10 I 2'1 A UILU-ENG-82-2014

~~2CIVIL ENGINEERING STUDIES STRUCTURAL RESEARCH SERIES NO. 502

RECEJVED

OCT 12 1982

AN ANALYTICAL STUDY OF THE INTERACTION OF FRAMES AND INFILL MASONRY WALLS

Metz Reference Room University of IllinolQ

BI06 NeEL 208N. Romine Street

Urbana, Illinoi~ 61801

by C. E. RIVERO

and

W. H. WALKER

A Technical Report of

Research Supported by the

NATIONAL SCIENCE FOUNDATION

under Grant Nos. ENV 77-07190

and

PFR 80-02582

DEPARTMENT OF CIVIL ENGINEERING

UNIVERSITY OF ILLINOIS

AT URBANA-CHAMPAIGN

URBANA, ILLINOIS

SEPTEMBER 1982

AN ANALYTICAL STUDY OF THE INTERACTION OF

FRAMES AND INFILL MASONRY WALLS

BY

C. E. RIVERO

and

W. H. WALKER

A Report on a Research Project Sponsored by the NATIONAL SCIENCE FOUNDATION

Research Grant Nos. ENV 77-07190 and PFR 80-02582

Metz Reference Room Un~versity of Illinois

Bl06 NeEL 208 N. Romine Street

Urbana, Illinois 61801

University of Illinois at Urbana-Champaign Urbana, Illinois

September 1982

50272 -101

_ ~~ORT. _~~~M E:~:~O_N t: ~~:_ ~~G-8~~lQH 4. Title and Subtitle

AN ANALYTICAL STUDY OF THE INTERACTION OF FRAtIES AND INFILL MASONRY WALLS

7. Author(s)

C. E. Rivero and W. H. Walker 1----------------.. -' ------.. --- --.. --- .. --- - .. ----

9. Performing Organization Name and Address

University of Illinois at Urbana-Champaign Department of Civil Engineering 208 N. Rom~ne Street Urbana, Illinois 61801

ii

J2 ___ _ 13. Recipient's Accession No.

-r---'-" 5. Report Date

.. _$ep.!:.~.!IlQ~~.J.9.§2 .. __ . 6.

8. Performing Organization Rept. No.

_ 5RS ... 5D2. 10. Project/Task/Work Unit No.

11. Contract(C) or Grant(G) No.

(C) ENV 77-07190

(G) PFR 80-02582 f----------- --------.--------------.-.--------.- -. -----.. ---.-.. --

12. Sponsoring Organization Name and Address

National Science Foundation Washington, D.C. 20550

13. Type of Report & Period Covered

14.

l------------------.-.----... ---.. -- .. - '--.-.- .--.---- ... -.-.- - .. ---- .. - '-"-' 15. Supplementary Notes

------_._---_._ ... -.- ._-- ---" . Hi. Abstract (Limit: 20e words)

In this study a nonlinear dynamic model to study the behavior of frames infilled by masonry walls is presented. The nonlinearities of the model include the interaction between the frame and wall, cracking and failure of the wall, the bracing effect that the wall has on the frame, the discontinuities between the frame and wall, and the inelastic behavior of the frame. The modeling of the masonry wall is based on the premise that the cracking mechanism in the wall may be separated from the material model assumed for the masonry.

Analytical results are presented for three one story, one bay frames and a three story, one bay frame infilled with masonry walls, under the action of a ground motion.

The analysis was able to represent the significant facets of behavior that have been observed experimentally. The wall was seen to develop damage and alter its behavior in accord with the boundary conditions imposed by the frame, the cracking patterns both inside and at the boundary of the wall are represented, and the wall could brace the frame both diagonally and induce short column behavior in the frame.

J----------------------------- -.--------.-.--. -.. --.--------- -----------i 17. Document Analysis a. Descriptors

Dynamic Response, Earthquake Resistant Design, Ground Motion, Masonry Walls, Frames, Seismic, Structural Dynamics, Interaction

b. Identifiers/Open-Ended Terms

c. COSATI Field/Group

18. Availability Statement

Release Unlimited

(See ANSI-Z39.18)

19. Security Class (This Report) 21. No. of Pages

~~nNrC~:1~,A~S~,S~T~F'·T~E~D--____ ~.----1-9-4-----~ 20. Security Class (This Page)

UNCLASSIFIED See InstructIons on Reverse

22. Price

OPTIONAL FORM 272 (4-77) (Formerly NTlS-35) Department of Commerce

iii

ACKNOWLEDGMENT

This investigation was part of a research program sponsored by

the National Science Foundation under Grant ENV 77-07190,

"Engineering Design For Natural Hazards". Any opinions, findings,

and conclusions or recommendations expressed in this publication

are those of the author and do not necessarily reflect the views

of the National Science Foundation. The report was prepared

as a doctoral dissertation by C. E. Rivero under the supervision

of W. H. Walker, Associate Professor of Civil Engineering, and

was submitted to the Graduate College of the University of

Illinois at Urbana-Champaign in partial fulfillment of the

requirements for the Ph.D. degree.

The development of the computer program and the numerical

results as presented in this report were obtained with the use

of the CDC Cyber 175 and IBM 4341 computers, and the Zeta 1453B

plotting device. These facilities are supported by the Computing

Services Office (CSO) of the University of Illinois. Partial

computer service funding was provided by the Research Board of

the Graduate College of the University of Illinois.

The authors wish to thank Professor D.A.W. Pecknold for his

constructive comments throughout the study. The authors are

also grateful for the assistance provided by the CSO Systems

Consulting staff and to Dr. J. M. Nau for providing the computer

programs for the spectrum plots and the plotting programs for

the time history response.

CHAPTER

1

2

3

4

5

iv

TABLE OF CONTENTS

INTRODUCTION. • • • • • • • 1.1 Introductory Remarks. 1.2 Object and Scope. • ••• 1.3 Review of Previous Research •• 1.4 Notation. • • •••••••

MECHANICAL MODEL. 2.1 Introduction •• 2.2 Development of Model •

2.2.1 Columns and Beams • 2.2.2 Wall. • •••

2.3 Assumptions ••

MATERIAL MODEL ••• 3.1 Introduct ion • • • • • 3.2 Frame Material Model 3.3 Joint Material Model 3.4 Gap Model ••••••. 3.5 Wall Material Model.

ANALYTICAL PROCEDURE. • • • • • . 4.1 Introduction •••• 4.2 Element Stiffness Matrix ••

4.2.1 Column and Beam Element.

• • e, •

'. .

Page

• 1 • 1 · 5 • 6

· . .11

.13

.13

.15

.15

.16

.18

.21

.21 · • .21

• .23 · • .24

.24

.27 • .27

.27

.27 4.2.2 Wall Element. • • •••.••• .28

.28

.29 4.2.3 Joint Element. 4.2.4 Gap Element ••••

4.3 Dynamic Analysis •• ••••.••• .30 4.3.1 Mass Matrix. • •••. 4.3.2 Damping Matrix. • ••• 4.3,.3 Stiffness Matrix •• 4.3.4 Ground Influence Vector.

4.4 Equations of Motion and Solution • • 4.4.1 Equations of Motion. 4.4.2 Integration ••• 4.4.3 Correction •••••••••

4.5 Computer Implementation.

COMPUTATIONS AND RESULTS •••

.31

.31

.35

.36 . • • • .38

.38

.39

.41 • • • • • .43

..46 5.1 Introduction •••••• 5.2 Proposed Computations ••

• • • • • • • • .46

5.3

5.2.1 Input Data •••• 5.2.2 Base Acceleration and Spectrum. Results •••••••

.46 · .48 · .50

.51 5.3.1 5.3.2 5.3.3 5.3.4 5.3.5

Open Frame. • • • Frame Wall ••••

. . . . . . . . . . .52

Frame Wall with Gaps(I) Frame Wall with Gaps(II) •• Three Story Frame Wall •••

.54 • • • . .61

.68

.73

v

Page

5.4 Discussion of Results and Design Implications ••• 75

6 SUMMARY AND CONCLUSIONS • 6 .1 Summary. • • • 6.2 Conclusions •••

TABLESe ••

FIGURES •

APPENDIX

A FAILURE SURFACE FOR MASONRY • •

B JOINT ELEMENT STIFFNESS MATRIX.

LIST OF REFERENCES ••

• .81 .81 .83

.85

.99

• 179

• • 183

• 186

vi

LIST OF TABLES

Table Page

5.1 Frame Data. • .86

5.2 Wall Data • • • .87

5.3 Joint Data. • . .87

5.4 Failure Surface Data. • • .88

5.5 Key Response Quantities and Events: Open Frame . · · .89

5.6 Key Response Quantities and Events: Frame Wall · .90

5.7 Key Response Quantities and Events: Frame Wall-Gaps(I) . . . . . . . . . . . . . . . . · · .91

5.8 Key Response Quantities and Events: Frame Wall-Gaps(II) • • • •• • ••• • •• 92

5.9 Frequencies - Mode Shapes: Three Story Structure · · .93

5.10 Key Response Quantities and Events: First Story. · · .94

5.11 Key Response Quantities and Events: Second Story · · .95

5.12 Key Response Quantities and Events: Third Story. · · .96

5.13 Response Quantities I: One Story Structure . · .97

5.14, . Response Quantities II: One Story Structure • · · .98

vii

LIST OF FIGURES

Figure Page

2.1 Schematic Representation of the Wall Model •••••• 100

2.2 Infilled Frame Specimen, Analytical Model Superimposed • • • • • • • • • • • • 101

2.3 Infilled Frame Specimen, Analytical Model Superimposed • • • • • • • • 102

2.4 Infilled Frame Specimen with Opening, Analytical Model Superimposed • • • • • • . • • 103

2.5 Idealization of a Three Story Frame Infill Wall ••• 104

3.1 Bilinear Moment Rotation Relation for the Columns and Beams • • 105

3.2 Joint Element Failure Surface • 106

3.3 Masonry Wall Failure Surface in Principal Stresses •• 107

3.4 Uniaxial Tensile Strength of Masonry vs Angle of Principal Stress with Respect to the Mortar Joints •• 108

3.5 Uniaxial Compressive Strength of Masonry vs Angle of Principal Stress with Respect to the Mortar Joints •• 109

5.1 Overall Dimensions of the Structure 110

5.2 Nodal Numbering Scheme ••. • 111

5.3 Frame, Wall and Gap Element Numbering Scheme. • 112

5.4 Joint Element Numbering Scheme. • 113

5.5 Elastic Spectrum for Sinusoidal Base Motion, 1 sec Duration, at 2% and 20% Damping • • • • • • • • 114

5.6 Relative Displacement, Velocity, and Absolute Acceleration of the Open Frame. • • • • • •• •• 115

5.7 Relative Displacement, Total Resistance, and Base Acceleration of the Open Frame. • • • • • • 116

5.8 Base Moment, Base Shear, and Axial Load for the Lef t Co 1 umn of the Open Frame • • • •• • • • 117

viii

Page 5.9 Center Shear, Top Moment, and Axial Load for

the Left Column of the Open Frame • • 118

5.10 Top Moment, Top Shear, and Axial Load for the Right Column of the Open Frame. • • ••• 119

5.11 Center Shear, Base Moment, and Axial Load for the Right Column of the Open Frame. • • • •••• 120

5.12 Load Displacement Response: Open Frame • 121

5.13 Relative Displacement, Velocity, and Absolute Acceleration of the Frame Wall •••••••••.•• 122

5.14 Relative Displacement, Total Resistance, and Base Acceleration of the Frame Wall • • •• • • • 123

5.15 Base Moment, Base Shear, and Axial Load for the Left Column of the Frame Wall • • • • • • • • 124

5.16 Center Moment, Top Moment, and Axial Load for the Left Column of the Frame Wall • • • ••• 125

5.17 Top Moment, Top Shear, and Axial Load for the Right Column of the Frame Wall. • • •• 126

5.18 Center Moment, Base Moment, and Axial Load for the Right Column of the Frame Wall. • • 127

5.19

5.20

5.21

5.22

5.23

5.24

5.25

Load Displacement Response: Frame Wall •

Cracking Sequence: Frame Wall from t=O.O to 0.155 sec.

Crack Pattern: Frame Wall at

Crack Pattern: Frame Wall at

Crack Pattern: Frame Wall at

Crack Pattern: Frame Wall at

Cracking Sequence: Frame Wall from t=0.275 to 0.555 sec

t=0.045 sec

t=0.060 sec

t=0.075 sec

t=O.110 sec

. . . . .

· · · · · ·

· · 5.26 Relative Displacement, Velocity, and Absolute

128

• 129

· · · · 130

· · 131

132

· · · · 133

· · · · 134

Acceleration of the Frame Wall-Gaps(I). • •• •• 135

5.27 Relative Displacement, Total Resistance, and Base Acceleration of the Frame Wall-Gaps(I). • • • • • 136

ix

Page 5.28 Base Moment, Base Shear, and Axial Load for

the Left Column of the Frame Wall-Gaps(I) • • • 137

5.29 Center Shear, Top Moment, and Axial Load for the Left Column of the Frame Wa11-Gaps(I) •••• 138

5.30 Top Moment, Top Shear, and Axial Load for the Right Column of the Frame Wal1-Gaps(I) •••••• 139

5.31 Center Shear, Base Moment, and Axial Load for the Right Column of the Frame Wal1-Gaps(I) ••• • 140

5.32 Load Displacement Response: Frame Wa1l-Gaps(I) ••• 141

5.33

5.34

5.35

5.36

5.37

5.38

Cracking Sequence: from t=O.O to 0.315

Frame Wal1-Gaps(I) sec •

Crack Pattern: Frame Wa1l-Gaps(I) at

Crack Pattern: Frame Wall-Gaps(I) at

Crack Pattern: Frame Wa1l-Gaps(I) at

Cracking Sequence: Frame Wal1-Gaps(I) from t=0.145 to 0.180 sec . . . . . Cracking Sequence: Frame Wal1-Gaps(I) from t=O.425 to 0.535 sec . . . . . .

t=0.095

t=O.100

t=O.110

. . . .

. 5.39 Relative Displacement, Velocity, and Absolute

• 142

sec · · 143

sec · · 144

sec · · 145

. . · · 146

. . · · 147

Acceleration of the Frame Wal1-Gaps(II) • • • •• 148

5.40 Relative Displacement, Total Resistance, and Base Acceleration of the Frame Wal1-Gaps(II) •• 149

5.41 Base Moment, Base Shear, and Axial Load for the Left Column of the Frame Wa11-Gaps(II) •••••• 150

5.42 Center Shear, Top Moment, and Axial Load for the Left Column of the Frame Wall-Gaps(II) •••••• 151

5.43 Top Moment, Top Shear, and Axial Load for the Right Column of the Frame Wall-Gaps(II) ••••• 152

5.44 Center Shear, Base Moment, and Axial Load for the Right Column of the Frame Wall-Gaps(II) • • • 153

5.45 Load Displacement Response: Frame Wall-Gaps(II) ••• 154

5.46 Cracking Sequence: Frame Wall-Gaps(II) from t=O.O to 0.260 sec ••••••••••.•••• 155

x

Page

5.47 Crack Pattern: Frame Wall-Gaps(II) at t=0.095 sec. · 156

5.48 Crack Pattern: Frame Wall-Gaps(II) at t=0.105 sec. · 157

5.49 Crack Pattern: Frame Wall-Gaps(II) at t=0.145 sec. · 158

5.50 Cracking Sequence: Frame Wall-Gaps(II) from t=0.290 to 0.455 sec . . . . . . . . . · · 159

5.51 Ground Motion for the El Centro Record, May 18, 1940, SOOE Component, Scaled to 0.4g ••••• 160

5.52 Elastic Spectrum at 2% and 20% Damping for the El Centro Record, May 18, 1940, SOOE Component, Scaled to 0.4g •••••••••••••••••••• 161

5.53 Relative Displacement, Velocity, and Absolute Acceleration: First Story. • • • • • ••• 162

5.54 Interstory Drift, Total Resistance, and Story Acceleration: First Story. • • • . • • • • • 163

5.55 Load Displacement Response: First Story. • 164

5.56 Relative Displacement, Velocity, and Absolute Acceleration: Second Story. • • • • .•• 165

5.57 Interstory Drift, Total Resistance, and Story Acceleration: Second Story • • • • • ••• 166

5.58 Load Displacement Response: Second Story •• • • 167

5.59 Relative Displacement, Velocity, and Absolute Acceleration: Third Story. • • . . • ••• 168

5.60 Interstory Drift, Total Resistance, and Story Acceleration: Third Story. • . • • 169

5.61 Load Displacement Response: Third Story. • • 170

5.62 Base Moment, Base Shear, and Axial Load for the Left Column: First Story. • • • •••• 171

5.63 Center ~~ent; Center Shear~ and Top Moment for the Left Column: First Story • • • • •••• 172

5.64 Base Moment, Base Shear, and Axial Load for the Right Column: First Story. • • • •••• 173

5.65 Center Moment, Center Shear, and Top Moment for the Right Column: First Story •••••••••• 174

xi

Page 5.66 Base Moment, Base Shear, and Axial toad

for the Left Column: Second Story ••••.••••• 175

5.67 Base Moment, Base Shear, and Axial Load for the Right Column: Second Story ••••••••• 176

5.68 Base Moment, Base Shear, and Axial Load for the Left Column: Third Story •••••••••• 177

5.69 Base Moment, Base Shear, and Axial Load for the Right Column: Third Story •••••••••. 178

1

CHAPTER 1

INTRODUCTION

1.1 Introductory Remarks

Infill walls have, for many years, formed an integral part of

buildings. Their use, as well as their size, shape, and material

composition have been largely based on their architectural

function. The use of these walls mainly as architectural elements

considerably simplified their structural design. This design was

usually done by the architect.

In the typical design process structural engineers seldom knew

where these partitions would go and since they served no

structural purpose, they were largely ignored in the design of

the main structural configuration. Yet engineers recognized the

fact that these partitions could influence the behavior of the

main structure especially when these walls were used to infill

the space between two columns. In order to take this effect into

account two basic assumptions were usually made about the

structural behavior of these walls or partitions:

1. The wall tended to behave as a short beam, mainly in

shear, being very stiff in its own plane and

relatively flexible normal to its own plane.

2. The wall was essentially a brittle element whose

lateral load capacity was small compared to the

2

lateral load capacity of the frame that surrounded

it.

The first of these assumptions was based mainly on the height

to width ratio of the wall which was usually less than one. The

second assumption was based on the material properties of the

wall which usually consisted of materials weak in tension and

shear. One of the.most widely used materials for these partitions

was, and still is, unreinforced masonry which has a very brittle

behavior in both tension and shear.

Although these assumptions seemed quite reasonable, they

ignored the confining effect that the frame had on the wall. This

fact, as will be seen, invalidated the assumptions and radically

altered the behavior of the frame that enclosed the wall. The

behavior of the frame-infil l-wal I system is relatively complex

and as yet not fully understood. Therefore, at the present time a

full explanation of its behavior cannot be given. Only the most

relevant and well understood facts that affect the frame wall

interaction will be discussed.

To understand the overall behavior of the frame-infill-wall

system one should look at the boundary conditions that the frame

imposes on the wall when a lateral load is applied. Before any

load is applied the infill wall may be assumed continuous with

the floor system while a gap or space may be assumed to exist

between the wall and columns that surround it. These assumptions

are based on widely used construction techniques for the

3

placement of these walls. As a first observation, these walls are

usually placed after the main structural elements have undergone

most of their initial dead load deformation. This usually

precludes any frame wall interaction due to the dead load of the

structure. In addition, as these walls are placed in the

structure a gap is usually left between the frame and wall to

allow for· ambient changes in both humidity and temperature.

Usually some material is placed in the gaps that exist between

the frame and wall. This material is usually weak in tension with

a relatively low modulus of elasticity.

Due to the existence of the gaps, the frame is initially free

to deform without any constraints. Had the frame been continuous

with the wall tension, compression, and shear stresses would

develop at their common boundary. This indicates that in regions

where tension would be developed, if the system were continuous,

the gap size would increase while in the areas of compression the

gaps either diminish or are totally closed creating an

interaction between the wall and frame.

The amount of contact between the frame and wall, if linear

elastic conditions are assumed, depends on the stresses at the

frame wall boundary and the amount of loading present. Properly

analyzed, the interaction between the frame and infill makes the

overall problem nonlinear even when a linear elastic, small

displacement theory is used. This nonlinearity may be explained

as follows. Boundary conditions imposed on the wall by the frame

4

depend to a great extent on the magnitude of the load while the

rigidity of the whole system depends on the boundary conditions

and contact stresses between the frame and wall. It is this

interdependence which makes the rigidity of the frame-infill-wall

system a function of the displacement.

As the load increases, additional nonlinearities may come into

playas the frame imposes a lateral load on the wall. Under

typical conditions the infill tends to behave as a short,

unreinforced masonry beam which usually results in a tensile

failure at the base of the wall. Under unconfined conditions the

wall would not be able to sustain any additional loading in fact,

the load level would drop, in spite of the fact that the wall is

virtually intact. When used as an infill the wall is not totally

free to move, the frame confines and constrains its movement and

usually forces the wall to change its mode of behavior. The wall

usually changes from a flexural-shear behavior to an

approximately axial compressive behavior across the diagonal of

the frame. At this point the wall has not only altered its

internal stress pattern but also its failure criteria from its

weakest mode in tension and shear to its most efficient form of

carrying load, compression. As a consequence of this change in

behavior, the infill wall, not only alters its own stress

pattern, but also modifies the fundamental frequency and behavior

of the frame that surrounds it.

5

The fact that the behavior of the frame may be substantially

modified by the presence of the infill points to the need of

studying this phenomenon both analytically and experimentally

under static and dynamic loads.

1.2 Object and Scope

One of the most severe loadings a frame-infill-wall system may

experience is that due to a large earthquake. There are numerous

examples of frames, although properly designed to resist

earthquake loading «hen acting alone, being severely damaged due

to the presence of infilled walls acting in the plane of the

frame. Therefore, to design rationally an infill frame against

earthquake loading the presence of these partitions should be

taken into account in the design process. To accomplish this goal

two basic steps should be taken:

1. Develop the techniques necessary to take explicitly

into account, during the analysis and design

process, the stiffening effect of these walls and

their localized effect on the frame that surrounds

it. These techniques should be developed so as to

use the same assumptions common to the design

process.

2. Under seismic loading, study the effect of these

partitions on the maximum distortions and forces in

the surrounding frame.

