RESEARCH
AND
DEVELOPMENT
BULLETIN
, r;*,
RDOS7.01D
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.‘-*‘:
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L-
Modeling Hysteretic
Behavior of
Coupled Walls for Dynamic Analysis
By M. Saatcioglu, A. T. Derecho,
and W. G. Corley
Keprinted wth permission from bimhyuuke
En,qnrerin~
and Strucrural
Dynamics, Vol. 11, 1983
PORTLAND CEMENT
Research
and Deve,o~men,
/ Constructro,,
ASSOCIATION
Technology
LaborafOrleS
.-
:
RESUME
On examine l%tude des techniques de simulation de la
t+ponse dynamique inelastique des structures en murs couples. L’etude
est ax~e sur les effets des param+tres d&finissant la relation forced~placement de la boucle d’hyster~sis. On examine plus particulierement les effets de l’interaction force axiale-moment,
de la diminution de
resistance, de la limite de resistance au cisaillement et des branches de
rechargement et de d~chargement de la boucle d’hyst&r&is ainsi que la
variation de l’inclinaison des branches de rechargement
caus6e. Les
effets des parambtres de la simulation sur 1(:s quantites choisies pour
represented la r~ponse sent &udi& et discut~s en dttail.
Un Ldifice de 20 Ltages avec murs couples a ete choisi pour I’analyse
dynamique. Des stries de param&tres caract~risant
la relation forcedeplacement des boucles d’hysteresis ont i?te obtenues a partir d’essais
r+alis+x en laboratoire en inversant lentement la charge statique. Des
accelerogrammes
de grand replacement
pr(xnregistres ont We utilises
comme d~placements initiaux.
Les resultats indiquent que les forces axiales s’exergant sur le muret la
diminution
de resistance
de la poutre peuvent avoir des effets
significatifs sur les envelopes
de rtponse. De Fegkres variations clans les
branches de rechargement et d~chargement des boucles d’hyst&r&is et la
variation de t’inclinaison des branches de rechargement semblent avoir
peu d’effet sur la reponse dynamique.
SUMMARY
Modelling techniques for dynamic inelastic response
analysis of coupled wall structures are investigated. Emphasis is placed
on effects of parameters defining the force-displacement
h ysteresis loop.
Specifically, effects of axial force-moment interaction, strength reduction, shear yielding, pinching, reloading, and unloading branches of
hysteresis loops are considered. Effects of modelling parameters on
selected response quantities are investigated and discussed in detail.
A 20-story coupled wall structure was selected for dynamic analysis.
Ranges of parameters
characterizing
force-displacement
hysteresis
loops were obtained from laboratory tests under slowly reversed static
loading. Previously recorded strong motion accelerograms were used as
input motions.
Results indicate that wall axial forces and beam strength reduction
can have significant effects on response envelopes. Moderate variations
in unloading and reloading branches of hysteresis loops and pinching
appear to have little effect on dynamic response.
ZUSAMMENFASSUNG
Es werden Modell-Techniken
fur eine
dynamische, unelastische Verformungs-Analyse
von Verbundwanden
untersucht. Dabei wird auf solche Parameter Wert gelegt, mit denen die
Hysterese-Schleife
fiir Kraftveranderungen
definiert werden kann.
Insbesondere werden Wirkungen axialer Belastungsmomente,
Druckverminderung, Scherbeanspruchung,
Quetschung und Wechselbelastung
auf den Verlauf von Hysterese-Kurven
in Etetracht gezogen. Die
Wirkung von Verformungs-Parametern
auf ausgewahlte Folgeeigenschaften werden untersucht und im einzelnen diskutiert.
Eine 20-stockige Verbund-Wandkonstruktion
wurde fttrdynamische
Analysen ausgewahlt. Parameter-Bereiche
zur Charakterisierung
von
Kraftverlagcrungs-Hy
stereseschleifen wurden in Laborversuchen durch
langsamen
statischen
Belastungswechsel
gewonnen.
Zuvor aufgenommene
Beschleunigungs-Diagramme
wrrrden als EingabeBewegung genutzt.
