generate the axial-

moment interaction diagrams (see Figures C.1.1 through C.1.3). These figures show that

each wall group has adequate biaxial bending capacities to resist the design forces. Shear

design of the wall pier is summarized in Table C.8.1 through C.8.3. The cross sectional

details are shown in Figures 4.3.1 through 4.3.3.

60

Table 4.1.1 Lateral Load Effects and Effective Moments in the X Direction Lateral Load Effects Beam Coupling Effects Effective Overturning Moments

Story OTM (k-ft)

Story Shear Vstr(kips)

Beam Shear Capacity Vby (two beams included)

(kips)

Coupling Moment=ΣVbyL

(k-ft)

ME=OTM-ΣVbyL (k-ft)

15 3060 335 229 4953 -189314 9048 655 229 9907 -85813 17381 913 229 14860 252112 27484 1110 229 19813 767011 38828 1254 229 24767 1406110 50979 1360 382 33023 17956

9 63634 1445 382 41278 223558 76642 1529 382 49534 271087 90003 1626 485 60019 299846 103833 1741 485 70504 333295 118301 1867 485 80988 373134 133566 1992 485 91473 420933 149710 2100 485 101958 477522 166705 2180 485 112442 542631 190340 2227 382 120698 69642

Notations: (1) Refer to Table C.1.3 for calculations of lateral load effects. (2) L, the coupling arm, is 21.6 ft shown in Fig. 4.1.1.

Table 4.1.2 X Direction Lateral Load Effect Distribution between Wall Piers Left Side Wall P100 in Tension Right Side Wall P200 in Compression

Story T=ΣVby (kips)

M1y =0.3ME (k-ft)

V1x =0.3Vstr (kips)

Vfx1 on P101=0.5V1x(kips)

Vfx1 on P102=0.5V1x(kips)

Vfx1 on P103 (kips)

P=ΣVby(kips)

M2y =0.7ME(k-ft)

V2x =0.7Vstr(kips)

Vfx2 on P201=0.5V2x (kips)

Vfx2 on P202=0.5V2x (kips)

Vfx2 on P203 (kips)

15 229 -568 100 50 50 0 229 -1325 234 117 117 014 459 -258 196 98 98 0 459 -601 458 229 229 013 688 756 274 137 137 0 688 1765 639 319 319 012 917 2301 333 166 166 0 917 5369 777 388 388 011 1147 4218 376 188 188 0 1147 9843 878 439 439 010 1529 5387 408 204 204 0 1529 12569 952 476 476 0

9 1911 6707 434 217 217 0 1911 15649 1012 506 506 08 2293 8132 459 229 229 0 2293 18976 1070 535 535 07 2779 8995 488 244 244 0 2779 20989 1138 569 569 06 3264 9999 522 261 261 0 3264 23330 1219 609 609 05 3749 11194 560 280 280 0 3749 26119 1307 654 654 04 4235 12628 598 299 299 0 4235 29465 1394 697 697 03 4720 14326 630 315 315 0 4720 33427 1470 735 735 02 5206 16279 654 327 327 0 5206 37984 1526 763 763 01 5588 20893 668 334 334 0 5588 48749 1559 779 779 0

Notation: See Fig. 4.1.1 for the locations of P100, P101, P102, P103, P200, P201, P202 and P203.

61

Table 4.2 X Direction Torsion Analysis Shear on Tension Wall Components

(kips) Shear on Compression Wall

Components (kips) Story Mtax (k-ft) Vtx1 on P101 Vtx1 on P102 Vtx1 on

P103 Vtx2 on P201 Vtx2 on P202

Vtx2 on P203

15 1668 2 -2 -34 5 -5 8014 3267 4 -4 -67 10 -10 15713 4552 6 -6 -94 13 -13 21912 5535 7 -7 -114 16 -16 26611 6256 8 -8 -129 19 -19 30110 6782 9 -9 -140 20 -20 326

9 7208 9 -9 -149 21 -21 3478 7627 10 -10 -157 23 -23 3677 8111 10 -10 -167 24 -24 3906 8683 11 -11 -179 26 -26 4185 9313 12 -12 -192 28 -28 4484 9934 13 -13 -205 29 -29 4783 10474 13 -13 -216 31 -31 5042 10871 14 -14 -224 32 -32 5231 11107 14 -14 -229 33 -33 534

Notation: See Fig. 4.1.2 for the locations of P101, P102, P103, P201, P202 and P203.

Table 4.3 Y Direction Lateral Load Effect Distribution between Wall Piers P100 P200

Story M1x=0.5 OTM (k-ft)

V1y=0.5 Vstr (kips)

Vfy1 on P101 (kips)

Vfy1on P102 (kips)

Vfy2 on P103 =V1y (kips)

M2x=0.5OTM (k-ft)

V2y=0.5Vstr (kips)

Vfy2 on P201 (kips)

Vfy2 on P202 (kips)

Vfy2 on P203 =V2y (kips)

15 1561 170 0 0 170 1561 170 0 0 17014 4573 329 0 0 329 4573 329 0 0 32913 8712 452 0 0 452 8712 452 0 0 45212 13673 544 0 0 544 13673 544 0 0 54411 19196 611 0 0 611 19196 611 0 0 61110 25078 660 0 0 660 25078 660 0 0 660

9 31194 702 0 0 702 31194 702 0 0 7028 37491 745 0 0 745 37491 745 0 0 7457 43990 796 0 0 796 43990 796 0 0 7966 50757 856 0 0 856 50757 856 0 0 8565 57877 921 0 0 921 57877 921 0 0 9214 65424 985 0 0 985 65424 985 0 0 9853 73433 1041 0 0 1041 73433 1041 0 0 10412 81890 1084 0 0 1084 81890 1084 0 0 10841 93690 1114 0 0 1114 93690 1114 0 0 1114

Notations: (1) See Fig. 4.2.1 for the locations of P100, P101, P102, P103, P200, P201, P202 and P203. (2) Refer to Table C.3.3 for calculations of lateral load effects, i.e, OTM and Vstr.

62

Table 4.4 Y Direction Torsion Analysis Shear on Left Side Wall Components

(kips) Shear on Right Side Wall Components

(kips) Story Mtay

(k-ft) Vty1 on P101 Vty1 on P102 Vty1 on P103 Vty2 on P201 Vty2 on P202 Vty2 on P203

15 1702 4 -4 58 4 -4 -5814 3287 7 -7 113 7 -7 -11313 4523 10 -10 155 10 -10 -15512 5443 12 -12 187 12 -12 -18711 6107 13 -13 210 13 -13 -21010 6598 14 -14 227 14 -14 -227

9 7017 15 -15 241 15 -15 -2418 7452 16 -16 256 16 -16 -2567 7964 17 -17 274 17 -17 -2746 8561 18 -18 294 18 -18 -2945 9208 19 -19 316 19 -19 -3164 9846 21 -21 338 21 -21 -3383 10408 22 -22 358 22 -22 -3582 10844 23 -23 372 23 -23 -3721 11135 24 -24 382 24 -24 -382

Notation: See Fig. 4.2.2 for the locations of P101, P102, P103, P201, P202 and P203.

Table 4.5.1 Design Demands for Biaxial Bending Design with 1.0X+0.3Y Combination

P100 (Biaxial Bending Case 1) P200 (Biaxial Bending Case 2)

Tension (kips) Mx (k-ft) My (k-ft) Compression (kips) Mx (k-ft) My (k-ft)Story (0.9D-0.2SDsD -

ρxT)/Φ 0.3ρxM1x/Φ ρxM1y/Φ (ρxP+0.2SDsD+1.2D+0.5L)/Φ 0.3ρxM2x/Φ ρxM2y/Φ

15 -192 756 -917 751 756 -213914 -385 2214 -416 1502 2214 -97013 -577 4218 1221 2253 4218 284912 -769 6620 3714 3004 6620 866611 -962 9294 6808 3755 9294 1588510 -1401 12142 8694 4753 12142 20286

9 -1840 15103 10824 5751 15103 252568 -2279 18152 13125 6749 18152 306257 -2885 21299 14518 7913 21299 338756 -3490 24575 16137 9077 24575 376535 -4096 28023 18066 10242 28023 421544 -4702 31677 20381 11406 31677 475553 -5307 35555 23121 12570 35555 539482 -5913 39649 26273 13735 39649 613031 -6322 45363 33719 14792 45363 78678

Notations: (1) Refer to Table C.4 in Appendix C for values of D and L (2) Refer to Table C.5.2 for the value of ρx (3) Φ is 0.9 for tension controlled failure.

63

Table 4.5.2 Design Demands for Biaxial Bending Design with 0.3X+1.0Y Combination

P100 (Biaxial Bending Case 3) P200 (Biaxial Bending Case 4)

Tension (kips) Mx (k-ft) My (k-ft) Compression (kips) Mx (k-ft) My (k-ft) Story (0.9D-0.2SDsD -

0.3ρyT)/Φ ρyM1x/Φ 0.3ρyM1y/Φ(0.3ρyP+0.2SDsD+1.2D+0.5L)/Φ ρyM2x/Φ 0.3ρyM2y/Φ

15 79 2230 -243 479 2230 -56814 159 6533 -110 958 6533 -25813 238 12446 324 1438 12446 75612 318 19533 986 1917 19533 230111 397 27422 1808 2396 27422 421810 411 35826 2309 2941 35826 5387

9 425 44562 2874 3486 44562 67078 439 53559 3485 4030 53559 81327 409 62843 3855 4619 62843 89956 379 72509 4285 5208 72509 99995 349 82681 4797 5797 82681 111944 318 93463 5412 6386 93463 126283 288 104905 6140 6975 104905 143262 258 116985 6977 7564 116985 162791 302 133843 8954 8169 133843 20893

Notations: (1) Refer to Table C.4 in Appendix C for values of D and L. (2) Refer to Table C.6.2 for the value of ρy (3) Φ is 0.9 for tension controlled failure.

Table 4.6.1 Design Demands for Shear Design with 1.0X+0.3Y Combination

Shear with 1.0X+0.3Y Effect Combination (kips) P101 P102 P103 P201 P202 P203

Storyρx(Vfx1+Vtx1Axavg

+0.3Vty1Ayavg) /Φ ρx(Vfx1+Vtx1Axavg

+0.3Vty1Ayavg) /Φ

ρx(Vtx1Axavg+ 0.3Vfy1+0.3Vty1A

yavg) /Φ

ρx(Vfx2+Vtx2Axavg+0.3Vty2 Ayavg)

/Φ

ρx(Vfx2+Vtx2Axavg+0.3Vty2 Ayavg)

/Φ

ρx(Vtx2Axavg+ 0.3Vfy2+0.3Vty2

Ayavg) /Φ 15 104 104 207 239 239 29914 203 203 402 468 468 58313 283 283 556 652 652 80712 344 344 671 793 793 97711 389 389 755 896 896 110010 422 422 816 972 972 1191

9 448 448 868 1032 1032 12668 474 474 921 1093 1093 13427 504 504 982 1162 1162 14306 540 540 1054 1244 1244 15345 579 579 1133 1334 1334 16474 618 618 1211 1423 1423 17593 652 652 1279 1501 1501 18572 676 676 1330 1558 1558 19311 691 691 1364 1591 1591 1977

Notations: (1) Refer to Table C.5.2 for the value of ρx. (2) Refer to Table C.5.1 for the value of Axavg. (3) Φ is 0.75 for shear design.

64

Table 4.6.2 Design Demands for Shear Design with 0.3X+1.0Y Combination Shear with 1.0X+0.3Y Effect Combination (kips)

P101 P102 P103 P201 P202 P203 Story ρy(Vty1Ayavg+0.3

Vfx1+0.3Vtx1Axavg

) /Φ

ρy(Vty1Ayavg+0.3Vfx1+0.3Vtx1Axavg

) /Φ

ρy(Vfy1+Vty1Ayavg

+0.3Vtx1Axavg) /Φ


) /Φ


) /Φ

ρy(Vfy2+Vty2Ayavg

+0.3Vtx2Axavg) /Φ

15 34 34 426 70 70 45014 66 66 823 137 137 87113 92 92 1133 190 190 120012 112 112 1364 231 231 144511 127 127 1531 261 261 162310 137 137 1655 283 283 17549 146 146 1759 301 301 18658 154 154 1868 319 319 19807 164 164 1996 339 339 21156 176 176 2145 363 363 22735 189 189 2307 389 389 24444 202 202 2467 415 415 26123 213 213 2607 438 438 27612 221 221 2716 455 455 28751 226 226 2788 465 465 2951

Notations: (1) Refer to Table C.6.2 for the value of.ρy. (2) Refer to Table C.6.1 for the value of Ayavg. (3) Φ is 0.75 for shear design.

65

V1x

M1y 2y

2x

M

P

X Modal Lateral LoadTension Wall P100 Compression Wall P200

TV

A A

P102

P101

P103

P202

P201

P203Vstr

0.5V1x

0.5V1x

0.5V2x

0.5V2x

V1x=0.3Vstr

strV 0.7V2x=

A-A

X

Y

L

Lines through Beam Inflection Points

Beam Axial Force

Bea

m S

hear

Ca p

acity

Bea

m S

hear

Cap

acity

Beam Axial Force

F.B.D

Notation: The stiffness ratio between P100 and P200 is 0.3/0.7 due to the coupling effect.

Fig. 4.1.1 X Direction Lateral Load Analysis

tx on P203VMtax

Tension Wall P100 Compression Wall P200tx on P202VV tx on P102

V tx on P103

VV tx on P101 tx on P201

X

Y

Notation: The stiffness ratio between P100 and P200 is 0.3/0.7 due to the coupling effect.

Fig.4.1.2 X Direction Torsion Analysis

66

Y Modal Lateral Load

1xMV1y,V2y

,M 2x

P100 (P200)

0.5VV 2y= str

V

P201

P202

V

A-A

P103

str P101

P10 2

2y

P203

X

Y

0.5V1y= V strV1y

AA

Notation: P100 and P200 have the same stiffness without the coupling effect.

Fig. 4.2.1 Y Direction Lateral Load Analysis

ty on P203VMtay

P200ty on P202VV ty on P102

V ty on P103

VV ty on P101 ty on P201

P100

X

Y

Notation: P100 and P200 have the same stiffness without the coupling effect.

Fig.4.2.2 Y Direction Torsion Analysis

67

6 .80

6.80

5.415.41

5.416.80

6.80

6.80

6 .80

85.

41

Distributed reinforcement in the flangetwo curtainsvertical:#7 @8" c-chorizontal: #7@8" c-c

horizontal: #7@ 10" c-cvertical:#7 @10" c-ctwo curtainsDistributed reinforcement in the web

transverse confinement bar:#7@5" c-c

transverse confinement bar:#7@5" c-clongitudinal bars:39 #11

longitudinal bars:24 #11

7

Group I Walls

Group II Walls

Group III Walls

Fig. 4.3.1 Section Details of Wall Group I

68

6.80

6 .80

5.41 5.415.415.41

6.806.80

6.80

6.80


horizontal: #7@10" c-cvertical:#7 @10" c-ctwo curtainsDistributed reinforcement in the web


transverse confinement bar:#7@5" c-clongitudinal bras:24 #11

longitudinal bras:9 #11

3x5.

41

Fig. 4.3.2 Section Details of Wall Group II

69

6.80

6.80

6.806.80

6.80

6.80

1 0.8

2

5 .41


horizontal: #7@18" c-cvertical:#7 @18" c-ctwo curtainsDistributed reinforcement in the web



10.8210.82

longitudinal bars:14 #11longitudinal bars:6 #11

Fig. 4.3.3 Section Details of Wall Group III

70

Chapter 5 Studies of Behaviors of Diagonally Reinforced Concrete

Coupling Beams

5.1 Notations

cA : Confined core area

sA : Area of the reinforcement in one diagonal core

b : Beam width

yC : Yield compression force of a diagonal core

uC : Ultimate compression force of a diagonal core

bd : Diameter of reinforcement

E : Elastic modulus

cE : Elastic modulus of concrete

secE : Secant modulus of concrete

rE : Post-yielding residual stiffness

cf ( yε ): Concrete stress at the steel yielding strain, yε represents steel yield strain

'ccf : Confined concrete strength

yf : Steel yield strength

uf : Steel ultimate strength

h : Beam depth

gI : Gross moment of inertia

eI : Effective moment of inertia

71

L : Beam span

yM : Beam yield moment capacity

uM : Beam ultimate moment capacity

yT : Yield tension force of a diagonal core

uT : Ultimate tension force of a diagonal core

yV : Beam yield shear capacity

uV : Beam ultimate shear capacity

α : Inclination of a diagonal core

T∆ : Elongation of the diagonal tension core

C∆ : Shortening of the diagonal compression core

y∆ : Beam yield vertical deformation between two ends

u∆ : Beam ultimate vertical deformation between two ends

cε : Unconfined concrete peak strain

ccε : Confined concrete peak strain

yε : Steel yield strain

uε : Steel ultimate strain

yθ : Beam yield chord rotation

uθ : Beam ultimate chord rotation

72

5.2 Objective

Tables 6.18 and 6.19 in FEMA 356 (Federal Emergency Management Agency,

2000) provide values for strength and rotation capacities of diagonally reinforced

concrete coupling beams. However, the stated capacities do not correspond to those

observed experimentally and are conservative. Nonlinear modeling of coupled core wall

structures requires accurate assessment of rotation capacity and strength of diagonally

reinforced coupling beams. Using experimental data from tests conducted between 1970s

and 2000s, two previously proposed analytical models and FEMA 356 were evaluated. A

new model was developed to overcome the deficiencies of the available models, and to

better predict the expected rotation capacity and strength of diagonally reinforced

coupling beams.

5.3 Test Data

The database (Shahrooz, 2005) contains data from 16 tests, as shown in Table 5.1.

The following tests were included in the data base.

(a) Three beams tested by Paulay in 1974 (Paulay, 1974). One of the beams had

an unconfined concrete core whereas the other two specimens were with

confined concrete.

(b) A diagonally reinforced concrete beam tested by Santhakumar in 1974 at the

University of Canterbury in New Zealand (Paulay, Santhakumar, 1976).

(c) Two specimens with unconfined concrete cores tested by Barney (Barney et

al., 1978).

(d) Two beams with confined cores tested by Tassios (Tassios, 1996).

73

(e) A series of tests containing seven specimens conducted by Galano (Galano,

2000).

(f) A specimen with confined core tested in 2001 by Hindi (Hindi, 2001).

5.4 Evaluation of Theoretical Models

5.4.1 Paulay’s Model (Paulay, 2002)

This model is structured with the assumption that the concrete in cores of the

coupling beam can stabilize the diagonal steels from buckling but has no contribution to

the beam capacity. At the time of steel bars yielding, the concrete has already cracked

severely and lost its capacity. Hence, the steel bars predominantly control the strength of

the beam.

The capacity of a diagonal coupling beam is calculated based on the statically

determined model illustrated in Fig. 5.1. The yield forces of diagonal steel bars are

calculated by Equation 5.1. The term represents tension force and the term

symbolizes the compression force. Due to the symmetric configuration of the beam in the

two diagonal directions, is equal to . Using equilibrium equations, the yield shear

capacity ( ) and yield moment capacity ( ) are computed by Equations 5.2 and 5.3.

yT yC

yT yC

yV yM

yT = = (5.1) yC sA yf

yV =2 sinyT α (5.2)

yM = yV L /2 (5.3)

74

Paulay’s model is based on a perfect elastic-plastic strain stress model for the

steel bars. Therefore, the beam ultimate strengths, which occur at the steel ultimate strain,

are equal to the yield strengths as Equation 5.4 shows.

uV = , = (5.4) yV uM yM

The second part of the model deals with the coupling beam deformations. The

elongation of the diagonal tension chord ( T∆ ) of a coupling beam at the time yielding of

bars can be expressed by Equation 5.5. The first term in the parentheses is the elongation

of the bars and the second term is the deformation due to the anchorage sliding.

T∆ = yε ( L /cosα +16 ) (5.5) bd

Experimental data suggest that the shortening ( C∆ ) of the diagonal compression

chord is 30 percent of . The average diagonal deformation is (T∆ T∆ + )/2. By

referring to Fig. 5.2, deformation

C∆

y∆ is the vertical component of the average diagonal

deformation. If the inclination of the diagonal is α , y∆ is calculated by Equation 5.6.

The chord rotation yθ , calculated as the ratio of y∆ to the beam span L , is obtained

from Equation 5.7.

y∆ =( +T∆ C∆ )/(2sinα )=1.3 T∆ /(2sinα )=0.65 yε ( L /cosα +16 )/sinbd α (5.6)

yθ = y∆ / L (5.7)

At the ultimate limit state, the strain of a bar reaches uε . An empirical equation

(Segui, 2003) considering 1 percent strain hardening effect is used to establish uε . This

relationship is shown in Equation 5.8. The vertical deformation and chord rotation at the

75

ultimate state ( and u∆ uθ ) is calculated by replacing yε in Equations 5.6 and 5.7

with uε .

uε =12 yε +( - )/(0.1uf yf E ) (5.8)

u∆ =0.65 uε ( L /cosα +16 )/sinbd α (5.9)

uθ = u∆ / L (5.10)

Using Equations 5.1 to 5.10, the capacities and chord rotations at the yield and

ultimate limit states of the 16 beams in the database were calculated. The results are listed

in Table 5.2.

5.4.2 Hindi’s Model (Hindi, 2001)

Riyadh Hindi proposed a truss model to analyze the capacity and deformation of

diagonally reinforced concrete coupling beams (Hindi and Sexsmith, 2001). This model

is similar to Paulay’s model except that confined diagonal concrete is assumed to develop

its compression strength and contribute toward capacity. This assumption is based on

axial restraining effects of floor slabs that can prevent formation of large cracks.

As shown in Fig 5.3, at the onset of yielding, the diagonal tension force is

equal to . The diagonal compression force includes yielding force of the

diagonal bars and the concrete compression force, as expressed in Equation 5.11.

yT

sA yf yC

yC = +sA yf cA )( ycf ε (5.11)

In this equation, is the confined concrete core area and cA )( ycf ε is the concrete stress

at the steel yielding strain yε . The shear capacity is the sum of the vertical components of

and , as shown in Equation 5.12 shows. yC yT

76

yV =( + )sinyT yC α (5.12)

At the ultimate limit state, the steel remains at its yielding strength and the

concrete reaches the peak strain. Hence, the tension force remains the same as ,

but the concrete compression force is changed to , as shown in Equation 5.13.

uT sA yf

cA'

ccf

uC = + (5.13) sA yf cA'

ccf

In this equation, is confined concrete strength at its peak strain 'ccf ccε . The ultimate

capacity is calculated from Equation 5.14.

uV =( + )sinuT uC α (5.14)

The deformations at yield and ultimate limit states are calculated based on the

assumption that shortening and elongation along the two diagonal directions are the

same. Moreover, deformations in the anchorage region are ignored. Therefore, Equation

5.6 is rewritten as Equation 5.15 and yθ is calculated from Equation 5.16.

y∆ = yε L /sinα (5.15)

yθ = y∆ / L = yε / sinα (5.16)

Similar to Paulay’s model, vertical deformation and chord rotation at the ultimate

limit state ( and u∆ uθ ) are obtained by replacing yε with ccε in the previous equations

to obtain Equations 5.17 and 5.18.

u∆ = ccε L /sinα (5.17)

uθ = u∆ / L = ccε / sinα (5.18)

The calculated deformations and capacities of the 16 beams in the database are

calculated based on the aforementioned equations. The results are organized in Table 5.3.

77

5.5 FEMA 356 (Federal Emergency Management Agency, 2000)

Section 6.8.2.3 of FEMA 356 states that the nominal flexure and shear strength of

a diagonally reinforced concrete beam shall be evaluated using principles and equations

from Chapter 21 of ACI 318-02. ACI 318-02 uses the same equation as Paulay’s model

to get the beam strength. Therefore, FEMA 356 and Paulay’s model are identical in how

strengths of diagonally reinforced concrete beams are calculated. However, the two

methods are different in so far as how deformation capacities are calculated.