R ference Room Metz e f Illinois

University 0 ~ 13106 NC~

. street 208 N. Ro,m1.n~~ s 6180~

Urbana, Il.J,.in ~

6

This research will deal mainly with the second step just

outlined. Since the present design philosophy in earthquake

engineering is based on allowing inelast.ic deformations to

dissipate energy, the study of the frame-infill-wall system,

under seismic loading, should include the inelastic domain.

The main objective of this research will be to develop an

analytical model of the frame-infill-wall system to study its

behavior versus that of the open frame. The scope of the model

will include nonlinear behavior of the frame, wall, and their

interaction under seismic loading. The research should include

the effect of the frequency content of the earthquake, and the

effect of the wall on the columns and girders that surround it.

1.3 Review of Previous Research

The literature that deals specifically with the frame-

infill-wall system is extensive, dating back to at least 1957[8].

A review of all the available literature on the subject will not

be done here, for most of the literature does not deal directly

with the interaction problem. A good review of the different

aspects of the overall problem may be found in Refs.[31,41,42].

Only that part of the literature which has a direct bearing on

the development of the model will be reviewed herein.

Apparently the first attempt at studying the frame-infill-wall

system while taking into account the change in behavior of the

7

wall was made by Stafford Smith [58,59,60]. By studying a simple

one story frame with an infilled wall, Smith developed what could

be called the "Equivalent Strut Method". This method consists of

the replacement of the infilled wall with an equivalent diagonal

bracing element. The area and therefore the stiffness of this

element is dependent on the relative rigidity of the frame versus

the infilled wall. The area of the bracing element was determined

by multiplying the thickness of the wall by an equivalent width

of wall which acts in compression along the diagonal of the

frame. To arrive at this equivalent width Smith developed

formulas that related the equivalent width to the amount of

contact area that is developed at the frame wall boundary. This

formula was expressed in terms of the structural characteristics

and dimensions of both the frame and infill wall.

The advent of the finite element method allowed several

researchers to make a more detailed model of the infilled wall

and to specify with greater accuracy the boundary conditions

imposed on the wall by the frame. This step was especially

important for unreinforced masonry walls.

Storm [61] represented the wall as an assembly of bricks but

he did not represent the mortar joints between the bricks or

between the infill wall and frame as an independent set of

elements. This brought on the problem of continuity. Whenever the

stresses between the bricks exceeded the failure stress for the

joints the two degrees of freedom would become independent

8

forcing a reformulation of the whole problem. In spite of this

drawback the crack pattern of the model seemed to follow the test

results reasonably well not only in the masonry infilled panel

but also at its interface with the frame. From this study it was

evident that the crack pattern in the masonry required a

different approach to that normally used in the study of

concrete. The need for a different approach is due in part to the

inhomogeneity of masonry which tends to create preferred planes

of failure along its mortar joints.

Franklin [19] represented the wall as an assembly of elements.

He provided equivalent stiffness values to represent the masonry

and used link elements to represent the contact stresses and

compatibility between the frame and wall. He approached the

modeling of the wall by dividing it into elements which were

larger than the bricks but smaller than the whole wall. He also

provided a brittle-type material model for the wall element to

represent the masonry. The link elements provided the interaction

between the frame and the wall. The failure model seemed too

stiff, for the wall remained continuous with the frame almost up

to failure. This seems to be in contradiction with the expected

behavior of the material that is usually placed in the gaps

between the frame and infill. The possible difficulty might be

found in the tensile versus bond failure of this type of

material. The concept of the overall model provided a promising

approach to the handling of the frame-infill-wall interaction.

9

Page [46,47] developed a model for masonry. Modeling the brick

and joints individually, he overcame most of the drawbacks of the

model used by Storm. He represented the mortar by using joint

elements similar to those developed in rock mechanics and allied

disciplines. This model allowed for cracking along mortar joints

to occur without having the problem of compatibility. The

proposed model makes it easy to handle most of the cracking

problems in masonry. This is especially true when reverse loading

is applied. Page used the model to predict the behavior of

several test specimens. The results given by the model seem to

follow fairly well the behavior of the test and the crack pattern

which was produced. Of all the proposed models this one has the

greatest versatility with respect to loading and boundary

conditions, and is the most sophisticated model of a masonry wall

available in the present literature.

All of the previously cited references have the common factor

of dealing specifically with monotonic loading. In spite of this,

some of the models have been used as the basis for the

development of new models to study static reverse loading as well

as dynamic loading.

Klinger [31] used the diagonal strut concept to try to predict

the behavior of the frame-infill-wall system under static

reversed loading. Based on his own experiments Klinger developed

a load displacement curve under reversed loading for the

10

equivalent diagonal element. He encountered the problem that to

represent the wall under reverse loading two elements were

needed, one across each diagonal. This related the damage to one

diagonal element when loading in one direction to the damage

suffered by the cross diagonal in the previous loading cycle. The

finite element method made the handling of this problem difficult

due to the independence of the two elements. He solved the

problem by making assumptions on how the damage suffered by the

diagonal element in one direction affected the behavior of the

other diagonal element. The assumptions made were based on the

maximum displacements suffered by the structure.

Kost [33] studied the gaps between the frame and wall under

dynamic loading for structures which always remained elastic. To

carry out the study Kost created gap elements to monitor the

behavior of the space between the infill wall and frame. He based

the model on nodal compatibility of the frame and infill wall.

Whenever the frame and wall came into contact at a specific node

he provided a relatively stiff spring as a link to guarantee

compatibility between the frame and infill. This approach,

although approximate in nature, facilitated the computations

because the original set of equations at the beginning of the

solution applied throughout the whole process. It is not clear

from the work how the author determines when the gaps that are

closed become open or how he keeps track of the gap size. This is

an important question if the structure undergoes permanent

deformations. Although the author considered only elastic

11

structures this method showed the effect of the gaps on the

dynamic behavior of the frame-infill-wall system.

1.4 Notation

The symbols used in this study are defined in the text when

they first appear. These symbols are listed here to provide an

easy reference of all the symbols defined.

{a}

{*a}

[A]

{B}

[c]

c. 1

{*d}

{*d } 2

F. 1

f'

[K']

[K .. ] 1J

k. 1

constants for Rayleigh type damping

acceleration vector

vector of the increment in acceleration

dynamic matrix

dynamic load vector

damping matrix

modal damping corresponding to mode i

vector of the increment in displacement

vector of the increment in displacements of the

structure degrees of freedom

vector of the increment in displacements of the

boundary degrees of freedom

control point for the failure surface

compressive strength of masonry

tensile strength of masonry

number of dynamic degrees of freedom

reduced stiffness matrix

submatrix, where i and j may be 1 or 2

modal stiffness corresponding to mode i

[M]

m. 1

M Y

m

n. 1.

n

p. 1

[Q]

q

{res}

{R}

{TIF}

{TDF}

{TEF}

{TIR}

*t

{v}

{*v}

e

12

mass matrix

modal mass corresponding to mode i

yield moment

constant

normal stress control point in joint failure surface

damping ratio corresponding to mode i

constant

circular frequency corresponding to mode i

modal matrix, matrix formed by the mode shapes

a constant in Newmark's integration scheme

total residual force vector

ground influence vector

shear stress control point in joint failure surface

total inertial force vector

total damping force vector

total external force vector

total internal force vector

time interval, or integration time step

velocity vector

vector of the increment in velocity

angle with respect to the mortar joints at which the

principal stresses occur

13

CRAPTER 2

MECHANICAL MODEL

2.1 Introduction

The modeling of the frame-infill-wall problem may be

classified into two broad categories: the equivalent diagonal

strut method, and the finite element method. Each one of these

methods has advantages and disadvantages which make each one well

suited for specific but different conditions. It must be observed

that the equivalent strut method only provides an analysis of the

interaction between the frame and infill wall. Once this analysis

is carried out the strut method must rely on a more general

technique such as the finite element method to analyze the

structure as a whole. This division in the analysis

distinguishes the equivalent strut method from the finite element

technique.

The equivalent diagonal strut has the great advantage of

simplicity. Without increasing the number of degrees of freedom

used in the analysis of the structure, it provides a simple

method of taking walls into account. However, the indirect

provision for handling the frame-wall interaction with this

method does. lack versatility. Unless additional validation

studies are done,· the method has a limited application whenever

the conditions under which it was derived are considerably

14

altered. Any attempt at using this approach for dynamic analysis

immediately runs into the problem of providing an overall load­

displacement curve, under reverse loading, for the diagonal

element. The development of this curve may be quite difficult,

especially for masonry walls, and specifically when the boundary

conditions of the frame-infill-wall system change considerably

under load reversal.

The finite element

disadvantages

flexibility

interaction

of the

especially

with the

technique easily

strut method.

in the modeling

frame. With

overcomes most of the

It allows for great

of the wall and its

material behavior of its various

adequate

parts the

modeling of the

frame-infill-walI

system may be studied with greater accuracy and detail. This

allows for more flexibility in the study of its behavior under

varied loading patterns, boundary conditions and structural

forms. However, the versatility gained in using the finite

element method may be overshadowed by the need to use a large

number of degrees of freedom to model the masonry wall

adequately. In the specific case of dynamic behavior the number

of degrees of freedom could be excessive.

The advantages and disadvantages of the two approaches for the

modeling of the frame-infill-wall system suggest the need to

develop a model to study this problem under nonlinear dynamic

loads. This model should have the versatility of the finite

element method while reducing the number of degrees of freedom

needed to model the frame-infill-wall system.

15

2.2 Development of Model

The development of the model and its characteristics may be

divided into two areas for discussion. The first area deals with

the topological development, specifically with the representation

of the structural configuration and the problems related to it.

The second area deals with the material model development to

describe the behavior and characteristics ·of the materials that

make up the structure. The topological development will be

treated in this chapter and the material model will be described

in Chapter 3.

2.2.1 Columns and Beams

Columns and beams will be represented by line elements, placed

along the centerlines of the members. Column elements will have

moment, shear, and axial load capacity. The axial load, shear,

and moment will be interdependent with respect to the stability

of the element. Beams will have the same characteristics as the

columns except that the axial behavior may be constrained when

the problem being analyzed allows this approximation or requires

it. All inelastic behavior will be concentrated at nonlinear

hinges located at the end of the elements. These hinges have zero

length and allow for plastic rotation to take place.

16

2.2.2 Wall

The modeling of the wall may be divided into three basic

areas:

1. Boundary between frame and wall.

2. Cracking in the wall itself.

3. Uncracked behavior of the specific wall segment.

The boundary between the frame and wall will be modeled by two

types of elements which may be called "gap elements" and "joint

elements", respectively. They will reflect, as their names

suggest, the conditions that exist at the frame wall boundary at

any given time.

The gap elements will model the space that exists between the

frame and wall, usually found on the sides and top of the wall.

The element will keep track of the gaps and determine when the

frame and wall come into contact, forcing continuity of the frame

and infill wall at that point.

The joint elements will model the boundary between the frame

and wall where continuity is initially assumed. This element

permits the representation of mortar joints at the base of the

wall. It allows continuity up to a certain stress then permits a

change 1n behavior similar to the gap element.

The uncracked wall itself is represented as an assemblage of

triangular elements. Since the study will focus mainly on masonry

walls, cracking along mortar joints becomes an important factor.

17

A joint element is placed at the edge of each wall element to

approximately represent the cracking in the masonry wall. Each

wall element represents several bricks and joints. It is assumed

that all cracking in the wall is concentrated along the

boundaries of the wall elements where the joint elements are

located. The shape of the wall element is taken to be a triangle.

When representing a rectangular section of the infill wall, this

shape allows for cracking across the diagonals as well as along

the boundaries of the rectangle.

A schematic representation of the wall model as applied to a

specific wall section is shown in Fig. 2.1. This figure shows how

the three elements which have already been defined are assembled

to form the overall model for the wall. A more detailed

application of the model is illustrated in Fig. 2.2 through 2.4

taken from Ref.[IB]. These figures represent the development of

the cracking patterns in the three specimens which were tested to

their ultimate capacity. On these figures the proposed wall model

has been superimposed as a series of triangles. Two of the

specimens, Fig. 2.2 and 2.3, consisted of a simple frame infilled

by a masonry wall while the third, Fig. 2.4, consisted of a

simple frame infilled by a masonry wall with an opening in the

center of the wall. The numbering scheme found in each figure

represents the order in which the cracks appeared on the specimen

as the lateral load was applied. It may be seen that cracking and

separation along the boundary in the test specimen may be modeled

by the proposed gap and joint elements. The cracking in the wall

18

itself may be approximately modeled by the cracking along the

boundary of the wall elements using the proposed joint element.

As may be seen from the figures some of the real cracks, as

represented by the numbering scheme, coincide with the

predetermined crack pattern of the proposed model while other

real crack patterns may only be modeled approximately.

The use of joint elements to simulate cracking in the wall is

based on the need for practicality and simplicity. The treatment

of cracking as a continuum property within a finite element is a

difficult problem, especially when reverse loading is cons~dered.

This treatment 1S even more difficult when a material like

masonry is being modeled. Masonry tends to crack along its mortar

joints with little damage to the surrounding bricks. When the

load is reversed these cracks close and the masonry strength is

essentially unaffected. This fact makes the cracking of masonry a

difficult problem to model as a continuum.

2.3 Assumptions

The postulated models for the beams, columns, and walls may

now be assembled to form the frame-infill-wall structure. To

accomplish this step certain assumptions and requirements should

be made. These assumptions should allow as much flexibility as

possible in representing the structure and should enable the

application of dynamic loads.

19

The assumptions are:

1. Torsional effects are neglected; therefore the

representation of the structure is limited to the

plane of the wall.

2. The connecting points or nodes of the structure will

have a maximum of three degrees of freedom, two

translations which are mutually perpendicular and

one rotation.

3. The relative angle of the members meeting at a node

is constant throughout the analysis.

4. Masses may be specified for each degree of freedom.

S. All dynamic loads on the structure are induced by

boundary accelerations.

6. Only one boundary acceleration may be a specified

but several points may have said acceleration.

7. Small deformations are assumed in the analysis so

the equations of equilibrium may be based on the

initial configuration.

8. The instantaneous stiffness matrix and nonlinear

properties of the structure are assumed consta~t

during each time step.

9. Only kinematic boundary conditions may be specified.

10. The kinematic boundary conditions may be specified

independently of the point at which the external

acceleration is specified.

20

Based on these assumptions and requirements, Fig. 2.5 shows an

example of the modeling of a three story structure. In this case

masses were assumed concentrated at the story .level, and specific

gaps were assumed to exist between the frame and infill. To

simulate cracking in the wall,

along the boundaries of

acceleration may be specified

joint elements would be specified

the wall elements. The ground

for one of the joints at the base

of the structure. Then, to simulate an earthquake type loading,

all degrees of freedom for the base of the structure would be

specified to move simultaneously. Although a solution to a three

story frame-infill-wall system will be demonstrated, the solution

to the specific three story structure shown in Fig. 2.5 will not

be done.

21

CHAPTER 3

MATERIAL MODEL

3.1 Introduction

The material models presented in this chapter and the physical

model presented in Chapter 2 will constitute the complete

inelastic nonlinear model to be used in this study. Just like the

topological aspects of the model which were divided into parts

representing different aspect s of the structure, the material

model has similar subdivisions which depend on the part of the

structure being represented. The material model may be divided

into four areas:

1. The frame material model.

2. The masonry material model.

3. The joint material model.

4. The gap model.

The words ''material model" are used here not only in the

context of stress strain relationships but also, as in the case

of the frame and gap model, to signify a load displacement

relationship.

3.2 Frrume Material Model

The beam and column elements of the frame are assumed to have

a bilinear moment rotation behavior concentrated in the nonlinear

22

hinges at the ends of the element. Figure 3.1 shows a typical

moment rotation curve. This type of moment rotation curve is well

suited for steel structures but lacks the degrading stiffness

behavior found in concrete structures. Yet ~ts simplicity will

help in the understanding of the proposed model.

In the case of the frame-infill-wall, the variation of the

curvature over the length of the column is, in general,

nonlinear. This is due to the interaction between the frame and

wall. Since the change in rotation between the ends of a beam

element is the integral of the curvature over its length, the

moment rotation relationship will not be bilinear as is being

assumed in the material model. However, in modeling the frame­

infill-wall problem, the interaction between the frame and wall

requires that the columns and beams be subdivided into smaller

segments. Over each of these segments the linear variation of the

curvature as is used in the proposed model should be a better

approximation of the real curvature.

In the case of columns no inelastic material interaction

be~ween the axial load and moment is assumed. The axial load has

a simple elastoplastic model for material behavior which is

independent of the moment rotation model shown in Fig. 3.1. This

means that yielding in the flexural mode does not necessarily

cause yielding in the axial mode and vice versa. The only

interaction of the axial load and moment comes indirectly through

the stability of the element. Although not in accord with the

21

CHAPTER 3

MATERIAL MODEL

3.1 Introduction

The material models presented in this chapter and the physical

model presented in Chapter 2 will constitute the complete

inelastic nonlinear model to be used in this study. Just like the

topological aspects of the model which were divided into parts

representing different aspects of the structure, the material

model has similar subdivisions which depend on the part of the

structure being represented. The material model may be divided

into four areas:

1. The frame material model.

2. The masonry material model.

3. The joint material model.

4. The gap model.

The words ''material model" are used here not only in the

context of stress strain relationships but also, as in the case

of the frame and gap model, to signify a load displacement

relationship.

3.2 Frame Material Model

The beam and column elements of the frame are assumed to have

a bilinear moment rotation behavior concentrated in the nonlinear

22

hinges at the ends of the element. Figure 3.1 shows a typical

moment rotation curve. This type of moment rotation curve is well

suited for steel structures but lacks the degrading stiffness

behavior found in concrete structures. Yet its simplicity will

help in the understanding of the proposed model.

In the case qf the frame-infill-wall, the variation of the

curvature over the length of the column is, in general,

nonlinear. This is due to the interaction between the frame and

wall. Since the change in rotation between the ends of a beam

element is the integral of the curvature over its length, the

moment rotation relationship will not be bilinear as is being

assumed in the material model. However, in modeling the frame­

infill-wall problem, the interaction between the frame and wall

requires that the columns and beams be subdivided into smaller

segments. Over each of these segments the linear variation of the

curvature as is used in the proposed model should be a better

approximation of the real curvature.

In the case of columns no inelastic material interaction

between the axial load and moment is assumed. The axial load has

a simple elastoplastic model for material behavior which is

independent of the moment rotation model shown in Fig. 3.1. This

means that yielding in the flexural mode does not necessarily

cause yielding in the axial mode and vice versa. The only

interaction of the axial load and moment comes indirectly through

the stability of the element. Although not in accord with the

23

moment-axial load interaction of columns this approach of

separating the axial behavior from the moment behavior was chosen

so as to simplify the interpretation of results of the overall

model.

3.3 Joint Material Model

In the model proposed in Chapter 2 cracking of the masonry

wall may be assumed to occur mainly along the mortar joints. Due

to the relative dimensions of a typical mortar joint one may

assume that only two of the six stress components are not zero.

They are taken to be:

1. Normal stress acting perpendicular to the direction

of the largest dimension in the joint.

2. Shear stress acting along the direction of the

largest dimension in the joint.

The joint is assumed to have linear elastic behavior up to

failure. After failure only compression stresses may be carried

across the joint. It is assumed that no shear stresses may be

carried across the joint after failure but incremental shear

stiffness relationships may exist when normal compressive

stresses are present in the joint. The assumed failure surface

for the joint element is shown in Fig. 3.2. Tbis figure relates

the maximum shear

specific normal

stress that may be carried by the joint to a

stress level. This figure is based on

experimental results which are presented in Ref.[41] and

discussed in Appendix A.

24

3.4 Gap Model

A gap element is a function which measures a space. It is in

relation to this function or measure that the material model may

be specified. The material model of the gap element may be

defined in accordance with a load displacement relationship.

The basic property of the gap element is the size of the

original gap. If the material on both sides of the gap remain

elastic the original size of the gap is a constant and may be

used as such. This assertion is based on the fact that when all

loads are removed or brought to their initial state the structure

returns to its original configuration. If the material on either

side of the gap is able to undergo inelastic deformations the

original gap size is no longer a constant and must be updated at

every step of the computation.

Once the two points across the gap come into contact the gap

element problem becomes a surface contact problem. This aspect of

the behavior will be discussed in the next chapter.

3.5 Wall Material Model

The wall model proposed in Chapter 2 is assumed to be

homogeneous, isotropic, and linearly elastic up to failure.

Although masonry is not a homogeneous material this assumption is

needed to simplify the modeling of the masonry. The assumption of

25

linear elastic behavior is based on the experimental evidence

available for masonry. This evidence seems to show that unlike

concrete, masonry behaves linearly almost up to failure. The

treatment of nonisotropic materials is simple to handle under the

finite element formulation but the present understanding of the

behavior of masonry does not warrant the use of such a model.

The assumed failure surface for masonry is shown in Fig. 3.3.

The surface is drawn using the principal stresses as the system

of coordinates and the assumption of a plane stress condition for

the masonry. Due to the inhomogeneous character of masonry the

failure surface may be assumed to depend not only on the

principal stresses but also on the angle that these principal

stresses make with the mortar joints. To make the failure surface

dependent on the principal stresses and the angle at which these

are applied the functional shape of the surface may be assumed

invariant. Then the five control points F1 , F2 , F3, F4 , and FS

may be assumed to be functions of the angle with respect to the

mortar joints at which the principal stresses act.

The control points F1, F2, in the tension side of the failure

surface may be assumed to depend on the principal stress angle in

accordance with the following equations:

f en x sin2

t

2 e + cos 8)

(3.1)

M~tz Reference Room University of Illinois

BI06 NCEL 208 N. Romine Stre6~

Urbana, Illinoia 61Scrm

where

26

n = constant greater than 1

e = angle with respect to the mortar joints at which

the principal stresses occur

f t masonry tensile strength

A plot of these two equations is shown in Fig. 3.4. These

equations are based on experimental results which may also be

found in Ref.[41] and discussed in Appendix A.

The assumed functional dependence of the failure surface

control points F3 , and F4 , on the pr incipal stress angle is shown

in Fig. 3.S. These functions are also based on experimental

results. These results may be found in Ref. [22] and are also

discussed in Appendix A. The failure surface control point FS is

assumed to be equal to the maximum of F3 or F4 multiplied by a

constant. This constant is assumed to be less than one. Based on

the existence of a biaxial state of stress, confinement may be

expected to make the control point FS greater than F3 or F4

rather than less as is being assumed. Due to the lack of

available information it was deemed a better approach to make FS

at most equal to the greater of F3 or F4 •

27

CHAPTER 4

ANALYTICAL PROCEDURE

4.1 Introduction

In the previous three chapters a model for the frame­

infill-wall system has been described. This chapter deals with

the specifics of the model which include, the choosing of the

finite elements to represent the structure, the development of

the equations of motion, the techniques for numerical integration

of these equations and the development of the computer program to

implement the model.