Die Ergebnisse zeigen, daf3 zentrale Belastungen von Wanden und
eine Verminderung
von Tragerfestigkeiten
bedeutungsvotle
Auswirkungen auf den Kurvenverlauf haben konnen. Maf3ige Veranderungen
der Be- und Entlastungs-Bereiche
von Hysterese-Schleifen
und
Quetschungen
scheinen nur geringe Auswirkungen
auf dynamische
Folgeerscheinungen
zu haben.
Modeling Hysteretic Behavior of Coupled
Walls for Dynamic Analysis
by Murat Saatcioglu,l
and W. Gene Corley3
Arnaldcl
T. Derecho,
z
SUMMARY
for dynamic inelastic response analysis of coupled wall structures are investigated. Emphasis is
placed on effects of parameters defining the force–displacement hysteresis loop. Specifically, effects of axial force–moment
Modelling
techniques
interaction,
strength
reduction,
shear yielding,
pinching,
reloading
and unloading
branches
of hysteresis
loops are
considered.
Effects ofmodelling
para.meters
on selected response
quantities
are investigated
and discussed
in detail.
A 20-storey
coupled
wall structure
was selected for dynamic
analysis.
Ranges of parameters
characterizing
force–
displacement
hysteresis
loops were obtained from laboratory tests under slowly reversed static loading. Previously
recorded strong motion accelerograms were used as input motions.
Results indicate that wall axial forces and beam strength reduction can have significant effects on response envelopes.
Moderate variations
dynamic response.
in unloading
and reloading branches of hysteresis loops and pinching appear to have little effect on
INTRODUCTION
Reinforced concrete coupled walls are often used to stiffen multistory
structures against seismic forces.
Under strong ground excitations., portions of a coupled wall system can be expected to deform beyond the
elastic limit. Therefore, dynamic response analysis of coupled walls should consider inelastic behaviour.
Computer time required for dynamic inelastic analysis is significantly affected by the complexity of the
force–deformation
hysteresis model used. Increased refinements in analytical models must be balanced
against the increased costs incurred in the analysis. The objective of this paper is to evaluate the significance
of a number of experimentally observed features of force–deformation
hysteresis loops in relation to dynamic
inelastic response of a 20-storey coupled wall structure.
Properties of a 20-storey coupled wall structure selected for dynamic analysis are given in Table I and
Figure 1. A modified version of computer program DRAIN-2D1 was used for the analysis. Each structural
member was idealized as a line element. Inelastic action was simulated by allowing formation of hinges at
member ends. Program capabilities, as modified by the Construction
Technology
Laboratories,
include
modelling
of inelastic moment–rotation
and shear–distortion
relationships.
Axial force–deformation
relationships
were considered
to be linearly elastic throughout
the investigation.
An inelastic force–
deformation
model for shear was used only when effects of shear yielding and pinching were being
investigated.
1Department
of Civil Engineering,
University
2 Wiss, Janney, Elstner & Assoc., Northbrook,
3 Director, Engineering Development Division,
of Toronto,
Ill.
Portland
Toronto,
Cement
Ontario
Association,
Skokie,
Ill.
Table I. Properties
Fundamental
1.0s
period
Number
of storeys
Height
Coupling
arm
Wall stiffrtess parameters
EI
GA
EA
Stiffness
Beam
taper*
stiffness
m
8.5m
parameters
EA
yield moment,
Strength
20
557
238,000,000 kN.m2
27,000,000 kN
632,000,000 kN
1.0 EI at base
08 EI at 6th floor
065 EI at 12th floor
~A
Wall
of the selected structure
MY
taper~
65,300 kN.m2
1,750,000 kN
4,100,000 kN
45,200kN.m
10OMY at base
0.75M,
Beam yield moment, My
Damping
Post-yield stiffness on
primary curve
W~ight
Weight for inertia forces
Base fixitv condition
Ground
motion
Intensity
of ground
motion$
Duration
at 6th floor
050M at 12th floor
339 k~m
5X of critical
5~; of elastic for walls
6%.. of elastic for beams
8,362 kN/wall
14,545 kN/waH
fully fixed
Pacoima Dam 1971, S16E~
15 El Centro
10s
1940, N–S
* The same taper also applies to ‘GA’ and CEA’.
t Yield moments are also adjusted at every floor based on the weight of the
structure.