FEMA 356 provides an effective stiffness value ( ) of 0.5 for the coupling

beam. Assuming the coupling beam is deformed in double curvature like a typical elastic

beam with two fixed ends, the vertical deformation of the beam can be derived from the

following equation.

eI CE gI

y∆ = /(6 ) = /(3 ) (5.19) yM2L CE eI yM

2L CE gI

The chord rotation at yield is calculated by dividing y∆ by the span length ( L ).

For the post-yield stage, a value of 0.030 is suggested by FEMA 356 as the

difference between the ultimate chord rotation and the yield rotation. Thus, the ultimate

chord rotation uθ is equal to yθ plus 0.030, as shown in Equation 5.20.

uθ = yθ +0.030 (5.20)

Table 5.4 lists the calculated capacities and deformations of the 16 beams in the database.

5.6 Statistical Analyses and Evaluation of Methods

The calculated yield strength, ultimate strength, yield chord rotation, and ultimate

chord rotation are compared against the corresponding measured values. The ratios of the

78

computed to the measured values are tabulated in Table 5.5. Clearly, the models capture

the observed capacities if this ratio is close to 1.0. Each of the aforementioned quantities

is evaluated in the following subsections.

5.6.1 Yield Strength

FEMA 356 and Paulay’s models produce capacities larger than the actual values

for only 4 cases out of 16 test specimens. The average yield strength from FEMA or

Paulay’s model is 0.84 times the measured value with a coefficient of variation of 28%.

The lower calculated capacity may be attributed to the fact that FEMA 356 and Paulay’s

model ignore the contribution of the core concrete to the beam capacity. For Hindi’s

model, the average value is 1.38 and coefficient of variation is 24%. Hence, Hindi’s

model tends to overestimate the yield capacity. Among the 16 specimens, the capacity for

only 1 case is less than the test value. The overestimation of capacity from Hindi’s model

is apparently because the contribution of confined core concrete is estimated excessively.

5.6.2 Ultimate Strength

Similar to yield strength, FEMA or Paulay’s model underestimates the coupling

beam ultimate strength. For all of the16 specimens, the ultimate capacities from FEMA

356 or Paulay’s model are less than the test values. The average value is 0.69 with a

coefficient of variation of 12%. For Hindi’s model, 3 cases out of the 16 specimens have

smaller ultimate capacities than the test values. The average value of calculated/measured

ultimate capacity is 1.32 and coefficient of variation is 30%. The results indicate again

79

that Hindi’s model overestimates the beam ultimate strength as it does for calculating the

yield strength.

5.6.3 Yield Chord Rotation

Compared with the experimental results, extremely small yield rotations are

produced by using FEMA method. None of the calculated yield rotations is larger than

the measured rotations, with the average and coefficient of variation of

calculated/measured ratio being 0.11 and 105%, respectively. The results imply that

FEMA method can not accurately represent the real behavior of coupling beams. The

major difference is attributed to the use of elastic beam analysis method in FEMA 356.

This method appears to be applicable to the conventional beams but inappropriate for

diagonally reinforced concrete beams as the deformation characteristics of the diagonally

reinforced beams and conventionally reinforced beams are significantly different (Park

and Paulay, 1975).

Hindi’s model results in the average value and coefficient of variation of

calculated/measured ratio of 0.99 and 46%, respectively. These values suggest that

Hindi’s model apparently provides good estimates of the yield rotation for diagonally

reinforced concrete beams.

The yield rotations as computed by Paulay’s method are smaller than the

experimental values for 13 out of 16 specimens in the database. The average value and

coefficient of variation of calculated/measured ratio is 0.82 and 45%, respectively.

Hence, Paulay’s method also tends to underestimate the yield deformation capacity.

5.6.4 Ultimate Chord Rotation

80

The tabulated results in Table 5.5 indicate that Hindi’s model can’t accurately

predict the ultimate chord rotations. The average value of calculated/measured is 0.31

with a coefficient of variation of 72%. The calculated ultimate chord rotations for all the

specimens are less than the measured values. The difference is due to how the ultimate

limit state is defined in Hindi’s model and test data. In Hindi’s model, the ultimate limit

state corresponds to when the core concrete reaches its peak strain ccε . However, the

ultimate limit state in the test data is taken as when the steel bars reach the ultimate

tensile strain uε , which is typically larger than ccε .

FEMA method estimates the ultimate chord rotation much better than it does the

yield chord rotation. The average value of calculated/measured ultimate chord rotation is

0.88 with a coefficient of variation of 32%. The better performance is partially attributed

to the incremental value of 0.030 used to relate the yield chord rotation to the ultimate

chord rotation (Fig. 5.4). The large value of 0.030 partly masks the effects of the

extremely small yield chord rotation calculated from FEMA method.

With the exception of one case, Paulay’s model results in larger ultimate chord

rotations than the test data. The average value of calculated/measured ultimate chord

rotation is 2.39 with the coefficient of variation of 36%.

5.7 Modified Model

Considering the values of yield strength, ultimate strength, yield rotation, and

ultimate rotation, Paulay’s model appears to be the most appropriate model to capture the

behavior of diagonally reinforced concrete coupling beams. This model can adequately

81

predict the behavior at the yield but not as well the ultimate responses. A modified

version of Paulay’s model is proposed herein.

In the modified model, Equation 5.2 is still used to calculate the yield strength of

the beam. To account for the strain hardening at the ultimate limit state, which is

neglected in Paulay’s model, the proposed model assumes a 25% increase in the yield

capacity, i.e.,

uV =1.25 (5.21) yV

For the yield chord rotation, Equation 5.6 is used in the modified model except

that the anchorage term is increased from 16 to 40 . The value of 40 is based on

the assumption of having 20 of anchorage length (as recommended by ACI-ASCE

Committee 352 Provisions for monolithic beam-column connections (ACI 352R-02)) on

either end of diagonal bar. Therefore, Equation 5.6 is changed to the following

expression.

bd bd bd

bd

y∆ =0.65 yε ( L /cosα +40 )/sinbd α (5.22)

Once the yield strength, ultimate strength, and yield chord rotation are

determined, the ultimate chord rotation is calculated by Equation 5.23 (see Fig. 5.5).

uθ = yθ +( - )/ (5.23) uV yV rE

In this equation, is the post yield modulus of elasticity. FEMA 356 suggests that

ranges between 0 and 10% of the elastic stiffness (

rE rE

E ). A value of 5% is used in the

proposed model.

The calculated values from the proposed model are summarized in Table 5.6. As

evident from the comparisons of the calculated versus the measured values, the proposed

82

model reproduces the test data appreciably better than existing models discussed in the

previous sections. The average values and coefficients of variation from the proposed

model of the calculated/measured ratio of yield capacity, ultimate capacity, yield rotation,

and ultimate rotation are respectively 0.84 and 28%, 0.87 and 12%, 1.09 and 44%, 1.20

and 44%. The proposed model will be used in the simulation studies described in Chapter

6.

83

Table 5.1 Diagonally Reinforced Concrete Beam Test Database (Organized By Drs. Shahrooz and Harries) L Depth b Diagonal

span h width SteelVy θy Vu θu

Researcher

ID(in) (in) (in)

L/h f'c (ksi)db (in) As (in2) fy (ksi) fu (ksi) α(deg)

Ac (in2) (kips) (rad) (kips) (rad)

0.88 2.40 41.8 316 40.00 31.00 6.00 1.29 4.825 1.00 2.37 41.7

34.9 18 125.00 0.004000 151.50 0.058000

0.88 2.40 44.4317 40.00

31.00 6.00 1.29 7.3481.00 2.37 39.2

34.9 18 110.00 0.003500 130.00 0.025500

0.88 2.40 37.6

Paulay & Binney, 1974

395 40.00

39.00 6.00 1.03 5.1501.00 2.37 41.6

41.9 18 110.00 0.003500 143.00 0.045000

CB-2A 19.69 19.69 5.12 1.00 4.133 0.39 0.49 73.10 110.80 37.2 15.5 48.11 0.014800 63.62 0.028500Tassios et al., 1996 CB-2B 19.69 11.81 5.12 1.66 3.814 0.39 0.49 73.10 110.80 19.8 15.5 25.85 0.017000 38.22 0.031250

P05 b1 23.62 15.75 5.91 1.50 5.787 0.39 0.49 82.23 95.72 25.5 15.5 53.84 0.008420 52.56 0.030000P06 b1 23.62 15.75 5.91 1.50 6.672 0.39 0.49 82.23 95.72 25.5 15.5 49.72 0.008584 54.14 NA P07 b1 23.62 15.75 5.91 1.50 7.832 0.39 0.49 82.23 95.72 25.5 15.5 48.14 0.007259 52.59 0.026600P08 b1 23.62 15.75 5.91 1.50 7.745 0.39 0.49 82.23 95.72 25.5 15.5 47.81 0.006878 51.44 NA P10 b2 23.62 15.75 5.91 1.50 6.788 0.39 0.49 82.23 95.72 25.5 15.5 54.24 0.007750 52.56 0.027700P11 b2 23.62 15.75 5.91 1.50 5.787 0.39 0.49 82.23 95.72 25.5 15.5 43.42 0.007860 52.81 NA

Galano and Vignoli, 2000

P12 b2 23.62 15.75 5.91 1.50 6.033 0.39 0.49 82.23 95.72 25.5 15.5 44.66 0.009762 53.51 0.0250000.38 0.22 70.70 104.70C6 16.67 6.67 4.00 2.50 2.620 0.50 0.20 59.20 103.20

19.1 0 7.80 0.006599 13.40 0.053389

0.38 0.22 82.50 125.10PCA, 1978

C8 33.33 6.67 4.00 5.00 3.470 0.50 0.20 62.80 102.50

9.8 0 4.30 0.006306 7.50 0.087582

Hindi et al., 2001 1 48.00 17.50 12.00 2.74 5.163 1.18 4.38 67.30 90.07 12.4 30.315 148.38 0.008197 204.59 0.049180

Santhakumar, 1974 B 15.00 12.00 3.00 1.25 4.352 0.375

0.25 0.32 47.07 65.09 35.6 1.27 56.50 0.002400 NA 0.035000

Notation: The shaded cells are for the specimens with unconfined cores. The non-shaded cells are for the specimens with confined cores.

84

Table 5.2 Strengths and Deformations Calculated According to Paulay’s Model

Yield Limit State Ultimate Limit State ID Vy=2Asfysinα

(kips) εy=fy/Es

∆y=1.3(L/cosα+16db)εy/(2sinα)

(in)

θy=∆y/L (rad)

εu=12εy+(fu-fy)/(0.1Es)

∆u=1.3(L/cosα+16db)εu/(2sinα)

(in)

θu=∆u/L (rad)

Vu=Vy(kips)

316 113.94 0.0014 0.104294 0.002607 0.02879 2.085878 0.052147 113.94

317 114.12 0.0014 0.104459 0.002611 0.02884 2.089177 0.052229 114.12

395 126.11 0.0014 0.091324 0.002283 0.02730 1.826478 0.045662 126.11

CB-2A 43.07 0.0025 0.084004 0.004267 0.04325 1.441408 0.073224 43.07CB-2B 24.11 0.0025 0.131670 0.006689 0.04325 2.259288 0.114772 24.11

P05 b1 34.46 0.0028 0.139081 0.005888 0.03868 1.897100 0.080311 34.46 P06 b1 34.46 0.0028 0.139081 0.005888 0.03868 1.897100 0.080311 34.46 P07 b1 34.46 0.0028 0.139081 0.005888 0.03868 1.897100 0.080311 34.46 P08 b1 34.46 0.0028 0.139081 0.005888 0.03868 1.897100 0.080311 34.46 P10 b2 34.46 0.0028 0.139081 0.005888 0.03868 1.897100 0.080311 34.46P11 b2 34.46 0.0028 0.139081 0.005888 0.03868 1.897100 0.080311 34.46P12 b2 34.46 0.0028 0.139081 0.005888 0.03868 1.897100 0.080311 34.46

C6 8.97 0.0022 0.109763 0.006584 0.04037 1.969989 0.118176 8.97

C8 5.25 0.0025 0.390819 0.011726 0.04481 6.946094 0.208404 5.25

1 126.66 0.0023 0.477967 0.009958 0.03570 7.352870 0.153185 126.66

B 17.45 0.0016 0.042481 0.002832 0.02569 0.672344 0.044823 17.45

85

Table 5.3 Strengths and Deformations Calculated According to Hindi’s Model Unconfined Concrete Confined Concrete

ID fc' (ksi) εc

Esec=fc'/εc (ksi)

fcc' (ksi) εcc=0.002(1+5(fcc'/fc'-1)) Ac

(in2)

Ec (ksi)

316 4.825 0.0020 2413 4.825 0.0020 18 3959

317 7.348 0.0020 3674 7.634 0.0024 18 4886

395 5.150 0.0020 2575 5.423 0.0025 18 4091

CB-2A 4.133 0.0020 2067 5.195 0.0046 15.5 3664CB-2B 3.814 0.0020 1907 4.855 0.0047 15.5 3520

P05 b1 5.787 0.0020 2893 5.787 0.0020 15.5 4336 P06 b1 6.672 0.0020 3336 6.672 0.0020 15.5 4656 P07 b1 7.832 0.0020 3916 7.832 0.0020 15.5 5044 P08 b1 7.745 0.0020 3872 7.745 0.0020 15.5 5016 P10 b2 6.788 0.0020 3394 10.29 0.0072 15.5 4696P11 b2 5.787 0.0020 2893 9.193 0.0079 15.5 4336P12 b2 6.033 0.0020 3017 9.465 0.0077 15.5 4427

C6 2.620 0.0020 1310 0

C8 3.470 0.0020 1735 0

1 5.163 0.0020 2582 6.009 0.0036 30.32 4096

B 4.352 0.0020 2176 5.658 0.0050 1.27 3760

86

Table 5.3 Strengths and Deformations Calculated According to Hindi’s Model (Continued) Yield Limit State Ultimate Limit State

ID εy x=εy/εc

r=Ec/(Ec-Esec)

fcon=fc'xr/(r-1+xr)Vy=(2Asfy+Acfcon)

sinα (kips)

∆y=εy×L/(cosα×sinα)

(in)

θy=∆y/L (rad)

Vu=(2Asfy+Acfcc')sinα(kips)

∆u=εcc×L/(cosα×sinα) (in)

θu=∆u/L (rad)

316 0.00144 0.7198 2.56 4.466 159.93 0.122723 0.003068 163.63 0.170488 0.004262

317 0.00144 0.7210 4.03 6.474 180.79 0.122917 0.003073 192.74 0.203667 0.005092

395 0.00137 0.6825 2.70 4.615 181.58 0.109850 0.002746 191.29 0.203600 0.005090

CB-2A 0.00252 1.2603 2.29 3.991 80.51 0.102997 0.005232 91.80 0.186670 0.009483CB-2B 0.00252 1.2603 2.18 3.695 43.51 0.155692 0.007909 49.60 0.292057 0.014836

P05 b1 0.00284 1.4178 3.01 5.073 68.29 0.172457 0.007301 62.90 0.121636 0.005149 P06 b1 0.00284 1.4178 3.53 5.605 71.84 0.172457 0.007301 68.80 0.121636 0.005149 P07 b1 0.00284 1.4178 4.47 6.029 74.67 0.172457 0.007301 76.54 0.121636 0.005149 P08 b1 0.00284 1.4178 4.39 6.013 74.56 0.172457 0.007301 75.96 0.121636 0.005149 P10 b2 0.00284 1.4178 3.61 5.663 72.23 0.172457 0.007301 103.09 0.435466 0.018435P11 b2 0.00284 1.4178 3.01 5.073 68.29 0.172457 0.007301 95.77 0.479619 0.020304P12 b2 0.00284 1.4178 3.14 5.233 69.36 0.172457 0.007301 97.58 0.467555 0.019793

C6 0.00225 8.97 0.121134 0.007267 8.97 0.121134 0.007267

C8 0.00252 5.25 0.498755 0.014964 5.25 0.498755 0.014964

1 0.00232 1.1603 2.70 5.064 159.62 0.531113 0.011065 165.77 0.832702 0.017348

B 0.00162 0.8116 2.37 4.228 20.57 0.051423 0.003428 21.62 0.158403 0.010560

Notation: (1) Mander’s confined concrete model is used in the calculation. (2) It is assumed that Member C6 reached its yield and ultimate states simultaneously because its core concrete area is 0 in the database. The same assumption is applied to Member C8.

87

Table 5.4 Strengths and Deformations Calculated According to FEMA 356 Method Element Properties Yield Limit State Ultimate Limit State

ID E=57000fc'0.5/1000

(ksi) Ig

(in4) L

(in) EIe=0.5EIg (kips-in2)

Vy=2fyAssinα (kips)

My=VyL/2 (kips-in)

∆y=MyL2/(3EIg) (in)

θy (rad)

Vu=2fuAssinα (kips)

θu (rad)

316 3959 14896 40.00 29488223 113.94 2278.79 0.02061 0.00052 113.94 0.03052

317 4886 14896 40.00 36390206 114.12 2282.39 0.01673 0.00042 114.12 0.03042

395 4091 29660 40.00 60661382 126.11 2522.10 0.01109 0.00028 126.11 0.03028

CB-2A 3665 3253 19.69 5961251 43.07 423.95 0.00459 0.00023 43.07 0.03023CB-2B 3520 703 19.69 1236934 24.11 237.32 0.01239 0.00063 24.11 0.03063

P05 b1 4336 1922 23.62 4166960 34.46 407.00 0.00908 0.00038 34.46 0.03038 P06 b1 4656 1922 23.62 4474163 34.46 407.00 0.00846 0.00036 34.46 0.03036 P07 b1 5044 1922 23.62 4847633 34.46 407.00 0.00781 0.00033 34.46 0.03033 P08 b1 5016 1922 23.62 4820626 34.46 407.00 0.00785 0.00033 34.46 0.03033 P10 b2 4696 1922 23.62 4512901 34.46 407.00 0.00839 0.00036 34.46 0.03036P11 b2 4336 1922 23.62 4166960 34.46 407.00 0.00908 0.00038 34.46 0.03038P12 b2 4427 1922 23.62 4254804 34.46 407.00 0.00890 0.00038 34.46 0.03038

C6 2918 99 16.67 144295 8.97 74.80 0.02401 0.00144 8.97 0.03144

C8 3358 99 33.33 166060 5.25 87.52 0.09758 0.00293 5.25 0.03293

1 4096 5359 48.00 10975312 126.66 3039.77 0.10635 0.00222 126.66 0.03222

B 3760 432 15.00 812219 17.45 130.86 0.00604 0.00040 17.45 0.03040

88

Table 5.5 Evaluation of All Models Yield Strength/ Test Yield Strength Yield Rotation/ Test Yield Rotation

ID Paulay's Hindi's FEMA Modified Paulay's Hindi's FEMA Modified

316 0.912 1.279 0.912 0.912 0.652 0.767 0.129 0.882

317 1.037 1.644 1.037 1.037 0.746 0.878 0.119 1.009

395 1.146 1.651 1.146 1.146 0.652 0.785 0.079 0.866

CB-2A 0.895 1.673 0.895 0.895 0.288 0.354 0.016 0.376CB-2B 0.933 1.683 0.933 0.933 0.393 0.465 0.037 0.530

P05 b1 0.640 1.268 0.640 0.640 0.699 0.867 0.046 0.903 P06 b1 0.693 1.445 0.693 0.693 0.686 0.851 0.042 0.886 P07 b1 0.716 1.551 0.716 0.716 0.811 1.006 0.046 1.047 P08 b1 0.721 1.560 0.721 0.721 0.856 1.061 0.048 1.105 P10 b2 0.635 1.332 0.635 0.635 0.760 0.942 0.046 0.981P11 b2 0.794 1.573 0.794 0.794 0.749 0.929 0.049 0.967P12 b2 0.772 1.553 0.772 0.772 0.603 0.748 0.039 0.779

C6 1.151 1.151 1.151 1.151 0.998 1.101 0.218 1.421

C8 1.221 1.221 1.221 1.221 1.859 2.373 0.464 2.335

1 0.854 1.076 0.854 0.854 1.215 1.350 0.270 1.721B 0.309 0.364 0.309 0.309 1.180 1.428 0.168 1.557

Average 0.839 1.376 0.839 0.839 0.822 0.994 0.113 1.085Standard Deviation 0.233 0.335 0.233 0.233 0.366 0.457 0.119 0.476

Coefficient of Variation 27.75% 24.32% 27.75% 27.75% 44.58% 46.00% 104.62% 43.88%

89

Table 5.5 Evaluation of All Models (Continued) Ultimate Strength/ Test Ultimate Strength Ultimate Rotation/ Test Ultimate Rotation

ID Paulay's Hindi's FEMA Modified Paulay's Hindi's FEMA Modified

316 0.752 1.080 0.752 0.940 0.899 0.073 0.526 0.365

317 0.878 1.483 0.878 1.097 2.048 0.200 1.193 0.831

395 0.882 1.338 0.882 1.102 1.015 0.113 0.673 0.404

CB-2A 0.677 1.443 0.677 0.846 2.569 0.333 1.061 1.172CB-2B 0.631 1.298 0.631 0.789 3.673 0.475 0.980 1.730

P05 b1 0.656 1.197 0.656 0.819 2.677 0.172 1.013 1.520 P06 b1 0.636 1.271 0.636 0.796 NA NA NA NA P07 b1 0.655 1.455 0.655 0.819 3.019 0.194 1.140 1.715 P08 b1 0.670 1.477 0.670 0.837 NA NA NA NA P10 b2 0.656 1.961 0.656 0.819 2.899 0.666 1.096 1.646P11 b2 0.652 1.813 0.652 0.816 NA NA NA NAP12 b2 0.644 1.824 0.644 0.805 3.212 0.792 1.215 1.824

C6 0.670 0.670 0.670 0.837 2.213 0.136 0.589 1.054

C8 0.700 0.700 0.700 0.875 2.380 0.171 0.376 1.009

1 0.619 0.810 0.619 0.774 3.115 0.353 0.655 1.721B NA NA NA NA 1.281 0.302 0.869 0.641

Average 0.692 1.321 0.692 0.865 2.385 0.306 0.876 1.202Standard Deviation 0.083 0.391 0.083 0.103 0.871 0.219 0.280 0.528

Coefficient of Variation 11.94% 29.56% 11.94% 11.94% 36.54% 71.66% 32.00% 43.92%Notation: The NA cell is due to unavailable test data.