4.2 Element Stiffness Matrix

In choosing the elements to represent the structural stiffness

it is desirable to choose elements that not only meet the

requirements of the previous chapters but are also as simple as

the model will permit. Each element stiffness constitutes a part

of the total structural stiffness. The assembly of the stiffness

matrix for the whole structure based on the element stiffness

presented in this section is covered in Sec. 4.3.3.

4.2.1 Column and Beam Element

The element chosen to represent the beams and columns of the

structure is the one developed by Giberson. The derivation of the

28

incremental stiffness matrix may be found in Ref.[20] and will

not be repeated here. To take into account the stability of the

element a geometric stiffness is added to the stiffness matrix.

This element was chosen because of its ability to have the

yield levels at each end specified independently of each other.

This characteristic may be needed because beams and columns

should be modeled as an assembly of shorter elements so as to

guarantee compatibility with the wall not only at the ends of the

columns and beams but also at intermediate points along their

length.

4.2.2 Wall Element

The element chosen to represent the wall is the constant

stress triangle. The derivation of the incremental stiffness

matrix for this element may be found in Ref.[68] and will not be

repeated here. This element was chosen for its simplicity and

widespread use. Although this element is usually considered to

give poor results when compared to other more sophisticated

elements it was deemed that a greater accuracy, at the present

time, may not be justified based on the present understanding of

some of the variables and assumptions made for the proposed

model.

4.2.3 Joint Element

Since most of the joint elements available in the literature

were deemed too complex for the present objectives of the study,

29

a simple element was derived. The details of the derivation of

the incremental stiffness matrix may be found in Appendix B.

The important characteristic of the joint element is that once

failure occurs incremental stiffness relations may exist only

when the stress across the joint is compressive. In addition only

when there is compression across the joint after failure will

there be an internal resistance developed across the joint. This

internal resistance will be due to the normal stresses only.

Incremental stiffness relations due to shear may exist after

failure only when there is a compressive normal stress, while the

internal resistance of the joint due to shear stresses is always

assumed to be zero after failure. The joint element has the

directional property of allowing joints to increase their

separation but not to decrease it. Therefore the joint element is

unidirectional and its incidences should be specified taking this

factor into account.

4.2.4 Gap Element

The stiffness of the gap element is zero as long as the gap

size is not reduced to zero. Once the gap is closed the problem

changes and it becomes a surface contact problem. In this study

the contact problem will be treated using an element which has a

change in stiffness when the gap closes.

Contact problems may be subdivided into three distinct phases.

These three phases may be denoted, "stick", "slip", and "gap

30

The stick mode is when points across the mode", respectively.

boundary have the same increments in displacements both

tangential to the surface. The slip mode is perpendicular and

when the increments in displacements tangential to the surface of

contact may be different while increments in displacements

perpendicular to the surface are the same. The gap mode is when

both normal and tangential increments in displacements across the

boundary are independent.

Whenever the slip or stick mode is the controlling phase, one

may derive a set of constraining equations for the increments in

displacements while the stresses across the boundary become

unknown. In this study this constraint is handled approximately

by placing a joint element between the two points which have come

into contact. By doi.ng this the constraint may only be satisfied

to within a certain order of magnitude which depends on the

material constants assumed for the joint element. The whole

process of gap detection and contact problem is placed under the

present gap element. The joint element used is the same as the

one derived in Appendix B but with the additional assumption that

the element has already reached the failure criteria. Therefore

the gap element is also unidirectional and its incidences must

also be specified taking this factor into account.

4.3 Dynamic Analysis

The equations of motion may be expressed as the sum of the

inertial forces, damping forces, structural forces, and external

31

loading for each degree of freedom. To evaluate the inertia,

damping, structural, and external forces one must calculate the

mass matrix, damping matrix, instantaneous stiffness matrix, and

the external influence vector for the whole structure.

4.3.1 Mass Matrix

In this study the lumped mass matrix technique will be used

exclusively. Masses may be lumped at any degree of freedom. This

generates a mass matrix whose off diagonal terms are all zero.

Since a mass may be lumped for any degree of freedom, rotational

as well as translational inertial forces may be present. The

overall structural mass matrix has the following form:

where

m1 0

[M] ,

(4.1) m. 1

0 ~

mr

[M] diagonal mass matrix

mi lumped masses at the dynamic degrees of freedom

I number of dynamic degrees of freedom

4.3.2 Damping Matrix

Linear viscous damping is adopted in this analysis because of

its mathematical simplicity and widespread use. Therefore the

damping forces are proportional to the velocities of the dynamic

32

degrees of freedom relative to the velocity of the boundary of

the structure. It was deemed desirable to have a damping matrix

which may have any or all of the following characteristics.

1. The damping ratio increases with the initial elastic

frequency.

2. The damping ratio decreases with the initial elastic

frequency.

3. The damping ratio may be specified arbitrarily for

any or all initial elastic frequencies.

The first two conditions may be accomplished by Rayleigh type

damping. The third condition may be achieved by transformation

using the natural mode shapes of the structure.

Rayleigh or proportional damping is expressed in the following

form:

where

[C]

[C] = damping matrix, same size as the mass matrix

[M] = mass matrix

[K'] reduced stiffness matrix

a1 ,a2 constants to be specified

(4.2)

Based on the orthogonality of the mode shapes with.respect to

the mass and stiffness matrix the damping matrix as defined in

Eq. 4.2 may be diagonalized. Using this property the following

33

equation is obtained:

or

where

C. l

a • 1

c. = modal damping corresponding to mode i l.

mi modal mass corresponding to mode i

k i = modal stiffness corresponding to mode i

n. = damping ratio corresponding to mode i 1

p. = circular frequency corresponding to mode i 1

(4.3)

(4.4)

The constants a1 and a2

in Eqs. 4.3 or 4.4 may be evaluated by

specifying one or two values of n. or c. 1. l.

depending on which

formula one uses. Given two values of ni then a 1 and a 2 may be

determined as follows:

where

2 p.p. (n.p. - n.p.) l J J l l ]

2 2 p. - p.

l J

2(n.p. - n.p.) l l J J

(4.5)

(4.6)

34

ni damping ratios specified

Pi = circular frequencies specified

Usually the damping ratios specified are those corresponding

to the first and second elastic frequencies of the structure. If

the damping matrix is considered proportional only to the

stiffness matrix then the damping ratio will increase with

frequency. If the damping matrix is considered proportional only

to the mass matrix then the damping ratio will decrease with

increasing frequency.

A different approach must be taken to provide an arbitrary

damping ratio for each frequency. Using the mode shape matrix as

an inverse transformation one may specify the damping ratios as

follows:

where

[C]

[C] = damping matrix

[Q] = modal matix, matrix formed by the mode shapes

c. = damping constant for mode i 1

(4. 7)

It is seen from Eq. 4.7 that only those modes shapes whose

values of c i are nonzero will be damped the rest of the modes

will have no damping unless proportional damping is specified. A

more detailed derivation of Eq. 4.7 may be found in Ref.[12]. The

damping matrix is assumed to remain constant.

35

4.3.3 Stiffness Matrix

The complete incremental stiffness matrix is formulated by

assembling all of the individual incremental stiffness matrices

from the element level. The full incremental stiffness matrix may

include both dynamic degrees of freedom and static degrees of

freedom. Therefore not all of the structural degrees of freedom

in the stiffness matrix may have corresponding damping and

inertia forces related to them.

In the usual approach the degrees of freedom for which no

masses have been specified are condensed out of the stiffness

matrix. In this study the need to have the displacements of all

the degrees of freedom readily available and the fact that, ~n

this case, no great computational advantage would be gained the

usual procedure of condensation was not followed. Therefore the

structural stiffness is as follows:

[ [Klll [K12l]

-*d I

'<d: J [K] {*d} (4.8)

[K21

] [K22

]

where

[KI1

] submatrix size I x I

[KI2 l submatrix size I x J

[K21 1 submatrix size J x I

[K22

l submatrix size J x J

36

I = number of dynamic degrees of freedom

J = number of static degrees of freedom

{*d } 1 = increment in displacements of the dynamic degrees of

freedom

{*d } 2 increment in displacements of the static degrees of

freedom

The instantaneous stiffness matrix may change, under inelastic

conditions, at every time step. This requires both a frequent

updating of the stiffness matrix and the evaluation of a residual

force vector which may be induced in the structure due to the

changes in the stiffness of the elements. This residual force

vector will be discussed in Sec. 4.4.3.

4.3.4 Ground Influence Vector

The ground influence vector is the set of structural

displacements induced in the structure when one applies a unit

displacement at one boundary point with zero displacement at all

other boundary points of the structure. For most structural

systems this vector may be derived from a consideration of

statics but in complex structures a more general approach is

needed.

Taking first the stiffness matrix of the structure times the

increments in displacements and setting them equal to zero

yields:

where

37

r -; - ! *d

1 [K] {*d}

l [Kll] [K12 ] j [:] (4.9)

[K21 ] [K22 ] *d 2 L. _J

submatrix corresponding to the structure degrees of

freedom

[K22 ] = submatrix corresponding to the boundary degrees of

{*d } 1

{*d } = 2

freedom

submatrix corresponding to the relation between

boundary degrees of freedom and structure degrees

of freedom

increment in displacements of the structure degrees

of freedom

increment in displacements of the boundary degrees

of freedom

Now applying the above definition one arrives at the following

equation:

(4.10)

This equation relates the boundary degrees of freedom to the

structure degrees of freedom. In the present study only one

boundary point may have a defined ground motion and Eq. 4.10 is

simplified to:

{*d } 1

*d • {R} 2

(4.11)

38

where

{R} = ground influence vector defined as -[K11 ]-1[K12

]

The ground influence vector will permit the equations of

motion to be written in terms of the relative displacements of

the structure. In general the influence vector depends on the

displacements of the structure as may be seen from Eq. 4.10. This

implies that under inelastic conditions it should be updated at

every time step. In the particular case that the ground influence

vector represents a rigid body mode, the vector becomes

independent of the displacements and a constant for all times.

The previous derivation may also be found in Ref.[12].

4.4 Equations of Motion and Solution

4.4.1 Equations of Motion

The equations of motion are stated in the incremental form

assumming that the properties of the structure are constant

within each time interval:

[C]

a

[

[M] 0·' [R .... a 0 a

..J

• *a g (4.12)

where

39

[M] = diagonal mass matrix

[c] damping matrix

[KII ] instantaneous structural stiffness matrix for the

dynamic degrees of freed·om evaluated at the end

of the previous time step

instantaneous structural stiffness matrix for the

static degrees of freedom evaluated at the end of

the previous time step

[K I2 ] = instantaneous structural stiffness matrix for the

{R}

*a *v *d 222

coupled degrees of freedom evaluated at the end

of the previous time step

ground influence vector

incremental accelerations, velocities, and

displacements of the dynamic degrees of freedom

incremental accelerations, velocities, and

displacements of the static degrees of freedom

*a incremental boundary acceleration g

4.4.2 Integration

To integrate the equations of motion the numerical scheme

developed by Newmark will be used. In Ref.[68] this method was

shown to be the most general second order integration scheme that

may be used to integrate the equations of motion. The incremental

acceleration and velocity are given as:

{*a} 1 {a} 2q (4.13 )

where

40

1 .(- - 1) (*t) {a} 4q

(4.14)

*t = time interval, or integration time step

q a constant which is usually chosen between 1/6 and

1/4 and controls the stability, convergence, and

accuracy of the integration scheme

{a} acceleration at the end of previous time step

{v} velocity at the end of previous time step

{*a} increment in acceleration

{*v} increment in velocity

{*d} increment in displacement

There are two basic ways to solve the equations of motion. One

is termed the explicit method and the other the implicit method.

The implicit method either assumes or calculates a value of the

,acceleration and then using the assumed relations between the

displacements, velocities, and accelerations integrates for the

velocities and displacements. The explicit method combines the

assumed relations between acceleration, velocity, and

displacements with the equation of motion to obtain an equation

from which the increments in displacements may be calculated

directly. In this study the explicit approach will be used.

Combining Eqs. 4.12, 4.13, and 4.14, the increment in

displacements may be expressed as a function of the response

values and stiffness properties at the end of the previous time

step. This is done as follows:

where

[A]

41

1 1 2 [M] + 2q(*t) [C] + [K]

q U,t)

1 1 1 [M][2q{a} + q(*t){v} - {R} • agl + [C][(4q ~ l)(*t){aJ

+ 21q {v}]

(4.15 )

(4.16 )

(4.17)

This approach has the advantage that if the stiffness

properties do not change over several time steps the inversion of

matrix [A] has to be done only once. In this study the value of

"q" will be taken as 1/4. This makes the solution scheme

unconditionally stable.

4.4.3 Correction

Since the conditions used to establish Eq. 4.15 are those

existing at the end of the previous time step, any change in the

properties of the structure during the present time interval are

not included ~n the equation. This implies that an iterative

solution process would be needed to arrive at the values of the

acceleration, velocity, and displacements. This approach may be

costly for the proposed model as Eq. 4.15 would have to be solved

several times in every time step. Therefore the iterative

approach will not be used in this study. Yet without an iteration

process errors can be shown to accumulate and considerably alter

the solution of the problem. Therefore some correction should be

applied to the response.

42

In this study the following correction is used: at the end of

each time step a total equilibrium equation is established. The

total inertial force must be in equilibrium with the total

damping force, total internal resistance of the structure, and

the total external force. If any changes in the structural

properties occurred during the time interval this

equilibrium equation would not be satisfied. This creates a

residual force which is added as an external load in the next

time interval. The residual load is formulated as follows:

where

{res}

{TIF}

{TDF}

{res} {TIF} + {TDF} + {TIR} + {TEF} (4.18)

total residual force vector of the structure

total inertial force vector of the structure

total damping force vector of the structure

{TEF} total external force vector of the structure

{TIR} = total internal force vector of the structure

The method just outlined has the advantage of making the

updating of the structural stiffness matrix independent of the

corrections which must be carried out at each time step. This is

accomplished by the use of the internal force vector which may be

calculated from the material models being used for the structure.

43

4.5 Computer Implementation

The model proposed in the previous sections was implemented

using a computer program called AWALL. The program AWALL is a

finite element program with the following capabilities.

The program has a main storage block where most of the data is

stored column-wise and in sequence by the use of pointers. All

computations are done through subroutines. The dynamic stiffness

matrix is stored in banded form with only half the band width

plus the diagonal being stored. At each joint three global

degrees of freedom are specified, two perpendicular translations,

and one rotation. The solution is in the plane of the structure

and no out-of-plane degrees of freedom may be specified. One may

specify a concentrated mass at each degree of freedom. The

solution has two time sequences which may be specifi~d: the time

sequence over which the ground motion is applied and the free

vibration sequence. Each may be specified independently. The

number of times the stiffness matrix is updated may be

specified. If the number of time steps to update the stiffness

matrix is greater than the total number of time steps available

for the solution then the structure is assumed to be elastic.

The structure is assumed to be initially unconstrained when

one specifies the joint coordinates and element incidences. One

may specify two types of constraints, absolute and relative. The

absolute constraints are those for which the degree of freedom is

either always zero or does not apply to the specific joint. The

44

relative constraints are those that force two or more degrees of

freedom to have the same response. If any of the degrees of

freedom in a relative constraint is in fact a degree of freedom

with an absolute constraint then all degrees of freedom with said

relative constraint will have an absolute constraint.

The program may generate three different types of ground

motions or it may 'read a specified input ground motion. The three

types of ground motions which the program generates may be

separated into two consecutive pulses with different scaling

factors for time and acceleration in each pulse Ref.[65]. The

basic pulse for each of the three ground motions it generates are

sinusoidal, linear, or quadratic. If a specific the ground motion

is read a linear interpolation is assumed between the points.

This condition may also be scaled in both time and acceleration.

The ground motion to be used may be specified for any degree of

freedom in the structure but only one degree of freedom may be

specified. At the present time the acceleration is generated and

stored before it is used and therefore this may require a large

amount of storage if the integration time step is small.

Based on the degree of freedom specified for the ground motion

the program generates a ground acceleration influence vector for

the masses of the structure. This vector is assumed constant

throughout the computation. In general, this will not effect the

results if said vector corresponds to a rigid body mode. Static

loads may be specified at any degree of freedom. No element loads

45

may be specified. The dynamic response is computed from the

equilibrium position generated by the static loads. All material

stress computations are based on the total displacements which

may include the static condition if static loads are present.

The program calculates all elastic frequencies and mode shapes

for the system but prints only the first ten. Damping may be

specified in two ways: Rayleigh damping and damping for a

specific mode shape. For Rayleigh damping one may specify the

mode shapes to determine the constants of proportionality. For

arbitrary damping of a specific mode shape one must specify a

damping ratio for each mass degree of freedom. Both types of

damping may be specified in a given problem and they will be

additive. The dimensions of the data provided must be consistent

as the program does not provide any transformations.

46

CHAPTER 5

COMPUTATIONS AND RESULTS

5.1 Introduction

The objective of this chapter is to assess the model proposed

in the previous four chapters. The first obstacle encountered is

the lack of experimental data to compare the results of any

computations that might be proposed. Only a limited number of

experiments under dynamic loading have been carried out and

although the results of these are very instructive in their

overall content they lack sufficient data to make a meaningful

comparison with the proposed analytical model. This lack of data

coupled with the nonlinearities of the problem suggest the need

to study the model on a simple frame wall system whose overall

behavior is sufficiently well understood. Under these conditions

the ground motion should also be simple so as to provide a

spectrum which is relatively smooth. This should permit an easier

interpretation of the results as nonlinearities will already make

the interpretation difficult.

5.2 Proposed Computations

In order to assess the proposed model four basic structures

are studied. The first is a one bay, one story frame which will

be used as a bench mark for the other three structures. The

47

second through fourth structure consists of the same frame as the

first structure to which a masonry infill wall is added. The

second structure will assume initial continuity between the frame

and wall, but the third and fourth structures have a specified

gap at the frame wall boundary. The third and fourth structures

will have different failure capacities specified for the masonry

wall.

In comparing these four structures the capabilities of the

model and the difference in behavior between the open frame and

the frame-infill-wall system should become clear. In addition the

effect of continuity at the frame wall boundary may be studied by

comparing the results of the second structure for which

continuity has been specified to the results of the third and

fourth structure for which a gap is specified.

In addition to the four basic structures a three story one bay

frame-infill-wall will be studied. The first story of the

structure will be essentially the same as the previous four

structures. The second and third stories will have a simpler

discretization as follows. The wall for the second and third

stories will be subdivided into only four elements while the

columns and beams will be made up of only one element and not two

elements as shown in Fig. 5.1. This structure will have a

specified gap size of zero but will not be assumed continuous

with the main frame. In addition the structure will be subjected

to a different ground motion than the previous four structures.

48

5.2.1 Input Data

The overall dimensions of the frame wall system to be used for

the first four structures is shown in Fig. 5.1. The span to

height ratio of the frame was taken as 2. This is an average

value for typical frames whose span to height ratio vary between

1.5 to 2.5. The height of the column, taken as 3.5 m, is also

believed to be typical. The size of the gap at the frame wall

boundary was taken as 5 mm. The space between the wall and the

ground shown in the figure is not a gap but signifies that the

wall may become discontinuous with the base during the loading

cycle. The masses have been specified at the three joints in the

beam with only the horizontal degree of freedom being active.

They are tied by a rigid link providing only one dynamic degree

of freedom 1n the horizontal direction. The value of the mass was

specified as 50 Mg making the total mass in the story 150 Mg.

This value was chosen based on the response spectrum of the

ground motion to be used for the first four structures. The

overall viscous damping was taken as 2.0 percent of critical. The

modulus of elasticity for the frame was taken as 25 GPa. The area

of the column was taken as 123 x 103 mm2 and the moment of

inertia was taken as 1.25 x 10 9 mm4. The area of the beam was

taken as 245 x 103 mm2 and the moment of inertia was taken as 5 x

109 mm4. The yield moment was taken as 214.0 kN-m for both the

beam and column. The slope of the yield branch in the moment

rotation relationship was taken as 1.0 percent of the original

stiffness for both the beam and column. The dead load was

49

concentrated at the corner masses. It was specified as 245 kN

each. The data for the frame is summarized in Table 5.1.

The wall was divided into 16 elements as this is the ~allest

number of elements that may reflect the bracing effect on the

surrounding columns. The modulus of elasticity of the masonry was

taken as 14 GPa. The equivalent thickness is based on an

ungrouted 200 mm brick and was taken as 100 mm. Poisson's ratio

was taken as 0.25. The masonry uniaxial compressive strength was

taken as 24.0 MPa for the third structure and 34.0 MPa for the

fourth structure; the masonry uniaxial tensile strength was taken

as 2.4 MFa for the third structure and 3.4 MPa for the fourth

structure. The dead load was distributed in proportion to the

area of the wall. It was specified as 1.25 kN at each independent

node 1n the wall. The data for the wall is summarized in Table

5.2.

A joint was specified at the boundaries of the wall elements

and at the base of the wall. Joint elements were specified at the

frame wall boundary for the structure where initial continuity

was assumed to exist between the wall and the frame. The modulus

of elasticity was taken as 7 GPa. The thickness of the joint was

taken as 10 mm and Poisson's ratio was taken as 0.15. The area of

the joint depends on its location and varied from 87.5 x 103 mm2

to 175 x 103 mm2 • The data for the joint element is summarized in

Table 5.3.

50

The control points for the different failure surfaces were

specified as follows. In Fig. 3.2 the value of sl and 01

were

given as 1360 kPa and 340 kPa, respectively "and the two slopes

"n" and ''mil were specified as 2 and 1, respectively. In Fig. 3.4,

part of the control points for Fig. 3.3, the value of lin" was

assumed as 1.5; io Fig. 3.5 the value of "n" was assumed as 0.42.

These values are summarized in Table 5.4.

For the first four structures Fig. 5.2 shows the joint

numbering corresponding to Fig. 5.1; Figs. 5.3 and 5.4 show the

element numbering for the same figure. The time step used for

integration of the equations of motion was taken as *t=O.OOs sec.

