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,
ASCE, vol. 94, ST1, January 1968.
4. Axley J.W., Bertero V.V., "Infilled Panels: Their Influence
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