90

Table 5.6 Strengths and Deformations Calculated According to Modified Model

ID Vy=2fyAssinα (kips) εy=fy/Es

db

(in) L

(in) α

(deg)

∆y=1.3(L/cosα+40db)εy/(2sinα)

(in)

θy=∆y/L (rad)

Vu=1.25Vy (kips)

θu=θy+(Vu-Vy)/(0.05E) (rad)

316 113.94 0.0014 0.937 40.00 34.9 0.141080 0.003527 142.42 0.021162

317 114.12 0.0014 0.937 40.00 34.9 0.141303 0.003533 142.65 0.021195

395 126.11 0.0014 0.937 40.00 41.9 0.121206 0.003030 157.63 0.018181

CB-2A 43.07 0.0025 0.394 19.69 37.2 0.109589 0.005567 53.84 0.033403CB-2B 24.11 0.0025 0.394 19.69 19.8 0.177375 0.009011 30.14 0.054064

P05 b1 34.46 0.0028 0.394 23.62 25.5 0.179558 0.007601 43.07 0.045608 P06 b1 34.46 0.0028 0.394 23.62 25.5 0.179558 0.007601 43.07 0.045608 P07 b1 34.46 0.0028 0.394 23.62 25.5 0.179558 0.007601 43.07 0.045608 P08 b1 34.46 0.0028 0.394 23.62 25.5 0.179558 0.007601 43.07 0.045608 P10 b2 34.46 0.0028 0.394 23.62 25.5 0.179558 0.007601 43.07 0.045608P11 b2 34.46 0.0028 0.394 23.62 25.5 0.179558 0.007601 43.07 0.045608P12 b2 34.46 0.0028 0.394 23.62 25.5 0.179558 0.007601 43.07 0.045608

C6 8.97 0.0022 0.435 16.67 19.123 0.156302 0.009376 11.22 0.056258

C8 5.25 0.0025 0.435 33.33 9.847 0.490760 0.014724 6.56 0.088346

1 126.66 0.0023 1.181 48.00 12.400 0.677084 0.014106 158.32 0.084635

B 17.45 0.0016 0.313 15.00 35.628 0.056066 0.003738 21.81 0.022426

91

Cy

Cy

yT

yT

My

Vy

My

Vy

L

Ty

yC

Vy

L

α

Fig. 5.1 Force Equilibrium of Paulay’s Model

∆

∆

α

y

h

Fig. 5.2 Coupling Beam Vertical Deformation of Paulay’s Model

92

Cy

yT

yT

yC

αTension Diagonal

Compression DiagonalL

A

A

B

B

A-A

B-B

Fig. 5.3 Force Equilibrium of Hindi’s Truss Model

0.030

Vy

θu

= Vu

yθθ

V

Fig. 5.4 Shear-Chord Rotation Relationship Defined by FEMA 356

Vy

θu

Vu

yθ

E

Er

θ

V

Fig. 5.5 Shear-Chord Rotation Relationship Defined by Modified Model

93

Chapter 6 Nonlinear Static and Dynamic Analyses

6.1 Notations

bC : Base shear amplification factor

e : Length of the rigid link

E : Elastic modulus

eI : Effective moment of inertia

L : Beam Span

yM : Beam yield moment capacity

r : Ratio of the post-yield stiffness to the elastic stiffness

tV : SRSS base shear in the modal spectrum response analysis

α : Stiffness degradation factor 1

β : Stiffness degradation factor 2

y∆ : Yield vertical deformation of the diagonally reinforced concrete beam

yϕ : Beam yield curvature

θ : Chord rotation of a link beam

1θ : Rigid link rotation at left side of the beam

2θ : Rigid link rotation at right side of the beam

yθ : Beam yield rotation capacity

uθ : Beam ultimate rotation capacity

94

6.2 Objective

The responses of the 15-story building designed based on performance-based

design (PBD) methodology were examined in this chapter to evaluate the adequacy of

PBD. For this purpose, nonlinear static and dynamic analyses were carried out. This

chapter summarizes these analyses and the corresponding results.

6.3 Pushover (Nonlinear Static) Analysis

6.3.1 Introduction

As discussed in Section 3.5.1, a well-designed building has to meet the following

performance requirements.

(1) At the Life Safety (LS) level earthquake, the beams are allowed to yield but the

wall piers are required to remain elastic. The maximum building story drift is to

be less than NEHRP-defined 2% limit.

(2) At the Collapse Prevention (CP) level earthquake, the wall piers are allowed to

yield after the beams have already yielded. The beams are allowed to reach their

ultimate limit capacities.

The Appendix to Chapter 5 of NEHRP 2000 provides guidelines for conducting

pushover analysis. For example, modeling aspects of components, force-deformation

characteristics, and selection of lateral loads are discussed in this appendix. The pushover

analysis of the 15-story building is based on these guidelines.

6.3.2 Computer Model

6.3.2.1 Geometry and Mass Configuration

95

The computer model along with the dimensions, member labels, and node numbers

is shown in Fig.6.1. The vertical members represent wall piers, and are located at the

centroid of each C shaped wall pier. The horizontal members are coupling beams. The

thin line in the middle represents the actual coupling beam and the two thick lines at the

ends are rigid links to reflect the large stiffness of the wall piers that are modeled by line

elements. The length of each rigid link is 7.78 ft, which is the distance between the

centroid of the wall section and the edge to which the beams are connected. Since the

core wall is modeled two dimensionally, one beam member represents two coupling

beams. The calculated masses used in the model are listed in Table A.4.1. At each level,

the mass is distributed equally to each node at that level.

6.3.2.2 Coupling Beam Member Properties

RUAUMOKO (Carr, 2000), the software employed to conduct the analyses, requires

the following quantities to formulate bilinear force-deformation characteristics for a beam

member: the yield moment capacity ( ), the effective moment of inertia ( ), and the

ratio of the residual stiffness to the effective stiffness (

yM eI

r ). The value of is calculated

based on Equation 5.3 as proposed by the modified model in Section 5.7. A linear

distribution of curvature along the beam is assumed. Based on the elastic analysis

concepts, the vertical deformation of the beam (

yM

y∆ ) is equal to 6

2Lyϕ when the ends of a

beam reach yield curvature ( yϕ ). The beam yield vertical deformation ( ) is computed

from the modified model (see Equation 5.22). The calculated deformations for various

y∆

96

coupling beam groups are listed in Table 6.1. The value of yield curvature ( yϕ ) is

subsequently computed from Equation 6.1.

yϕ = 2

6Ly∆ (6.1)

The effective moment of inertia ( ) is calculated based on the fundamental elastic

analysis concepts as shown in Equation 6.2. The values for these three coupling beam

groups are provided in Table 6.1.

eI

eI

eI = /yM E yϕ (6.2)

The value of the residual stiffness ratio r is taken as 0.02, which is an empirical value

generally accepted for reinforced concrete members (Harries et al., 1998).

6.3.2.3 Wall Member Properties

The quadratic beam-column element is selected to model the wall piers. This

element takes into account the interaction between the axial force capacity and moment

capacity by using a parabolic capacity curve (see Fig. 6.2). The four control points of the

curve represent the maximum compression capacity, maximum tension capacity,

maximum positive flexure capacity, and maximum negative flexure capacity. A cross

sectional analysis computer program called XTRACT was used to calculate the values of

these four points for the C-shaped wall piers. Per ACI 21.7.5.1 the reinforcement bars

concentrated in the boundary elements and distributed in the flanges and web were

included. Table 6.2 lists the calculated values of the control points in each wall group.

Appendix D provides the details of the XTRACT calculation of the control point values

in different groups. The signs in the PM interaction curve from XTRACT need to be

97

reversed before using them to establish control points in RUAUMOKO quadratic beam-

column elements because of the difference in the sign conventions of these two programs.

The effective moment of inertia ( ) and residual stiffness ratio (eI r ) are also

required by RUAUMOKO to set up the bilinear force-deformation characteristics for the

quadratic beam-column member. The effective stiffness (E eI ) as calculated by

XTRACT is divided by E (taken as 57000 'cf ) to obtain . The residual stiffness ratio

(

eI

r ) is directly taken from the XTARCT output (which is called “Bilinear Hardening

Slope”). These values for the tension and compression walls were obtained from

XTRACT analyses (see Appendix D). The average values of and eI r are used because

RUAUMOKO allows only single values of and eI r regardless of the direction of

loading. For static pushover analyses, it is possible to predetermine which wall pier will

be in tension or in compression and different values of and eI r can be assigned to the

two wall piers. However, in dynamic analyses each wall pier will be subjected to load

reversals. For consistency between the static and dynamic analyses, it was decided to use

average values of and eI r in all the analyses reported in this chapter.

6.3.2.4 Applied Lateral Loads

According to Section 5A.1.2 of NEHRP, in the pushover analysis the pattern of

lateral loads applied at the mass center of each story should follow the distribution of

fundamental mode as obtained from a modal analysis. For the 15-story building, the

fundamental mode is in the coupled direction. Section 5A.1.2 of NEHRP also states that

the increment of the lateral loads should be sufficiently small to permit capturing of

98

significant changes in individual components, such as yielding, buckling, or failure in the

model. In the reported study, the lateral load increment is set to 1 percent of the modal

base shear ( ). The modal base shear is calculated as the SRSS of the base shears of the

first two modes in the coupled direction (see Table 3.2 for the base shear of each mode).

As Section 5.5.7 of NEHRP requires, the SRSS value needs to be scaled to reach 85

percent of the base shear of equivalent lateral force (ELF) method (see Table 3.2 for 85%

of ELF base shear). For the prototype structure, the value of SRSS base shear was

increased to reach the target value. As Section 5A.1.4.3 of NEHRP stipulates, the

analyses are terminated when the story drift exceeds 125 percent of the design drift limit,

which is 2% percent of the story height. Hence, the pushover analysis was terminated

when the story drift exceeded 2.5% of the story height.

tV

6.3.3 Results and Discussions

Fig. 6.3 (a) shows the development of the roof displacement with the increment of

the lateral loads. The abscissa is the roof displacement in feet and the ordinate is the ratio

of the base shear to the model base shear ( ). The response curve in Fig. 6.3 (a) is

approximately a tri-linear curve. The large slope of the initial part of the curve implies

that the structure is in the elastic state. The slope of the curve changes at a base shear

level equal to 0.70 . This change represents the initiation of yielding of the coupling

beams. Due to reduced effectiveness of the coupling beams after yielding, the stiffness of

the structure is reduced and the roof displacement increases more rapidly. The second

change of the slope occurs at the base shear level equal to 1.22 , at which both wall

piers at the ground story form hinges and the stiffness of the building is reduced further.

tV

tV

tV

99

Fig. 6.3 (c) shows the sequence of yielding. The first yielding occurs in the 8th and

11th floor coupling beams at the base shear equal to 0.70 . Subsequently, yielding

spreads to other coupling beams. The last yielding of coupling beams occurs in the first

level at the base shear level of 1.01 . The range of base shear and displacements

corresponding to yielding of the coupling beams are marked in Fig. 6.3 (a), and the exact

values are shown in Fig. 6.3 (b). The first wall to yield is the tension wall at the ground

level at the base shear level of 1.07 , followed by yielding of the tension walls at the 4

tV

tV

tVth

level yields (at 1.14 ) and the 2tVnd level (at 1.20 ). The first yielding of the

compression wall happens at the ground level at 1.22 . Before exceeding the 2.5% story

drift limit (at which the analysis was terminated), the 5

tV

tV

th level tension yielded at 1.30

followed by yielding of the 4

tV

th level compression wall at 1.33 . The range of yielding in

the wall piers is shown in Fig. 6.3 (a) with the exact sequences of yielding (11 to 16)

denoted in Fig. 6.3 (b) and 6.3 (c).

tV

The roof displacements corresponding to LS and CP level earthquakes are needed to

evaluate the performance of the prototype structure. The SRSS of the roof displacements

of the first two modes in the coupled direction is 0.79 ft. After multiplying the base shear

amplification factor ( =1.73) to scale up to reach 85% of the ELF base shear level, the

roof displacement at LS level is 1.37 ft. The design spectrum response spectrum in

NEHRP, which was used to conduct the reported modal analyses, corresponds to LS level

earthquake acceleration. Therefore, the calculated roof displacement of 1.37 ft is the

displacement at LS level. As seen from Fig. 6.3 (a), this displacement occurs at the base

shear level of 1.02 . The displacement at the CP level is 1.5 times that at the LS level

bC

tV

100

(see Section 3.1). Hence, the roof displacement at the CP level is 2.06ft, which happens at

the base shear of 1.20 (see Fig 6.3 (a)). tV

The previous results verify that the beams yield gradually and at the LS level all the

beams have yielded while wall piers remain elastic. The yielding of walls happens prior

to the CP level. At the CP level, three tension wall piers have yielded. The coupling

beams undergo additional inelastic deformations beyond the LS level. These responses

meet the predefined design requirements described in Section 6.3.1.

The beam chord rotations at the LS and CP levels were obtained based on the

rotations of rigid links computed by RUAUMOKO. As the rigid links, which are between

the column elements representing wall piers and the coupling beams, rotate the beam

undergoes vertical deformations as shown in Fig. 6.4. The total deformation of the beam

is the sum of the absolute values of 1∆ and 2∆ . Note that 1∆ and 2∆ are equal to 1θ e

and 2θ e , respectively, where 1θ and 2θ are the rotations calculated by RUAUMOKO

and e is the length of the rigid link. The beam chord rotation (θ ) can then be found from

Equation 6.3.

θ =( 1∆ + 2∆ )/ L =( 1θ + 2θ )e / L (6.3)

This equation does not account for the deformation of the beam, i.e., it assumes the beam

is a rigid link. Therefore, the beam rotations ( 1θ + 2θ ) need to be added into the value

computed from Equation 6.3. The beam rotation is taken as the average of the absolute

values of 1θ and 2θ . Therefore, Equation 6.4 is used to establish the beam chord rotation.

θ =( 1θ + 2θ ) e / L +( 1θ + 2θ )/2 (6.4)

The beam chord rotations along the building height for the LS and CP levels are

plotted in Fig. 6.5. At the LS level, all the beams yield but do not reach the ultimate chord

101

rotation capacity. On the other hand, at the CP level the beams above the 6th floor exceed

the ultimate chord rotation capacity. The level of chord rotations at the LS and CP levels

meet the performance criteria discussed in Section 6.3.1.

6.4 Nonlinear Dynamic Analysis

6.4.1 Computer Model

The model used for pushover analyses was also used for the dynamic analyses to

ensure some level of consistency between the results from two methods. The stiffness and

mass distributions were the same as those used for pushover analyses. However, dynamic

analyses require additional input parameters, as discussed in the following.

The modified Takeda model (Carr, 2000), as shown in Figure 6.6, was employed to

model the hysteretic force-deformation relationships of the coupling beams and wall piers.

This model accounts for stiffness and strength degradations, which are two important

characteristics of reinforced concrete members under cyclic loads. The factorsα and β

are used to control the level of stiffness degradation. The selected α for wall piers and

beams are 0 and 0.1, respectively. The selected β for walls and beams are 0.6 and 0.5,

respectively. These values were selected based on a prior research (Harries et al., 1998).

The strength degradation in RUAUMOKO program is expressed in terms of a reduction

factor. The value of the reduction factor is related to the member ductility (see Fig. 6.7).

The selected values for beams and wall piers are listed in Table 6.4. Five ground motions

ranging from slight intensity to severe intensity were selected, three of which were

recorded acceleration records (El Centro1940 NS, Northridge 1994 at Pacoima Dam NS,

and Northridge 1994 at Slymar Hospital NS) and two were artificial accelerograms. The

102

artificial records were generated by SIMQKE, which is a component of RUAUMOKO.

The selected ground motions are shown in Fig. 6.8 with their acceleration response

spectra illustrated in Fig. 6.9. Based on the shown response spectra, El Centro 1940 NS

record is deemed to be a service level earthquake, and the remaining records are taken as

collapse prevention level earthquake.

6.4.2 Results and Discussions

The histories of the roof displacement for the five seismic ground excitations are

shown in Fig. 6.10. The maximum roof drifts are 0.82 ft, 0.86 ft, 0.91 ft, 1.23 ft, and 1.83

ft for simulated LS level, El Centro (NS), Northridge at Pacoima Dam (NS), simulated

CP level, and Northridge at Slymar (NS), respectively. The artificial ground motions at

LS and CP levels continue to impart energy to the building as evident by continued roof

drift reversals throughout the ground motion records, whereas the selected recorded

ground motions stop producing significant deformations during approximately the last

half of the records

The maximum inter-story drifts along the building height are displayed in Fig. 6.11.

The maximum values (as a percentage of story height) are 1.0%, 1.12%, 1.62%, 1.86%

and 2.49% for El Centro, simulated LS level, simulated CP level, Northridge at Pacoima,

and Northridge at Slymar, respectively. These magnitudes match the expected input

energies of the different records. For example, at 1.8 sec., which is the period of the first

mode, Northridge at Slymar has the largest acceleration response spectrum value and the

lowest value is for El Centro record. The maximum inter-story drift of 1.12% at LS level

103

excitation is less than 2%, which meets one of the predefined performance criteria

discussed in Section 6.3.1.

The complete histories of chord rotation under the five selected ground excitations

are displayed in Fig. 6.12 through 6.16. The maximum chord rotations with the time at

which and locations where the maximum rotations occur are tabulated in Table 6.5. For

service level ground motions (i.e., El Centro and simulated LS), the beam chord rotations

exceed the yield rotation capacity ( yθ ) but are less than the ultimate capacity ( uθ ). This

behavior coincides with the predefined LS level requirement of Section 6.3.1, where

beams are expected to yield but not exceed the ultimate state. Under the excitations

considered to correspond to collapse prevention level (i.e., simulated CP record,

Northridge at Pacoima, and Northridge at Slymar), the beam chord rotations exceed both

the yield and ultimate rotation capacities. This performance also meets the performance

criteria at CP level where beams are allowed to reach the ultimate state, as discussed in

Section 6.3.1.

An index, referred to as Wall Damage Index (WDI), is used to quantify the wall

state during seismic ground excitations. A value of zero indicates that the wall pier is

elastic, i.e., the axial load-moment demands are within the capacity interaction diagram.

On the other hand, WDI equal to 1 indicates that the axial load-moment demands are

outside of the interaction surface, and the wall has experienced inelastic deformations.

Figures 6.12 and 6.13 show that all wall piers remain elastic under El Centro and

simulated LS level ground motion, which satisfies the performance criteria defined in

Section 6.3.1. Figures 6.14 through 6.16 show the histories of WDI under simulated CP

level, Northridge at Pacoima, and Northridge at Slymar, respectively. Under all of these

104

three ground motions, the ground level wall piers develop inelastic deformations and

plastic hinges form. Wall piers at other levels near the middle or close to the top of

structure also experience inelastic deformations. Levels 4, 10, 11, and 12 develop

inelastic deformations under simulated CP motion. Levels 6, 7, 11, and 12 under

Northridge at Pacoima motion, and levels 4, 5, 8, 9, 10, and 11 under Northridge at

Slymar motion also experience inelastic deformations. Although inelastic deformation is

more likely to be produced in the ground level, other floors also experience inelastic

deformations albeit less frequently. The predefined criteria in Section 6.3.1 allow wall

piers in any level to undergo inelastic deformation under motions corresponding to

collapse prevention level. Hence, the behaviors of wall piers are in accordance with the

performance criteria for which the structure was designed. The occurrence of plastic

deformations in these upper levels indicates a gradual reduction of the stiffness as the

coupling beams reach their ultimate limit state. Similar observations have also been made

by others (McNiece, 2004).

105

Table 6.1 Beam Member Properties

L

(ft) My (k-ft)

∆y

(ft) φy=6∆y/L2

(/ft) Ie=My/Eφy

(ft4) γ

Group I (Lv 2-7) 6 1456 0.0530 0.00883 0.260 0.02 Group II(Lv 8-10,1) 6 1147 0.0506 0.00843 0.214 0.02 Group III(Lv 11-15) 6 688 0.0464 0.00773 0.140 0.02

Table 6.2 Values of Four Control Points for Quadratic Beam-Column Elements (refer to Fig. 6.2)

PYC (kips)

PB (kips)

MB (k-ft)

PC (kips)

MC (k-ft)

PYT (kips)

Group I (Lv 1-3) -79630 -43700 133800 -23400 -115800 13970 Group II(Lv 4-7) -72880 -42810 112200 -21250 -100600 9433 Group III(Lv 8-15) -68180 -40900 100400 -20290 -91360 5553

Table 6.3 Wall Member Properties Tension Wall Compression Wall Average Value

Ie

(ft4) γ

Ie

(ft4) γ

Ie

(ft4) γ

Group I (Lv 1-3) 230 0.023 368 0.007 299 0.015

Group II(Lv 4-7) 162 0.007 267 0.007 215 0.007

Group III(Lv 8-15) 116 0.004 181 0.002 149 0.003

106

Table 6.4 Strength Degradation Factors (refer to Fig. 6.7)

DUCT1 DUCT2 RDUCT DUCT3 Beam Member 20 30 0.5 0

Shear Wall Member 14 20 0.5 0

Table 6.5 Maximum Chord Rotations under Five Selected Ground Motions Rotation Capacity*

(rad) Ground Motion Maximum Chord

Rotation (rad)

Location Time (sec)

θy θu

El Centro 0.035 Floor 13 5.6 0.0077 0.0464 Simulated LS 0.039 Floor 14 14 0.0077 0.0464 Simulated CP 0.059+ Floor 14 11.7 0.0077 0.0464

Northridge at Pacoima 0.068+ Floor 15 4.1 0.0077 0.0464 Northridge at Slymar 0.088+ Floor 15 4 0.0077 0.0464

Notation: * Rotation capacity corresponding to the floor where the maximum chord rotation was obtained. + Although the coupling beams have exceeded their ultimate rotation capacities, the wall piers can still provide resistance

because they have not reached their ultimate capacities.

107

Fig. 6.1 Nonlinear Analyses Model 21.56 ft

12.1

7 ft

9.17

ft9.

17ft

9.17

ft9.

17ft

9.17

ft9.

17ft

9.17

ft9.

17ft

9.17

ft9.

17ft

9.17

ft9.

17ft

9.17

ft9.

17ft

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

25

26

30

27

28

29

31

32

17

18

19

20

22

21

23

24

1 16

312 17

3 1832

433

19

534

20

635

21

7

15

41

31

54

4403

3492

2428

2141

72

0411 62

0193

25

983

42

873

23

3622

e=7.78 ft L=6 ft

Node Number

Member Number

108

Fig. 6.2 Axial Load-Moment Interaction Diagram for Quadratic Beam-column

Element

109

CP

1.20

LS

1.02

1.01

0.70

1.07

1.33

0.00

(a) Base Shear vs. Roof Displacement

0.00 0.50 1.00 1.50 2.00 2.50 3.00 3.50

1615

109

87

65

43

21

11

1213

14

0.00

0.20

0.40

0.60

0.80

1.00

1.20

1.40

1.60

0.00 0.50 1.00 1.50 2.00 2.50 3.00 3.50

Roof Displacement (ft)

Base Shear Ratio (V/Vt)

Roof Displacem t (ft)

(b) Member Yield Sequence along the Curve

Beam Yield Range

Wall Yield Range

Refer to (c) for the position of the member that each n ber represents.