The data for the three story structure is essentially the same

as the data for the previous four structures with the notable

exception of the mass specified at each story. The mass for the

three story structure was taken as 30 Mg for each story. This

value is also presented in Table 5.1.

5.2.2 Base Acceleration and Spectrum

The input base acceleration used for the first four structures

was taken as a sine wave with a period of 0.5 sec. The total

duration of the base motion was taken as 1.0 sec. This gives a

sinusoidal motion which has two complete sine waves. The peak

acceleration for the ground motion was taken as 0.4g. The

spectrum for this base motion at 2 and 20 percent of critical

51

damping is shown in Fig. 5.5. As would be expected the spectrum

has a peak value at approximately 2 Hz (0.5 sec); a second

maximum occurs at about 0.5 Hz (2.0 sec). At very high and low

frequencies the spectrum tends to the maximum ground acceleration

of 0.4g and the maximum ground

respectively.

displacement of 312 mm,

The input base acceleration used for the three story structure

consisted of the first ten seconds of the South-East component of

the Imperial Valley earthquake recorded at El Centro California

on May 18, 1940. The ground acceleration was scaled such that the

maximum acceleration was 0.4g. In addition a 2 sec prefixed pulse

was added at the beginning of the ground motion. The complete

ground motion is shown in Fig. 5.51. The elastic spectrum for the

ground motion at 2 and 20 percent damping is shown in Fig. 5.52.

5.3 Results

The results of the computations are presented in three

different forms. The first is a time history response of the

system, which will include plots of the relative displacement,

relative velocity, and absolute acceleration of the mass of the

system. Time history plots of the moment shear and axial load of

the columns will be presented for comparison between the

different structures. In addition, a time history plot of the

relative displacement, total resistance, and base acceleration

will be included. The second method for presenting the results is

in the form of a load displacement plot for the mass of the

52

system. The third method of presenting the results will be in the

form of sketches of damage sequences for the structure. This last

method will only be presented for the frame-infill-wall model and

not for the open frame.

5.3.1 Open Frame

The results of the computations for the open frame are

presented in Figs. 5.6 through 5.12. Figures 5.6 through 5.11 are

time history plots and Fig. 5.12 is the load displacement curve

for the time history response. A brief summary of the key events

that occurred during the response is presented in Table 5.5.

The elastic fundamental frequency of the open frame is 1.53

Hz. Looking at the spectrum for the base acceleration in Fig. 5.5

this sets the open frame just to the left of the maximum response

at 2.0 Hz. As the frame yields the equivalent frequency should

decrease while the equivalent damping increases forcing the

response to move away from the maximum value at 2.0 Hz. This

observation with respect to the open frame is important in

interpreting some of the overall results for the frame wall

system. The relative position of the initial frequency with

respect to the peak value of the spectrum at 2.0 Hz may serve as

an approximate guide for this comparison.

Figure 5.6 shows the time history plots of the relative

displacement, the relative velocity and absolute acceleration. As

expected for a bilinear system, the total acceleration reaches a

53

maximum value in accordance with the yield resistance of the

frame. The relative displacement has a maximum value of 77.3 mm

and the total acceleration has a maximum value of O.188g. Figure

5.7 shows the time histories for the relative displacement total

force and ground acceleration.

Figures 5.8 through 5.11 correspond to the time history

response of the columns. The shear and moment presented in Fig.

5.8 correspond to the base of the left column while the shear and

moment in Fig. 5.9 correspond to the top of the same column. The

shear and moment presented in Fig. 5.10 correspond to the top of

the right column and the moment and shear presented in Fig. 5.11

correspond to the bottom of the same column. The time history

response of the axial behavior for each column is presented in

each figure, respectively.

Figure 5.12 shows the load displacement response for the open

frame. The response has three branches: the first branch is the

elastic behavior. The second branch corresponds to yielding at

the bottom of the columns while the third branch corresponds to

full yielding in the frame at the top and bottom of the columns

as well as yielding in the beams. Figure 5.12 shows some erratic

behavior in the vicinity of the yield point. This is due mainly

to the fact that the force that is plotted is the one at the end

of the time step to which no corrections for yielding have been

added. When a typical overshoot of the yield point occurs this

value is corrected in the next time step since no iteration is

used to correct the error in the same time step.

54

Figures 5.6 through 5.12 show, as should be expected, a

response which approximates the behavior of a typical one story

ductile frame. These results will serve as a basis for comparison

with the three frame wall solutions to be" presented in the

following sections.

5.3.2 Frame Wall

The response of the frame wall system for which no gaps have

been specified is shown in Figs. 5.13 through 5.25. Figures 5.13

through 5.19 have a correspondence with Figs. 5.6 th~ough 5.12 of

the open frame. Figures 5.20 through 5.25 represent the cracking

pattern for the frame wall. A brief summary of the key events

that occurred during the response is presented in Table 5.6.

Two types of comparisons will be made. The first will be a

qualitative comparison of the time history responses of the frame

wall to those of the open frame; and of the corresponding load

displacement curves. The second will be a qualitative correlation

of the time history responses with both the cracking pattern

shown in Figs. 5.20 through 5.25 and the load displacement curve

of Fig. 5.19.

The elastic fundamental frequency of the frame wall system is

approximately 13 Hz. The elastic fundamental frequency of the

open frame was 1.53 Hz which gives a ratio of frequencies of the

wall to the frame of 8.6. Since the total mass is unchanged the

55

stiffness must change in proportion to the square of the

frequency ratio. Therefore the frame wall structure is 74 times

stiffer than the open frame. This very large increase in

stiffness is in accord with what has been observed

experimentally. As long as the frame is in full contact with the

wall large increases in the stiffness may be expected.

Looking at the spectrum in Fig. 5.5 it may be seen that the

frequency of the frame wall system,

the right of the maximum response

13 Hz, falls considerably to

which occurs at 2 Hz. This

shift in frequency results, in this case, in a reduction of the

maximum response quantities. As the equivalent frequency

decreases with the damage suffered, the system should tend to

move towards the maximum response but at a higher damping ratio

as the equivalent damping should increase with the damage.

Looking at the time history of the relative displacement shown

in Fig. 5.6 for the frame and Fig. 5.13 for the frame wall two

basic differences may be noticed. The open frame tends to vibrate

around a permanent displacement of about 50 to 55 mm in the

forced vibration part of the response. This is very typical of

bilinear systems. No permanent set is detected in the frame wall

response as the frame is not allowed to reach its yield

displacement. Yet, as will be seen, considerable inelastic

response has taken place. The second difference lies in the free

vibration response. While the open frame vibrates around a

permanent displacement of about 3 mm the frame wall shows a

56

heavily damped response, decreasing almost linearly to zero. This

type of response is similar to that encountered in friction type

damping. In the same figures one may compare the absolute

acceleration of both systems. While the open frame shows the

distinctive upper limit to the absolute acceleration due to the

yielding of the frame, the frame wall system shows no yielding at

all and the maximum acceleration is approximately 2.7 times

greater than that of the open frame. Comparing the total

resistance of both systems Fig. 5.7 and Fig. 5.14 show that the

open frame has a maximum resistance of 245 kN while the response

of the frame wall reaches a maximum of about 680 kN.

It seems apparent that a fundamental change in the behavior of

the frame has come about. This change in.the overall response

quantaties may not be solely justified on the basis of a large

change in the fundamental frequency due to the presence of the

wall. Further evidence of this fact may be seen in Figs. 5.15

through 5.18 which show the response of the moment, shear and

axial load for the infilled frame. From these figures one may

conclude that the load carried by the system may not be assigned

as if the wall and frame were acting in parallel. Figures 5.15

through 5.18 show time histories for these columns which are

radically different from the time histories of a frame that has

not yielded. Additional evidence that the behavior of the open

frame has been altered is shown in Fig. 5.19. This figure shows a

load displacement curve that reflects neither an open frame type

behavior nor that of a frame and wall acting strictly in

57

parallel. Since the frame has not yielded, the inelastic behavior

shown in Fig. 5.19 must be due solely to the interaction between

the frame and wall and to cracking within the wall.

In order to understand the behavior of the frame-infill-wall

system one should take a closer look at Figs. 5.13 through 5.25.

Figures 5.20 through 5.25 represent the sequence of cracking in

the wall as well as the crack pattern. They are divided in the

following manner. Figure 5.20 represents the summary of crack

patterns from time t=O.O to 0.155 sec. The numbering sequence

indicates at which time step the specific portion of the crack

occurred. Figures 5.21 through 5.24 are based on Fig. 5.20. This

set of figures represent the existing crack pattern at the time

specified in each figure. In doing this a better picture of the

sequence of cracking may be observed and studied. Figure 5.25 is

similar to Fig. 5.20, but is presented for the time interval from

t=0.275 to 0.555 sec.

The load displacement curve shown in Fig. 5.19 depicts a

change in behavior of the frame wall system from linear to

nonlinear at a displacement of about 0.4 mm. This displacement

corresponds approximately to a time t=0.045 sec. At this time the

displacement time history shown in Fig. 5.13 shows a change in

the response.

Up to this point the wall and frame act as a monolithic unit

to carry the horizontal force of 410 kN which has been developed.

In this interaction the system tends to behave as the cross }13tZ RefereIlce Room

LD.i.vers~- t.y of' Illinois. :2106 NGEL

2GB ~. P~mi~s 2tre9~ ~~~~.:J.~.i.d.~ ::::':;,..:;:::~ 'E<:.3·a

5B

section of a beam with the columns tending to act as flanges and

the wall tending to act as a web. This may be verified by looking

at the response of the columns shown in Figs. 5.15 through 5.1B.

In these figures one may see that frame action as represented by

the shears and moments in the columns may be considered small in

comparison to the axial behavior of these same elements. The

axial response of the elements shown in Figs. 5.15 and 5.1B

correspond to about 100 kN in tension and compression,

respectively. This seems to indicate that the columns tend to act

as axial elements in the interaction with the wall, much like the

flanges of a beam.

The above interpretation is partially based on the

compatibility between the frame and wall and the compatibility

between the ground and the base of the wall. Figure 5.21

represents the cracking pattern at time t=O.045 sec. This figure

shows that cracks have already developed between the frame and

wall. Since there is no detectable change in the stiffness, as

may be seen in the load displacement response of Fig. 5.19, it

seems that full compatibility between columns and wall is not

necessary as long as a certain amount of shear can be transmitted

between the column and wall.

This condition does not seem to hold true for the

compatibility between the walland ground where the stresses to

be transmitted across the crack are normal tensile stresses. This

may be seen in Fig. 5.20 at time t=O.05 sec where a crack has

59

developed at the base of the wall producing a radical change in

both the stiffness and behavior of the frame wall system. This is

depicted in the load displacement response of Fig. 5.19 as a drop

in the load at a displacement of about 0.5 mm.

Figure 5.22 shows the crack pattern at time t=0.060 sec. As

may be seen a continuous crack has developed between the right

column and the wall in addition to the crack which has developed

between the wall and its boundary on the lower left side of the

wall. The cracks along the boundary alter the behavior of the

frame wall system from a shear wall to a frame that is braced

across its diagonal.

The load displacement curve in Fig. 5.19 seems to indicate

that the change in behavior is marked by a transition which

transition begins at time t=0.05 sec with a displacement of 0.5

mm and continues up to time t=0.125 sec at a displacement of

about 2.3 mm. In this transition two different phases may be

identified. The first occurs at a displacement of about 0.75 mm

and the second occurs at a displacement of about 2.0 mm. Both

phases are related to additional cracking in the wall both inside

and at its boundary.

Figure 5.23 represents the crack pattern at time t=0.075 sec.

This crack pattern corresponds to the end of the first phase of

the transition at a displacement of about 0.75 mm. Figure 5.23

seems to indicate that the reduction in load shown in Fig. 5.19

may be due mainly to extensive cracking inside the wall.

60

Especially noticeable is the horizontal crack on the right side

of the wall at midheight.

Figure 5.24 represents the crack pattern at time t=0.110 sec.

As may be seen, additional cracking has occurred along the

diagonal of the wall. This crack pattern corresponds to a

displacement of about 1 mm in the load displacement response of

Fig. 5.19. The response shows an increase in load from time

t=O.075 sec. This fact shows the importance of the development of

a horizontal crack in the transition phase between shear wall

and braced frame behavior.

The second phase of the transition occurs at time t=O.125 sec

and a displacement of 2 rom. The crack pattern in Fig. 5.20

indicates that this mainly corresponds to additional cracking

along the base of the wall. The load displacement curve ,in Fig.

5.19 shows a drop in the load as the wall moves into its final

condition of acting as a bracing element across the diagonal of

the frame.

It may be seen from Fig. 5.19 that the change in behavior

brings about a greater energy dissipation in the system without

any yielding having occurred in the frame.

Figure 5.25 shows the additional cracking corresponding to the

first load reversal in Fig. 5.19. As the load is reversed some

cracks that were opened in the first cycle are now closed. These

cracks are not shown in Fig. 5.25 but the load displacement curve

61

shows that as the load is reversed the frame becomes braced

across the alternate diagonal.

That a full transition to a braced frame has occurred may be

seen in the time histories of the columns shown in Figs. 5.15

through 5.18. After time t=O.130 sec, the moments and shears are

no longer ~all, and the axial loads in the columns alternate

from tension to compression as would be expected in a braced

frame under reversed loading.

5.3.3 Frame Wall with Gaps(I)

The response of the frame wall system for which 5 mm gaps have

been specified is shown in Figs. 5.26 through 5.38. These figures

represent the same group of response quantities presented for the

frame wall without gaps. The masonry's ultimate compressive

strength was taken as 24 MPa and the ultimate tensile strength

was taken as 2.4 MPa. A brief summary of the key events that

occurred during the response is presented in Table 5.7.

The elastic fundamental frequency of the structure is the same

as that of the open frame since the wall is not in contact with

the frame and therefore does not contribute to the stiffness

under small displacement theory. The elastic fundamental

frequency of 1.53 Hz for this frame wall system, like the open

frame, is thus just to the left maximum response point on the

spectrum, Fig. 5.5.

62

The time histories of the relative displacement and absolute

acceleration shown in Fig. 5.26 seems to show that the behavior

of the frame wall with a gap is in this case essentially

elastoplastic. The absolute acceleration shows, with the

exception of the two spikes, the typical flat response at a

maximum value which is characteristic of an elastoplastic frame.

The maximum value of the absolute acceleration, again ignoring

the two spikes, is about 0.25g, that is, about 30 percent greater

than the open frame. This increase is directly related to the two

spikes which represent, as will be seen, a change in the behavior

of the frame. The relative displacement also shown in the same

figure shows that the maximum relative displacement is 52 mm

which is about 40 percent less than that corresponding to the

open frame. Also, the relative displacement tends tO,vibrate

about a permanent displacement as may be seen in the free

vibration phase of the response. This again is characteristic of

elastoplastic systems.

Although the behavior reflected in these response quantities

is approximately elastoplastic, there are certain discrepancies

which distinguish it from the open frame. The first are the

obvious spikes which are due to the failure of the upper corners

of the wall. The second is the greater yield load. Since the wall

has no elastoplastic behavior and the open frame has a lower

yield load, the increase in the load at which the yield plateau

is reached can only be due to a change in the behavior of the

63

frame due to the presence of the wall. The third discrepancy may

be seen at time t=0.8 and 1.05 sec in Fig. 5.26 where an increase

in the accelerations above the yield plateau implies again a

change in the behavior in this case ·away from the preceding

elastoplastic mode. These differences are due to the presence of

the wall and its bracing effect on the frame. These changes in

behavior may be studied by looking at the load displacement curve

in Fig. 5.32 and relating its changes to the time histories and

cracking patterns.

The cracking, yield, and failure pattern up to time t=0.315

sec is presented in Fig. 5.33. This figure indicates that the 5

mm gap closes at time t=0.09 sec. Up until that time the frame is

free to deform. This is reflected in a change in stiffness in the

load displacement curve at a displacement of 5 mm. At time

t=0.095 sec the center of the wall, due to tensile stresses,

develops a horizontal crack at midheight on its left side; this

may be seen in the crack pattern in Fig. 5.34. This figure also

shows horizontal cracking along the wall base. This cracking is

also reflected in a slight change in stiffness in the load

displacement curve before the maximum load is reached.

At time t=O.l sec the frame wall system reaches its maximum

capacity and the upper left corner of the wall begins to fail.

This may be seen in Fig. 5.35 where the wall has cracked along

the horizontal plane at midheight ~n addition to extensive

cracking in the lower half of the wall. The condition of the wall

64

and frame at time t=0.110 sec is shown in Fig. 5.36. It may be

seen that the upper left corner of the wall has failed. This

condition corresponds in the load displacement curve of Fig. 5.32

to the right side of the bottom part of the· spike. The system

should again behave as an open frame; and this may be verified in

Fig. 5.32 by the fact that the stiffness of the system after the

failure of the wall is approximately the same as the original

stiffness.

From time t=0.11 to 0.125 sec the frame deforms with little

interference from the wall, but at time t=0.125 sec the frame

comes into contact with the wall at midheight. This is depicted

in Fig. 5.33. This contact induces a change in stiffness in the

structure which may be seen in the load . displacement curve of

Fig. 5.32 at a displacement of about 12 mm. At this point the

left column becomes braced inducing a change in the moment and

shear. This may be detected in Figs. 5.28 and 5.29. Here, as

expected, the shear and moment in the lower half of the column,

Fig. 5.28, are reduced while these same quantities increase in

the upper half of the column, Fig. 5.29. No such behavior is seen

in the right column, Figs. 5.30 and 5.31, as this column deforms

without any constraints.

At time t=O.145 sec part of the frame yields. This is depicted

in Fig. 5.32 by the reduction in stiffness at a displacement of

about 16 mm. The yielding in the frame occurs in the top part of

the left column including the beam and at the base of the right

65

column. This may be seen in the column response shown in Figs.

5.29 and 5.31 where the moment has reached its yield value.

Figure 5.29 shows that the shear, induced by the short column

effect, has not yet reached its maximum value. Therefore between

time t=0.145 and O.lBO sec the shear in the column increases to a

value greater than that of the open frame.

At time t=O.lBO sec, the frame again yields. This time at the

center of the left column and the top of the right column forming

a yield mechanism. This is depicted in Fig. 5.37 where both time

steps t=0.145 and O.lBO sec are shown. This step is depicted in

the load displacement curve at about 25 mm where the yield

plateau is reached. Also at this time the shear and moment at the

center and base of the left column reaches a plateau as may be

seen in Figs. 5.2B and 5.29.

At time t=0.20 sec, the wall becomes fully cracked along its

base. The only change that occurs is in the behavior of the lower

left column shown in Fig. 5.2B. As the wall, still bracing the

column at midheight, moves to the right, the base moment of the

left column increases up to yield at time t=0.305 sec. At time

t=0.315 sec. the gap at the lower right side of the wall closes.

At time t=0.305 sec, the maximum displacement is reached and

the system begins to unload. As the frame wall begins to unload

the frame remains braced and therefore the system unloads with a

higher stiffness than the open frame. This may be seen in Fig.

5.32 at a displacement of 52 mm. Once the center of the left

66

column moves away from the wall, the system returns to its

original open frame stiffness. This occurs at a displacement of

about 45 mm. Figure 5.28 shows that the bottom of the braced

column on the left side of the frame does not unload until the

frame loses contact with the wall at time t=0.35 sec. At this

time the shear drops to reflect the fact that the wall no longer

carries any shear, then it reloads. This jump in the shear is due

to the fact that no iteration within a time step is carried out

and therefore no redistribution of the unbalanced forces may be

accomplished within a given time step.

The load displacement curve in Fig. 5.32 shows that between

the displacement of about 42 mm and 20 mm the frame, which is

free to deform, behaves elastically. At time t=O.425 sec the

upper part of the left column and the base of the right column

yield. This depicted in the load displacement response as a

change 1n stiffness at a displacment of 18 mm. At time t=O.45 sec

the frame yields at the base of the left column and at the top of

the right column. This sequence corresponds in Fig. 5.32 to the

yield plateau that occurs at a displacement of about 5 mm. This

whole sequence is presented in Fig. 5.38.

At time t=O.46 sec, the frame comes into contact with the

wall. The load displacement curve in Fig. 5.32 shows that this

contact does not occur at a zero displacement but at a

displacement of about 1 mm. This is due to the fact that the

wall, which has cracked at midheight, has been displaced slightly

67

more to the right than its base whose total movement is

restricted to 10 mm. During the time interval t=0.46 to 0.465 sec

the interaction with the wall increases the resistance of the

structure by more than 4 times the open frame value. This is seen

in Fig. 5.32 as the spike in the lower half of the plot.

At time t=0.465 sec the upper right corner of the wall fails.

At the same time part of the lower right corner also fails. This

is depicted in Fig. 5.38 at time t=0.465 sec and seen as a

reduction in the load in Fig. 5.32. The load displacement curve

of Fig. 5.32 shows that the load does not reduce to the yield

plateau that existed at a displacement of 1 mm. This is due to

the bracing effect of the wall on the right column. This is shown

in the column response of Fig. 5.30 as an increase in the shear

in the upper half of the column at time t=0.47 sec. From time

t=0.465 to 0.5 sec, the shear at the top of the column increases

until the column yields at its midsection. It should be noted, as

seen in Fig. 5.31, that the moment at the base of the right

column has dropped below its yield value just as the moment at

the center of the column increases due to the bracing effect.

Figure 5.32 shows that the system undergoes two additional

cycles of loading. The principal difference between these two

additional cycles and the two previous ones is that the frame

yields before it comes into contact with the wall at midheight.

Once the wall and frame come into contact, the shear starts to

increase again but this time in a nonlinear fashion due to the

68

fact that the wall may slide along its base. This is depicted in

the column behavior of Figs. 5.28 through 5.31 as' an increase or

decrease in the shear at times t=0.8 and 1.05 sec. The load

displacement curve of Fig. 5.32 also shows this change at a load

level of 245 kN and a displacement of about 25 mm for the third

cycle and a load level of 245 kN and a displacement of about -25

mm for the fourth cycle.

The fundamental difference in behavior between the frame wall

with no gaps and the present structure is the fact that no

diagonal compressive behavior has been observed. Although the

wall alters the behavior of the frame by bracing the columns, the

behavior is still that of a frame. This may be seen in the axial

load behavior of the columns in this structure. The behavior of

the columns in this structure is similar to the columns of an

open frame. Their behavior lacks any of the large axial load

changes encountered in the previous structure changes which were

mainly due to the diagonal bracing effect of the wall.