1 (0.70)

2 (0.71)

3 (0.72)

4 (0.73)

2 (0.71)

5 (0.74)

3 (0.72)

5 (0.74)

3 (0.72)

6 (0.75)

8 (0.80)

1 (0.70)

7 (0.77)

9 (0.87)

10 (1.01)

12 (1.14)

11 (1.07)

16 (1.33)

14 (1.22)

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 Notation: 14 (1.22)

Yield sequence Base shear (unit Vt )

13 (1.20)

15 (1.30)

1st yielding in wall piers

1st yielding in coupling beams

1.60Base Shear Ratio (V/Vt)

(c) Locations of Yield Members

Fig. 6.3 Pus ver Analysis Result 110

en

um

ho

0.20

0.40

0.60

0.80

1.00

1.20

1.40

e L e

θ1

2θ

∆ 1

2∆Node

Wall

Rigid Link

Fig. 6.4 Beam Vertical Deformation Caused by Rigid Link Rotations

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

0 0.01 0.02 0.03 0.04 0.05

LS Chord Rotation from Rigid Link Method CP Chord R

Yield Chord Rotation Capacity Ult imate C

Story

Fig.6.5 Chord Rotation Distributions at L

111

Chord Rotation (rad)

0.06 0.07 0.08

otation from Rigid Link Method

hord Rotation Capacity

S and CP States

Fig 6.6 Modified Takeda Hysteresis Model

Fig. 6.7 Strength Degradation Model Used in RUAUMOKO

112

-1.20-1.00-0.80-0.60-0.40-0.200.000.200.400.600.80

0 2 4 6 8 10 12 14 16 18 20

-1.20-1.00-0.80-0.60-0.40-0.200.000.200.400.600.80

0 2 4 6 8 10 12 14 16 18 20

-1.20-1.00-0.80-0.60-0.40-0.200.000.200.400.600.80

0 2 4 6 8 10 12 14 16 18 20

-1.20-1.00-0.80-0.60-0.40-0.200.000.200.400.600.80

0 2 4 6 8 10 12 14 16 18 20

-1.20-1.00-0.80-0.60-0.40-0.200.000.200.400.600.80

0 2 4 6 8 10 12 14 16 18 20

Fig. 6.8 Selected Earthquake Ground Motions

Time (s)

CP Simulated

Northridge Pacoima Dam 1994 NS

Northridge Slymar Hospital 1994 NS

LS Simulated

El Centro 1940 NS

Gro

und

Acc

eler

atio

n (g

)

113

Acceleration Response (g)

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

2.2

2.4

2.6

2.8

3.0

3.2

3.4

3.6

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

Northridge at Pacoima Dam (NS)

Northridge at Sylmar (NS)

Simulated CP Level

Design Spectrum at LS Level

Design Spectrum at CP Level

Simulated LS Level

El Centro (NS) Period (s)

Fig. 6.9 Structural Acceleration Response Spectra Induced by 5 Selected Ground

Motions

114

Roo

f Dis

plac

emen

t (ft

)

El Centro (NS)

0.86

-2-1.5

-1-0.5

00.5

11.5

2

0 2 4 6 8 10 12 14 16 18 20

Simulated LS Level

0.82

-2-1.5

-1-0.5

00.5

11.5

2

0 2 4 6 8 10 12 14 16 18 20

Northridge Pacoima (NS)

-0.91

-2-1

-0

.5-1.50

0.51

1.52

0 2 4 6 8 10 12 14 16 18 20

Simulated CP Level

1.23

-2-1.5

-1-0.5

00.5

11.5

2

0 2 4 6 8 10 12 14 16 18 20

Northridge Sylmar (NS) 1.83

-2-1.5

-1-0.5

00.5

11.5

2

0 2 4 6 8 10 12 14 16 18 20

Fig. 6.10 Roof Displacement History

Time (s)

115

Story

El Centro (NS) 1.0%

Simulated LS Level 1.12% Simulated CP Level

1.62%

Northridge Pacoima (NS) 1.86%

Story Drift (% )

Northridge Slymar (NS) 2.49%

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

0.00% 0.50% 1.00% 1.50% 2.00% 2.50%

Story Drift (%)

Code Limit for Inter-story Drift

Fig. 6.11 Story Drift Envelope

116

θy=0 .0 0 77

θu=0 .0 4 6 4

-0 .0 6-0 .05-0 .0 4-0 .0 3-0 .0 2-0 .0 10 .0 00 .0 10 .0 20 .0 30 .0 40 .050 .0 6

0 2 4 6 8 10 12 14 16 18 20

0

1

0 2 4 6 8 10 12 14 16 18 200

1

0 2 4 6 8 10 12 14 16 18 20

θy=0 .0 0 77

θu=0 .0 4 6 4

-0 .0 6-0 .05-0 .0 4-0 .0 3-0 .0 2-0 .0 10 .0 00 .0 10 .0 20 .0 30 .0 40 .050 .0 6

0 2 4 6 8 10 12 14 16 18 20

0

1

0 2 4 6 8 10 12 14 16 18 200

1

0 2 4 6 8 10 12 14 16 18 20

θy=0 .0 0 77

θu=0 .0 4 6 4

-0 .0 6-0 .05-0 .0 4-0 .0 3-0 .0 2-0 .0 10 .0 00 .0 10 .0 20 .0 30 .0 40 .050 .0 6

0 2 4 6 8 10 12 14 16 18 20

0

Right Side Walls Left Side Walls Beams

Level 14

Level 15

1

0 2 4 6 8 10 12 14 16 18 200

1

0 2 4 6 8 10 12 14 16 18 20

θy=0 .0 0 77

θu=0 .0 4 6 4

-0 .0 6-0 .05-0 .0 4-0 .0 3-0 .0 2-0 .0 10 .0 00 .0 10 .0 20 .0 30 .0 40 .050 .0 6

0 2 4 6 8 10 12 14 16 18 20

0

1

0 2 4 6 8 10 12 14 16 18 200

1

0 2 4 6 8 10 12 14 16 18 20

θy=0 .00 77

θu=0 .0 4 6 4

-0 .0 6-0 .05-0 .0 4-0 .0 3-0 .0 2-0 .0 10 .0 00 .0 10 .0 20 .0 30 .0 40 .050 .0 6

0 2 4 6 8 10 12 14 16 18 20

0

1

0 2 4 6 8 10 12 14 16 18 200

1

0 2 4 6 8 10 12 14 16 18 20

Time (s)

WD

I

Cho

rd R

otat

ion

(rad

)

Level 13

Level 12

Level 11

Time (s) Time (s)

Fig. 6.12 Member Responses under El Centro Ground Motion (Lv11-Lv15)

Notations: 1. θy and θu are the yield and ultimate chord rotation capacities, see Section 6.3.2.2 2. WDI is the abbreviation of Wall Damage Index; WDI=0: Elastic behavior; WDI=1: Inelastic

Behavior; see Section 6.4.2

117

Beams Left Side Walls Right Side Walls

θy=0 .00 8 4

θu=0 .0 50 6

-0 .0 6-0 .05-0 .0 4-0 .0 3-0 .0 2-0 .0 10 .0 00 .0 10 .0 20 .0 30 .0 40 .050 .0 6

0 2 4 6 8 10 12 14 16 18 20

0

1

0 2 4 6 8 10 12 14 16 18 200

1

0 2 4 6 8 10 12 14 16 18 20

θy=0 .0 0 8 4

θu=0 .0 50 6

-0 .0 6-0 .05-0 .0 4-0 .0 3-0 .0 2-0 .0 10 .0 00 .0 10 .0 20 .0 30 .0 40 .050 .0 6

0 2 4 6 8 10 12 14 16 18 20

0

1

0 2 4 6 8 10 12 14 16 18 200

1

0 2 4 6 8 10 12 14 16 18 20

θy=0 .0 0 8 4

θu=0 .0 50 6

-0 .05-0 .0 4-0 .0 3-0 .0 2-0 .0 10 .0 00 .0 10 .0 20 .0 30 .0 40 .050 .0 6

0 2 4 6 8 10 12 14 16 18 20

0

Level 9

Level 10

1

0 2 4 6 8 10 12 14 16 18 200

1

0 2 4 6 8 10 12 14 16 18 20

θy=0 .0 0 8 8

θu=0 .0 53 0

-0 .0 6-0 .05-0 .0 4-0 .0 3-0 .0 2-0 .0 10 .0 00 .0 10 .0 20 .0 30 .0 40 .050 .0 6

0 2 4 6 8 10 12 14 16 18 20

0

1

0 2 4 6 8 10 12 14 16 18 200

1

0 2 4 6 8 10 12 14 16 18 20

θy=0 .0 0 8 8

θu=0 .0 53 0

-0 .0 6-0 .05-0 .0 4-0 .0 3-0 .0 2-0 .0 10 .0 00 .0 10 .0 20 .0 30 .0 40 .050 .0 6

0 2 4 6 8 10 12 14 16 18 20

0

1

0 2 4 6 8 10 12 14 16 18 200

1

0 2 4 6 8 10 12 14 16 18 20

Time (s)

Level 8

Level 7

Level 6

WD

I

Cho

rd R

otat

ion

(rad

)

Time (s) Time (s)

Fig. 6.12 Member Responses under El Centro Ground Motion (Lv6-Lv10) Notations: 1. θy and θu are the yield and ultimate chord rotation capacities, see Section 6.3.2.2

2. WDI is the abbreviation of Wall Damage Index; WDI=0: Elastic behavior; WDI=1: Inelastic Behavior; see Section 6.4.2

118


θy=0 .0 0 8 8

θu=0 .0 53 0

-0 .0 6-0 .05-0 .0 4-0 .0 3-0 .0 2-0 .0 10 .0 00 .0 10 .0 20 .0 30 .0 40 .050 .0 6

0 2 4 6 8 10 12 14 16 18 20

0

1

0 2 4 6 8 10 12 14 16 18 200

1

0 2 4 6 8 10 12 14 16 18 20

θy=0 .0 0 8 8

θu=0 .0 53 0

-0 .0 6-0 .05-0 .0 4-0 .0 3-0 .0 2-0 .0 10 .0 00 .0 10 .0 20 .0 30 .0 40 .050 .0 6

0 2 4 6 8 10 12 14 16 18 20

0

1

0 2 4 6 8 10 12 14 16 18 200

1

0 2 4 6 8 10 12 14 16 18 20

θy=0 .0 0 8 8

θu=0 .0 53 0

-0 .0 6-0 .05-0 .0 4-0 .0 3-0 .0 2-0 .0 10 .0 00 .0 10 .0 20 .0 30 .0 40 .050 .0 6

0 2 4 6 8 10 12 14 16 18 20

0

Level 4

Level 5

1

0 2 4 6 8 10 12 14 16 18 200

1

0 2 4 6 8 10 12 14 16 18 20

θy=0 .0 0 8 8

θu=0 .0 53 0

-0 .0 6-0 .05-0 .0 4-0 .0 3-0 .0 2-0 .0 10 .0 00 .0 10 .0 20 .0 30 .0 40 .050 .0 6

0 2 4 6 8 10 12 14 16 18 20

0

1

0 2 4 6 8 10 12 14 16 18 200

1

0 2 4 6 8 10 12 14 16 18 20

θy=0 .0 0 8 4

θu=0 .0 50 6

-0 .0 6-0 .05-0 .0 4-0 .0 3-0 .0 2-0 .0 10 .0 00 .0 10 .0 20 .0 30 .0 40 .050 .0 6

0 2 4 6 8 10 12 14 16 18 20

0

1

0 2 4 6 8 10 12 14 16 18 200

1

0 2 4 6 8 10 12 14 16 18 20

Level 3

Level 2

Level 1

WD

I

Cho

rd R

otat

ion

(rad

)

Time (s) Time (s) Time (s)

Fig. 6.12 Member Responses under El Centro Ground Motion (Lv1-Lv5)



119

θy=0 .0 0 77

θu=0 .0 4 6 4

-0 .05-0 .0 4-0 .0 3-0 .0 2-0 .0 10 .0 00 .0 10 .0 20 .0 30 .0 40 .050 .0 6

0 2 4 6 8 10 12 14 16 18 20

0

1

0 2 4 6 8 10 12 14 16 18 200

1

0 2 4 6 8 10 12 14 16 18 20

θy=0 .0 0 77

θu=0 .0 4 6 4

-0 .05-0 .0 4-0 .0 3-0 .0 2-0 .0 10 .0 00 .0 10 .0 20 .0 30 .0 40 .050 .0 6

0 2 4 6 8 10 12 14 16 18 20

0

1

0 2 4 6 8 10 12 14 16 18 200

1

0 2 4 6 8 10 12 14 16 18 20

θy=0 .0 0 77

θu=0 .0 4 6 4

-0 .0 6-0 .05-0 .0 4-0 .0 3-0 .0 2-0 .0 10 .0 00 .0 10 .0 20 .0 30 .0 40 .050 .0 6

0 2 4 6 8 10 12 14 16 18 20

0


Level 14

Level 15

1

0 2 4 6 8 10 12 14 16 18 200

1

0 2 4 6 8 10 12 14 16 18 20

θy=0 .0 0 77

θu=0 .0 4 6 4

-0 .0 6-0 .05-0 .0 4-0 .0 3-0 .0 2-0 .0 10 .0 00 .0 10 .0 20 .0 30 .0 40 .050 .0 6

0 2 4 6 8 10 12 14 16 18 20

0

1

0 2 4 6 8 10 12 14 16 18 200

1

0 2 4 6 8 10 12 14 16 18 20

θy=0 .00 77

θu=0 .0 4 6 4

-0 .0 6-0 .05-0 .0 4-0 .0 3-0 .0 2-0 .0 10 .0 00 .0 10 .0 20 .0 30 .0 40 .050 .0 6

0 2 4 6 8 10 12 14 16 18 20

0

1

0 2 4 6 8 10 12 14 16 18 200

1

0 2 4 6 8 10 12 14 16 18 20

Time (s)

WD

I

Cho

rd R

otat

ion

(rad

)

Level 13

Level 12

Level 11

Time (s) Time (s)

Fig. 6.13 Member Responses under Simulated LS Ground Motion (Lv11-Lv15) Notations: 1. θy and θu are the yield and ultimate chord rotation capacities, see Section 6.3.2.2


120


θy=0 .00 8 4

θu=0 .0 50 6

-0 .0 6-0 .05-0 .0 4-0 .0 3-0 .0 2-0 .0 10 .0 00 .0 10 .0 20 .0 30 .0 40 .050 .0 6

0 2 4 6 8 10 12 14 16 18 20

0

1

0 2 4 6 8 10 12 14 16 18 200

1

0 2 4 6 8 10 12 14 16 18 20

θy=0 .0 0 8 4

θu=0 .0 50 6

-0 .0 6-0 .05-0 .0 4-0 .0 3-0 .0 2-0 .0 10 .0 00 .0 10 .0 20 .0 30 .0 40 .050 .0 6

0 2 4 6 8 10 12 14 16 18 20

0

1

0 2 4 6 8 10 12 14 16 18 200

1

0 2 4 6 8 10 12 14 16 18 20

θy=0 .0 0 8 4

θu=0 .0 50 6

-0 .05-0 .0 4-0 .0 3-0 .0 2-0 .0 10 .0 00 .0 10 .0 20 .0 30 .0 40 .050 .0 6

0 2 4 6 8 10 12 14 16 18 20

0

Level 9

Level 10

1

0 2 4 6 8 10 12 14 16 18 200

1

0 2 4 6 8 10 12 14 16 18 20

θy=0 .0 0 8 8

θu=0 .0 53 0

-0 .0 6-0 .05-0 .0 4-0 .0 3-0 .0 2-0 .0 10 .0 00 .0 10 .0 20 .0 30 .0 40 .050 .0 6

0 2 4 6 8 10 12 14 16 18 20

0

1

0 2 4 6 8 10 12 14 16 18 200

1

0 2 4 6 8 10 12 14 16 18 20

θy=0 .0 0 8 8

θu=0 .0 53 0

-0 .0 6-0 .05-0 .0 4-0 .0 3-0 .0 2-0 .0 10 .0 00 .0 10 .0 20 .0 30 .0 40 .050 .0 6

0 2 4 6 8 10 12 14 16 18 20

0

1

0 2 4 6 8 10 12 14 16 18 200

1

0 2 4 6 8 10 12 14 16 18 20

Time (s)

Level 8

Level 7

Level 6

WD

I

Cho

rd R

otat

ion

(rad

)

Time (s) Time (s)



121


θy=0 .0 0 8 8

θu=0 .0 53 0

-0 .0 6-0 .05-0 .0 4-0 .0 3-0 .0 2-0 .0 10 .0 00 .0 10 .0 20 .0 30 .0 40 .050 .0 6

0 2 4 6 8 10 12 14 16 18 20

0

1

0 2 4 6 8 10 12 14 16 18 200

1

0 2 4 6 8 10 12 14 16 18 20

θy=0 .0 0 8 8

θu=0 .0 53 0

-0 .0 6-0 .05-0 .0 4-0 .0 3-0 .0 2-0 .0 10 .0 00 .0 10 .0 20 .0 30 .0 40 .050 .0 6

0 2 4 6 8 10 12 14 16 18 20

0

1

0 2 4 6 8 10 12 14 16 18 200

1

0 2 4 6 8 10 12 14 16 18 20

θy=0 .0 0 8 8

θu=0 .0 53 0

-0 .0 6-0 .05-0 .0 4-0 .0 3-0 .0 2-0 .0 10 .0 00 .0 10 .0 20 .0 30 .0 40 .050 .0 6

0 2 4 6 8 10 12 14 16 18 20

0

Level 4

Level 5

1

0 2 4 6 8 10 12 14 16 18 200

1

0 2 4 6 8 10 12 14 16 18 20

θy=0 .0 0 8 8

θu=0 .0 53 0

-0 .0 6-0 .05-0 .0 4-0 .0 3-0 .0 2-0 .0 10 .0 00 .0 10 .0 20 .0 30 .0 40 .050 .0 6

0 2 4 6 8 10 12 14 16 18 20

0

1

0 2 4 6 8 10 12 14 16 18 200

1

0 2 4 6 8 10 12 14 16 18 20

θy=0 .0 0 8 4

θu=0 .0 50 6

-0 .0 6-0 .05-0 .0 4-0 .0 3-0 .0 2-0 .0 10 .0 00 .0 10 .0 20 .0 30 .0 40 .050 .0 6

0 2 4 6 8 10 12 14 16 18 20

0

1

0 2 4 6 8 10 12 14 16 18 200

1

0 2 4 6 8 10 12 14 16 18 20

Time (s)

Level 3

Level 2

Level 1

WD

I

Cho

rd R

otat

ion

(rad

)

Time (s) Time (s)



122

θy=0 .0 0 77

θu=0 .0 4 6 4

-0 .0 6-0 .0 5-0 .0 4-0 .0 3-0 .0 2-0 .0 10 .0 00 .0 1

0 .0 20 .0 30 .0 40 .0 50 .0 6

0 2 4 6 8 10 12 14 16 18 2 0

0

1

0 2 4 6 8 10 12 14 16 18 200

1

0 2 4 6 8 10 12 14 16 18 20

θy=0 .0 0 77

θu=0 .0 4 6 4

-0 .0 6-0 .0 5-0 .0 4-0 .0 3-0 .0 2-0 .0 10 .0 00 .0 1

0 .0 20 .0 30 .0 40 .0 50 .0 6

0 2 4 6 8 10 12 14 16 18 2 0

0

1

0 2 4 6 8 10 12 14 16 18 200

1

0 2 4 6 8 10 12 14 16 18 20

θy=0 .0 0 77

θu=0 .0 4 6 4

-0 .0 6-0 .0 5-0 .0 4-0 .0 3-0 .0 2-0 .0 10 .0 00 .0 1

0 .0 20 .0 30 .0 40 .0 50 .0 6

0 2 4 6 8 10 12 14 16 18 2 0

0


Level 14

Level 15

1

0 2 4 6 8 10 12 14 16 18 200

1

0 2 4 6 8 10 12 14 16 18 20

θy=0 .0 0 77

θu=0 .0 4 6 4

-0 .0 6-0 .0 5-0 .0 4-0 .0 3-0 .0 2-0 .0 10 .0 00 .0 1

0 .0 20 .0 30 .0 40 .0 50 .0 6

0 2 4 6 8 10 12 14 16 18 2 0

0

1

0 2 4 6 8 10 12 14 16 18 20

θy=0 .00 77

θu=0 .0 4 6 4

-0 .0 6-0 .0 5-0 .0 4-0 .0 3-0 .0 2-0 .0 10 .0 00 .0 1

0 .0 20 .0 30 .0 40 .0 50 .0 6

0 2 4 6 8 10 12 14 16 18 2 0

0

1

0 2 4 6 8 10 12 14 16 18 20

0

1

0 2 4 6 8 10 12 14 16 18 20

0

1

0 2 4 6 8 10 12 14 16 18 20

Time (s)

WD

I

Cho

rd R

otat

ion

(rad

)

Level 13

Level 12

Level 11

Time (s) Time (s)

Fig. 6.14 Member Responses under Simulated CP Ground Motion (Lv11-Lv15) Notations: 1. θy and θu are the yield and ultimate chord rotation capacities, see Section 6.3.2.2


123


θy=0 .00 8 4

θu=0 .0 50 6

-0 .0 6-0 .05-0 .0 4-0 .0 3-0 .0 2-0 .0 10 .0 00 .0 10 .0 20 .0 30 .0 40 .050 .0 6

0 2 4 6 8 10 12 14 16 18 20

0

1

0 2 4 6 8 10 12 14 16 18 20

θy=0 .0 0 8 4

θu=0 .0 50 6

-0 .0 6-0 .05-0 .0 4-0 .0 3-0 .0 2-0 .0 10 .0 00 .0 10 .0 20 .0 30 .0 40 .050 .0 6

0 2 4 6 8 10 12 14 16 18 20

0

1

0 2 4 6 8 10 12 14 16 18 200

1

0 2 4 6 8 10 12 14 16 18 20

θy=0 .0 0 8 4

θu=0 .0 50 6

-0 .05-0 .0 4-0 .0 3-0 .0 2-0 .0 10 .0 00 .0 10 .0 20 .0 30 .0 40 .050 .0 6

0 2 4 6 8 10 12 14 16 18 20

0

1

0 2 4 6 8 10 12 14 16 18 200

1

0 2 4 6 8 10 12 14 16 18 20

θy=0 .0 0 8 8

θu=0 .0 53 0

-0 .0 6-0 .05-0 .0 4-0 .0 3-0 .0 2-0 .0 10 .0 00 .0 10 .0 20 .0 30 .0 40 .050 .0 6

0 2 4 6 8 10 12 14 16 18 20

0

1

0 2 4 6 8 10 12 14 16 18 200

1

0 2 4 6 8 10 12 14 16 18 20

θy=0 .0 0 8 8

θu=0 .0 53 0

-0 .0 6-0 .05-0 .0 4-0 .0 3-0 .0 2-0 .0 10 .0 00 .0 10 .0 20 .0 30 .0 40 .050 .0 6

0 2 4 6 8 10 12 14 16 18 20

0

1

0 2 4 6 8 10 12 14 16 18 200

1

0 2 4 6 8 10 12 14 16 18 20

0

1

0 2 4 6 8 10 12 14 16 18 20

Level 9

Level 8

Level 7

Level 6

Level 10

WD

I

Cho

rd R

otat

ion

(rad

)




124


θy=0 .00 8 8

θu=0 .0 53 0

-0 .0 6-0 .0 5-0 .0 4-0 .0 3-0 .0 2-0 .010 .0 00 .01

0 .0 20 .0 30 .0 40 .0 50 .0 6

0 2 4 6 8 10 12 14 16 18 20

0

1

0 2 4 6 8 10 12 14 16 18 200

1

0 2 4 6 8 10 12 14 16 18 20

θy=0 .00 8 8

θu=0 .0 53 0

-0 .0 6-0 .0 5-0 .0 4-0 .0 3-0 .0 2-0 .010 .0 00 .01

0 .0 20 .0 30 .0 40 .0 50 .0 6

0 2 4 6 8 10 12 14 16 18 20

θy=0 .00 8 8

θu=0 .0 53 0

-0 .0 6-0 .0 5-0 .0 4-0 .0 3-0 .0 2-0 .010 .0 00 .01

0 .0 20 .0 30 .0 40 .0 50 .0 6

0 2 4 6 8 10 12 14 16 18 20

0

Level 5

1

0 2 4 6 8 10 12 14 16 18 200

1

0 2 4 6 8 10 12 14 16 18 20

θy=0 .00 8 8

θu=0 .0 53 0

-0 .0 6-0 .0 5-0 .0 4-0 .0 3-0 .0 2-0 .010 .0 00 .01

0 .0 20 .0 30 .0 40 .0 50 .0 6

0 2 4 6 8 10 12 14 16 18 20

0

1

0 2 4 6 8 10 12 14 16 18 200

1

0 2 4 6 8 10 12 14 16 18 20

θy=0 .00 8 4

θu=0 .0 50 6

-0 .0 6-0 .0 5-0 .0 4-0 .0 3-0 .0 2-0 .010 .0 00 .01

0 .0 20 .0 30 .0 40 .0 50 .0 6

0 2 4 6 8 10 12 14 16 18 20

0

1

0 2 4 6 8 10 12 14 16 18 200

1

0 2 4 6 8 10 12 14 16 18 20

0

1

0 2 4 6 8 10 12 14 16 18 200

1

0 2 4 6 8 10 12 14 16 18 20

Time (s)

Level 4

Level 3

Level 2

Level 1

WD

I

Cho

rd R

otat

ion

(rad

)

Time (s) Time (s)



125

θy=0 .0 0 77

θu=0 .0 4 6 4

-0 .0 6-0 .0 5-0 .0 4-0 .0 3-0 .0 2-0 .0 10 .0 00 .0 1

0 .0 20 .0 30 .0 40 .0 50 .0 60 .0 70 .0 8

0 2 4 6 8 10 12 14 16 18 2 0

θy=0 .0 077

θu=0 .0 4 6 4

-0 .0 6-0 .0 5-0 .0 4-0 .0 3-0 .0 2-0 .0 10 .0 00 .0 1

0 .0 20 .0 30 .0 40 .0 50 .0 60 .0 70 .0 8

0 2 4 6 8 10 12 14 16 18 2 0

θy=0 .0 0 77

θu=0 .0 4 6 4

-0 .0 6-0 .0 5-0 .0 4-0 .0 3-0 .0 2-0 .0 10 .0 00 .0 1

0 .0 20 .0 30 .0 40 .0 50 .0 60 .0 70 .0 8

0 2 4 6 8 10 12 14 16 18 2 0

θy=0 .0 0 77

θu=0 .0 4 6 4

-0 .0 6-0 .0 5-0 .0 4-0 .0 3-0 .0 2-0 .0 10 .0 00 .0 1

0 .0 20 .0 30 .0 40 .0 50 .0 60 .0 70 .0 8

0 2 4 6 8 10 12 14 16 18 2 0

θy=0 .0 0 77

θu=0 .0 4 6 4

-0 .0 6-0 .0 5-0 .0 4-0 .0 3-0 .0 2-0 .0 10 .0 00 .0 1

0 .0 20 .0 30 .0 40 .0 50 .0 60 .0 70 .0 8

0 2 4 6 8 10 12 14 16 18 2 0

0

1

0 2 4 6 8 10 12 14 16 18 200

1

0 2 4 6 8 10 12 14 16 18 20

0

1

0 2 4 6 8 10 12 14 16 18 200

1

0 2 4 6 8 10 12 14 16 18 20

0

1

0 2 4 6 8 10 12 14 16 18 200

1

0 2 4 6 8 10 12 14 16 18 20

0

1

0 2 4 6 8 10 12 14 16 18 20

0

1

0 2 4 6 8 10 12 14 16 18 20

0

1

0 2 4 6 8 10 12 14 16 18 20

0

1

0 2 4 6 8 10 12 14 16 18 20

Time (s)

WD

I

Cho

rd R

otat

ion

(rad

) Right Side Walls Left Side Walls Beams

Level 14

Level 13

Level 12

Level 11

Level 15

Time (s) Time (s)

Fig. 6.15 Member Responses under Northridge Pacoima Ground Motion (Lv11-Lv15)



126


θy=0 .0 0 77

θu=0 .0 4 6 4

-0 .0 6-0 .0 5-0 .0 4-0 .0 3-0 .0 2-0 .0 10 .0 00 .0 1

0 .0 20 .0 30 .0 40 .0 50 .0 60 .0 70 .0 8

0 2 4 6 8 10 12 14 16 18 2 0

θy=0 .0 077

θu=0 .0 4 6 4

-0 .0 6-0 .0 5-0 .0 4-0 .0 3-0 .0 2-0 .0 10 .0 00 .0 1

0 .0 20 .0 30 .0 40 .0 50 .0 60 .0 70 .0 8

0 2 4 6 8 10 12 14 16 18 2 0

θy=0 .0 0 77

θu=0 .0 4 6 4

-0 .0 6-0 .0 5-0 .0 4-0 .0 3-0 .0 2-0 .0 10 .0 00 .0 1

0 .0 20 .0 30 .0 40 .0 50 .0 60 .0 70 .0 8

0 2 4 6 8 10 12 14 16 18 2 0

θy=0 .0 0 77

θu=0 .0 4 6 4

-0 .0 6-0 .0 5-0 .0 4-0 .0 3-0 .0 2-0 .0 10 .0 00 .0 1

0 .0 20 .0 30 .0 40 .0 50 .0 60 .0 70 .0 8

0 2 4 6 8 10 12 14 16 18 2 0

θy=0 .0 0 77

θu=0 .0 4 6 4

-0 .0 6-0 .0 5-0 .0 4-0 .0 3-0 .0 2-0 .0 10 .0 00 .0 1

0 .0 20 .0 30 .0 40 .0 50 .0 60 .0 70 .0 8

0 2 4 6 8 10 12 14 16 18 2 0

0

1

0 2 4 6 8 10 12 14 16 18 200

1

0 2 4 6 8 10 12 14 16 18 20

0

1

0 2 4 6 8 10 12 14 16 18 200

1

0 2 4 6 8 10 12 14 16 18 20

0

Time (s)

Level 9

Level 10

Cho

rd R

otat

ion

(rad

)

Time (s) Time (s)

1

0 2 4 6 8 10 12 14 16 18 200

1

0 2 4 6 8 10 12 14 16 18 20

0

1

0 2 4 6 8 10 12 14 16 18 20

0

1

0 2 4 6 8 10 12 14 16 18 20

0

1

0 2 4 6 8 10 12 14 16 18 20

0

1

0 2 4 6 8 10 12 14 16 18 20

Level 8

Level 7

Level 6

WD

I

Fig. 6.15 Member Responses under Northridge Pacoima Ground Motion (Lv6-Lv10)



127


θy=0 .0 0 8 8

θu=0 .0 53 0

-0 .0 6-0 .0 5-0 .0 4-0 .0 3-0 .0 2-0 .0 10 .0 00 .0 10 .0 20 .0 30 .0 40 .050 .0 60 .070 .0 8

0 2 4 6 8 10 12 14 16 18 2 0

0

1

0 2 4 6 8 10 12 14 16 18 200

1

0 2 4 6 8 10 12 14 16 18 20

θy=0 .0 0 8 8

θu=0 .0 53 0

-0 .0 6-0 .0 5-0 .0 4-0 .0 3-0 .0 2-0 .0 10 .0 00 .0 10 .0 20 .0 30 .0 40 .050 .0 60 .070 .0 8

0 2 4 6 8 10 12 14 16 18 2 0

0

1

0 2 4 6 8 10 12 14 16 18 200

1

0 2 4 6 8 10 12 14 16 18 20

θy=0 .0 0 8 8

θu=0 .0 53 0

-0 .0 6-0 .0 5-0 .0 4-0 .0 3-0 .0 2-0 .0 10 .0 00 .0 10 .0 20 .0 30 .0 40 .050 .0 60 .070 .0 8

0 2 4 6 8 10 12 14 16 18 2 0

0

Level 4

Level 5

1

0 2 4 6 8 10 12 14 16 18 200

1

0 2 4 6 8 10 12 14 16 18 20

θy=0 .0 0 8 8

θu=0 .0 53 0

-0 .0 6-0 .0 5-0 .0 4-0 .0 3-0 .0 2-0 .0 10 .0 00 .0 10 .0 20 .0 30 .0 40 .050 .0 60 .070 .0 8

0 2 4 6 8 10 12 14 16 18 2 0

0

1

0 2 4 6 8 10 12 14 16 18 200

1

0 2 4 6 8 10 12 14 16 18 20

θy=0 .0 0 8 4

θu=0 .0 50 6

-0 .0 6-0 .0 5-0 .0 4-0 .0 3-0 .0 2-0 .0 10 .0 00 .0 10 .0 20 .0 30 .0 40 .050 .0 60 .070 .0 8

0 2 4 6 8 10 12 14 16 18 2 0

0

1

0 2 4 6 8 10 12 14 16 18 200

1

0 2 4 6 8 10 12 14 16 18 20

Level 3

Level 2

Level 1

WD

I

Cho

rd R

otat

ion

(rad

)


Fig. 6.15 Member Responses under Northridge Pacoima Ground Motion (Lv1-Lv5) Notations: 1. θy and θu are the yield and ultimate chord rotation capacities, see Section 6.3.2.2


128

θy=0 .0 0 77

θu=0 .04 6 4

-0 .0 8-0 .07-0 .0 6-0 .05-0 .0 4-0 .0 3-0 .0 2-0 .0 10 .0 00 .0 10 .0 20 .0 30 .0 40 .050 .0 6

0 2 4 6 8 10 12 14 16 18 2 0

0

1

0 2 4 6 8 10 12 14 16 18 200

1

0 2 4 6 8 10 12 14 16 18 20

θy=0 .0 0 77

θu=0 .04 6 4

-0 .0 8-0 .07-0 .0 6-0 .05-0 .0 4-0 .0 3-0 .0 2-0 .0 10 .0 00 .0 10 .0 20 .0 30 .0 40 .050 .0 6

0 2 4 6 8 10 12 14 16 18 2 0

0

1

0 2 4 6 8 10 12 14 16 18 200

1

0 2 4 6 8 10 12 14 16 18 20

θy=0 .00 77

θu=0 .04 6 4

-0 .0 8-0 .07-0 .0 6-0 .05-0 .0 4-0 .0 3-0 .0 2-0 .0 10 .0 00 .0 10 .0 20 .0 30 .0 40 .050 .0 6

0 2 4 6 8 10 12 14 16 18 2 0

0


Level 14

Level 15

1

0 2 4 6 8 10 12 14 16 18 200

1

0 2 4 6 8 10 12 14 16 18 20

θy=0 .00 77

θu=0 .04 6 4

-0 .0 8-0 .07-0 .0 6-0 .05-0 .0 4-0 .0 3-0 .0 2-0 .0 10 .0 00 .0 10 .0 20 .0 30 .0 40 .050 .0 6

0 2 4 6 8 10 12 14 16 18 2 0

0

1

0 2 4 6 8 10 12 14 16 18 20

θy=0 .0 077

θu=0 .04 6 4

-0 .0 8-0 .07-0 .0 6-0 .05-0 .0 4-0 .0 3-0 .0 2-0 .0 10 .0 00 .0 10 .0 20 .0 30 .0 40 .050 .0 6

0 2 4 6 8 10 12 14 16 18 2 0

0

1

0 2 4 6 8 10 12 14 16 18 20

0

1

0 2 4 6 8 10 12 14 16 18 20

0

1

0 2 4 6 8 10 12 14 16 18 20

Time (s)

WD

I

Cho

rd R

otat

ion

(rad

)

Level 13

Level 12

Level 11

Time (s) Time (s)

Fig. 6.16 Member Responses under Northridge Sylmar Ground Motion (Lv11-Lv15) Notations: 1. θy and θu are the yield and ultimate chord rotation capacities, see Section 6.3.2.2


129


θy=0 .0 0 8 4

θu=0 .0 50 6

-0 .0 8-0 .0 7-0 .0 6-0 .0 5-0 .0 4-0 .0 3-0 .0 2-0 .0 10 .0 00 .01

0 .0 20 .0 30 .0 40 .0 50 .0 6

0 2 4 6 8 10 12 14 16 18 20

0

1

0 2 4 6 8 10 12 14 16 18 20

θy=0 .0 08 4

θu=0 .0 50 6

-0 .0 8-0 .0 7-0 .0 6-0 .0 5-0 .0 4-0 .0 3-0 .0 2-0 .0 10 .0 00 .01

0 .0 20 .0 30 .0 40 .0 50 .0 6

0 2 4 6 8 10 12 14 16 18 20

θy=0 .0 08 4

θu=0 .0 50 6

-0 .0 8-0 .0 7-0 .0 6-0 .0 5-0 .0 4-0 .0 3-0 .0 2-0 .0 10 .0 00 .01

0 .0 20 .0 30 .0 40 .0 50 .0 6

0 2 4 6 8 10 12 14 16 18 20

θy=0 .0 08 8

θu=0 .0 53 0

-0 .0 8-0 .0 7-0 .0 6-0 .0 5-0 .0 4-0 .0 3-0 .0 2-0 .0 10 .0 00 .01

0 .0 20 .0 30 .0 40 .0 50 .0 6

0 2 4 6 8 10 12 14 16 18 20

0

1

0 2 4 6 8 10 12 14 16 18 200

1

0 2 4 6 8 10 12 14 16 18 20

θy=0 .0 08 8

θu=0 .0 53 0

-0 .0 8-0 .0 7-0 .0 6-0 .0 5-0 .0 4-0 .0 3-0 .0 2-0 .0 10 .0 00 .01

0 .0 20 .0 30 .0 40 .0 50 .0 6

0 2 4 6 8 10 12 14 16 18 20

0

1

0 2 4 6 8 10 12 14 16 18 200

1

0 2 4 6 8 10 12 14 16 18 20

0

1

0 2 4 6 8 10 12 14 16 18 20

0

1

0 2 4 6 8 10 12 14 16 18 200

1

0 2 4 6 8 10 12 14 16 18 20

0

1

0 2 4 6 8 10 12 14 16 18 200

1

0 2 4 6 8 10 12 14 16 18 20

Time (s)

Level 9

Level 8

Level 7

Level 6

Level 10

WD

I

Cho

rd R

otat

ion

(rad

)

Time (s) Time (s)



130


θy=0 .00 8 8

θu=0 .0 53 0

-0 .0 8-0 .0 7-0 .0 6-0 .0 5-0 .0 4-0 .0 3-0 .0 2-0 .010 .0 00 .01

0 .0 20 .0 30 .0 40 .0 50 .0 6

0 2 4 6 8 10 12 14 16 18 2 0

0

1

0 2 4 6 8 10 12 14 16 18 20

θy=0 .00 8 8

θu=0 .0 53 0

-0 .0 8-0 .0 7-0 .0 6-0 .0 5-0 .0 4-0 .0 3-0 .0 2-0 .010 .0 00 .01

0 .0 20 .0 30 .0 40 .0 50 .0 6

0 2 4 6 8 10 12 14 16 18 2 0

θy=0 .00 8 8

θu=0 .0 53 0

-0 .0 8-0 .0 7-0 .0 6-0 .0 5-0 .0 4-0 .0 3-0 .0 2-0 .010 .0 00 .01

0 .0 20 .0 30 .0 40 .0 50 .0 6

0 2 4 6 8 10 12 14 16 18 2 0

0

1

0 2 4 6 8 10 12 14 16 18 200

1

0 2 4 6 8 10 12 14 16 18 20

θy=0 .00 8 8

θu=0 .0 53 0

-0 .0 8-0 .0 7-0 .0 6-0 .0 5-0 .0 4-0 .0 3-0 .0 2-0 .010 .0 00 .01

0 .0 20 .0 30 .0 40 .0 50 .0 6

0 2 4 6 8 10 12 14 16 18 2 0

0

1

0 2 4 6 8 10 12 14 16 18 200

1

0 2 4 6 8 10 12 14 16 18 20

θy=0 .00 8 4

θu=0 .0 50 6

-0 .0 8-0 .0 7-0 .0 6-0 .0 5-0 .0 4-0 .0 3-0 .0 2-0 .010 .0 00 .01

0 .0 20 .0 30 .0 40 .0 50 .0 6

0 2 4 6 8 10 12 14 16 18 2 0

0

1

0 2 4 6 8 10 12 14 16 18 20

0

1

0 2 4 6 8 10 12 14 16 18 200

1

0 2 4 6 8 10 12 14 16 18 20

0

1

0 2 4 6 8 10 12

Time (s)

Level 4

WD

I

Cho

rd R

otat

ion

(rad

)

14 16 18 200

1

0 2 4 6 8 10 12 14

Level 3

Level 2

Level 1

Level 5

Time (s) 16 18 20

Time (s)



131

Chapter 7 Conclusions and Recommendations for Future Research 7.1 Summary

The thesis presents the details of an insightful investigation of the behavior of a

15-story coupled core wall (CCW) building designed by a performance-based design

(PBD) methodology. ETABS elastic analyses of four prototype structures were carried

out initially to establish the critical geometry of the structure used in this research. The

high shear stresses in the coupling beams show the difficulties of using strength-based

design method of current building codes. Performance-based design method is introduced

as a viable alternative design approach. In this study, PBD criteria are proposed for

performance at the life safety (LS) and collapse prevention (CP) levels. Coupling beams

are allowed to yield, the wall piers are required to remain elastic, and the maximum story

drift is limited to less than 2% under LS seismic loads. Under CP seismic loads, wall

piers are permitted to yield and beams are allowed to reach their ultimate limit state.

Other performance criteria may be selected to capture the expected behavior of CCW.

Different shear capacities, within the limits of capacities of constructible coupling beams,

are assigned to coupling beams at different levels to simulate approximately the

distribution of shear forces computed from elastic analysis. Within the context of PBD, a

simplified method is suggested for computing the wall pier design forces without

cumbersome iterations. Using the data from previous tests, a new model was developed

to more accurately capture the expected rotational capacities and strengths of diagonally

reinforced concrete coupling beams at yield and ultimate limit states. This model was

132

used as part of modeling of the prototype structure. Detailed nonlinear static and dynamic

analyses were conducted to evaluate the adequacy of the PBD methodology used herein.

7.2 Conclusions

The following conclusions may be drawn from the results and discussions presented

in this report.

(1) The critical dimensions of typical CCW buildings are determined by two

factors: (i) 2% story drift limit specified in the US code (NEHRP) and (ii) a minimum

66% degree of coupling as specified in Canadian code (NBCC). Usually, the first

parameter controls, and the second parameter is checked.

(2) High shear stresses in coupling beams are produced by following the traditional

strength-based design method. The largest shear stress exceeds the maximum value

specified in ACI building code (10 'cf ) and the maximum value for constructible beams

(6 'cf ). The high shear stresses indicate that the prototype building can not be designed

by strength-based method.

(3) The assumption of strength-based design method, which enforces the coupling

beams to yield simultaneously with the wall piers, is not valid. This assumption is the

underlying cause of high shear stresses in coupling beams.

(4) FEMA 356 provides conservative estimations of deformation and strength

capacities of diagonally reinforced concrete coupling beams, especially for yield chord

rotation. The proposed new model provides a more precise method for predicting the

133

behaviors of diagonally reinforced concrete beams, which are needed for nonlinear

analyses.

(5) The performance-based design method greatly reduces the shear stresses in

coupling beams, and increases the wall pier forces which can be handled by traditional

design measures. The ACI building code and practical constructability requirements can

easily be met by using this method.

(6) The pushover analysis indicates that the coupling beams and wall piers yield at

different base shear levels.

(7) The beams in the upper levels undergo larger chord rotations than those close to

the lower parts. The nonlinear dynamic analyses also demonstrate that the maximum

chord rotation occurs within the top three stories. As a result, the beams in the upper

levels need to be designed with more deformation capacities.

(8) The inelastic deformation of wall pier is not restricted to the ground floor. It

spreads to wall piers in other floors both in the pushover and nonlinear dynamic analyses.

The occurrence of inelastic deformations in the upper floors indicates reduction of

structural stiffness and loss of degree of coupling.

(9) The pushover analysis indicates that at drift corresponding to the life safety (LS)

limit state the walls have remained elastic while all the coupling beams have undergone

inelastic deformations. Under ground motions equal to (or less than) the LS level, a

similar observation can be made. The wall piers do not experience inelastic deformation

until the collapse prevention (CP) limit state or ground motions equal to (or more than)

134

the CP level are considered. These observations regarding the behavior of the prototype

structure under static and dynamic loads suggest the adequacy of the PBD methodology

as presented herein for design of coupled core walls.

7.3 Recommendations for Future Research

This research is an attempt to implement PBD methodology for design of CCW

structures. The following future research topics are recommended to further develop this

new design method.

(1) The accuracy of the new model (proposed in Chapter 5) for predicting of

response of diagonally reinforced concrete beams needs to be verified through additional

tests, particularly those with ACI 318-02 compliant specimens.

(2) The PBD design with performance criteria other than those implemented in this

research (see Section 6.3.1) should be tried out. Thus, it will be possible to optimize PBD

design by comparing the results from different performance criteria.

(3) The contributions of structural members surrounding the coupled core walls

should be taken into account in the analyses. In particular, the effects of slab stiffness and

the participation of perimeter and other gravity load columns need to be investigated.

(4) Performance based design of different types of coupling beams, such as steel,

hybrid, and fused coupling beams, should be conducted.

(5) Additional CCW systems with complex configurations (e.g. asymmetric systems

and wall piers with openings) should be investigated to more extensively evaluate the

135

applicability of PBD method.

136

Reference

American Concrete Institute Committee 318, 2002, ACI 318-02 Building Code

Requirements for Reinforced Concrete and Commentary (ACI 318-02/ACI 318R-02),

American Concrete Institute, Farmington Hills, MI

ACI-ASCE Committee 352 (Joint), 2002, Recommendations for Design of Beam-Column

Connections in Monolithic Reinforced Concrete Structures (ACI 352R-02), American

Concrete Institute, Farmington Hills, MI

Barney, G.B., Shiu, K.N., Rabbat, B.G., Fiorako, A.E., Russell, H.G., and Corley, W.G.,

1978, Earthquake Resistant Structural Walls-Test of Coupling Beams, Report to National

Science Foundation, Portland Cement Association, Skokie, IL

Bertero, V., 1997, Performance-based Seismic Engineering: A Critical Review of

Proposed Guidelines, Seismic Design Methodologies for the Next Generation of Codes,

A.A. Balkema Publisher, pp1-32

Brienen, P., 2002, Spreadsheets (electronic version) of a 10-Story CCW Building Design

Carr, J, 2000, Manual of Ruaumoko-Computer Program Library, Department of Civil

Engineering, University of Canterbury

137

Computers and Structures Inc (CSI), 1997, ETABS v6.2 Three Dimensional Analysis of

Building Systems User Manual

Federal Emergency Management Agency, 2000, NEHRP Recommendation Provisions for

Seismic Regulations for New Buildings and Other Structures (NEHRP 2000)

Federal Emergency Management Agency, 2000, Prestandard and Commentary for the

Seismic Rehabilitation of Buildings (FEMA 356)

Fortney, P., 2005, Next Generation Coupling Beams, Ph.D Dissertation, Civil and

Environmental Engineering Dept., University of Cincinnati

Galano,L., Vignoli, A., 2000, Seismic Behavior of Short Coupling Beams with Different

Reinforcement Layouts, ACI Structural Journal, Vol 97, No. 6, pp 171-179

Hindi, A., Sexsmith, R., 2001, A Proposed Damage Model for R/C Bridge Columns

under Cyclic Loading, Earthquake Spectra, EERI, Vol. 17, No. 2

Harries, K.A., Mitchell, D., Redwood, R.G., and Cook, W.D., 1998, Nonlinear Seismic

Response Predictions of Walls Coupled with Steel and Concrete Beams, Canadian

Journal of Civil Engineering, Vol 25, pp803-818

138

Harries, K.A., Fortney P.J., Shahrooz B.M., and Brienen P., 2003, Design of Practical

Diagonally Reinforced Concrete Coupling Beams—A Critical Review of ACI 318

Requirements, ACI Structural Journal, in press

Harries, K. A., Shahrooz, B. M., Brienen, P., Fortney, P. J., 2004, Performance-Based

Design of Coupled Wall Systems, Composite Construction V, July 18-23, 2004, Kruger

National Park, South Africa, in press

Imbsen Software Systems, 2002, Release Notes of XTRACT v 2.6.2—Cross Sectional Xs

Structural Analysis of Components

McNeice, D., 2004, Performance Based Design of a 30 Story Coupled Core Wall

Structure, Master Thesis, Department of Civil Engineering, University of South Carolina

National Research Council of Canada, 1995, National Building Code of Canada, (NBCC)

Park, R., Paulay, T., 1975, Reinforced Concrete Structures, John Wiley&Sons Inc.,

pp645-655

Paulay, T., and Binney, J. R., 1974, Diagonally Reinforced Coupling Beams of Shear

Walls, ACI Special Publication, Shear In Reinforced Concrete, Vol 2, pp579-598

139

Paulay, T., and Santhakumar, A.R., 1976, Ductile Behavior of Coupled Shear Walls,

Journal of Structural Division, ASCE, Vol. 102, No. ST1, pp.93-108

Paulay, T., Priestley, M.J.N., 1992, Seismic Design of Reinforced Concrete and Masonry

Buildings, John Wiley&Sons Inc., pp376-377, pp381-383

Paulay, T., 2002, A Displacement-Focused Seismic Design of Mixed Building Systems,

Earthquake Spectra, 18(4), pp689-718.