5.3.4 Frame Wall with Gaps(II)

In this frame wall structure the ultimate compressive strength

for the masonry was increased to 34 MPa and the tensile strength

to 3.4 MPa. A gap of 5 mm was specified between the wall and the

frame just as in the previous structure. The response of the

system is shown in Figs. 5.39 through 5.50. They represent the

same response quantities as the previous structures. A brief

summary of the key events that occurred during the response is

presented in Table 5.8.

69

As with the previous structure, the wall is not in contact

with the frame and does not contribute to its stiffness under the

small displacement theory therefore its fundamental frequency of

1.53 Hz coincides with that of the open frame. This again sets

the frame wall system just to the left of the spectrum's maximum

response.

The time histories of the relative displacement and absolute

acceleration are shown in Fig. 5.39. Unlike the previous test

this response shows no elastoplastic behavior. The relative

displacement response shows no permanent deformation and the free

vibration response seems to be that of an open frame. The maximum

displacement is about 19 mm which is about 2.7 times less than

the previous structure and 4 times less than the open frame. The

limitation on the maximum displacement is due, as will be seen,

to the development of the wall as a diagonal element across the

frame and not, as in the previous structure, to the effective

shortening of the column caused by the bracing effect of the wall

when its upper half failed.

The load displacement curve is shown in Fig. 5.4,5; in which a

sharp change in stiffness is seen at a displacement of about 5 mm

as the frame comes into contact with the wall. This condition is

depicted in Fig. 5.39 as a sharp rise in the absolute

acceleration at a time t=O.l sec. The closing of the gap is

depicted in Fig. 5.46 at time t=O.09 sec.

70

The first change in stiffness comes at time t=0.095 sec due to

cracking in the wall. This is seen in the load displacement plot

of Fig. 5.45 and in the absolute acceleration of Fig.

crack pattern for time t=0.095 sec is shown in Fig.

5.39. The

5.47. The

second change in stiffness is seen in Fig. 5.45 at a displacement

of about 7 mm. This corresponds to a time t=0.105 sec in Fig.

5.48; a full horizontal crack develops at the midbeight of the

wall and almost a full crack at the base of the wafl in addition

to some diagonal cracking.

The third change in stiffness in Fig. 5.45 takes place at

about 11 rom and is the most important of the three that occur. At

this point the wall has failed along its base as may be seen in

Fig. 5.49 at time t=0.145 sec. The wall may now slide along its

base and, as expected, the load level declines as seen in the

load displacement response, Fig. 5.45. This decline begins at a

displacement of 11 mm and ends, as expected, at a displacement of

16 mm when the wall comes into contact with the right column at

its lower right corner. This is depicted in Fig. 5.46 at time

t=0.185 sec.

In the load displacement curve the load begins to increase

again at a displacement of 16 mm. But the increase is small as

the base acceleration has already begun to decline. The base of

the columns yields at time t=O.190 sec this may be seen in Figs.

5.41 and 5.44. At this time no appreciable change has occurred in

71

the axial behavior of the columns indicating that the wall is not

yet acting as a diagonal element. This is due to the fact that no

contact has occurred between the beam and wall, an apparent

requisite for this type of behavior.

Once unloading occurs, Fig. 5.45 shows that the structure

returns to the stiffness of the open frame. This unloading begins

at a displacement of about 17 mm and remains unchanged up to a

displacement of 5 mm. This is to be expected as the wall is now 5

rom to the right of its original position. The crack pattern in

Fig. 5.50 depicts the behavior of the system during the load

reversal.

At time t=0.29 sec the frame comes into contact with the wall

at midheight. Since the wall has had a greater relative

displacement in its upper half only the gap elements at midheight

which correspond to this condition are closed. This contact as

well as the one that occurs at time t=O.32 sec seems to have

little effect on the stiffness of the system. This may be

verified by looking at the absolute acceleration in Fig. 5.39 and

load displacement curve in Fig. 5.45.

At time t=0.335 sec the frame comes into contact with the wall

at the upper right corner. This corresponds in the load

displacement curve with the increase in the stiffness at a

displacement of 5 mm in the lower right quadrant. The increase in

stiffness is less than in the first loading cycle as the wall may

slide along its midheight as well as along its base. The change

72

in stiffness at 4 mm is due to the sliding effect. Tbis condition

lasts, while the load is increasing, until a displacement of

-4 mm.

At this point a drop in the load occurs. This corresponds to

the failure of part of the wall at time t=0.37 sec as depicted in

Fig. 5.50. In this same figure in the next time step, the beam

comes into contact with the wall, setting the condition for the

wall to behave as a diagonal element. From a displacement of-4

mm which occurs at time t=0.37 sec to a displacement of -12 mm,

at t=0.405 sec, no appreciable increase in load is detected.

At time t=O.405 sec the wall comes into contact with the base

of the left column and begins to behave as a diagonal element. In

addition the frame yields at the base of both columns. Toe load

increases as the behavior of the system is altered from an open

frame behavior to a braced frame behavior. The right column is

put in tension as may be seen in the column response histories

shown in Figs. 5.43 and 5.44; confirming the fact that a large

compressive force is being carried across the diagonal of the

frame. This bracing of the frame across its diagonal is repeated

in the next two cycles of loading as may be seen by looking at

Figs. 5.41 through 5.45.

An important factor that differentiates this response from

that of the frame wall system which had no gaps is the level of

the moments and shears when the system is acting as a braced

frame. In the present case, the moments and shears are larger

73

than the ones developed in the frame wall system for which

initial continuity was assumed.

5.3.5 Three Story Frame Wall

The response of the three story structure is shown in Figs.

5.51 through 5.69. Figure 5.51 is the ground motion and Fig. 5.52

is the elastic spectrum for the ground motion. Figures 5.53

through 5.69 represent the same response quantaties as the

previous structures. A brief summary of the key events that

occurred during the response of each story is presented in Tables

5.10 through 5.12. The first three seconds of the motion are not

included in Figs. 5.53 through 5.69 because the response is

undetectable when compared to the rest of the time history.

The size of the gap for all three stories was taken as zero

but the wall was not assumed continuous with the frame as in the

case of the frame wall of Sec. 5.3.2. Under the small

displacement theory this gives a set of frequencies which

correspond to the open frame. In order to assess the difference

between specifying a gap size of zero and a continuous system two

sets of frequencies were calculated; these are shown in Table

5.9. Again the great difference in the frequencies between the

open frame, as represented by the gap size of zero, and the frame

wall is evident. Although the fundamental elastic frequency of

the three story frame wall is 1.33 Hz the response should

correspond to a frequency of about 10.0 Hz due to the gap size of

zero.

74

Due to the complexity of the ground motion a detailed

discussion as provided for the previous structures will not be

presented. However, there are some changes in behavior that are

due to the occurrence of a very specific event.

In the first story there are three very specific changes in

behavior as may he seen in the load displacement curve, Fig.

5.55. At first the frame acts -monolithically with the wall giving

the first story a relatively high stiffness. This condition lasts

for the first 3.5 sec providing for small displacements, -shears

and moments while giving large changes in the axial loads of the

columns. Between t=3.5 and 4.1 sec considerable cracking is

developed within the wall until a complete crack across the

diagonal of the wall is developed and a large change in stiffness

occurs; Fig. 5.55. The third major change in the behavior of the

first story occurs at t=6.37 sec when the frame yields at the

base of the columns. This condition is depicted in Fig. 5.55 as

two large loops which provide the maximum load developed in the

story.

In the third story up to t=3.5 sec the system behaves as a

monolithic unit with the same consequences as the first story of

having small displacements, shears and moments with a relatively

high stiffness. The first change in behavior occurs at t=4.66 sec

with a slight reduction in the stiffness due to some cracking

within the wall, this may be seen in Fig. 5.61. The second and

75

largest change in behavior occurs at t=5.32 sec. At this time a

full crack across the diagonal of the wall is developed. The

development of this crack results in a sharp reduction in the

stiffness of the story, as shown in Fig. 5.61. After this time

the displacements, moments and shears are no longer small.

In the second story the most interesting event occurs at t=4.1

sec. At this point the interstory drift, Fig. 5.57, has a radical

change in behavior where the response frequency of the story is

altered. The story then vibrates at a frequency of about 30 Hz

but its base motion slowly fluctuates. Since anyone of the three

stories has a frequency, if treated as a single degree of

freedom, of about 30 Hz the second story then behaves as a

relatively stiff system which sits on a more flexible one.

Something similar is reflected in the third story response, Fig.

5.60, up to t=5.32 sec at which point the wall becomes fully

cracked across its diagonal. Little or no change in stiffness

occurs for the second story throughout the ground motion as may

be seen in Fig. 5.58.

5.4 Discussion of Results and Design Implications

The most important result that may be deduced from the

previous studies is the tremendous influence that the initial

conditions of strength and gap size have on the behavior of the

system. The radical change in behavior between the second, third,

and fourth structure is a case in point.

76

The second structure consisted of a frame with an infill

masonry wall but the wall was assumed continuous with the frame

at the beginning of the ground motion. The third structure

differed from the second one in that a gap was specified at the

frame wall boundary. The size of this gap was taken as 5 mm. The

significant difference in the results of these two structures has

already been shown. The fourth structure differed from the third

structure only in the compressive strength assumed for the

masonry. Again the difference in behavior between the fourth

structure and the second and third structures was evident.

The differences in behavior between the second and third

structures may be in part explained by looking at the spectrum

for the base acceleration shown in Fig. 5.5. The gap specified

for the third structure causes the initial frequency of that

system to be the same as that of the open frame, 1.53 Hz. This

places the third structure close to the maximum response of the

spectrum at a frequency of 2 Hz. The second structure has an

initial frequency of 13 Hz due to the continuity with the wall.

This places the second structure at the right side of the

spectrum at a much lower response than the third structure. The

spectrum indicates that if these two systems remain at their

original frequencies they will reach different maximums. One is

in fact looking at two different structures. In the normal design

process this is not evident.

77

Although the location of the initial natural frequency is

important, this condition is not wholly satisfactory in

explaining the great change in behavior that a 5 mm gap may

induce. This is due to the fact that: (1) neither system retains

its original natural frequency; and (2) the spectrum in Fig. 5.5

provides no information about the time domain.

If the specified gap of 5 mm were reduced, one may reasonably

assume that the response of the third structure would tend

towards the response of the second structure. This fact would be

independent of the original frequency of the system as this

frequency would remain at 1.53 Hz until the gap size was reduced

to zero. This argument is supported by the results of the three

story structure where a gap of size zero was specified. This

structure in fact behaved initially at the higher frequency of

10.0 Hz rather than at 1.33 Hz, the frequency of the open frame.

This fact coupled with the two previous observations points to

the size of the gap as the key factor that determines the

behavior of the frame-infill-wall system. If the gap is small

enough, the frame-infill-wall will tend to behave as if the

system were originally continuous. In the present case, the

system with the frequency of 13 Hz. would approximately determine

the response of the third structure. This is in spite of the fact

that the initial frequency of this system would be 1.53 Hz. If

the gap is not small enough, then the behavior of the system

78

depends not only on the gap size but also on the strength of the

wall and on the location of the initial frequency with respect to

the response spectrum. This last observation is based on the

results of the third and fourth structures.

In order for the wall to fail as it did in the third

structure, the 'frame must develop sufficient momentum such that

the wall cannot resist the load being applied. In order to

develop this momentum, the gap size must not only be sufficiently

large, but the response of the frame to the ground motion must be

such that it can develop the momentum necessary to make the wall

fail.

The spectrum in Fig. 5.5 gives the maximum response for a

given frequency but says nothing about the time response of the

system. In spite of this, the maximum response of the open frame

may serve as an indication of whether sufficient momentum can be

developed to make the wall fail. In this case a 5 mm gap with the

present ground motion is sufficient to fail the wall in the third

structure but not enough for the fourth structure.

The difference in behavior between the third and fourth

structures point to the fact that contact between the beam and

wall is a necessary condition to develop bracing across the

diagonal of the frame. The fourth structure shows that unless

the contact between the beam and wall occur, no significant

changes in the axial behavior of the columns will result. This

condition is important since many structures have a large space

79

between the top of the wall and the beam. Hence, there are

structures which may only be able to shorten the effective length

of the surrounding columns, as in the third structure, but never

diagonally brace the frame, as in the fourth structure.

Although specific design recommendations must await further

research, several observations may be made:

1. Most of the frames infilled by masonry walls do have

specific separation between the columns and wall

placing them in the same category as the third and

fourth structures. This is supported by the type of

damage observed for these systems under earthquake

loadings.

2. This research indicates that for the study of the

frame masonry wall system, the use of a cracking

mechanism as well as the ability to model the gaps

between the frame and wall may be indispensable.

3. The use of the Equivalent Strut Method to represent

these walls in a design process requires judgment,

experience and an understanding of the behavior of

the system. A case in point is the marked difference

in behavior between the second and third structures.

In the second structure the straightforward use of

this method should yield good results. The use of

this method in the third structure is not as evident

and its use would require a clear understanding of

the limitations of the method as well as good

engineering judgment. }Let:::; TIsfs:ran08 R::,·~ml

U;:,iV8:rS::-:Y of' I11.::':"G.cts BIOS =~;-~-'::Ej

~-~ ~. P:~~~~ :'~~~V L~·" .~r-:.

80

4. Contact between the beam and wall is indespensable

to develop a diagonal bracing across the frame. If

contact between the beam and wall cannot occur, as

in the case of walls which are much shorter than the

columns, diagonal bracing will not occur.

S. The difference in behavior of the structures that

were studied points to the inadequacy of using the

open' frame frequency in designing frames to which

infill walls have been added.

81

CHAPTER 6

SUMMARY AND CONCLUSIONS

6.1 Summary

A nonlinear dynamic model to study the behavior of frames

infilled by masonry walls has been presented. The nonlinearities

of the model include the interaction between'the frame and wall,

cracking and failure of the wall, the bracing effect that the

wall has on the frame, the discontinuities between the frame and

wall, and the inelastic behavior of the frame.

The fundamental concept for modeling the masonry wall was

based on the premise that the cracking mechanism in the wall may

be separated from the material model assumed for the masonry.

This premise was based on the fact that masonry tends to have

preferred planes of failure along its mortar joints. The

separation of the cracking mechanism not only simplified the

modeling of the masonry but in the end seemed to be indispensable

in predicting the different modes of behavior of the frame­

infill-wall system. '

Each part of the proposed model was represented by elements

which were as simple as the assumptions would permit. This was

done to better determine the capabilities of the model.

82

The model was used to study three one story, one bay frame­

infill-wall systems for which different initial properties were

specified and to which a sinusoidal ground motion was applied. In

addition the model was used to study a three story, one bay

frame-infil l-wal I to which an earthquake ground motion was

applied.

The analytical

different facets

model

of

showed the ability to

behavior that have

represent the

been observed

experimentally. Specifically, the wall was able to alter its

behavior in accord with the boundary conditions

frame. The model was also able to effectively

imposed- by the

represent the

cracking pattern both inside and at the boundary of the wall. It

also showed the ability to brace the frame both diagonally and as

a short column.

Discrepancies that showed up in the computations were mainly

due to the very simple material and element models used.

Specifically, the material model for masonry after failure

created the greatest discrepancies.

The studies that were done pointed to the inadequacy of some

of the most common assumptions made with regard to a frame­

infill-wall system. The size of the initial separation between

the frame and wall may be the principal factor in determining the

behavior of the system while a combination of the initial

frequency, gap size, and wall strength may be the second most

important condition.

83

6.2 Conclusions

The following conclusions are based on the development and use

of the proposed model.

1. The crack mechanism proposed or same equivalent

system which allows the masonry wall to become

discontinuous seems to be indispensable in modeling

the frame-infill-wall system. The Equivalent Strut

Method seems to be extremely limited in this respect

while full discretization through the Finite Element

Method would seem too expensive at the present time.

2. The proposed model seems to be able to represent the

different modes of behavior observed experimentally.

3. The behavior of the frame-infill-wall system is

strongly depedent on the gap size specified. Any

attempt at modeling this system must include the

ability to specify gaps at the frame wall boundary.

4. The three most important variables in this system

are the gap size, the strength of the infill wall,

and the time of the maximum response of the open

frame. The last condition may be measured by the

initial frequency and the response spectrum leaving

only the time of occurrence as a variable.

S. Diagonal bracing of the frame by the wall depends

upon the wall coming into contact with the beam at

opposite diagonal corners.

84

6. The fundamental elastic frequency of the open frame

is not an adequate measure of the frequency or

behavior of the frame-infill-wall system.

7. Further research is necessary to better understand

the behavior of the frame-infill-wall system and

develop adequate design recommendations.

85

TABLES

86

Table 5.1 Frame Data

Modulus of Elasticity •••••••••••••••••••••• 25 GPa

Beam Cross Section Area ••••••••••••• 245 x 103 mm2

Beam Moment of Inertia •••••••••••••••• 5 x 10 9 mm4

Column Cross Section Area ••••••••••• 123 x 103 mm2

Column Moment of Inertia ••••••••••• 1.25 x 10 9 mm4

Yield Moment ••••••••••••••••••••••••••• 214.0 kN-m

Total Mass •.••••••••••••••••••••••••••••••• 150 Mg.

Total Dead Load ••••••••••••.••••••••••••••• 490 kN

Damping Ratio •••••••••••••••••••••••••••••••••• 2%

Mass Three Story Frame •••••••••• 30 Mg (per story)

87

Table 5.2 Wall Data

Modulus of Elasticity •••••••••••••••••••••• 14 GPa

Wall Thickn.ess ••••••••••••••••••••••••••••• 100 IIlIIl

Poisson's Ratio •••••••••••••••••••••••••••••• 0.25

Masonry Compressive Strength •••••••• 24 to 34 MPa

Masonry Tensile Strength ••••••••••• 2.4 to 3.4 MPa

Total Dead Load ••••••••••••••••••••••••••••• 60 kN

Table 5.3 Joint Data

Modulus of Elasticity ••••••••••••••••••••••• 7 GPa

Joint Thickness •.•.••••••••••••••••••••••••• 10 IIlIIl

Poisson's Ratio •••••••••••••••••••••••••••••• 0.15

Cross Sectional Area ••••••••••• 87.5-175 x 103 mm2

88

T~ble 5.4 Failure Surface Data

Joint Failure Surface (Fig. 3.2)

Shear Stress Control Point ••••••••••••••• 1360 kPa

Normal Stress Control Point ••••••••••••••• 340 kPa

Slope Tension Branch •••••••••••••••••••••••••••• 2

Slope Compression Branch •••••••••••••••••••••••• 1

Masonry Failure Surface

Ratio of Tensile Strengths •••••••• 1.5 (Fig. 3.4)

Ratio of Compressive Strengths •••• 0.42 (Fig. 3.5)

Col.

Left

Right

Freq. (Hz)

1.53

Max. Base Mom. (kN-m)

215.0

215.0

Table 5.5

Max. Dis. (mm)

77.3

Key Response Quantities and Events: Open Frame

Max. Vel. (mrn/ sec)

-445.6

Max. Abs. Ace. (g)

0.188

Max. Res. (kN)

245.0

Max. Double Amp. (mm)

83.0

Max. Base Shear (kN)

Max. Center lJom. (kN-m)

Max. Top Shear (kN)

Max. Top Mom. (kN-m)

Min. Axial Load (kN)+

Axial Dead Load (kN)

126.0 123.0 215.0 180.0 245.0

126.0 123.0 215.0 182.0 245.0

+Axial response for bottom section of column.

Max. Axial Load (kN)

310.0

308.0

(Xl

\0

Col.

Left

Right

Time (sec)

0.45

0.075

0.125

Freq.

~

13.0

Table 5.6

Max. Dis. (mm)

3.5

Key Response Quantities and Events: Frame Wall

Max. Vel. (mrn/ sec)

-43.2

Max. Abs. Ace. (g)

0.462

Max. Res. (kN)

680.0

Max. Double Amp. (tum)

6.8

Max. Base Mom. (kN-m)

Max. Base Shear (kN)

Max. Center Mom. (kN-m)

Max. Top Shear (kN)

Max. Top Mom. (kN-m)

Min. Axial Axial Dead Load (kN)+ Load (kN)

110.8

118.0

Disp. (nun)

0.04

O. 75

2.0

100.0 64.0 69.0 -29.0 130.0

68.4 59.0 63.5 -24.0 130.0

Summary of Key Events

Event

Change in response from linear to nonlinear.

Partial transition to braced frame; horizontal crack at midheight and boundary of wall.

Full transition to braced frame; crack at midheight of wall; almost full crack along base and left side of wall.

+Axial response for bottom section of column.

Max. Axial Load (kN)

219.0

282.0

See Fig.

5.19

5.23

5.20

\.0 0

Col.

Left

Right

Time (sec)

0.090

0.110

0.125

0.145

0.180

0.305

Table 5.7 Key Response Quantities and Events: Frame Wall-Gaps (I)

Freq.

~ Max. Dis.

(mrn) Max. Vel. (mrn/sec)

Max. Abs. Ace. (g)

Max. Res. (kN)

Max. Double Amp. (nun)

1.53 52.1 -522.2 0.272 (0.767)++

400.0 (1100)++

99.0

Max. Base Max. Base Max. Center Hax. Top Max. Top Min. Axial Axial Dead Max. Axial Mom. (kN-m) Shear (kN) Mom. (kN-m) Shear (kN) Mom. (kN-m) Load (kN)+ Load (kN) Load (kN)

215.0 124.0 215.0 245.0 215.0 176.0 245.0 308.0 (95.2)++

215.0 126.0 215.0 245.0 215.0 176.0 245.0 307.0 ( ... 112)++

Summary of Key Events

Disp. (mm) Event See Fig.

5 Upper left corner gap closes; wall and frame interact; large change in stiffness. 5.33

9 Upper left corner of wall fails. 5.36

12 Column braced at midheight. 5.33

16 Partial yielding of frame; bottom right column, top left column and beam. 5.37

25 Full yielding of braced frame top right column and beam, center left column. 5.37

52 Wall slides along base; gap closes at bottom of right column; maximum displacement 5.33 reached.

+Axial response for bottom section of column.

++Values at spike.

\D I--'

Col.

Left

Right

Time (sec)

0.090

0.145

0.185

0.335

0.405

Table 5.8 Key Response Quantities and Events: Frame Wall-Gaps (II)

Freq.

~ Max. Dis.