Shahrooz, B., Harries, K., 2005, Spreadsheets (electronic version) of Coupling Beam

Experiment Database

Tassios, P., Moretti, M., Bezas, A., 1996, On the Behavior and Ductility of Reinforced

Concrete Coupling Beams of Shear Walls, ACI Structural Journal, November-December,

pp711-720

Segui, T., 2003, LRFD Steel Design(3rd Edition), Thomson, pp7-10

140

Appendix A Preliminary Design Calculations

Notations:

xA : Torsion amplification factor in the X direction

xavgA : Average value of of all floors xA

yA : Torsion amplification factor in the Y direction

xavgA : Average value of of all floors yA

dC : Deflection amplification factor

sC : Seismic response coefficient of the ELF method

cvxC : Vertical distribution factor of ELF

F : Lateral force from the ELF method

h : Story height

ih : Height from measured from the base to level i

I : Importance factor of the building, equal to 1

L : Coupling arm, distance between two centroids of adjacent walls

: Accidental torsion associated with taM F

: Overturning moment at the base otmM

: Beam shear V

bV : Design base shear

sumV : Sum of beam shears

: Building total weight W

A-1

: Weight of story i iw

δ : Total displacement

eδ : Elastic displacement

avgδ : Average displacement

maxδ : Maximum displacement

taδ : Displacement due to accidental torsion

: Story drift ∆

A-2

Table A.1.1 Gravity Load of Prototype I Top Floor

Unit Weight Volume or Area Weight Walls 150 pcf 496 ft3 74 (kips)Beams 150 pcf 40 ft3 6 (kips)Columns 150 pcf 408 ft3 61(kips)Slab 150 pcf 6667 ft3 1000 (kips)Partitions 10 psf 10000 ft2 100 (kips)Claddings 15 psf 1836 ft2 28 (kips)Mechanics 5 psf 10000 ft2 50 (kips)Total Loads of Top Floor 1319 (kips)

Middle Floors Unit Weight Volume or Area Weight Walls 150 pcf 991 ft3 149 (kips)Beams 150 pcf 40 ft3 6 (kips)Columns 150 pcf 815 ft3 122 (kips)Slab 150 pcf 6667 ft3 1000 (kips)Partitions 10 psf 10000 ft2 100 (kips)Claddings 15 psf 3668 ft2 55 (kips)Mechanics 5 psf 10000 ft2 50 (kips)Total Loads of Middle Floor 1482 (kips)

Ground Floor Unit Weight Volume or Area Weight Walls 150 pcf 1153 ft3 173 (kips)Beams 150 pcf 40 ft3 6 (kips)Columns 150 pcf 948 ft3 142 (kips)Slab 150 pcf 6667 ft3 1000 (kips)Partitions 10 psf 10000 ft2 100 (kips)Claddings 15 psf 4268 ft2 64 (kips)Mechanics 5 psf 10000 ft2 50 (kips)Total Loads of Ground Floor 1535 (kips)Total Building Weight 22120 (kips)

A-3

Table A.1.2 ELF Lateral Load of Prototype I

Story Story Weight, wi

(kips) Height above Grade, hi

(ft) Vertical Distribution

Factor, Cvx

Lateral Load at Stories, F

(kips)

Accidental Torsion at Stories, Mta

(k-ft) 15 1319 140.55 0.162 407.4 2037.0 14 1482 131.38 0.159 399.9 1999.5 13 1482 122.21 0.137 346.0 1730.0 12 1482 113.04 0.117 296.0 1480.0 11 1482 103.87 0.099 250.0 1250.0 10 1482 94.70 0.082 207.8 1039.0

9 1482 85.53 0.067 169.5 847.5 8 1482 76.36 0.054 135.1 675.5 7 1482 67.19 0.041 104.6 523.0 6 1482 58.02 0.031 78.0 390.0 5 1482 48.85 0.022 55.3 276.5 4 1482 39.68 0.014 36.5 182.5 3 1482 30.51 0.009 21.6 108.0 2 1482 21.34 0.004 10.6 53.0 1 1535 12.17 0.001 3.6 18.0

Sum 22120 1.000 2521.7

Lateral Load Distribution at Stories

Story

Fx(kips)123456789

101112131415

0.0 100.0 200.0 300.0 400.0 500.0

A-4

Table A.1.3 Max Story Drift of Prototype I in the X Direction

Story δavg

(ft) δta (ft)

δmax= δavg + δta (ft)

Ax=[ δmax /(1.2 δavg)]2

δe= δavg +Axavg δta(ft)

δ= δe Cd/I(ft)

∆ (ft)

h (ft)

∆/ h (%)

15 0.7412 0.1742 0.9155 1.06 0.9433 4.7164 0.2861 9.17 3.12%14 0.6934 0.1662 0.8595 1.07 0.8861 4.4303 0.3003 9.17 3.27%13 0.6437 0.1572 0.8009 1.07 0.8260 4.1299 0.3159 9.17 3.45%12 0.5920 0.1473 0.7393 1.08 0.7628 3.8140 0.3311 9.17 3.61%11 0.5383 0.1365 0.6748 1.09 0.6966 3.4830 0.3440 9.17 3.75%10 0.4830 0.1249 0.6079 1.10 0.6278 3.1390 0.3538 9.17 3.86%

9 0.4265 0.1126 0.5391 1.11 0.5570 2.7851 0.3595 9.17 3.92%8 0.3695 0.0997 0.4692 1.12 0.4851 2.4256 0.3607 9.17 3.93%7 0.3127 0.0865 0.3992 1.13 0.4130 2.0649 0.3566 9.17 3.89%6 0.2569 0.0731 0.3300 1.15 0.3417 1.7083 0.3468 9.17 3.78%5 0.2030 0.0598 0.2628 1.16 0.2723 1.3615 0.3306 9.17 3.61%4 0.1519 0.0468 0.1987 1.19 0.2062 1.0309 0.3070 9.17 3.35%3 0.1047 0.0346 0.1393 1.23 0.1448 0.7239 0.2752 9.17 3.00%2 0.0628 0.0233 0.0860 1.30 0.0897 0.4487 0.2330 9.17 2.54%1 0.0277 0.0133 0.0410 1.53 0.0431 0.2157 0.2157 12.17 1.77%

Axavg 1.16 Max ∆/h 3.93%

Table A.1.4 Max Story Drift of Prototype I in the Y Direction

Story δavg

(ft) δta (ft)


Ay=[ δmax /(1.2 δavg)]2

δe= δavg +Ayavg δta(ft)

δ= δe Cd/I (ft)

∆ (ft)

h (ft)

∆/ h (%)

15 0.7880 0.1742 0.9622 1.04 1.0010 5.0050 0.3742 9.17 4.08%14 0.7230 0.1662 0.8892 1.05 0.9262 4.6308 0.3814 9.17 4.16%13 0.6577 0.1572 0.8149 1.07 0.8499 4.2494 0.3873 9.17 4.22%12 0.5924 0.1473 0.7397 1.08 0.7724 3.8621 0.3912 9.17 4.27%11 0.5273 0.1365 0.6638 1.10 0.6942 3.4709 0.3924 9.17 4.28%10 0.4630 0.1249 0.5879 1.12 0.6157 3.0784 0.3901 9.17 4.25%

9 0.4000 0.1126 0.5126 1.14 0.5377 2.6884 0.3839 9.17 4.19%8 0.3390 0.0997 0.4387 1.16 0.4609 2.3045 0.3734 9.17 4.07%7 0.2805 0.0865 0.3670 1.19 0.3862 1.9311 0.3581 9.17 3.91%6 0.2252 0.0731 0.2983 1.22 0.3146 1.5730 0.3380 9.17 3.69%5 0.1739 0.0598 0.2337 1.25 0.2470 1.2351 0.3125 9.17 3.41%4 0.1273 0.0468 0.1741 1.30 0.1845 0.9226 0.2815 9.17 3.07%3 0.0860 0.0346 0.1205 1.36 0.1282 0.6411 0.2448 9.17 2.67%2 0.0508 0.0233 0.0741 1.47 0.0793 0.3963 0.2029 9.17 2.21%1 0.0225 0.0133 0.0358 1.78 0.0387 0.1934 0.1934 12.17 1.59%

Ayavg 1.22 Max ∆/h 4.28%Notations for Tables A.1.3 and A.1.4 : (1) Values in columns of δavg and δta are from ETABS calculations. (2) Column of δe represents

the elastic displacement at each level including the amplified torsion deformation. (3) Column of δ is the displacement considering the

inelastic effect. Cd=5 and I=1 (4) ∆, the story drift, is the difference of δ for two adjacent stories.

A-5

Table A.1.5 Degree of Coupling (DOC) of Prototype I

Story Beam Shear V

(kips) 15 195.614 220.113 255.512 296.411 337.810 376.69 410.68 438.67 459.26 471.45 473.74 464.13 439.42 395.11 324.0

Shear Sum Vsum (kips) 5557.9

Coupling Arm L(ft) 19.0

Coupling Moment Mc=2VsumL (k-ft) (2 beams at one level) 211201.3

Overturning Moment Motm (k-ft) 265062.9

DOC=Mc/Motm 79.7%

A-6

Table A.2.1 Gravity Load of Prototype II Top Floor

Unit Weight Volume or Area Weight Walls 150 pcf 612 ft3 92 (kips)Beams 150 pcf 40 ft3 6 (kips)Columns 150 pcf 408 ft3 61 (kips)Slab 150 pcf 6667 ft3 1000 (kips)Partitions 10 psf 10000 ft2 100 (kips)Claddings 15 psf 1836 ft2 28 (kips)Mechanics 5 psf 10000 ft2 50 (kips)Total Loads of Top Floor 1337 (kips)



A-7

Table A.2.2 ELF Lateral Load of Prototype II




Factor, Cvx

Lateral Loadat Stories, F

(kips)


(k-ft) 15 1337 140.55 0.160 413.4 2067.0 14 1517 131.38 0.159 409.8 2049.0 13 1517 122.21 0.137 354.6 1773.0 12 1517 113.04 0.118 303.4 1517.0 11 1517 103.87 0.099 256.2 1281.0 10 1517 94.70 0.083 212.9 1064.5

9 1517 85.53 0.067 173.7 868.5 8 1517 76.36 0.054 138.4 692.0 7 1517 67.19 0.042 107.2 536.0 6 1517 58.02 0.031 79.9 399.5 5 1517 48.85 0.022 56.7 283.5 4 1517 39.68 0.014 37.4 187.0 3 1517 30.51 0.009 22.1 110.5 2 1517 21.34 0.004 10.8 54.0 1 1575 12.17 0.001 3.7 18.5

Sum 22633 1.000 2580.2


Story

Fx(kips)123456789

101112131415

0.0 100.0 200.0 300.0 400.0 500.0

A-8

Table A.2.3 Max Story Drift of Prototype II in the X Direction

Story δavg

(ft) δta (ft)




δ= δe Cd/I(ft)

∆ (ft)

h (ft)

∆/ h (%)

15 0.5459 0.1167 0.6626 1.02 0.6734 3.3670 0.2036 9.17 2.22%14 0.5114 0.1109 0.6224 1.03 0.6327 3.1634 0.2137 9.17 2.33%13 0.4756 0.1047 0.5802 1.03 0.5899 2.9497 0.2249 9.17 2.45%12 0.4381 0.0978 0.5359 1.04 0.5449 2.7247 0.2358 9.17 2.57%11 0.3990 0.0904 0.4894 1.04 0.4978 2.4889 0.2454 9.17 2.68%10 0.3586 0.0825 0.4410 1.05 0.4487 2.2435 0.2526 9.17 2.75%

9 0.3172 0.0741 0.3913 1.06 0.3982 1.9909 0.2569 9.17 2.80%8 0.2753 0.0655 0.3407 1.06 0.3468 1.7340 0.2580 9.17 2.81%7 0.2333 0.0566 0.2899 1.07 0.2952 1.4760 0.2552 9.17 2.78%6 0.1921 0.0477 0.2397 1.08 0.2442 1.2208 0.2482 9.17 2.71%5 0.1520 0.0389 0.1909 1.09 0.1945 0.9726 0.2367 9.17 2.58%4 0.1140 0.0303 0.1444 1.11 0.1472 0.7359 0.2198 9.17 2.40%3 0.0789 0.0223 0.1012 1.14 0.1032 0.5161 0.1967 9.17 2.14%2 0.0476 0.0149 0.0625 1.20 0.0639 0.3195 0.1664 9.17 1.81%1 0.0214 0.0085 0.0298 1.35 0.0306 0.1531 0.1531 12.17 1.26%

Axavg 1.09 Max ∆/h 2.81%

Table A.2.4 Max Story Drift of Prototype II in the Y Direction

Story δavg

(ft) δta (ft)




δ= δe Cd/I (ft)

∆ (ft)

h (ft)

∆/ h (%)

15 0.5545 0.1167 0.6712 1.02 0.6868 3.4338 0.2607 9.17 2.84%14 0.5089 0.1109 0.6198 1.03 0.6346 3.1731 0.2651 9.17 2.89%13 0.4630 0.1047 0.5676 1.04 0.5816 2.9080 0.2686 9.17 2.93%12 0.4170 0.0978 0.5148 1.06 0.5279 2.6394 0.2708 9.17 2.95%11 0.3713 0.0904 0.4617 1.07 0.4737 2.3686 0.2709 9.17 2.95%10 0.3261 0.0825 0.4085 1.09 0.4195 2.0977 0.2687 9.17 2.93%

9 0.2818 0.0741 0.3559 1.11 0.3658 1.8290 0.2638 9.17 2.88%8 0.2388 0.0655 0.3043 1.13 0.3130 1.5652 0.2560 9.17 2.79%7 0.1977 0.0566 0.2543 1.15 0.2618 1.3092 0.2450 9.17 2.67%6 0.1588 0.0477 0.2065 1.17 0.2129 1.0643 0.2306 9.17 2.51%5 0.1227 0.0389 0.1615 1.20 0.1667 0.8337 0.2127 9.17 2.32%4 0.0898 0.0303 0.1201 1.24 0.1242 0.6210 0.1911 9.17 2.08%3 0.0607 0.0223 0.0830 1.30 0.0860 0.4299 0.1657 9.17 1.81%2 0.0360 0.0149 0.0508 1.39 0.0528 0.2641 0.1043 9.17 1.14%1 0.0159 0.0085 0.0244 1.00 0.0320 0.1599 0.1599 12.17 1.31%




A-9

Table A.2.5 Degree of Coupling (DOC) of Prototype II

Story Beam Shear V

(kips) 15 190.814 211.213 240.712 275.011 310.110 343.09 371.88 395.37 412.16 421.25 421.24 410.43 386.12 344.61 280.3





DOC=Mc/Motm 75.5%

A-10

Table A.3.1 Gravity Load of Prototype III Top Floor

Unit Weight Volume or Area Weight Walls 150 pcf 643 ft3 96 (kips)Beams 150 pcf 40 ft3 6 (kips)Columns 150 pcf 408 ft3 61 (kips)Slab 150 pcf 6667 ft3 1000 (kips)Partitions 10 psf 10000 ft2 100 (kips)Claddings 15 psf 1836 ft2 28 (kips)Mechanics 5 psf 10000 ft2 50 (kips)Total Loads of Top Floor 1341 (kips)



A-11

Table A.3.2 ELF Lateral Load of Prototype III




Factor, Cvx

Lateral Loadat Stories, F

(kips)


(k-ft) 15 1341 140.55 0.160 414.8 2074.0 14 1526 131.38 0.159 412.4 2062.0 13 1526 122.21 0.138 356.8 1784.0 12 1526 113.04 0.118 305.3 1526.5 11 1526 103.87 0.099 257.8 1289.0 10 1526 94.70 0.083 214.3 1071.5

9 1526 85.53 0.067 174.8 874.0 8 1526 76.36 0.054 139.3 696.5 7 1526 67.19 0.042 107.9 539.5 6 1526 58.02 0.031 80.4 402.0 5 1526 48.85 0.022 57.0 285.0 4 1526 39.68 0.014 37.6 188.0 3 1526 30.51 0.009 22.2 111.0 2 1526 21.34 0.004 10.9 54.5 1 1586 12.17 0.001 3.7 18.5

Sum 22765 1.000 2595.2


Story

Fx(kips)123456789

101112131415

0.0 100.0 200.0 300.0 400.0 500.0

A-12

Table A.3.3 Max Story Drift of Prototype III in the X Direction

Story δavg

(ft) δta (ft)




δ= δe Cd/I(ft)

∆ (ft)

h (ft)

∆/ h (%)

15 0.5259 0.0975 0.6234 1.00 0.6273 3.1364 0.1889 9.17 2.06%14 0.4930 0.0928 0.5858 1.00 0.5895 2.9475 0.1985 9.17 2.16%13 0.4588 0.0875 0.5463 1.00 0.5498 2.7490 0.2090 9.17 2.28%12 0.4229 0.0818 0.5047 1.00 0.5080 2.5400 0.2193 9.17 2.39%11 0.3855 0.0756 0.4611 1.00 0.4641 2.3206 0.2284 9.17 2.49%10 0.3466 0.0690 0.4157 1.00 0.4185 2.0923 0.2351 9.17 2.56%

9 0.3068 0.0621 0.3689 1.00 0.3714 1.8571 0.2393 9.17 2.61%8 0.2665 0.0549 0.3214 1.01 0.3236 1.6178 0.2404 9.17 2.62%7 0.2261 0.0475 0.2736 1.02 0.2755 1.3773 0.2379 9.17 2.59%6 0.1862 0.0400 0.2263 1.03 0.2279 1.1394 0.2315 9.17 2.52%5 0.1476 0.0327 0.1803 1.04 0.1816 0.9079 0.2208 9.17 2.41%4 0.1108 0.0256 0.1364 1.05 0.1374 0.6870 0.2052 9.17 2.24%3 0.0768 0.0188 0.0956 1.08 0.0964 0.4819 0.1838 9.17 2.00%2 0.0465 0.0127 0.0591 1.12 0.0596 0.2981 0.1555 9.17 1.70%1 0.0210 0.0073 0.0282 1.26 0.0285 0.1426 0.1426 12.17 1.17%

Axavg 1.04 Max ∆/h 2.62%Table A.3.4 Max Story Drift of Prototype III in the Y Direction

Story δavg

(ft) δta (ft)




δ= δe Cd/I (ft)

∆ (ft)

h (ft)

∆/ h (%)

15 0.4504 0.0975 0.5479 1.03 0.5622 2.8112 0.2116 9.17 2.31%14 0.4135 0.0928 0.5063 1.04 0.5199 2.5996 0.2156 9.17 2.35%13 0.3764 0.0875 0.4639 1.05 0.4768 2.3839 0.2188 9.17 2.39%12 0.3392 0.0818 0.4210 1.07 0.4330 2.1651 0.2207 9.17 2.41%11 0.3021 0.0756 0.3778 1.09 0.3889 1.9444 0.2211 9.17 2.41%10 0.2655 0.0690 0.3345 1.10 0.3447 1.7233 0.2195 9.17 2.39%

9 0.2296 0.0621 0.2916 1.12 0.3008 1.5039 0.2156 9.17 2.35%8 0.1947 0.0549 0.2496 1.14 0.2577 1.2883 0.2094 9.17 2.28%7 0.1613 0.0475 0.2088 1.16 0.2158 1.0788 0.2006 9.17 2.19%6 0.1297 0.0400 0.1698 1.19 0.1756 0.8782 0.1890 9.17 2.06%5 0.1004 0.0327 0.1330 1.22 0.1378 0.6892 0.1745 9.17 1.90%4 0.0736 0.0256 0.0992 1.26 0.1030 0.5148 0.1571 9.17 1.71%3 0.0499 0.0188 0.0688 1.32 0.0715 0.3577 0.1364 9.17 1.49%2 0.0297 0.0127 0.0424 1.41 0.0443 0.2213 0.1128 9.17 1.23%1 0.0134 0.0073 0.0206 1.00 0.0217 0.1085 0.1085 12.17 0.89%




A-13

Table A.3.5 Degree of Coupling (DOC) of Prototype III

Story Beam Shear V

(kips) 15 190.714 211.013 240.312 274.411 309.110 341.89 370.58 393.87 410.46 419.45 419.34 408.53 384.22 342.81 278.7





DOC=Mc/Motm 75.5%

A-14

Table A.4.1 Gravity Load of Prototype IV Top Floor

Unit Weight Volume or Area Weight Walls 150 pcf 689ft3 103 (kips)Beams 150 pcf 50 ft3 8 (kips)Columns 150 pcf 408 ft3 61 (kips)Slab 150 pcf 6667 ft3 1000 (kips)Partitions 10 psf 10000 ft2 100 (kips)Claddings 15 psf 1836 ft2 28 (kips)Mechanics 5 psf 10000 ft2 50 (kips)Total Loads of Top Floor 1350 (kips)

Middle Floors Unit Weight Volume or Area Weight Walls 150 pcf 1376 ft3 206 (kips)Beams 150 pcf 50 ft3 8(kips)Columns 150 pcf 815 ft3 122 (kips)Slab 150 pcf 6667 ft3 1000 (kips)Partitions 10 psf 10000 ft2 100 (kips)Claddings 15 psf 3668 ft2 55 (kips)Mechanics 5 psf 6667 ft2 50 (kips)Total Loads of Middle Floor 1541 (kips)


A-15

Table A.4.2 ELF Lateral Load of Prototype IV

Story Story Weight

wi (kips) Height above Grade

hi (ft)

Vertical Distribution Factor

Cvx

Lateral Load at Stories Fx (kips)

Accidental Torsion at Stories Mta (k-ft)

15 1350 140.55 0.159 417.7 2088.5 14 1541 131.38 0.159 416.6 2083.0 13 1541 122.21 0.138 360.5 1802.5 12 1541 113.04 0.118 308.4 1542.0 11 1541 103.87 0.099 260.4 1302.0 10 1541 94.70 0.083 216.5 1082.5

9 1541 85.53 0.067 176.6 883.0 8 1541 76.36 0.054 140.7 703.5 7 1541 67.19 0.042 109.0 545.0 6 1541 58.02 0.031 81.3 406.5 5 1541 48.85 0.022 57.6 288.0 4 1541 39.68 0.015 38.0 190.0 3 1541 30.51 0.009 22.5 112.5 2 1541 21.34 0.004 11.0 55.0 1 1604 12.17 0.001 3.7 18.5

Sum 22987 1.000 2620.0


Story

Fx(kips)123456789

101112131415

0.0 100.0 200.0 300.0 400.0 500.0

A-16

Table A.4.3 Max Story Drift of Prototype IV in the X Direction

Story δavg

(ft) δta (ft)




δ= δe Cd/I(ft)

∆ (ft)

h (ft)

∆/ h (%)

15 0.4166 0.0574 0.4739 1.00 0.4749 2.3747 0.1429 9.17 1.56%14 0.3902 0.0552 0.4454 1.00 0.4464 2.2318 0.1506 9.17 1.64%13 0.3627 0.0526 0.4153 1.00 0.4162 2.0812 0.1589 9.17 1.73%12 0.3339 0.0497 0.3836 1.00 0.3845 1.9223 0.1665 9.17 1.82%11 0.3040 0.0464 0.3503 1.00 0.3512 1.7558 0.1729 9.17 1.89%10 0.2731 0.0427 0.3158 1.00 0.3166 1.5830 0.1774 9.17 1.94%

9 0.2416 0.0388 0.2804 1.00 0.2811 1.4055 0.1800 9.17 1.96%8 0.2098 0.0347 0.2445 1.00 0.2451 1.2255 0.1802 9.17 1.97%7 0.1782 0.0304 0.2085 1.00 0.2091 1.0453 0.1779 9.17 1.94%6 0.1470 0.0260 0.1730 1.00 0.1735 0.8673 0.1730 9.17 1.89%5 0.1169 0.0215 0.1385 1.00 0.1389 0.6943 0.1650 9.17 1.80%4 0.0884 0.0172 0.1055 1.00 0.1058 0.5292 0.1539 9.17 1.68%3 0.0618 0.0130 0.0748 1.02 0.0751 0.3753 0.1388 9.17 1.51%2 0.0381 0.0091 0.0472 1.06 0.0473 0.2366 0.1188 9.17 1.30%1 0.0179 0.0055 0.0235 1.19 0.0235 0.1177 0.1177 12.17 0.97%

Axavg 1.02 Max ∆/h 1.97%

Table A.4.4 Max Story Drift of Prototype IV in the Y Direction

Story δavg

(ft) δta (ft)




δ= δe Cd/I (ft)

∆ (ft)

h (ft)

∆/ h (%)

15 0.3414 0.0574 0.3988 1.00 0.4054 2.0270 0.1511 9.17 1.65%14 0.3137 0.0552 0.3688 1.00 0.3752 1.8759 0.1542 9.17 1.68%13 0.2857 0.0526 0.3383 1.00 0.3443 1.7217 0.1567 9.17 1.71%12 0.2576 0.0497 0.3073 1.00 0.3130 1.5650 0.1584 9.17 1.73%11 0.2297 0.0464 0.2760 1.00 0.2813 1.4066 0.1587 9.17 1.73%10 0.2020 0.0427 0.2447 1.02 0.2496 1.2480 0.1576 9.17 1.72%

9 0.1748 0.0388 0.2136 1.04 0.2181 1.0904 0.1550 9.17 1.69%8 0.1484 0.0347 0.1831 1.06 0.1871 0.9354 0.1505 9.17 1.64%7 0.1231 0.0304 0.1535 1.08 0.1570 0.7849 0.1512 9.17 1.65%6 0.0992 0.0247 0.1239 1.08 0.1267 0.6337 0.1353 9.17 1.48%5 0.0769 0.0204 0.0973 1.11 0.0997 0.4984 0.1249 9.17 1.36%4 0.0566 0.0162 0.0728 1.15 0.0747 0.3735 0.1082 9.17 1.18%3 0.0386 0.0130 0.0516 1.24 0.0531 0.2653 0.0990 9.17 1.08%2 0.0232 0.0091 0.0322 1.34 0.0333 0.1664 0.0824 9.17 0.90%1 0.0107 0.0055 0.0162 1.60 0.0168 0.0840 0.0840 12.17 0.69%




A-17

Table A.4.5 Degree of Coupling (DOC) of Prototype IV

Story Beam Shear V

(kips) 15 159.214 186.413 224.312 267.111 309.610 349.19 383.68 412.17 433.86 447.95 453.44 448.33 429.62 392.11 327.2





DOC=Mc/Motm 79.7%

A-18

Appendix B Beam Design Calculations

Notations:

A : Floor area

: Diagonal core effective area chA

: Diagonal core gross area gA

hhA : Area of horizontal reinforcement

shA : Area of transverse reinforcement

vdA : Area of diagonal reinforcement

vhA : Area of vertical reinforcement

xA : Torsion amplification factor

xavgA : Average of of all floors xA

wb : Beam width

: Width of diagonal core xb

: Concrete cover c

: Vertical distribution factor of Mode m vxmC

: Height of diagonal core xd

: Lateral load of Mode m xmF


: Longitudinal reinforcement yield strength yf

: Transverse reinforcement yield strength yhf

h : Height of beam section

B-1

: Reinforcement development length dl

: Clear span of link beam nl

: Length of rectangular wall pier wl

: Accidental torsion associated with of Mode m taxmM xmF

: Number of transverse reinforcement legs in beam section shn

: Center to center distance of transverse reinforcement or distribution

reinforcement

s

: Shear on wall pier caused by pfV xmF

: Shear on wall pier caused by ptV taxmM

: Story Shear strV

: Nominal shear demand for coupling beam nV

: Shear on wall pier after torsion amplification and wall length adjustment wV

: Weight of story i iw

α : Inclination of diagonal reinforcement

maxγ : Maximum ratio of wall pier shear to story shear

avgδ : Floor average displacement

taδ : Displacement caused by taxmM

iφ : Model shape factor of story i

ρ : Redundancy factor or steel ratio

B-2

Table B.1.1 Lateral Loads and Accidental Torsion of Mode 1 in the Coupled Direction Story Story Weight wi

(kips) Mode Shape φi Vertical Distribution Load CvxmLateral Load at Stories Fxm

(kips) Accidental Torsion at Stories Mtaxm

(k-ft) 15 1350 0.063 0.114 127.0 634.814 1541 0.059 0.123 136.1 680.613 1541 0.055 0.114 127.0 634.812 1541 0.051 0.106 117.4 586.811 1541 0.046 0.097 107.3 536.710 1541 0.042 0.087 96.9 484.7

9 1541 0.037 0.078 86.2 431.28 1541 0.033 0.068 75.3 376.67 1541 0.028 0.058 64.3 321.76 1541 0.023 0.048 53.4 267.15 1541 0.019 0.038 42.7 213.74 1541 0.014 0.029 32.5 162.53 1541 0.010 0.021 22.9 114.42 1541 0.006 0.013 14.2 71.01 1604 0.003 0.006 7.0 35.2

Sum 22987 1 1110.4

B-3

Table B.1.2 Lateral Loads and Accidental Torsion of Mode 2 in the Coupled Direction Story Story Weight wi

(kips) Mode Shape φi Vertical Distribution Load CvxmLateral Load at Stories Fxm

(kips) Accidental Torsion at Stories Mtaxm

(k-ft) 15 1350 0.059 -0.225 -145.2 -726.014 1541 0.045 -0.195 -125.7 -628.613 1541 0.029 -0.128 -82.3 -411.412 1541 0.013 -0.057 -36.6 -183.211 1541 -0.003 0.014 9.0 44.910 1541 -0.018 0.080 51.8 259.0

9 1541 -0.032 0.138 89.0 445.28 1541 -0.042 0.183 118.1 590.77 1541 -0.049 0.213 137.1 685.46 1541 -0.051 0.224 144.7 723.65 1541 -0.050 0.218 140.8 703.84 1541 -0.045 0.195 126.0 630.13 1541 -0.036 0.159 102.4 511.82 1541 -0.026 0.113 72.7 363.71 1604 -0.015 0.067 43.0 215.0

Sum 22987 1 644.8

B-4

Table B.2.1 Torsion Amplification of Mode 1 in the Coupled Direction xA

Story avgδ (ft) taδ (ft) maxδ = avgδ + taδ (ft) Ax=[ maxδ /(1.2 avgδ )]2

15 0.1582 0.0221 0.1803 1.0014 0.1486 0.0213 0.1699 1.0013 0.1386 0.0204 0.1590 1.0012 0.1281 0.0194 0.1475 1.0011 0.1172 0.0182 0.1354 1.0010 0.1058 0.0169 0.1227 1.00

9 0.0941 0.0155 0.1096 1.008 0.0822 0.0139 0.0961 1.007 0.0702 0.0123 0.0825 1.006 0.0583 0.0106 0.0689 1.005 0.0467 0.0088 0.0555 1.004 0.0355 0.0071 0.0425 1.003 0.0250 0.0054 0.0304 1.032 0.0155 0.0038 0.0193 1.071 0.0074 0.0023 0.0097 1.20

Axavg 1.02Notations: (1) Column of avgδ (the average displacement) and taδ (the displacement due to accidental torsion), are from ETABS calculation (2) The value of Ax is limited between the range from 1.0 to 3.0

B-5

Table B.2.2 Torsion Amplification of Mode 2 in the Coupled Direction xA

Story avgδ (ft) taδ (ft) maxδ = avgδ + taδ (ft) Ax=[ maxδ /(1.2 avgδ )]2

15 -0.0118 0.0001 0.0119 1.0014 -0.0089 0.0005 0.0094 1.0013 -0.0059 0.0010 0.0068 1.0012 -0.0026 0.0015 0.0041 1.7111 0.0006 0.0020 0.0027 3.0010 0.0037 0.0025 0.0062 1.98

9 0.0063 0.0030 0.0093 1.498 0.0084 0.0033 0.0116 1.347 0.0097 0.0034 0.0131 1.276 0.0103 0.0034 0.0137 1.235 0.0100 0.0032 0.0132 1.214 0.0090 0.0029 0.0118 1.213 0.0073 0.0024 0.0097 1.232 0.0052 0.0018 0.0070 1.271 0.0029 0.0012 0.0119 1.39

Axavg 1.42Notations: (1) Column of avgδ (the average displacement) and taδ (the displacement due to accidental torsion), are from ETABS calculation (2) The value of Ax is limited between the range from 1.0 to 3.

B-6

Table B.3.1 Redundancy Factor ρ of Mode 1 in the Coupled Direction P101 (Group I)

Story Vf (kips) Vt (kips) Vf+AxavgVt(kips) Lw(ft) Vw=(Vf+AxavgVt)10/Lw(kips) Vstr(kips) γ=|Vw/Vstr|15 31.6 16.4 48.3 10 48.3 127.0 0.3814 65.6 18.1 84.1 10 84.1 263.1 0.3213 97.3 21.8 119.6 10 119.6 390.1 0.3112 126.8 25.7 153.0 10 153.0 507.4 0.3011 153.7 29.5 183.8 10 183.8 614.8 0.3010 178.1 32.9 211.7 10 211.7 711.7 0.30

9 199.9 36.0 236.6 10 236.6 797.9 0.308 219.0 38.6 258.3 10 258.3 873.3 0.307 235.3 40.6 276.7 10 276.7 937.6 0.306 248.8 42.0 291.7 10 291.7 991.0 0.295 259.6 42.7 303.1 10 303.1 1033.8 0.294 267.8 42.6 311.2 10 311.2 1066.3 0.293 273.3 41.6 315.7 10 315.7 1089.2 0.292 276.4 39.7 316.9 10 316.9 1103.4 0.291 278.2 38.7 317.6 10 317.6 1110.4 0.29

P103 (Group II)Story Vf (kips) Vt (kips) Vf+AxavgVt(kips) Lw(ft) Vw=(Vf+AxavgVt)10/Lw(kips) Vstr(kips) γ=|Vw/Vstr|

15 0 7.7 7.9 25 3.1 127.0 0.0214 0 15.0 15.3 25 6.1 263.1 0.0213 0 32.3 32.9 25 13.2 390.1 0.0312 0 47.4 48.3 25 19.3 507.4 0.0411 0 60.9 62.1 25 24.8 614.8 0.0410 0 73.0 74.4 25 29.8 711.7 0.04

9 0 83.9 85.6 25 34.2 797.9 0.048 0 93.7 95.5 25 38.2 873.3 0.047 0 102.4 104.5 25 41.8 937.6 0.046 0 110.3 112.5 25 45.0 991.0 0.055 0 117.4 119.8 25 47.9 1033.8 0.054 0 124.1 126.5 25 50.6 1066.3 0.053 0 130.4 133.0 25 53.2 1089.2 0.052 0 136.9 139.6 25 55.8 1103.4 0.051 0 141.7 144.5 25 57.8 1110.4 0.05

γmax= 0.38Notations: (1) Vf is the wall pier shear due to lateral loads Fxm. (2) Vt is the wall pier shear due to torsion Mtaxm.(3) Axavg is the average of Ax (refer to Table B.2.1). (4) Lw is the wall length. (5) Vstr is the story shear. ρ=2-20/(γmaxA0.5) 1.47

B-7

Table B.3.2 Redundancy Factor ρ of Mode 2 in the Coupled Direction P101 (Group I)

Story Vf (kips) Vt (kips) Vf+AxavgVt(kips) Lw(ft) Vw=(Vf+AxavgVt)10/Lw(kips) Vstr(kips) γ=|Vw/Vstr|15 -35.87 -9.14 -48.86 10 -48.86 -145.21 0.3414 -66.97 -11.82 -83.77 10 -83.77 -270.92 0.3113 -87.38 -13.82 -107.02 10 -107.02 -353.20 0.3012 -96.53 -14.56 -117.23 10 -117.23 -389.83 0.3011 -94.41 -13.88 -114.14 10 -114.14 -380.86 0.3010 -81.68 -11.83 -98.50 10 -98.50 -329.05 0.30

9 -59.7 -8.58 -71.90 10 -71.90 -240.02 0.308 -30.46 -4.39 -36.70 10 -36.70 -121.89 0.307 3.54 0.39 4.09 10 4.09 15.20 0.276 39.5 5.35 47.10 10 47.10 159.92 0.295 74.54 10.06 88.84 10 88.84 300.68 0.304 105.93 14.1 125.97 10 125.97 426.70 0.303 131.44 17.1 155.75 10 155.75 529.05 0.292 149.51 18.6 175.95 10 175.95 601.80 0.291 160.91 21.81 191.91 10 191.91 644.80 0.30

P103 (Group II)Story Vf (kips) Vt (kips) Vf+AxavgVt(kips) Lw(ft) Vw=(Vf+AxavgVt)10/Lw(kips) Vstr(kips) γ=|Vw/Vstr|

15 0 9.89 14.06 25 5.62 -145.21 0.0414 0 28.73 40.84 25 16.33 -270.92 0.0613 0 40.64 57.77 25 23.11 -353.20 0.0712 0 46.31 65.83 25 26.33 -389.83 0.0711 0 46 65.38 25 26.15 -380.86 0.0710 0 40.13 57.04 25 22.82 -329.05 0.07

9 0 29.43 41.83 25 16.73 -240.02 0.078 0 14.91 21.19 25 8.48 -121.89 0.077 0 -2.16 -3.07 25 -1.23 15.20 0.086 0 -20.42 -29.03 25 -11.61 159.92 0.075 0 -38.44 -54.64 25 -21.86 300.68 0.074 0 -54.95 -78.11 25 -31.24 426.70 0.073 0 -68.95 -98.01 25 -39.20 529.05 0.072 0 -80.11 -113.87 25 -45.55 601.80 0.081 0 -83.38 -118.52 25 -47.41 644.80 0.07

γmax= 0.34Notations: (1) Vf is the wall pier shear due to lateral loads Fxm. (2) Vt is the wall pier shear due to torsion Mtaxm. (3) Axavg is the average of Ax (refer to Table B.2.2). (4) Lw is the wall length. (5) Vstr is the story shear. ρ=2-20/(γmaxA0.5) 1.41

B-8

B-8

Table B.4.1 Design of Coupling Beam Group I Design Parameter Comments

bw (in)--beam width 20 h (in)--beam height 30 ln (in)--clear span 72 fc

' (ksi) 6 fy,fyh (ksi) 60 Vn (kips)--design shear capacity demand 279 Nominal shear stress=6root(fc

')Diagonal Reinforcement

ln/h 2.4 When ln/h<4.0, a diagonal beam is recommended by bx (in)--width of diagonal element 11 At least bw/2, referring to ACI 21.7.7.4dx (in)--depth pf diagonal element 6 At least bw/5, referring to ACI 21.7.7.4 c (in)--concrete coverage 1.5 Per ACI 7.7.1α (degree) 15.4 Inclination of diagonal bars,Calculated Avd (in2) 8.7 Avd=Vn/(2fysin(α))Choose 6 #10 in one diagonal core At least 4 longitudinal bars needed, per ACI 21.7.7.4Actual Avd (in2) 7.6 actual capacity (kips) 243 Actual shear stress=5.2 root(fc

')ρ (steel ratio) 2.5% 1%<ρ<6%, per ACI 21.4.3.1

Check Diagonal Section Dimension Transverse Dimension of Core

Min clear distance of bars (in) 1.9 Greater(1.5,1.5db),db=diameter of #10,per ACI 7.6.3Actual clear distance (in) 3.1 (bx-2dt-3db)/2,dt = diameter of core transverseChecking result OK

Vertical Dimension of Core Min clear distance of bars (in) 1.9 Max(1.5,1.5db),db=diameter of #10, per ACI 7.6.3Actual clear distance (in) 2.5 dx-2dt-2db, Checking result OK

Development Length Per ACI 21.7.7.4ld (in)--Required 54.6 Per ACI 12.2.2Actual ld (in) 55

Diagonal Transverse Reinforcement nsh 3 Number of transverse legshx (in) 5.25 c-c space between legs in core width directionsx (in) 6 sx=4+(14-hx)/3,4<sx<6,per ACI 21.4.4.2Max allowed s(in)--c-c space of transverse 5 Lesser (bw/4, 6db,sx), per ACI 21.4.4.2Actual s (in) 4 Ag (in2)--gross area of diagonal core 126 (dx+2×1.5) × (bx+2×1.5), Coverage is 1.5in Ach (in2)--out to out cross section 66 bx×dx

hc (in)--c-c height between transverse 5.5 dx-dtAsh (in2)--Required 0.6Ash=0.3s×hc×fc'/fyh× [(Ag/Ach)-1], per ACI Eq. 21-3 0.2 Ash=0.09×s×hc×fc

'/fyh, per ACI Eq. 21-4Actual Ash (in2) 0.6 3 #4 legsChoose 3 legs of #4@4

Distribution Reinforcement Horizontal Steel

Max s (in)--c-c space between horizontal steels 6 Lesser (h/5,12 ), per ACI 11.8.5Actual s (in) 6 Required Ahh (in2) 0.2 Ahh=0.0015bw×sChoose #4@5 in the side faces and #[email protected] in the top/bottom faces

Distance between horizontal steels is adjusted tofit the dimension of the beam face

Actual Avh (in2) 0.4 2 legs of #4Vertical Steel

Max s (in)--c-c space between horizontal steels 6 Lesser (h/5,12 ), per ACI 11.8.4Actual s (in) 6 Required Avh (in2) 0.3 Avh=0.025bw×sChoose #4@6 Actual Avh (in2) 0.4 2 legs of #4

B-9

Table B.4.2 Design of Coupling Beam Group II Design Parameter Comments


' (ksi) 6 fy,fyh (ksi) 60 Vn (kips)--design shear capacity demand 177 Nominal shear stress=3.8 root(fc


ln/h 2.4 When ln/h<4.0, a diagonal beam is recommended by bx (in)--width of diagonal element 11 At least bw/2, referring to ACI 21.7.7.4dx (in)--depth pf diagonal element 6 At least bw/5, referring to ACI 21.7.7.4 c (in)--concrete coverage 1.5 Per ACI 7.7.1α (degree) 15.4 Inclination of diagonal bars,Calculated Avd (in2) 5.6 Avd=Vn/(2fysin(α))Choose 6 #9 in one diagonal core At least 4 longitudinal bars needed, per ACI 21.7.7.4Actual Avd (in2) 6 Actual capacity (kips) 191 Actual shear stress=4.1 root(fc

')ρ (steel ratio) 2.0% 1%<ρ<6%, per ACI 21.4.3.1

Check Diagonal Section Dimension Transverse Dimension

Min clear distance of bars (in) 1.7 Greater(1.5,1.5db),db=diameter of #9,per ACI 7.6.3Actual clear distance (in) 3.3 (bx-2dt-3db)/2,dt = diameter of core transverseChecking result OK

Vertical Dimension Min clear distance of bars (in) 1.7 Greater(1.5,1.5db),db=diameter of #9,per ACI 7.6.3Actual clear distance (in) 2.7 dx-2dt-2db, Checking result OK

Development Length Per ACI 21.7.7.4ld (in)--Required 39 Per ACI 12.2.2Actual ld (in) 40


hc (in)--c-c height between transverse 5.5 dx-dtAsh (in2)--Required 0.6 Ash=0.3s×hc×fc'/fyh× [(Ag/Ach)-1], per ACI Eq. 21-3 0.2 Ash=0.09×s×hc×fc



Max s (in)--c-c space between horizontal steels 6 Lesser (h/5,12 ), per ACI 11.8.5Actual s (in) 6 Required Ahh (in2) 0.2 Ahh=0.0015bw×sChoose #4@5 in the side faces and #[email protected] in the top/bottom faces




B-10

Table B.4.3 Design of Coupling Beam Group III Design Parameter Comments


' (ksi) 6 fy,fyh (ksi) 60 Vn (kips)--design shear capacity demand 98 Nominal shear stress=2.1root(fc


ln/h 2.4 When ln/h<4.0, a diagonal beam is recommended by bx (in)--width of diagonal element 11 At least bw/2, referring to ACI 21.7.7.4dx (in)--depth pf diagonal element 6 At least bw/5, referring to ACI 21.7.7.4 c (in)--concrete coverage 2 Per ACI 7.7.1α (degree) 15.4 Inclination of diagonal bars,Calculated Avd (in2) 3.1 Avd=Vn/(2fysin(α))Choose 6 #7 in one diagonal core At least 4 longitudinal bars needed, per ACI 21.7.7.4Actual Avd (in2) 3.6 Actual capacity (kips) 115 Actual shear stress=2.4root(fc

')ρ (steel ratio) 1.2% 1%<r<6%, per ACI 21.4.3.1

Check Diagonal Section Dimension Transverse Dimension

Min clear distance of bars (in) 1.5 Greater(1.5,1.5db),db=diameter of #7,per ACI 7.6.3Actual clear distance (in) 3.7 (bx-2dt-3db)/2,dt =diameter of core transverseChecking result OK

Vertical Dimension Min clear distance of bars (in) 1.5 Max(1.5,1.5db),db=diameter of #7 per ACI 7.6.3Actual clear distance (in) 3.3 dx-2dt-2db, Checking result OK

Development Length Per ACI 21.7.7.4ld (in)--Required 34 Per ACI 12.2.2Actual ld (in) 35


hc (in)--c-c height between transverse 5.5 dx-dtAsh (in2)--Required 0.6 Ash=0.3s×hc×fc'/fyh× [(Ag/Ach)-1], per ACI Eq. 21-3 0.2 Ash=0.09×s×hc×fc



Max s (in)--c-c space between horizontal steels 6 Lesser (h/5,12 ), per ACI 11.8.5Actual s (in) 6 Required Ahh (in2) 0.2 Ahh=0.0015bw×s

Choose #4@5 in the side faces and #[email protected] in the top/bottom faces




Appendix C Wall Design Calculations

Notations:

A : Area

cvA : Section gross area

shA : Area of transverse reinforcement

xA : Torsion amplification factor in the X direction

xavgA : Average value of of all levels xA

yA : Torsion amplification factor in the Y direction

yavgA : Average value of of all levels yA

: Section thickness b

: Base shear amplification factor bC

: Vertical distribution factor in the X direction vxC

: Vertical distribution factor in the Y direction vyC

: Accumulated dead load corresponding to dead loads at that story and above D

: Dead load of story i iD


: Transverse steel yield strength yhf

: Lateral loads in the X direction xF

: Lateral loads in the Y direction yF

: Center to center distance between two transverse reinforcement bars located

at the edge of the section

ch

C-1

L : Accumulated live load corresponding to live loads at that story and above

: Live load of the story i iL

: Wall length wL

taxM : Accidental torsion associated with the X direction lateral loads

: Accidental torsion associated with the Y direction lateral loads tayM

OTM: Overturning moment

: Uniformly distributed live load Lp

: Uniformly distributed dead load, accounting for weight of the slab, column,

partition, and cladding

Dp

s : Vertical space between transverse steel bars or distributed steel bars

DSS : Design spectral response acceleration at short period

: Base shear from ELF bV

strV : Story Shear

wV : Shear on the wall sub-section

'wV : modified by the factor of 10/ wV wl

yV : Base shear of each mode in the Y direction calculated by modal response

spectrum analysis

W : Total weight of the building

iW : Level i weight

: Concrete core weight, including one C-shaped wall section and one beam cW

yW : Effective weight of each mode in the Y direction

C-2

cα : Factor in calculating nV

xγ : Ratio of to in the X direction 'wV strV

maxxγ : Maximum xγ

yγ : Ratio of to in the Y direction 'wV strV

maxyγ : Maximum yγ

xavgδ : SRSS of average displacements of all considered modes in the X direction

maxxδ : SRSS of maximum displacements of all considered modes in the X

direction

yavgδ : SRSS of average displacements of all considered modes in the Y direction

maxyδ : SRSS of maximum displacements of all considered modes in the Y

direction


iφ : Shape factor of level i

ρ : Longitudinal reinforcement ratio

nρ : Horizontal reinforcement ratio

vρ : Vertical reinforcement ratio

xρ : Redundancy factor in the X direction

yρ : Redundancy factor in the Y direction

C-3

Table C.1.1 Lateral Loads and Their Effects for Mode 1 in the X Direction

Lateral Load Calculations Load Effects

Story Story Weight

wi (kips) Mode Shape

φi

Vertical Distribution Factor, Cvx

Lateral Load,Fx (kips)