(mm) Max. Vel. (mm/sec)

Max. Abs. Ace. (g)

Max. Res. (kN)

Max. Double Amp. (mm)

1.53 19.0 -287.0 0.841 1260.0 36.0

Max. Base Mom. (kN-m)

214.0

214.0

Disp. (mm)

5

11

16

5

12

Max. Base Max. Center Max. Top Max. Top Min. Axial Axial Dead Shear (kN) Mom. ~kN-m2 Shear (kN) Mom. (kN-m) Load (kN)+ Load (kN)

118.0 130.0 207.0 -210.0 245.0

149.0 119.0 200.0 -250.0 245.0

Summary of Key Events

Event

Upper left corner gap closes; wall and frame interact; large change in stiffness.

Full crack developed at midheight and at base; large reduction in load and stiffness.

Max. Axial Load (kN)

300.0

303.0

See Fig.

5.46

'5.49

Lower right corner gap closes; load increases; no change in column load yet. 5.46

Upper right corner gap closes; wall and frame interact; large change in stiffness; 5.50 wall displaced 5 mm from original position.

Lower left corner gap closes; upper right corner gap between beam and wall closes; 5.50 frame becomes braced; large change in axial load of column.

+Axial response for bottom section of column.

'-0 N

93

Table 5.9 Frequencies - Mode Shapes: Three

Continuous

Hode No. 1 2 3

Nat. Freq. (Hz) 9.92 32.2 54.2

Participation Factor ++ B.71 3.62 -1.04

Mode Shapes+++

Story 3 0.1473 -0.0937 -0.1138

Story 2 0.0981 0.0787 0.1324

Story 1 0.0449 0.1355 -0.0534

+ Gap size O.

++Participation Factor

T {q.} [M) {R} 1.

1.

Story Structure

Gap Specified+

Mode No. 1 2 3

1.33 4.07 6.60

8.89 3.01 -1.41

0.1398 -0.1050 -0.0527

0.1062 0.0779 0.1264

0.0503 0.1274 -0.1207

{q.} = mode shape. 1.

Col.

Left

Right

Time (sec)

3.5

4.1

6.37

7.2

Max. Dis. (nun)

-36.2

Max. Base Morn. (kN-m)

214.5

214.5

Drift (rnrn)

0.-0.5

0.-0.5

17.0

36.0

Table 5.10

Max. Vel. (rum/sec)

-327.0

Max. Base Shear (kN)

123.0

123.0

Key Response Quantities and Events: First Story

Max. Abs. Ace. (g)

-2.75

Max. Center Morn. (kN-m)

25.3

69.5

Max. Res. (kN)

478.0

Max. Top Shear (kN)

123.1

162.1

Summary of Key Events

Max. Interstory Drift (mm)

-36.2

Max. Top Morn. (kN-m)

214.3

214.4

Max. Story Acc. (g)

-2.82

Min. Axial Load (kN)+

267.0

233.0

Max. Axial Load (kN)+

778.0

794.0

Event See Fig.

Frame and wall act as a unit; large changes in axial force; small moments 5.62 and shears.

Full crack across diagonal of wall from top right to bottom left; large change 5.55 in stiffness; moment and shears no longer small.

Frame yields at base of columns; larger loops in load displacement curve. 5.55

Max. displacement; permanent set of 20 rnm. 5.54

+ Axial response for bottom section of column.

\.0 ~

Hax. Dis. ~rn) __

-36.6

Col.

Left

Table 5.11 Key Response Quantities and Events: Second Story

Max. Vel.

Jrnrnlsec) Max. Abs. Ace. (g)

Max. Res. (kN)

-350.0

Max. Base Mom. (kN-m)

76.3

3.22 765.0

Max. Base Shear (kN)

38.9

Max. Interstory Drift (mm)

-1.39

Max. Story Ace. (g)

3.62

Min. Axial Load (kN)

Max. Axial Load (kN)

70 324

Right 75.0 37.2 60 319

Time (Sec)

4.1

4.7

Drift (mm)

0.5

1.4

Summary of Key Events

Event

Full crack in first story detected as a change in response.

Story is very stiff compared to first story; story vibrates at a much higher frequency fluctuating about displacement of first story; frequency ~30 Hz.

See Fig.

5.57

5.57

\.0 \.n

Max. Dis. (rum)

-46.0

Col.

Left

Right

Time Drift (sec) (mm)

3.5 0.3

4.66 1.0

5.32 1.2

Table 5.12

Max. Vel. (rum/ sec)

-497.0

Key Response Quantities and Events: Third Story

Max. Abs. Acc. (g)

-1.96

Max. Res. (kN)

579.0

Max. Interstory Drift (rom)

-14.8

Max. Story Acc. (g)

-1.99

Max. Base Mom. (kN-m)

Max. Base Shear (kN)

Min. Axial Load (kN)

Max. Axial Load (kN)

158.0 88.0 8 162

162.0 90.0 -20

Summary of Key Events

Event

Frame and wall act as a unit; small moments and shears; large stiffness.

Some cracking across diagonal; some change in stiffness.

Full crack across diagonal; major change in stiffness.

171

See Fig.

5.61

5.61

5.61

\0 0'\

Table 5.13 Response Quantities I: One Story Structure

Gap Freq. Case (rum) ~

A N.A. 1.53

B N.G. 13.0

C 5 1.53

D 5 1.53

1.53 Fig. 5.5

13.0

Hax. Dis. (nun)

77.3

3.5

52.1

19.0

128.0

0.70

+Value at spike

Notes: A - Open Frame

B - Frame Wall

C - Frame Wall-Gaps(I)

D - Frame Wall-Gaps(II)

Max. Abs. Ace. (g)

0.188

0.462

0.272 (0.767)+

0.841

1.21

0.48

Permanent Set (nun)

4.0

O.

30.0

O.

N.A.

N.A.

Max. Res. (kN)

245.0

680.0

400.0 (1100)+

1260.0

1780.0

706.0

N.A. - Not Applicable

N.G. - No Gaps

Comments on Behavior

Ductile frame; yield at base and top of columns; yield in beam.

Initial monolithic; columns as flanges, wall as web; change to bracing across diagonal of frame.

Top left and right corner of wall fail; bracing and yielding of columns at midheight.

Failure of joint at base of wall; shift in wall; ~ bracing across diagonal of frame. ~

Table 5.14 Response Quantities II: One Story Structure

Left Column 13:!.ght Column

Case Case A B C D A B C D Comments -

Max. Base 215.0 110.8 2l5.0 214.0 215.0 118.0 215.0 214.0 Mom. (kN-m)

Max. Base 126.0 100.0 124.0 118.0 126.0 126.0 149.0 Shear (kN)

Max. Center 64.0 215.0 68.4 215.0 Yielding in Case C due to bracing Mom. (kN-m) of column.

Max. Top 123.0 245.0 130.0 123.0 59.0 245.0 119.0 Difference between Case A and Case Shear (kN) C due to bracing of column.

\.!)

00

Max. Top 215.0 69.0 215.0 207.0 215.0 6:3.5 215.0 200.0 Mom. (kN-m)

Min. Axia1+ 180.0 -29.0 176.0 -210.0 182.0 -2,4.0 176.0 -250.0 Load (kN) (95.2)++ (-112)++

I Maximum differences in axial

Axial Dead 245.0 130.0 245.0 245.0 245.0 l30~0 245.0 245.0 i response exhibited by fr~mes which became braced across -thelr

Load (kN) diagonals.

Max. Axial 310.0 219.0 308.0 300.0 308.0 282.0 307.0 303. 0 l Load (kN)

+ Axial response for bottom section of column.

++Value at spike

Notes: A - Open Frame C - Frame Wall-Gaps (I)

B - Frame Wall D - Frame Wa1l-Gaps(II)

99

FIGURES

100

I

~ -Ir--Horizontal Joint Gap

I 1-T

Vert ica I Joint Gap

Wall Element ~

Figure 2.1 Schematic Representation of the Wall Model

LoodinO Direction

5

Figure 2.2 Infilled Frame Specimen, Analytical Model Superimposed (after Fiorato, Sozen, and Gamble, Ref. [18])

20

4

4 I--' o I--'

10

Loodino Direction

,~ ~. - F~ 6 • •

Figure 2.3 Infilled Frame Specimen, Analytical Model Superimposed (after Fiorato, Sozen, and Gamble, Ref. [18])

I-' o N

16

4

12

a--......12

Figure 2.4

Loodino Direction

2 ..

13

20

4~---II -11

Infilled Frame Specimen with Opening, Analytical Model Superimposed (after Fiorato, Sozen, and Gamble, Ref. [18])

f-J o w

104

w

Figure 2.5 Idealization of a Three Story Frame Infi11 Wall

Figure 3.1

M ::: Moment B ::: Rototion My ::: Yield Moment

105

M

Bilinear Moment Rotation Relation for the Columns and Beams

. .,. \.,~ \', ,'\0 .

,I ..... , .... , ,.... ~.

'~

.,. As

,.~

SI

NI

Un = Normal Stress

As = Shear Stress

NI ; SI ; n ; m = Control Points For The Fai lure

Figure 3.2 Joint Element Failure Surface

m

<Tn

~ o 0'\

/ /

/

107

/

0'"1 10'"2 = Principa I Stresses

/ /

/ /

F, I F2 ; F3 j F4 ; F5 = Control Points For The Fai lure Surface

F~ = Max (F3 ; F4 )

/ /

/

Figure 3.3 Masonry Wall Failure Surface in Principal Stresses

Metz !e~erence Room University o~ Il1~o1s

Bl06 NeEL 208 N. Romine Str8et

Urbana, Illinois 61801

n x fT

FI

fT

8 - Angle of Principal Stress with Respect to Mortar Joints

0°

Figure 3.4

22.5° 45°

6

67.5° 90°

Uniaxial Tensile Strength of Masonry vs Angle of Principal Stress wi~h Respect to the Mortar Joints

n X f T

F2

fT

~ o 00

flm

F3

n x flm

8 - Angle of Principal Stress with Respect to Mortar Joints

I I --,- ---+- -- - -1', ---J flm­,

/ /

/

0°

Figure 3.5

, , " " " /

I /

'" ~I / " / " / " / I "1/

F4

'.J/ n x flm-

15° 30° 60° 75° 90°

8

Uniaxial Compressive Strength of Masonry vs Angle of Principal Stress with Respect to the Mortar Joints

I-' o 1..0

E E to

E E to V r--

E E o to I'-

,5mm I 3495 mm 3495mm 5mm - II. -4.. -+_ -{

M rigid link

M risrid link M

beam beam

gap gap

fj I gap gap §

r-l o U

Figure 5.1

Wall-Base May Become Discontinuous-No Gap-

Overall Dimensions of the Structure

I-' I-' o

50 60 57

24 37

Ei 15

2 3 4 ___ L /7k..\"7;<W7.I<.\,vY;<\.'(//\W7"'~~!II<\Yl7\\\(/;A\\Yn:\VJA\Y/A\V7I<\\(I;<'\.\(M.W/~Yll\W~II\'Y7J8\(1/<.\Y7AWI A~ Vl7.::'~.(7;~V"'WlM '(/ ~"'f:lh\ \. '\( / A,\. '( //

Figure 5.2 Nodal Numbering Scheme

J--I f-I f-I

@

CD

30 D-29

28 b. 27 26 b. 25

24 ~ 23

!

6) 32 34 D- b. 31 33

@

CL?)

@

0) ~ @

~-Gap Elements

® • 36 38 ~ ~ 35 37

@

! @ ~ @

39 /j.

40

41 ~ 42

43 ~ 44

4S I.'l 46

®

I ®

>sA t' j xc, 'HcAx ("n.,)( ( >,)( (» j( (j' j «, > >t (), A (j')( <" H(j >X(>;>.O'2CC» ..... C})hVjJCV; 1( 0;1( ... ;>1(0> 1 C O;LC<';X<;;x<;;x<}}}t", 2CO;X'" 'XCO; H C;;;

Figure 5.3 Frame, Wall and Gap Element Numbering Scheme

J-I ...... N

647 481 B 8122

8 Horizontal Joint [D Vertical Joint • Horizontal a Vertical Joint

49B

//AV71A:\·VA.\.vY;<\\{//\'\'VA.\,(7;<\:.ro;.\\(//\\\(/)(.\V~X\Y/~Y/7-W~YT~~V7~V7"W-/:A\,(/A"''Tr~~~7~TR\VT~~T~V7A'\. '{/;c.,.'\. V77

Figure 5.4 Joint Element Numbering Scheme

f--J f-1 W

5000.0

2000.0

1000.0

,-.. 500.0 0 w (f)

! 200.0

(3

~ 100.0 I XI X YXI XI *001 XI KI X 1_ XIIXI X I XI 0

9 w ~ 50.0

20.0

10.0

5.0

a~

0.01 0.02 0.05 0.1 0.2 0.5 1.0 2.0 5.0 10.0 20.0

FREQUENCY (H Z )

Figure 5.5 Elastic Spectrum for Sinusoidal Base Motion, 1 sec Duration, at 2% and 20% Damping

50.0 100.0

J-I J-I .J::--

:E: :too.

~ ~.3:1 t--Z

50. l..LJ ~ l..LJ C> -< ---1 D-en o. ~

1.6 ~ . 8 :1.8

• 4 c:::l 01· 1 . 1..4 . 2 .6 .

.--1 l..LJ 0:::

-50. TIME [SECONDS)

250.

:z o. ~ 01. .4\

.6 / .8 \1- :1.2 ).4 1.6

~ 2.

.2 / I--' I--' . 0\

U)

l..LJ 0::: -250. .--1 -cs::::: t--c::.:l J-

-SOD. TIME (SECONDS]

c..!) . 5

:z:. c::.:l t-t I---ct::: 0::: LW .--1 o. l..LJ c.....:> °1\ .eI .. 4 \ / . 8 :1.2 :1 .. 6 2 . c....:> .6 i. i.4 i.8 -cs:::::

l..LJ OJ -cs::::: en

-.5

Figure 5.7 Relative Displacement, Total Resistance, and Base Acceleration of the Open Frame

400.

:c 200. I :z ~

r- o. :z LU

t \ .8 / :1.2 :1.~ :1.6 / 2. :E:

~ Cl .2 .6 :1. 1.8 :::E:

-200. -21.5.3

-400. TI~1E [SECONDS]

200.

::z: ~

~ I I / I ~ I ,.....

0:::: o. \i. 74 ~8 ,.....

< o. 1.2 :1.6 2. -.....J LU ::r: Cf)

.6

-200. TI~1E [SECONDS]

400. T I

:z I~ / \ / ~ ~ ~

200. ---1 -c:::t:: !-i >< -c:::t::

O. I +-- I ~ --l O. .4 .8 1..2 :1.6 2.

. 2 .6 :1 . :1.4 :1.8

Figure 5.8 Base Moment, Base Shear, and Axial Load for the Left Column of the Open Frame

z: ~

0:: <: w ::c en

:E I

:z ~

I--:z L1...J :::::E c::::l :::c

:z:: ~

---1 -< ~ >< <

200.

o . 0,.

-200.

400.

200.

o. 0 ..

-200.

-400.

400 ",

200.

o. I

o.

Figure 5.9

22.9

1..2 :1.6 2. 1.. .4 1.~8

TIME [SECONDS]

~ ~

.8 1..2 :1.6 2. 00

.2 . 6 1. . 1..4 1..8

-21.4.8

TIME [SECONDS]

~aOg.~

I I I

.4 .8 1..2 1..6 2. .2 . 6 1. . 1..4 1..8

Center Shear, Top Moment, and Axial Load for the Left Column of the Open Frame

400.

:L: 200. I :z: ~

t-- o. ::z: LJ..J

Of \ .8

/ 1. 1..2 ~ 1.~ 2. :L:

c::::l ~.2 ~ .6 1.4 1.8 ::c

-200. --- -214.8

-400. TIME [SECONDS)

200. I ~22.9 r :z: ~

o. 01· ./

I~ L I ~ I I / I\: I I--' 0:::: .4\

.6 / \1- 14 I--'

-c:c: .8 1.2 2. \0

LJ..J .2

1.6 1 8 :I:

en

-200. TIME [SECONDS)

400.

::z: ~

200. '~ ---1 -c:c: H >< -c:c:

o. I -, I --~------~---~----~

o. . 4 .8 1.2 1..6 2. .2 . 6 1. • 1.4 1..8

Figure 5.10 Top Moment, Top Shear, and Axial Load for the Right Column of the Open Frame

200.

z ~

0::: o. ~ o. .4W .8

\1. :1.2 1.4 1.6 hs 2. LU

::r:: .2 .6 Cf)

- .6

-200. TIME [SECONDS)

400.

::E 200. I ::z ~

"' J-- I-' :::::z:: o. N LL.1 o. 1..2 :1.6 2. 0 :L: c::l .2 .6 1.4 :1.8 ::E

-200. -2:1.5.3

-400. TIME [SECONDS)

400.

:z: ~

200. --1 -c:t:: H >< -<

o. I I

o. .. 4 .8 1..2 :1.6 2. .2 . 6 1. • :1. .. 4 :1.8

Figure 5.11 Center Shear, Base Moment, and Axial Load for the Right Column of the Open Frame

R E S I S T A N C E

K N

300.

200.

:Lao.

-80. :to. -20. a,.

-200.

REL. DISPL .. ~~oo_

Figure 5.12 Load Displacement Response: Open Frame

70. 80.

I-' N I-'

5. ~

r-\5~2 ~

~ :z LU ~ LU c....:> o. -< -1 0\. \.4

/ .6 .8\ 1.6 2. 0-

Cf) . 2 ~ . 1.4 :1.8 t-i c:::J

· --1 LU 0::: -5.

TIME [SECONDS) 50.

c.....:> LU en '-.. :£: ~

>-.-- o. Or. .t I I \ I t. I ~L I I I j--I

t-i ·1 71- N c...:> .8 1.2 :1.6 2. N c::::l --1 .6 ~.4 ~.8 LU ::::-

· --1 W

-50.1 "'8"-48. :15 a:::

TIME [SECONDS) . 5 k),.4622

C!)

:z Cl t-i .---< a:::: L1.J O. --1

°1·\ .el .4 \ .6

/_8 i.2 i.6 2. L1.J c....:> 1. • i.4 1.8 c...:> -<

· en en -<

-.5

Figure 5.13 Relative Displacement, Velocity, and Absolute Acceleration of the Frame Wall

::c 5 • ::E ~512 ~ ::z: LLJ ~ lLJ C) o. 1/ \~ <C ---.J

/.6 .8\ :1.6 2. CL o. (f) .2 1. 1.B J---i Cl . ---' LJ..J 0::::

-5. TIME (SECONDS)

1000.

z ~

oJ

r I ~ I / \8 ,I I I ~ o. .2 \

.4 / N .

:1.6 2. I.J,) (J.) : = 1.2 L1.J

o. .6 1..4 1.B ~

--1 <C I--CJ I--

-:1000. TIME [SECONDS)

C!) . 5

:z c::::l H ~ <C 0::: W --1 o. lLJ c.....:> 01\ .eI . 4

\ .6 / .8 1.2 1.6 2. c.....:> 1. 1.4 1.8 -=:t:::

LJ..J en <C co

-.5

Figure 5.14 Relative Displacement, Total Resistance, and Base Acceleration of the Frame Wall

::z:: ~

~ -< t---i >< <C

-100.

400.

T I

200. ~1.8.5

_-99.57 TIME [SECONDS]

~ o. \,", /1 h / I I I I I

Or .8 i .. 2 :1.6 2 . .. 4

.2 .6 1. :1.4 1.8

-200.

Figure 5.15 Base Moment, Base Shear, and Axial Load for the Left Column of the Frame Wall

~ I

:::z: ~

J-­::z: l.L..l :::L: Cl ~

:z:: ~

fr:: -c: l..L.J ::c C/':)

z ~

---l ~ t-t >< -c:

:1..00.

50.

o. 0 ..

.2

-50.

50.

D.

Of '\.~ -50.

-:100.

400.

I 200.

a. o.

.2

-200.

Figure 5.17

3.5:1..

I I I I . 4 .8 1..2 1..6 2.

.6 1. . 1..4 :L.8

TIME (SECONDS]

I I I .4 .8 :1..2

/\ A /'\. :1... ~ 1..8 I-'

N (j\

-------." ~58.75

TIME (SECONDS]

--6-21.6.

.4 . 8 :L.2 :1..6 2 .. • 6 :1. . :1..4 1..8

Top Moment, Top Shear, and Axial Load for the Right Column of the Frame Wall

50.

~ , :z: o. ~

01· \ ~ .8~ 2. f-- .2 . 6 :1 . :1.8 z LJ.J :::E: c:::::::l -50. :::E

Y6S.S8

-:100. TIME [SECONDS]

:100.

E I

:z: o. ~

°1· \ .2 / .4 ~ ! ~2 :1.e 2.

f-- :1. :1.4 :1.8 I-' z N LLJ '-l

E Cl -1.00. :::E

~1.1.8.2

-200. TIME (SECONDS)

400. I ~~:1.5 :z: ~ 200.

-1 ct:: l-i >< o. ~

01. .2t' .8 :1.2 :1.6 2. . 2 .6 :1 . :1.4 :1.8

-200.

Figure 5.18 Center Moment, Base l1oment, and Axial Load for the Right Column of the Frame Wall

R E S I S T A N C E

K N

800.

600.

r-- p;; .......... PI }l }'

-4. -2. 0 .. 2. -3. -:1. :1.

-200.

-400.

-600.

REL .. DISPL .. MM -800.

Figure 5.19 Load Displacement Response: Frame Wall

4. 3.

f-I N 00

1 - t=0.030 sec 2 - t=O.035 sec 3 - t=O.040 sec 4 - t=O.045 sec 5 - t=O.050 sec 6 - t=0.055 sec

2

3

4

5

4

3

IL 5 15

7 - t=0.060 sec 8 - t=O.065 sec 9 - t=0.075 sec

10 - t=O.085 sec 11 - t=O.095 sec 12 - t=0.100 sec

16

13 - t=O.110 sec 14 - t=0.ll5 sec 15 - t=0.120 sec 16 - t=0.125 sec 17 - t=0.130 sec 18 - t=0.155 sec

Cracking

14

13

6

5

///I:\\(7;<\Y7./\\YY;<\'(/7\\'V;<.\Y77\\y7A\\(//\\V7/{,\~~'</AW70!<\,</~r~vT1.W1~V rA.~V77-,\W7,\'\rT~/.~,,·YT~'l7~VTR\VTA'(T~VTAW lA.'VTl

Figure 5.20 Cracking Sequence: Frame Wall from t=O.O to 0.155 sec

J-I N \0

t =0.045 sec. -Cracking

//Av/;'(\Y~~\V/<\VIx\\(JJ\\Y/;<WM\V/)\\Yl/WVmy)x..,,</)\w~/)"''l/)&nA<.V;MY;X'"JX\Y1AWJ;;:::.&~/;<SvIX~\W~;X\<'(7 A.\<'(JI

Figure 5.21 Crack Pattern: Frame Wall at t=O.045 sec

I-' W o

t = 0.060 sec. -Crocking

/M~/?\\f;).\V/<\WX\\(/J\.\Y /)(wJX\YIX\Y/XWmVM,\?I/<:'N~II"''<;k\vJ~ vIAy/XWJX\'\!1 AWJ;;;:~';v~1 AW7J.\\!tll}N-()~IX\«/ ;<.."V/l

Figure 5.22 Crack Pattern: Frame Wall at t=O.060 sec

I-' W I-'

','1.,

,-

t = 0.075 sec. --- Cracking

/~V~Y77\\Y"''';{\Y//\''~/'{\Y/~\VT7-'Y/A\ y/~v/ ..(\v/ A\'<'7~V /AV/A''(/ ,.<.,''<''Jf0V I AW/~V" A' V/A'V/ ,,'V" A' V/A"V/~Y /AVT~VT~~~VT7\\~'l71

Figure 5.23 Crack Pattern: Frame Wall at t=O.075 sec

f-I LA> N

0'1

U ~ ~ g o ~ _u

~ I

133

Metz Re~erence Room University of Illinois

:BlD6 HeEL 208 N. ,Romine straet

Urbq.na. Illinois 61601

c

oW

oW ,...

..J

;.:

1 - t=0.275 sec ·8 - t=0.330 sec 2 - t=0.280 sec 9 - t=0.335 sec 3 - t=0.290 sec 10 - t=0.340 sec 4 - t=0.295 sec 11 - t=0.350 sec 5 - t=O.300 sec 12 - t=O.355 sec 6 - t=0.305 sec 13 - t=0.380 sec 7 - t=O.325 sec 14 - t==0.555 sec

-Cracking

8

7

3

2

3

//Av/)t;\y»\~).\\(/A\f;)'\y/"'\YI;<\\(J}\\'7I/<\ymy/)\\Y7A\Ym:w;{,\(Ji«'(~Yl)..wlM'nx:, VIA ,'VX, 'lIMY! AWJ)$9~) A-. ...... v7~)\S.8)~M:WI ~,yl/

Figure 5.25 Cracking Sequence: Frame Wall from t=0.275 to 0.555 sec

I-' UJ +:--

100. ::E ::E

l---l- ~2.12 ::z:: 50. L1.J

::E LJ..J <:.....)

< --.J 0.... or +-U) o. l--i c=l • 4 \ / .8 :1~ :1.2 ~ 2 • . .2 .6 :1.8 --.J LJ..J 0:: -50.

TIME [SECONDS] 500.

<:.....) LJ..J U) ........... :c ::E o. >- OJ_ \4 /6 .8 \ /1_2 ~ 2 • t-- . 2 :1 . :1.4 :1.8 f-'

w l--i VI <:.....)

C:l _J

-500 _ ! "'Ef-522.2 LJ..J ::::-

--.J LLJ a::::: -:1000.

TIME (SECONDS) :1. T c:..!) Q.7673

:z: Cl ~ l--< n::: LJ..J o. --.J

°1· \~.4 ~ :1.2 ----~.6 2. LLJ <:.....) .6 :1 . :1.4 :1.8 <:.....)

< . U) en -<C

-:1.

Figure 5.26 Relative Displacement, Velocity, and Absolute Acceleration of the Frame Wall-Gaps(I)

L: 1.00.

L:

..-:z:

50. ~ .r&E2.1.2 w L: LLl C'> < -.J D-C/) a. H Cl

0\. . 4 \ / .8 1.~ 1..2 ~ 2. . .2 .6 .4 :1.8 --1 w 0:::

-50. TIME (SECONDS)

1.000.

::z o. . 8 ~ 1.6 ----- 2. ~ 01. .~ 1. . :1.4 1..8 !-' . 2 w .

0'1 C/) LLJ n::: -1.000. --1 <t: i--Cl i--

-2000. TIME [SECONDS)

c...!:> . 5

:z: Cl H i--<t: n::: LLJ ---1 o. LLJ O:\;J \ / .8 1..2 1..6 2. c.....:> . 4 c....:> .6 1. . 1..4 :L.8 -ct::

LLJ (/)

-<C en

-.5

Figure 5.27 Relative Displacement, Total Resistance, and Base Acceleration of the Frame Wall-Gaps(r)

400.

:::E 1 O~:15 • I 200 .. ""

,----::z: ~

..- o. z:: LW Or L:

\/ .. ~ / .6 \ / 1.. 1.". 4 / 1..8 c::::l :::E

-200.

-400. TIME [SECONDS)

200.

:::z:: ~

o. ofLbJ\

I L I \ I I /1 I~ I 4 I-' 0::::

.6 / \i. i.2 / "" /2. w -ex:: .8 '" lJ..J 1. .. 6 ~ ::r:: :1..4 .8 en

-200. TIME (SECONDS]

400.

::z: ~ ~,

200. --' -ex:: ...-; >< ~

o. .- . -I . ,- - J - , ,--o. . 4 .8 1..2 1..6 2 .

.2 .6 1.. 1..4 1..8

Figure 5.28 Base Moment, Base Shear, and Axial Load for the Left Column of the Frame Wall-Gaps(I)

:z: ~

0::: -< UJ ::r: Cf)

E I

:z ~

r-:z: L1.J :'E c:l :E:

:z ~

-.I -< J--i >< --<

400.

200.

o.

Or -200.

TIME [SECONDS] 400.

200.

o.

Of \ .2 j2i4.9

.6

-200.

:L.8 1.2 2. :L.

:1. :1.

-400. TIME· [SECONDS]

400.

200. fi

o. L L ______ I _____ 1 ______ -'-____ ----' I I -----r------ ---1 -- I - ---I

o. .4 .8 :1..2 :1.6 2. . 2 .6 :1 . :1..4 :1..8

Figure 5.29 Center Shear, Top Moment, and Axial Load for the Left Column of the Frame Wall-Gaps(I)

..... w ex>

400.

~ 1 O,z:t4.8 J 200. / '\. I

::z:: ~

J- o. :z: LJ..J 01· '\ /4 \ .8 / 1..2 ~.4 i) ::E C) .2 .6 i. :1..8 ~

-200.

-400. TIME (SECONDS)

200.

::z:: o. ~

0\_ -~ _61 .8

~ i.2/ :l.6 ~2_ ~ a:::: .2 :1..4 .8 w

\.0 <C LJ..J ::z:: -200. CJ:)

-400. TIME [SECONDS)

400.

:z ~

200. ---1 < f--i >< <C

o. o. .4 .8 :1.2 :1.6 2.

. 2 .6 i . 1..4 :1.8

Figure 5.30 Top Moment, Top Shear, and Axial Load for the Right Column of the Frame Wall-Gaps(I)

200.

:z ~

n:::: o. -< o. 'V .~

.8 \:1. / i'/ 1.4 i.6 ~:L /2. L1.J

:I: .2 U)

-:125.8

-200. TIME (SECONDS]

400.

x: 200. I :z: ~

t--

O. t~' I I :z:

\ J ~ 1-1

LL.J o. .4 . 8 1.2 2 . .p... :l::: 0 Cl

-200. .2 -214.7 . 6 1 . 1 . :L.8

~

-400. TIME (SECONDS)

400.

:z: ~ 200.

---1 -c: ...., >< o.

01·

I I I I I -c: .4 . 8 :L.2 i.6 2 .

.2 .6 i . :1.4 i.8

-200.

Figure 5.31 Center Shear, Base Moment, and Axial Load for the Right Column of the Frame Wall-Gaps(I)

R E S I S T A N C E

K N

-60.

/

REL. DISPL ... MM

Figure 5.32

:1000.

750.

500.

2 50~.

-20. 0,.

-500.

\ \

- 750. \

\ I

-1000.~ -:1250.

Load Displacement Response:

? -; 40.

20. / 60.

i I

I I

I I

___ J Frame Wall-Gaps(I)

I--' ~ I--'

7

9

13

7

IA 1'&

1 - t=0.090 sec 2 - t=0.095 sec 3 - t=0.100 sec 4 - t=0.105 sec 5 - t=0.110 sec

6 - t=0.125 sec 7 - t=0.145 sec 8 - t=0.170 sec 9 - t=0.180 sec

10 - t=0.190 sec

8A 8. 6.& tc 64 =..-=

11 - t=0.200 sec 12 - t=0.205 sec 13 - t=0.305 sec 14 - t=0.315 sec

.A Closed Gap j Yield Joint - Crocking

E23 Foi led

~

8

If"[ - .... - .&14 2 4 3 II 31 .14

8

7

//1&;:;:;:<\vx~W?\~w.l<\VJx\WI<\y;;<\V/.1:\YI).\\VX\Vmm\.\(IX.W~Y7/\w/m/~vlMVJXW1XW1AWJ~Y~1AWIX.~W/~7X\;'(I~"Y/1

Figure 5.33 Cracking Sequence: Frame Wa11-Gaps(I) from t=O.O to 0.315 sec

I-' ,J::-. N

A A

t = 0.095 sec A Closed Gap j Yield Joint

-Cracking

//AV1)(\y;~\{//\\fJ;<\m\W?\<\(J}\\'7/AWJ~v/)\\YJXWmv/,{,<{;X'N~1J/\'''<m:R}XSJJMV;X,yJX\Y1A<:N~<.(/J$8;kNlX.~~M'¢"<7 A.WlI

Figure 5.34 Crack Pattern: Frame Wall-Gaps(I) at t=O.095 sec

f-I +:-­w

A ...

t = 0.100 sec.

• Closed Gap; Yield Joint - Cracking o Failed

//,I(\v7AY/J\.\'IIJ-;<\\{~\Vl\\YII<\\(Ii<\Y//(wns.v/)\\\(/k&m<'(/7..\\(IXW7M7XwJk'9IX.'0JIA\VlX' «1M\1JA~YIJ:.'YI,<..,' V7X~"X" Y.r~~1 ",,,«/I

Figure 5.35 Crack Pattern: Frame Wall-Gaps (I) at t=O.lOO sec

j-1 .f:'­~

A A

t = 0.110 sec.

• Closed Gop i Yield Joint -Crocking

~ Foiled·

~V/..;~"\";<\\{~I(\YJ)!\V71<\\(J}\\Y/I<\V7&IX\Y//(\ymVM,'<7X'0.(~\,IAWIJ0N/~V)A\v/)\.,<.(/)0..Y7A~)A'lI/"v7X<!f..1J/\.\<'(~7X\'\()""YJ)

Figure 5.36 Crack Pattern: Frame Wall-Gaps(r) at t=O.110 sec

~ +:-­Ln

2

AI AI

A2 A2

AI AI

A -~

1- t = 0.145 sec. 2- t =0.180 sec. Closed Gap j Yield Joint Cracking Failed

2

2

/M\\>;;<\VJ\~Y~W7;<\V/I<\YM\Y//(\VlMJ;(\\(/XW1;e\.y);{,<!IX\«/~I/.s\(/k0flJ.\.,VI/,\\(lX',<7I«vlA.WmY~/A.'V/X<!tol\W~JX."'.;'(/~

Figure 5.37 Cracking Sequence: Frame Wall-Gaps(I) from t=0.14S to 0.180 sec

....... +:-­(j"I

r) t:.

1 - t=O.425 sec 2 - t=O.450 sec 3 - t=O.460 sec I, - t == 0 . 46 5 sec

.5 - t=0.480 sec 6 - t=O.500 sec 7 - t=O.510 sec 8 - t=0.535 sec

.. Closed Gap; Yi eld Joint - Cracking

~ (4 )

Failed

Gap Has Opened

7£ 7£

2

2

6

1(8)

/~V7AY~~A\WI(\y7;<\Y7I\W7A\Y7AW77;\V~WAWTA\'{7;{~'7A\'(~~7A..W7J-.W77"\(7MY/AW/~770...V~~'Xr~/\,\'(/~''(//

Figure 5.38 Cracking Sequence: Frame Wall-Gaps(I) from t=0.425 to 0.535 sec

r-o .po "-J

20. ~9.4 ::::E:: ::::E::

r-:z: LlJ ::::E:: LlJ '-..) o. --< .-...J

01· \4 16 .8 \ / 1..2 ~ 2. C-. 2 :L • 1..8 en 1.4

J--f c::::l

· .-...J LJ.J 0:::: -20.

TIME [SECONDS] 500.

'-..) LJ.J Cf)

'-- 250. ::::E:: :::c

>-+~ I :::::=:=-=- --=r= =r===- =::::::::::J t--a r- 1/ I~ I / \~ (. I

t--i o. .2 '\ ·41 /1. .,J::--

'-..) 1.2 :L.6 2. 00 c:::l .8

1..4 1..8 o.

.6 .-...J LJ.J ::::.>- -250. · '8'-286.8

.-...J LJ.J n::: -500.

TIME· [SECONDS) 1.. T 0..841.2

c.!)

:z: Cl J--f r-<t:: 0:::: LJ.J o. .-...J

01· .2r .4 ~6 r· s ~.2 1..6 2. LJ.J '-..) i. 1.4 i.B C-.:)

<t::

· Cf:J CJ:l <t::

-1..

Figure 5.39 Relative Displacement, Velocity, and Absolute Acceleration of the Frame Wall-Gaps (II)

::E: :E:

I--:z: w L: W C)

<X:: ---.J (L Cf) I--f c::::l

---1 W n:::::

:z: ~

U) lL.J 0:::::

---1 <C r-C)

r-

c..!)

::;z: c:::l J---1 }--<X:: n:::: l..LJ ---1 l..LJ C) C) -<t::

lL.J U)

<C co

20. ~9.4

o. 01_ \4 /.6 .8 \ / ~.2 ~~ 2.

.2 :1 • ~.4 ~.8

-20. TIME (SECONDS)

2000.

O. Or . 2 \ / .6

~ ~I ----.f '\Fu __ n-1--u- (U______.- '--r----~

.8 ~.2 ~.6 2. ~. ~.4 ~.8

-2000. TIME (SECONDS)

. 5

o. 01\ (\ I I I I I

,-- r ,- , ,

.8 ~.2 :1.6 2. 1.. :1.4 ~.8

-.5

Figure 5.40 Relative Displacement, Total Resistance, and Base Acceleration of the Frame Wall-Gaps(II)

,...... ~ \0

400.

:L: 200. I z: ~

t- O • z: lJ...J

Of ~ . 4 ·V..1 ~ 1.6 ---------2 . :£:

c::J 1. 1.4 :1.8 ~

-200.

-400. TIME [SECONDS]

200.

T ~ .r€>.1.1.7.7 100.

z: :::::::..::::

o. 01. \. ~ I' . ~ \ 7 1.12 ~ J Ck::

,..... -< 2. V1 lJ...J 0 ::c en

-100.

-200. TIME [SECONDS)

500. I z --------- ~ .. g ~ 250.

~ ~ l--i >< o. --<

0 .. . 4

.6 \/

.8 :1.2 :1.6 2 . .2 1. 1. .. 4 :1.8

-250.

Figure 5.41 Base Moment, Base Shear, and Axial Load for the Left Column of the Frame Wall-Gaps(II)

::z: ::::..c:::

0:::: -r.: l..J.J ::r: Cf)

:z: ::::..c:::

--1 -t:: J-i >< -t::

200 ..

o. a,.

-400.

500.

250.

o.

01" -250.

Figure 5.42

29.8

.8 1..2 2. 1. • ~.4 :1.8

TIME [SECONDS]

~.!-I.g

Ir------______ ---

. 4 .8 1..2 1..6 2. .2 .6 ~. 1. .. 4 1..8

Center Shear, Top Moment, and Axial Load for the Left Column of the Frame Wall-Gaps(II)

:z: ~

-400.

200.

iOO.

~ o. LU :r: U)

:z ~

-:100.

-200.

500.

250.

~ o. t--; >< -<

-250.

-500.

a ..

D .. .2

:1.6 1..4 i.B

TIME [SECONDS]

:1.8.5

:1..2 :1. :1.4 :1..8

TIME (SECONDS)

~~.2

1..2 :1.6 .6 :1.4 :1.8

Figure 5.43 Top Moment, Top Shear, and Axial Load for the Right Column of the Frame Wall-Gaps(II)

2.

2.

f-A In I'-)

:;::::: ~

n:::: -< l..LJ ::r:: en

::::E: I

Z ~

I--::z: l..LJ ::E: Cl :::E:

z ~

--.J <t:: 1--1 >< <t::

200.

o. 0,.

-200.

400.

200.

o. 0,.

-200.

-400.

500.

250.

o. 0,.

-250.

-500.

Figure 5.44

L __ ~~ ___ ..4 .~-----~o ----.

:1.8 2.

TIME (SECONDS)

1 .. 6 --2. 1 .. 1.4 :1.8

-214.1

TIME [SECONDS)

.2

:1..2 1.6 2. .6 :1.4 1.8

Center Shear, Base Moment, and Axial Load for the Right Column of the Frame Wall-Gaps(II)

I-' In LV

R E S I S T A N C E

K N

-20.

REL. DISPL .. MM

Figure 5.45

1.500.

1.000.

500.

0,. 1.0. -5.

-500.

-1.000.

-1.500.

Load Displacement Response: Frame Wall-Gaps(II)

20. 1.5.

I-' IJl ~

12

I. I.

1 - t=O.090 sec 8 - t=O.130 sec 2 - t=0.095 sec 9 - t=0.145 sec 3 - t=O.lOO sec 10 - t=0.150 sec 4 - t=O.10S sec 11 - t=O.185 sec 5 - t=O.110 sec 12 - t=O.190 sec 6 - t=O.115 sec 13 - t=O.235 sec 7 - t=O.120 sec 14 - t=O.260 sec

A Closed Gop; Yield Joint - Crocking

All ..... .... ... .... =-tAli 12

4 3

//AVJAYIA~~\V7'\VI)...\\(I/(\'7//<.\Y7J:.\Y7J\\'l.(?;<W7I«'<(,%<.'\\(7X"N~n.w/k.."I"'0JI/,\\()X' «IXW? AWm;.«~j~, V/X~MR~JX\"<) A..<.:'/ll

Figure 5.46 Cracking Sequence: Frame Wal1-Gaps(II) from t=O.O to 0.260 sec

t-' Ln Ln

A A

t = 0.095 sec A Closed Gap; Yield Joint

-Cracking

//'AVlAVx~?\\f?I<W;<\V/:x\m\\(I?IWlMJ/2\\VXWmWA"'..(jX'N~Mw;MIl\\. vjAvjx\.'\(IXS\!IA"&m-Yik8IAWlXj~!m'W)~~I/X\<{1 ~w»

Figure 5.47 Crack Pattern: Frame Wall-Gaps(II) at t=O.095 sec

r..a 1I1 0\

u Cl,) en Cl.

157

lC)~0'I 2~·~r----------~--------~ c5 ~ ~ II E ~ -uU

~I

Metz Reference Room University of Illinois

Bl06 NeEL 208 N. Romine street'

Urbana, Ill~oia. 618~

(j Q)

CJJ

l.I)

o r-i

o II

.j...J

C l-l Q)

+..J +..J co

p..

~ (j

co l-l

U

• A

t=O.145sec • Closed Gap; Yield Joint

-Cracking

/~\VA Y7A\V";K«~\VA\V/X,\(l/\\y //\\VI:«\VJ)\'YMWll<\y;;{,<'(IX:'N~/I\WI)" .... n", vlM'l))"', «17\\\1) Aw/J&'.(/J&.(Jk\.v7X<!'tnx\~7I\\«/ ~,,«»

Figure 5.49 Crack Pattern: Frame Wall-Gaps(II) at t=O.145 sec

~ \Jl 00

1 - t=0.290 sec 6 - t=0.335 sec 2 - t=0.310 sec 7 - t=0.340 sec 3 - t=O.315 sec 8 - t=0.345 sec 4 - t=O.320 sec 9 - t=0.350 sec 5 - t=0.330 sec 10 - t=0.355 sec

~

11 - t=0.360 sec 12 - t=0.365 sec 13 - t=0.370 sec 14 - t=0.380 sec 15 - t=0.390 sec

• -~ ( 9)

16 - t=0.405 sec 17 - t=0.455 sec

Closed Gap; Yield Joint Cracking

Failed

Gap Has Opened

14. 14.

6 6

=w=: AI (9) ~ A4(II)

16. 16....... "..... ( ( ( ( ( ( (I ( ( ( ( ( ( ~ 16

~V7Ay~y/?\\f;;<\Y7/.\VIX\YJ}\\Y/)(W7M/)\\Y/xwm\(//{\'<7J0N)~N/}"'VMJ)\'..v7A\\(;A\\(1MYIA"&m<'(;~1 ~SvlX~7}\\~7X,\"(T~VIl

Figure 5.50 Cracking Sequence: Frame Wall-Gaps(II) from t=0.290 to 0.455 sec

,...., In \0

c..n

z: c:l ~ r­< ~ L.U ---1 L.U c.....:> C)

< c:l :z: :::::> c:l

. 5

a .

~ -.5

C) 500. L.U en ............ ::E: ::E:

>­I---

~ o. c:::l ---' LLI :::::>-

c:::t :z:: :::::> c:::l

~ -500.

::E 200 . ::E

J­:z:: LL.l ::E L.U c.....:>

:s o. CL (f) ~ c:::l

c:::l :z:: :::::> c:::l

~ -200.

0,. 40. 50.

TIME (SECONDS)

0,. 35. 45. 55.

-384.1.

TIME [SECONDS)

Figure 5.51 Ground Motion for the E1 Centro Record, May 18, 1940, SOOE Component, Scaled to 0.4g

60.

I--' ~ o

5000.0

2000.0

1000.0

r-.. 500.0 0

~

l ~ 200.0

~ 100.0 I ;XJ A IttXJ AI x£?ol XI tx:1 X I~XI \Xl X I XI 0

~ '-00 ~ .).

EOIO

tOIO

5.0 0.01 0.02 0.05 0.1 0.2 0.5 1.0 2.0 5.0 ~o.o 20.0 50.0 100.0

FREQUENCY (HZ)

Figure 5.52 Elastic Spectrum at 2% and 20% Damping for the E1 Centro Record, May 18, 1940, SOOE Component, Scaled to 0.4g

I-' (J\

I-'

:::E: :::E:

J-­::z:: l.J..J :E: L.&..J (....) a: -.J (l-en ......... Cl . -.J l.LJ ~

'-.:l l.J..J en .......... ~ :E:

>­J-­......... (....) c::l -.J l.J..J ::::-. ---l La.J ~

'-!l

::z: Cl ......... J--0: ~ LW ---l LU (..j (..j cc

(j) r::n 0:

2"'. 15. lQJ. 5. 0.