AccidentalTorsion

M

Overturning Moment,

OTM tax (k-ft) (k-ft)

Story Shear Vstr

(kips)

15 1350 0.063 0.114 127.0 634.8 1164 12714 1541 0.059 0.123 136.1 680.6 3577 26313 1541 0.055 0.114 127.0 634.8 7154 39012 1541 0.051 0.106 117.4 586.8 11807 50711 1541 0.046 0.097 107.3 536.7 17444 61510 1541 0.042 0.087 96.9 484.7 23970 712

9 1541 0.037 0.078 86.2 431.2 31287 7988 1541 0.033 0.068 75.3 376.6 39295 8737 1541 0.028 0.058 64.3 321.7 47893 9386 1541 0.023 0.048 53.4 267.1 56981 9915 1541 0.019 0.038 42.7 213.7 66461 10344 1541 0.014 0.029 32.5 162.5 76239 10663 1541 0.010 0.021 22.9 114.4 86226 10892 1541 0.006 0.013 14.2 71.0 96344 11031 1604 0.003 0.006 7.0 35.2 109858 1110

Table C.1.2 Lateral Loads and Their Effects for Mode 2 in the X Direction Lateral Load Calculations Load Effects

Story Story Weight wi (kips)

Mode Shape φi

Vertical Distribution Factor, Cvx

Lateral Load,Fx (kips)

AccidentalTorsion,

M

Overturning Moment,

OTM tax (k-ft) (k-ft)

Story Shear Vstr,

(kips)

15 1350 0.059 -0.225 -145.2 -726.0 -1332 -14514 1541 0.045 -0.195 -125.7 -628.6 -3816 -27113 1541 0.029 -0.128 -82.3 -411.4 -7055 -35312 1541 0.013 -0.057 -36.6 -183.2 -10629 -39011 1541 -0.003 0.014 9.0 44.9 -14122 -38110 1541 -0.018 0.080 51.8 259.0 -17139 -329

9 1541 -0.032 0.138 89.0 445.2 -19340 -2408 1541 -0.042 0.183 118.1 590.7 -20458 -1227 1541 -0.049 0.213 137.1 685.4 -20319 156 1541 -0.051 0.224 144.7 723.6 -18852 1605 1541 -0.050 0.218 140.8 703.8 -16095 3014 1541 -0.045 0.195 126.0 630.1 -12182 4273 1541 -0.036 0.159 102.4 511.8 -7331 5292 1541 -0.026 0.113 72.7 363.7 -1812 6021 1604 -0.015 0.067 43.0 215.0 6035 645

C-4

Table C.1.3 SRSS of Load Effects in the X Direction SRSS of Load Effects from Table

C.1.1&C.1.2 SRSS Results Multiplied by CbStory OTM

(k-ft) Story Shear Vstr

(kips) OTM (k-ft)

Story Shear Vstr (kips)

15 1769 193 3060 335 14 5230 379 9048 655 13 10047 528 17381 913 12 15886 641 27484 1110 11 22444 725 38828 1254 10 29468 786 50979 1360

9 36783 835 63634 1445 8 44302 884 76642 1529 7 52025 940 90003 1626 6 60019 1006 103833 1741 5 68382 1079 118301 1867 4 77206 1151 133566 1992 3 86537 1214 149710 2100 2 96361 1260 166705 2180 1 110023 1287 190340 2227

Notation: See Table 3.2 for Cb value

Table C.2.1 Modal Mass Participation in the Y Direction Mode 1 Mode 2 Total

Modal Mass (kips) yW 15984 4973 Building Total Mass W (kips) 22987 22987 Mass Participation = /W yW 70% 22%

92%

Table C.2.2 Base Shear Amplification Factor for the Y Direction bC

Mode 1 Mode 2 Vy (kips) 1190 829Vt SRSS of both Vy (kips) 14500.85Vb from ELF (kips) 2227Cb =0.85Vb/Vt 1.54

C-5

Table C.3.1 Lateral Loads and Their Effects for Mode 1 in the Y Direction



Mode Shape φi

Vertical Distribution Factor, Cvy

Lateral Load,Fy

(kips)

AccidentalTorsion,

M

Overturning Moment,

OTM tay (k-ft) (k-ft)

Story Shear, Vstr

(kips)

15 1350 0.067 0.127 151.0 755.0 1385 15114 1541 0.062 0.133 158.5 792.6 4223 31013 1541 0.056 0.121 144.6 722.9 8387 45412 1541 0.051 0.110 130.6 653.0 13749 58511 1541 0.045 0.098 116.6 583.2 20180 70110 1541 0.040 0.086 102.8 514.0 27554 804

9 1541 0.035 0.075 89.2 445.9 35746 8938 1541 0.029 0.064 75.9 379.5 44634 9697 1541 0.025 0.053 63.1 315.7 54100 10326 1541 0.020 0.043 51.0 255.0 64034 10835 1541 0.015 0.033 39.7 198.3 74333 11234 1541 0.011 0.025 29.3 146.5 84899 11523 1541 0.008 0.017 20.1 100.3 95650 11722 1541 0.005 0.010 12.1 60.5 106511 11841 1604 0.002 0.005 5.9 29.3 120998 1190

Table C.3.2 Lateral Loads and Their Effects for Mode 2 in the Y Direction



Mode Shape φi

Vertical Distribution Factor, Cvy

Lateral Load,Fy

(kips)

AccidentalTorsion,

M

Overturning Moment,

OTM tay (k-ft) (k-ft)

Story Shear, Vstr

(kips)

15 1350 0.058 -0.196 -162.4 -811.8 -1489 -16214 1541 0.041 -0.161 -133.5 -667.6 -4202 -29613 1541 0.025 -0.096 -79.6 -397.8 -7645 -37512 1541 0.008 -0.031 -25.6 -128.1 -11323 -40111 1541 -0.008 0.031 25.7 128.5 -14765 -37510 1541 -0.022 0.087 71.8 358.9 -17549 -304

9 1541 -0.034 0.133 110.1 550.7 -19323 -1938 1541 -0.043 0.167 138.8 694.2 -19824 -557 1541 -0.049 0.189 156.5 782.3 -18889 1026 1541 -0.050 0.196 162.5 812.4 -16466 2645 1541 -0.049 0.190 157.1 785.7 -12601 4214 1541 -0.044 0.171 141.6 708.1 -7437 5633 1541 -0.037 0.142 117.9 589.4 -1192 6812 1541 -0.028 0.107 88.6 443.2 5865 7701 1604 -0.018 0.071 59.3 296.3 15953 829

C-6

Table C.3.3 SRSS of Load Effects in the Y Direction

SRSS of Load Effects from Table C.3.1&C.3.2 SRSS Results Multiplied by Cb

Story OTM (k-ft)


OTM (k-ft)


15 2033 222 3122 340 14 5957 428 9147 657 13 11349 589 17424 905 12 17811 709 27346 1089 11 25005 795 38391 1221 10 32668 860 50156 1320

9 40634 914 62387 1403 8 48838 971 74983 1490 7 57303 1037 87980 1593 6 66118 1115 101513 1712 5 75393 1200 115754 1842 4 85224 1283 130849 1969 3 95657 1356 146867 2082 2 106673 1413 163779 2169 1 122045 1450 187380 2227

Notation: See Table C.2.2 for Cb value

Table C.4 Dead and Live Loads for Wall Pier Design

Story Tributary Area, A

(ft2)

Distributed Dead Load,

pD (psf)

Distributed Live Load,

pL (psf)

Core Weight,

Wc (kips)

Dead Load at Each Story,

Di=pDA+Wc(kips)

Live Load at Each Story,

Li=pLA (kips)

Total L=ΣLi (kips)

Total D=ΣDi (kips)

15 914 133 50 107 229 46 46 22914 914 133 50 107 229 46 91 45713 914 133 50 107 229 46 137 68612 914 133 50 107 229 46 183 91411 914 133 50 107 229 46 229 114310 914 133 50 107 229 46 274 1371

9 914 133 50 107 229 46 320 16008 914 133 50 107 229 46 366 18287 914 133 50 107 229 46 411 20576 914 133 50 107 229 46 457 22865 914 133 50 107 229 46 503 25144 914 133 50 107 229 46 548 27433 914 133 50 107 229 46 594 29712 914 133 50 107 229 46 640 32001 914 138 50 141 267 46 686 3467

C-7

Table C.5.1 Redundancy Factor for the X Direction

Story

Shear on Wall Component P201,

Vw (kips)

Wall Length, Lw (ft)V’

w = Vw10/Lw (kips)

Story Shear,Vstr(kips) γx= V’

w /Vstr

15 122 10 122 334 0.3714 239 10 239 653 0.3713 333 10 333 910 0.3712 404 10 404 1107 0.3711 457 10 457 1251 0.3710 496 10 496 1356 0.37

9 527 10 527 1442 0.378 557 10 557 1525 0.377 593 10 593 1622 0.376 635 10 635 1737 0.375 681 10 681 1863 0.374 726 10 726 1987 0.373 765 10 765 2095 0.372 794 10 794 2174 0.371 812 10 812 2221 0.37

γxmax 0.37 ρx=2-20/(γxmax100) 1.45

Table C.5.2 Torsion Amplification Factor for the X Direction

Story δxavg (ft)

δxmax (ft)

Amplification Factor Ax=(δxmax/1.2δxavg)2

15 0.3214 0.3656 1.0014 0.2997 0.3425 1.0013 0.2772 0.3185 1.0012 0.2538 0.2933 1.0011 0.2298 0.2669 1.0010 0.2052 0.2396 1.00

9 0.1804 0.2120 1.008 0.1556 0.1841 1.007 0.1311 0.1563 1.006 0.1071 0.1288 1.005 0.0840 0.1021 1.034 0.0621 0.0764 1.053 0.0420 0.0525 1.092 0.0244 0.0313 1.141 0.0104 0.0140 1.25

Axavg 1.04

C-8

Table C.6.1 Redundancy Factor for the Y Direction

Story

Shear on Wall Component P203,

Vw (kips)

Wall Length, Lw

(ft) V’

w=Vw10/Lw (kips)

Story Shear, Vstr (kips) γy= V’

w /Vstr

15 238 25 95 341 0.2814 459 25 184 658 0.2813 632 25 253 905 0.2812 761 25 304 1089 0.2811 853 25 341 1222 0.2810 922 25 369 1320 0.28

9 980 25 392 1404 0.288 1041 25 417 1491 0.287 1113 25 445 1593 0.286 1196 25 478 1713 0.285 1287 25 515 1842 0.284 1376 25 550 1970 0.283 1454 25 582 2082 0.282 1515 25 606 2170 0.281 1556 25 622 2228 0.28

γymax 0.28 ρy=2-20/(γymax100) 1.29

Table C.6.2 Torsion Amplification Factor for the Y Direction

Story δyavg

(ft) δymax (ft)

Amplification Factor Ay=(δymax/1.2δyavg)2

15 0.2238 0.2612 1.0014 0.2055 0.2417 1.0013 0.1871 0.2219 1.0012 0.1687 0.2018 1.0011 0.1503 0.1815 1.0110 0.1322 0.1612 1.03

9 0.1144 0.1410 1.068 0.0971 0.1211 1.087 0.0806 0.1018 1.116 0.0650 0.0832 1.145 0.0505 0.0656 1.174 0.0373 0.0495 1.223 0.0256 0.0350 1.302 0.0155 0.0223 1.441 0.0073 0.0116 1.77

Ayavg 1.15

C-9

C-10

Table C.7.1 Boundary Element Design for Group I Boundary Element at Flange End

Transverse Steel Perpendicular to X hc(in) C-C distance of two transverse steels at the edge of confined area

s(in) 5 Vertical spacing per ACI 21.4.4.2

Min Ash(in2) 1.6 0.09shcfc'/fyh per ACI 21.4.4.1(b)

Actual Ash(in2) 3.0 Choose 5 Legs of No. 7 @5

Check max distance (c-c) between legs (in) 13

Distance OK or Not OK Less than 14 inches is OK per ACI 21.4.4.3Transverse Steel Perpendicular to Y

hc(in) 17 C-C distance of two transverse steels at the edge of confined area



Actual Ash(in2) 1.8 Choose 3 Legs of No. 7@5

Check max distance (c-c) between legs (in) 9.5

Distance OK or Not OK Less than 14 inches is OK per ACI 21.4.4.3Boundary Element at Intersection Extending into Flange

Transverse Steel Perpendicular to X

hc(in) 27






hc(in) 17





Distance OK or Not OK Less than 14 inches is OK per ACI 21.4.4.3Boundary Element at Intersection Extending into Web


hc(in) 17






hc(in) 71





Distance OK or Not OK Less than 14 inches is OK per ACI 21.4.4.3

C-11

Table C.7.2 Boundary Element Design for Group II Boundary Element at Flange End





Actual Ash(in2) 1.2 Choose 2 Legs of No. 7 @5







Check max distance (c-c) between legs (in) 9.5 Distance OK or Not OK Less than 14 inches is OK per ACI 21.4.4.3

Boundary Element at Intersection Extending into Flange Transverse Steel Perpendicular to X

hc(in) 27






hc(in) 17





Distance OK or Not OK Less than 14 inches is OK per ACI 21.4.4.3Boundary Element at Intersection Extending into Web


hc(in) 17






hc(in) 60






C-12

Table C.7.3 Boundary Element Design for Group III Boundary Element at Flange End

ρ 2.8% Actual longitudinal steel ratio

400/fy 0.7% Minimum steel ratio as checking limit

Check 21.7.6.5 Yes If ρ>400/fy, check ACI 21.7.6.5

Check Transverse Bars Perpendicular to X

Using 2 No 7 Legs@8 Vertical spacing s=8 inches per ACI 21.7.6.5



Check Transverse Bars Perpendicular to Y

Using 3 No 7 Legs@8 Vertical spacing s=8 inches per ACI 21.7.6.5Check max distance (c-c) between legs (in) 9.5 Distance OK or Not OK Less than 14 inches is OK per ACI 21.4.4.3

Boundary Element at Intersection Extending into Flange




Check Transverse Bars Perpendicular to X

Using 4 No 7 Legs@8 Vertical spacing s=8 inches per ACI 21.7.6.5Check max distance (c-c) between legs (in) 13 Distance OK or Not OK Less than 14 inches is OK per ACI 21.4.4.3

Check Transverse Bars Perpendicular to Y




Boundary Element at Intersection Extending into Web




Check Transverse Legs Perpendicular to X



Distance OK or not OK Less than 14 inches is OK per ACI 21.4.4.3

Check Transverse Legs Perpendicular to Y

Using 5 No 7 Legs@8 Vertical spacing s=8 inches per ACI 21.7.6.5Check max distance (c-c) between legs (in) 13 Distance OK or Not OK Less than 14 inches is OK per ACI 21.4.4.3

C-13

Table C.8.1 Shear Design for Group I Flange P101, P102, P201, P202

Dimensions and Factors b (in) 20 Thickness

lw (in) 130 Flange length

Section Dimension Check

Acv (in2) 2600 b× lw

Root (fc’)Acv (kips) 201

Vu (kips) 1591 Design Shear value from Table 4.7.1

ρn,ρv low limit 0.0025 If(Vu>root(fc’)Acv) per ACI 21.7.2.1

Shear Strength

αc 2 Per ACI 21.7.4.1

ρn 0.0075 Eq (21-7) per ACI 21.7.4.1

Final ρn,ρv 0.0075 Shall be greater than 0.0025, the low limit

Select 2 Layers of No. 7 @8 for Vertical and Horizontal Steel s=8 less than the max allowable spacing 18 inches

Actual ρn,ρv 0.0075

Check Shear Strength Up limit Per ACI 21.7.4.4

Vn(kips) 1963 Eq 21-7 per ACI 21.7.4.1

10root(fc')Acv 2014 Up limit value per ACI 21.7.4.4

Checking OK If Vn <10root(fc')Acv, OK

Web P103, P203

Dimension and Factors b (in) 20 Thickness

lw (in) 320 Web length



Root(fc')Acv (kips) 496



Shear Strength

αc 2 Per ACI 21.7.4.1

ρn 0.0051 Eq 21-7 per ACI 21.7.4.1





Vn(kips) 3295 Eq 21-7 per ACI 21.7.4.1



C-14

Table C.8.2 Shear Reinforcement Design for Group II

Flange P101, P102, P202, P202

Dimensions and Factors

b (in) 20 Thickness




Root(fc’)Acv (kips) 201



Shear Strength

αc 2 Per ACI 21.7.4.1

ρn 0.0065 Eq (21-7) per ACI 21.7.4.1





Vn(kips) 1573 Eq 21-7 per ACI 21.7.4.1



Web P103, P203

Dimension and Factors

b (in) 20 Thickness







Shear Strength

αc 2 Per ACI 21.7.4.1

ρn 0.0042 Eq 21-7 per ACI 21.7.4.1





Vn(kips) 3295 Eq 21-7 per ACI 21.7.4.1



C-15

Table C.8.3 Shear Reinforcement Design for Group III Flange P101, P102, P202, P202

Dimensions and Factors

b (in) 20 Thickness




Root(fc’)Acv (kips) 201



Shear Strength

αc 2 Per ACI 21.7.4.1

ρn 0.0044 Eq (21-7) per ACI 21.7.4.1





Vn(kips) 1183 Eq 21-7 per ACI 21.7.4.1



Web P103, P203

Dimension and Factors

b (in) 20 Thickness







Shear Strength

αc 2 Per ACI 21.7.4.1

ρn 0.0026 Eq 21-7 per ACI 21.7.4.1


Select 2 Layers of No. 7 @18 for Vertical and Horizontal Steel s=18 equal to the max allowable spacing 18 inches



Vn(kips) 2271 Eq 21-7 per ACI 21.7.4.1



-100000

0

100000

-250000 0

-60000

0

60000

-150000 0 150000

Notation: The capacity curves were generated by XTRACT. The difference between the curvesame diagram is due to the difference of axial loads at difference levels.

Fig. C.1.1 Biaxial Bending Capacity Check for Wall Group I

-120000

0

120000

-400000 0

)

My (k-ft)

-120000

0

120000 My (k-ft)

Mx (k-ft) -600000 0 600000

Lv 2 Demand

Lv 1 Demand

Lv 3 Demand Lv

LvCurves from outside to inside represent the capacities of Lv 1, Lv 2 and Lv 3

Curves from outside to inside represent the capacities of Lv 1, Lv 2 and Lv 3

Lv 1

1.0X+0.3Y Combination in Compression 0.3X+1.0Y Combination in Co

Curves from outside to inside represent the capacities of Lv 3, Lv 2 and Lv 1

Mx (k-ft)

My (k-ft)

d

d

My (k-ft)

Curves inside recapacitiand Lv 3

1.0X+0.3Y Combination in Tension 0.3X+1.0Y Combination i

C-16

Mx (k-ft

s in

400000

3 Demand

2 Demand

Demand

mpression

)

dd

frompres

es of

n Te

Mx (k-ft

Lv 3 Deman

Lv 3 Demand
Lv 2 Demand
Lv 2 DemanLv

Lv 1 Deman

1 Deman

250000

the

outside to ent the Lv 1, Lv 2

nsion

Curves from outside to inside represent the capacities of Lv 4, Lv 5, Lv 6, Lv 7

-120000

0

120000

-300000 0 300000

My (k-ft)

d

d

d

-120000

0

120000

-300000 0 300000

Mx (k-ft)

d

dd

d


Mx (k-ft)

My (k-ft)

d 1.0X+0.3Y Combination in Compression 0.3X+1.0Y Combination in Compression

-60000

0

60000

-120000 0 120000

My (k-ft)

-90000

0

90000

-180000 0

dd

Mx (k-ft)


My (k-ft)

)

Curves from outside to inside represent the capacities of Lv 7, Lv 6, Lv 5, Lv 4 d

d 1.0X+0.3Y Combination in Tension 0.3X+1.0Y Combination in Tensi Notation: The capacity curves were generated by XTRACT. The difference between the curves in the same diagram is due to the difference of axial loads at difference levels.

Fig. C.1.2 Biaxial Bending Capacity Check for Wall Group II

C-17

Mx (k-ft

Lv 5 Deman

Lv 5 Deman

180000

d
Lv 5 Deman Lv 5 Deman
Lv 6 Deman

Lv 6 Deman

Lv 6 Demand
Lv 6 Deman
Lv 7 Deman

Lv 7 Deman

Lv 7 Deman
Lv 7 Deman
Lv 4 Deman

Lv 4 Deman

Lv 4 Demand
Lv 4 Demand
on

-90000

0

90000

-180000 0 180000

Curves from outside to inside represent the capacities of Lv 8, Lv9, Lv 10, Lv 11

Mx (k-ft)

My (k-ft)

d

0

70000

-180000 0 180000

dd

Curves from outside to

My (k-ft)

Mx (k-ft)

1.0X+0.3Y Combination in C My (k-ft)

-50000

0

50000

-80000 0


1.0X+0.3Y Combination

Notation: The capacity curves were genesame diagram is due to the difference of Fig. C.1.3 Biaxial Bending Capa

Lv 9 Deman
d -70000inside represent the
capacities of Lv 8, Lv9, Lv 10, Lv 11

ompression 0.3X+1.0Y Combinatio

80000

-50000

0

50000

-120000 0

Mx (k-ft)

in Tension

d


0.3X+1.0Y Combi

My (k-

rated by XTRACT. The difference between the caxial loads at difference levels.

city Check for First Four Levels in WIII

C-18

Lv 9 Demand
d
n in Compression

120000

d

d

Mx (k-ft)

nation

ft)

urves i

all G

Lv 9 Deman
Lv 9 Demand
Lv 10 DemandLv 11 De

Lv 10 DemanLv 11 De

Lv 10 Demand

Lv 10 Demand

Lv 8 Deman
Lv 8 Deman
Lv 8 Deman

in

n th

ro

Lv 8 Deman

man

Lv 11 Demand
Lv 11 Demand
mand

Tension

e

up

-70000

0

70000

150000 1500000

Mx (k-ft)

My (k-ft)

Lv 13 Demand -50000

0

50000

-120000 0


My (k-ft)

)-


1.0X+0.3Y Combination in Compression 0.3X+1.0Y Combination in Compr

-


40000

0

40000

-100000 100000

d

My (k-ft)

0

Mx (k-ft)

Lv 14 Demand

My (k-ft)

-50000

0

50000

-100000 0


Lv 14 Dem

1.0X+0.3Y Combination in Tension 0.3X+1.0Y Combination in Tensio Notation: The capacity curves were generated by XTRACT. The difference between the curves in the same diagram is due to the difference of axial loads at difference levels. Fig. C.1.3 (Continued) Biaxial Bending Capacity Check for Last Four Levels in

Wall Group III

C-19

Mx (k-ft

120000

d
Lv 14 Demand d Lv 14 Deman
Lv 15 Demand
Lv 15 Deman
and

ession

100000

Mx (k-ft)

and

d

Lv 15 Deman
Lv 15 Demand
Lv 12 Demand
Lv 13 Dem
Lv 12 Demand

Lv 12 Demand
Lv 12 Deman Lv 13 Demand Lv 13 Demand
n

Appendix D Calculated Wall Pier Parameters from XTRACT for RUAUMOKO Modeling

D-1

Coupling Shear Wall

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