-5. - 1 C2'J • - 15. -20. -25. -30 . -35. -4QJ.

30"'. 25C2J. 2QH2J • 150. 100. 50.

CZl. -50.

-100. -150. - 2£2"2' • -25QJ • -300. -350.

2.5 2.

1 • 5 1 • .5 0.

- • 5 - 1 •

- 1 .5 -2.

-2.5 -3.

4. 8. ~. 3.5 8.5

- 36 . 15

TIME (SECONDS)

TIME (SECONDS)

8. 9. 3.5 7.5 8.5

-2.75

Figure 5.53 Relative Displacement, Velocity, and Absolute Accele~ation: First Story

1 ~. 9.5

10. 9.5

I-' (J'.

N

::E: E

I-­La-...-~ c:::J

>­a.::: C) ...­U) Q:: LW I-­::::.2:: t--t

:z: ~

. U) l.J..J ~

-..:l cc t-­c:::l t--

(..!J

:z: C) ...­t-­cc t:l£: l.J....J -.J l.J....J ~ ~ cc >­~ c:::::::l t-­t:.r.:J

20. 15. 1~. 5. 12J.

-5. -10. -15. -20 . -25. -3121. -35. -40.

51210. 4121121. 300. 200. 100.

0. -1QH?l. -200. -30121. -400. -500.

2.5 2.

1 .5 1 . .5

12J. - .5 - 1 •

-1 .5 -2.

-2.5 -3.

4. 8. 9. 3.5 8.5

-36.15

TIME (SECONDS)

9. 8.5

TIME (SECONDS)

8. 9. 8.5

Figure 5.54 Interstory Drift, Total Resistance, and Story Acceleration: First Story

9.5

9.5

9.5

10.

10.

10.

f--J (J\

LV

f;;~ E C' .... ) I C' .... ) T

500.

400.

500.

200.

~ '-;7- 7,/' ~' - 40 . t~ -30 . ~ · - 1 (ZJ • 10. 20 . E-35. -2 -. - . 5. 15.

I< N

-400.

STOR

-500.

Figure 5.55 Load Displacement Response: First Story

j-I Ci' +:--

:E: ~

.­:z: LW ~ LW '-.) a: -..:I CL C.f) .......... c::::J

-.J LLJ c:L!::

'-.) LLJ CJ) .......... ~ ~

:::­r­.......... L) c::::l -.J WJ :::>

-.J LLJ c::u:

c...!J

::z: C) t--I

I-­a: ~ WJ -..:I l.....W '-.) '-.) a:

CJ) co a:

2CZJ. 15 • 1CZJ • 5. CZJ.

-5. -10. -15. -20. -25. -30. -35. -40.

3C2JC2J. 25CZJ. 2CZJ0 • 150. 1 QHZJ •

5C2J. 0.

-50. -100. -150. -200. -250. -300. -350.

3.5 3.

2.5 2.

1 .5 1 • .5 0.

- .5 - 1 •

- 1 .5 -2.

-2.5 -3.

-3.5

3.5

3.5

4.

TIME (SECONDS)

-349.5

TIME (SECONDS)

8.

-36.58

8. 7.5

g. 8.5

8.5

9. 8.5

Figure 5.56 Relative Displacement, Velocity, and Absolute Acceleration: Second Story

g.s

9.5

1 0 •

10.

I--' 0'1 111

::c ~

t-­w.... I--f

~ CJ

>­~ Cl t-­(f) ~ W t-­:z .......

:z: ~

. C./"J l..J....J c.c: --..:J a: t­o I--

c::..o

:z: c:::a ....... t­o: Co! l..a..J --..:J l..a..J U U a: >­~ Cl t-­C/)

1 .4 1 .2

1 • .8 .6 .4 .2 l2J.

- .:2 -.4 -.6 -.8 - 1 •

-1.2 -1.4

700. 600. 500. 400. 3CZJ0. 200. 1QJQJ.

0 . -lQJ0. -200. -3QJCZJ. -40CZJ. -S01Z). -61Z)0. -7QJQJ.

3.5 3.

2.5 2 .

1 .5 1 • .5 0.

- .5 - 1 •

-1 .5 - 2 •

-2.5 -3.

-3.5

g. 8.5

-1.393

TIME (SECONDS)

9.

TIME (SECONDS)

8. 9. 3.5 7.5 8.5

Figure 5.57 Interstory Drift, Total Resistance, and Story Acceleration: Second Story

g.S

9.5

g.5

10.

1 !ZI •

lQJ.

I-' (J\ (J\

R E S I S T

-1.4C -1. -.6 E- 1 • 2 - .8

K N

7QJQJ.

6QJQJ.

500.

400.

3QJ0.

-400.

-500.

-600.

STORY D~IFTr MM -700.

.6 .4 .8

Figure 5.58 Load Displacement Response: Second Story

1 • 1 .2

1 • 4

I-' 0\ '-.J

~ 4CZJ • ::E: 30 .

..- 20. :z: ~ 10. t3 0 . S - 10 3l-. 4 . V \ / 5 . \.- / 6. V \ I 7\ I ~ 8. 9 · 1 Ql • 8=s -20: + 3.5 4.5V 5·V 6.5 .5 \ ~.5/,\ /~ t--f

.C) -30 . .-.:i - 4 0 . ~ -50. e-4S.98

TIME (SECONDS) C) 400. LLJ ~ 300 . ~ 200.

100.

>- ~ l "n o· I C! on \ N ~ 1 W \ . ~ I ~ I V\J3 ! , ! • , ,! ~ ~ IU.~VVUVV"JVU I V 'qJ \7 t .-g -100. 4.~.. · J •• 0. ~ d -200. >- - 3CZHZJ • ~ -400. LLJ c:::l.C - 500 .

TIME

c....!J 1 .75 1 • 5

1 .25 ~ 1. t--f .75 ..- .5 & .25 L.L.J 0 . d - .: ~~ 3f. V - - .• ~\I~'~l~UIIII'II"IIRI'l" It r'tV' Vi . \ I \.J . 8 . 9 • 1 Ql • ~ _ . 75 t 3 . 5 ~ 4 lH1!111' 5 . 5 f5 • 5 7 • 5 8 . 5 9 . 5 CC - 1 .

. - 1 .25 c:n -1.5 22 -1 .:~~ J.. (!)-1 .962

Figure 5.59 Relative Displacement, Velocity, and Absolute Acceleration: Third Story

d "" B. i~

:::E :::E

.­LL. J---I

~ CJ

>­~ Cl I-­(f) ~ WJ )--:z: t--1

:z ~

(f) WJ ~

~ a: I-­C)

I--

i~ <! (() CD ,1-

,"', ~5 t;:J U:l W f--'. ~:d

)"j ~"J <;""\_ CD __ " ~_J c.q \4) :z: :"~ (J) CD c::J • ' , I 0 Ii J---I " :2; H;, CD I-­:t", C1 t;J a:

l?'.1 H Q Cl.£: t- 1 t-t 10-.1 (I) WJ ; ~-' t:d ~

do t;gt3 '1> 0 at:...) d: tt a:

" >-~ Cl I-­(J.J

14. 12. 10.

8 . 6. 4. 2. 0.

-2. -4. -6. -8.

-10. - 12. -14. -16.

600. 500. 400. 300. 200. 100.

0. -100. -200. -300. -400. -500.

1 .75 1 .5

1 .25 1 •

.75 .5

.25 0.

-.25 - .5

-.75 - 1 •

-1 .25 -1 .5

-1.75 -2.

3.5

Figure 5.60

4. 4.5

TIME (SECONDS)

8. 9. 7.5 8.5

TIME (SECONDS)

8. g. 8.5

Interstory Drift, Total Resistance, and Story Acceleration: Third Story

~. 9.5

10. g.s

10. g.s

........ 0\ 1.0

R E S I S T A N

600.

500.

400.

300.

200.

E -15. -10. -5. ~ '-12.5 -7~~.5 K N

-2QJQJ

-3QJ0

-4QJQJ

STORY -500.

5. 2.5 7.5

1FT, MM

Figure 5.61 Load Displacement Response: Third Story

10. 12.5

15.

f-l -....,J

o

:c J

:z:: ~

t-­:z: LW :::E C) :c

:z: ~

~ <:r: LL.J =r: Cf:J

~ ~

--..J cc ~

>< a:

250. 200. 150. 100. 50.

0. -50.

-100. -150. -200. -250.

140. 120. 100. 80. 60. 40. 20.

0. -20. -40. -60. -80.

-100. -120. -14QJ.

800. 750. 700. 650. 600. 550. 500. 450. 400. 350. 300. 250.

3.5

3.5

3. 3.5

4.

TIME (SECONDS)

4.

TIME (SECONDS)

4. 5. 6. 7. 8. 9. 4.5 .5.5 6.5 7.5 8.5

Figure 5.62 Base Moment, Base Shear, and Axial Load for the Left Column: First Story

1 QJ • 9.5

f-I -.....J f-I

30. 25. 20.

:E 15. I

:z 10. ~

5. I-- 0. :z:: -5. LW =e:: -10. C) =e:: -15.

-20. -25.

140. 120. 100. 80.

:z: 60. ~ 40.

20. ~ 0. a: -20. LJ.J ::r: -40. CJ') -60.

-80. -100. -120. -140.

260. 200.

~ 15QJ.

I li2'0. :z: ~ 6QJ.

I-- 0. ::z: -50. LLJ ~ - lQJ0. C) ~ -150.

-200. -250.

Figure 5.63

8. g. 8 ti.s

TIME (SECONDS)

3.5

TIME· (SECONDS)

b.5

Center Moment, Center Shear, and Top ~1oment for the Left Column: First Story

1 QJ •

~ --..J tv

L: I

:z ~

t­:z: l.LJ L: c::) L:

:z: ~

~ a: LJ...J :::r:: CJ'J

:z: ~

---l a: I--t

>< a:

250. 200. 150. 100. 50.

0. -50.

-100. -150. -200. -250.

141Zl. 120. 100. 80. 60. 40. 20.

0. -20. -40. -60. -80.

-lQJ0. -120. -140.

800. 750. 700. 650. 600. 550. 500. 450. 400. 350. 300. 250. 200.

3.5

3.5

3. 3.5

4.

TIME (SECONDS)

4.

TIME (SECONDS)

4. 5. 6. 7. 8. C1. 4.5 5.5 6.5 7.5 8.5

Figure 5.64 Base Moment, Base Shear, and Axial Load for the Right Column: First Story

10. 9.5

f-l -....,J w

70. 60. 50.

L: 40. I

:z 30. ~

20. I-- 1QJ. :z: LJ-I 0. ::E c::)

- 1 QJ . ~

-20. -30.

i4~.

I :z: 4 ~ 2 ~ -2 cc -4 LL..J

:I~ ::.c U)

= i 250. 200.

L: 150.

I 100. ::z: ~ 5QJ.

t-- 0. :z -5QJ. L&...J :::E: -10121. 0 L: -15121.

-20121. -25QJ.

Figure 5.65

8. 7. 8.5 g.5

TIME (SECONDS)

3.5 9.5

- 162 • 1

TIME· (SECONDS)

Center Moment, Center Shear, and Top Moment for the Right Column: First Story

J-I

'" +:--

::E. I

:z ~

I--z:. La...J ::L: C) ::L:

:z: ~

~ a:: LW :::x:: U)

:z: ::::..L

-.J a: .-->< a::

50. 40. 30. 20. 10. 0.

-10. -20. 3.5 -50. -40. -50. -60. -70. -80.

TIME (SECONDS) 40. 35. 30. 25. 20. 15. 1 0 • 5. 0.

-5. -10. -15.

4. 3.5

-20. -25. -50.

TIME (SECONDS)

323..7

3. 4. 5. 6. 7. 8. 9. 3.5 4.5 5.5 6.5 7.5 8.5

Figure 5.66 Base Moment, Base Shear, and Axial Load for the Left Column: Second Story

10. 9.5

..... '" lJ1

~ I

::z: ~

~ ::z: LLJ ::E: C) :E:

::z: ~

at: a: w....J ::r: CJ';J

::z: ~

-.J a: J---t

>< cr:

80. 70. 60. 50. 40. 30. 20. 1 ~ • 0.

- 1 QJ • ~. 4. -2QJ. 3.5 -30. -40. -50.

TIME (SECONDS) 30. 25. 20. 15 • 1 0 • 5.

-~: ~. 3.5~~ ~ J _~67Tl-=-r 71· 1~~--/ 8~\ --Z~-;;;. \_ L\ 1~ · I-'

" (J'\ -1tZ1. -15. -2tZ1. -25. -30. -35. -40.

TIME (SECONDS)

3. 4. 5. 6. 7. 8. 9. 10. 3.5 4.5 5.5 6.5 7.5 8.5 9.5

Figure 5.67 Base Moment, Base Shear, and Axial Load for the Right Column: Second Story

::E: I I

z: ~

t--

~ il~: 5£. 4. 5.

" I \ fj.' \ I \ 71. \ IV 8. ...........,

~9. '¥QI • :z: l..W 3.5 4.5 :::E c:::;J :.E.:

TIME (SECONDS) 100.

8Q1. 6Q1.

::z: 4Q1. ~ 20.

-2~: { '11 • • wrI"~ & til- \ I ~ I / 'v' \ / "'\d'::>~ / \ f--J ~

1 I ~\ ? "" a::: 4 • ·"'~"-5:pn- ~ 7 ~v v. .... v v .;,. "" LW ::c 3.5 (f) -4Q1. 4.5

-6Q1. -8Q1.

-10Q1.

TIME (SECONDS) 18Q1. 160. 140.

::z: 120. ~ 100. -..J

80. cr: 60. ~

40. >< a:: 20. 0.

-20. ~I' 4. 5. 6. 7. 8. 9. 10. -40. 3.5 4.5 5.5 6.5 7.5 8.5 9.5

Figure 5.68 Base Moment, Base Shear, and Axial Load for the Left Column: Third Story

t~~· 7 I ~ ~~.

r- -2~'3f 4 :z -4' · ~ :§.: 3.5 4.5

:E: :J~ - 4 - 6

lQJ0. 8~.

60. :z: 40 .

TIME (SECONDS)

~

2121. 4 .. , Ai \ (I"., ( \ (\ I V\ /'\.. c----" ~ !..\ ~ 5 -2~: 4 ..... 4'~ ~ 7 ~v 11 -\ (, v.~ -\]V -~ 00

~ -40 .. - 3.5 4.5 -6~.

-8f2J. -1f2Jf2J.

18~. 16f2J. 14f2J.

::z:: 120. ~ 1 '2HZ' • -...:l 80 · cc 6QJ • X 4f2.1 • a: 2QJ •

0.

TIME (SECONDS)

3. 4. 5. 6. 7. 8. ~. 1~. 3.5 4.5 5.5 6.5 7.5 8.5 ~.5

Figure 5.69 Base Moment, Base Shear, and Axial Load for the Right Column: Third Story

179

APPENDIX A

FAILURE SURFACE FOR MASONRY

A.l Wall Element Failure Surface

Masonry is a nonhomogeneous, nonisotropic, material; and as

stated in Chapter 3 this is difficult to handle analytically. To

simplify the problem the assumption was made that masonry could

be treated as a homogeneous isotropic material. However,

experimental results show that this assumption may not be

extended to the failure mechanism for masonry. This implies that

the failure surface for masonry is in general not symmetric.

The shape of the failure surface shown in Fig. 3.3 is an

assumption based on a linear interpolation between the control

points F1 , F2 , F3 , F4 , and FS. These five control points are in

turn a function of the angle that the principal stress makes with

the mortar joints. This condition implies that the specific shape

of the failure surface shown is not constant and in general

depends on the magnitude of the normal and shear stresses

applied.

There are two different aspects of the failure surface for

masonry which must be coupled to form the failure surface shown

in Fig. 3.3. The first is the overall shape of the failure

surface for the given angle that the principal stresses make with

the mortar joints. The second is the change in the shape of the

failure surface due to a change in the angle that the principal

180

stresses make with mortar joints. These two aspects are in

general not independent of each other but an assumption of

independence is made to make the problem tractable.

Although there is adequate experimental evidence to support

the functional dependence of the control points Fl through F4 on

the principal stress angle there is little or no direct

experimental evidence to support a specific interpolation between

these control points except as might be determined in an analogy

with plain concrete. Due to the lack of experimental evidence a

simple linear interpolation between points FI through" FS is

assumed. This is believed to be a conservative assumption.

The control points F1 , F2 , F3 , and F4 represent a simple

uniaxial loading condition. In making these control points a

function of the angle that the principal stress makes with the

mortar joints a dual purpose is served. First, it allows the

failure surface to depend on the principal stress angle without

having to undergo a complex analytical or experimental study and

second it allows the matching of these conditions to the

experimental evidence already available for uniaxial stress

tests.

The dependence of the control points FI and F2 on the angle

that the principal stress makes with the mortar joints is shown

in Fig. 3.4. This figure is based on a test developed by F. B.

Johnson and J. N. Thompson to determine the tensile strength of

masonry. The test consists of applying a compressive load along

181

the diameter of a circular masonry disk. This induces a tensile

stress in the disk normal to the direction in which the load is

applied. Therefore this test provides an indirect measure of the

tensile strength of the masonry assemblage. By rotating the disk

the angle that the principal stress makes with the mortar joint

may be varied. The authors carried out a series of tests and

presented their data in a paper at the First International

Conference on Masonry Structural Systems in November 1967. Part

of this data along with a discussion of the test was later

presented in Ref.[41]. Figure 3.4 is an approximation of the

experimental results presented in Fig. 1-20 of Ref.[41].

The dependence of the control points F3 and F4 on the angle

that the principal stress makes with the mortar joints is shown

in Fig. 3.5. This figure is based on the experimental results

presented by Hamid and Drysdale in Ref.[22]. Hamid and Drysdale

carried out a series of uniaxial compressive tests where the

angle that the mortar joint made with the compressive load was

varied_ They accomplished this by cutting out from a larger

specimen the prism elements to be tested. Figure 3.5 is an

approximation of the experimental results presented by the

authors in Fig. 4 of Ref.[22].

No experimental evidence is available to define the control

point F5; therefore, the assumption was made to take the largest

of the values of either F3 or F4 - It is believed, based on an

analogy with concrete, and possible confining effects, that this

is a conservative assumption.

BIOS KeEL 208 N. Rcmi:J.8 ~·~;:reB·t:

TT,...h~r.~ T11 ., ,!",\,...~ .:; ~.-r~[,';"11

182

A.2 Joint Element Failure Surface

The joint failure surface presented in Fig. 3.2 is based on

the experimental results presented by Benjamin and Williams in

Ref.[7]. The authors tested a series of frames with an infilled

masonry wall to study their load displacement relation as

compared to an o'pen frame.

In their effort to better understand the behavior of the

masonry wall they carried out a series of tests on specimens

which consisted of two bricks bonded by a mortar joint. These

tests consisted in applying normal stresses which were oriented

at different angles with respect to the mortar joints. This

caused different combinations of shear and normal stress across

the mortar joint. These series of tests allowed the authors to

determine the mortar joint failure mechanism under combined

stresses. Part of the results presented in Ref.[7] were later

reprinted in Ref.[41]. Figure 3.2 is an attempt at approximating

the experimental results presented in Fig. 6 of Ref.[7].

Figure 3.2 is not confined to failure along the mortar joints

as the experiments in Ref.[7] might imply. A series of tests on

full masonry panels where the cracking pattern also passed

through the bricks showed a similar failure surface as that shown

in Fig. 3.2. These results are presented in Refs.[18] and [41].

183

APPENDIX B

JOINT ELEMENT STIFFNESS MATRIX

The derivation of the stiffness matrix for the joint element

is expressed in a straightforward finite element formulation

based on Fig. B-1.

(I t V2

~ • • ~ u2 u,

l~ .. I

Figure B-1 Unidimensional Joint Element

The interpolation functions relating the generalized

displacements to the nodal displacements are expressed as

follows:

where

{u} = generalized displacements

Nl=N2 [1 - X/t], [X/t]; interpolation functions

t = joint thickness

{u"} nodal displacements

Based on these functions the strains are given as follows

-, a

0 ax {e e } {u} x xy a

0 ax L --I

(B-1)

(B-2)

184

where

{ex;e } normal and shear strain xy

{u} = generalized displacements

Subtitution of Eq. B-1 into B-2 gives

j

dN1 0 ~

{e e } x xy

dN2 0 ~

{u"} (B-3)

Using Eq. B-3 and a stress strain relationship for uniaxial

compression and pure shear one arrives at the following equation

using the principle of virtual work

where

{p}

-r o

{p} & nodal loads on the joint

A K area of the joint

t = thickness of the joint

{u"} (B-4)

Kl & a 2x2 stiffness matrix whose elements are the

elastic modulus

K2 a 2x2 stiffness matrix whose elements are the shear

modulus

185

The relationship needed to substitute in the equations

formulated in Chapter 4 are of the incremental type. Equation B-4

can be made incremental by simply allowing the nodal loads and

displacements to represent incremental quantities rather than the

total loads and displacements as derived. The changes in the

incremental stiffness which introduce the nonlinearities are

provided by the changes in the stress strain relationship of the

material. Before failure of the joint occurs, the relationship

between stress and strain is assumed linear. This provides both

an internal resistance as well as an incremental stiffness. After

failure the incremental stiffness as well as the internal

resistance depend on the state of stress. If the normal stress is

tension, the incremental stiffness as well as the internal

resistance will be zero. If the normal stress is compression then

an incremental stiffness will exist as in the elastic case before

failure. However, the internal resistance will only be provided

by the normal stress present in the joint.

186

LIST OF REFERENCES

1. Anaguostopaulos S.A., Haviland R. , Biggs J. , "Use of

Inelastic Spectra in Aseismic Design", Journal of the

Structural Division, ASCE, vol. 104, ST1, >January 1978.

2. Anderson J.C., Gupta R.P., "Earthquake Resistance Design of

Unbraced Frames", Journal of the Structural Division, ASCE,

vol. 94, ST1, January 1968.

3. Arnold P., Adams D., Le-Wu Lu, "Strength and Behavior of an

Inelastic Hybrid Frame", Journal of the Structural Division,

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