$ Unless otherwise noted in the text.
\ Based on spectrum intensity.
Six different accelerograms were examined to select an input motion which would be critical in terms of the
frequency content. Response spectra of si]~gle-degree-of-freedom
systems were used as basis for the
preliminary
(lengthening
selection.
The
initial
fundamental
period
of the
selected
structure
and
possible
softening
in period) in the structure due to yielding were considered in selecting three potentially critical
accelerograms.
These were 1940 El Centro, E–W record; 1971 Holiday Orion, E–W record; and 1971
Pacoima Dam, S16E record. The structure was analysed under 10s of each input motion. Based on the
comparison of response envelopes created by each input motion, the 1971 Pacoima Dam, S 16E record was
selected for use in most cases. However, the response histories under this input motion indicated that the
maximum response oc(;urred early in the analysis, Therefore, in cases where this feature of the response did
not allow the effect of a particular parameter to show up clearly, the E–W component of the 1940 El Centro
record
was used.
Rotational ductility factor defined as the ratio of maximum to yield rotations is used as a measure of
inelastic deformation.
For the purpose of analysis, members were assumed to have unlimited ductility
capacities. This allowed an assessment
of the ductility requirements
in each member for a specific
combination of parameters.
Unrealistically
high ductility demands imply undesirable response, indicating
that the parameters leading to such behaviour should be avoided.
DEGRADING
STIFFNESS
MODEL
Reinforced concrete members generally exhibit loss of stiffness during unloading
and reloading when
subjected to inelastic load cycles. Takeda’s mode12 as shown in Figure 2 was adopted as the basic form of
2
~
‘
r-
0 at
5.5 m = 44.0
m
1
PLAN
II
m\\\\\\\\\\\\\w
i
-Fi
ELEVATION
Figure
I. Coupled
wall structure
selected for investigation
M,V
.,
MY,VY
//
e,y
/ ‘1
/
*
Figure 2. Takeda’s
hysteretic
loop
hysteretic loop for this investigation. A. bilinear idealization of the primary curve was used in the model. The
first line segment prior to yielding characterizes the effective elas~ic stiffness of a member. The second line
segment characterizes the post-yield stiffness and starts at yield. Determination
of the effective elastic stiffness
based on sectional properties is straightforward.
Reduction in stiffness due to cracking can be incorporated in
the effective elastic stiffness.
Inelastic stiffness during loading, unloading
and reloading is generally more difficult to determine.
Practical ranges for these post-yield parameters can, however, be obtained from examination of experimental
test results.
Tests of concrete members under slow load reversals show some additional decrease in member stiffness
due to slip of reinforcement. Although there is no separate mechanism in the analytical model used in this
investigation for bond slip action, its effect can be accounted for by assigning.appropriate
slopes to loading
branches.
As a first step in the investigation,
variation in post-yield stiffness was studied. Examination
of a large
number of moment–rotation
relationships for different concrete members revealed that the post-yield slope
of the primary curve normally lies in the range of 3 to 10 per cent of the effective elastic slope.
Ratio of post-yield flexural stiffness to effective elastic flexural stiffness is greater for coupling beam
elements than for wall elements. For wall elements loaded into the post-yield range, an approximately
uniform moment along the element length results in yielding along a significant portion, if not all, of the
element length. Coupling beams, bent in double curvature, are subjected to a steep moment gradient with
yielding localized at member ends. The portion of member between localized hinges remains elastic at all
stages of loading.
Table II. Post-yield slope as a
percentage of elastic slope
Coupling
beams
Walls
‘BM
Case
(%
W)
1
2
2
3
:
10
4
5
10
:
20
20
—.—
—
A
w.
aW=5*10,aem=107e
aW=50/~,ae~= 60/o
‘W=2”’0’aBM
““”’”””””
=3 %
d
z
-J
z
-1
>
10 -;
>
a
p
a
o
tm
.. .
O-J
I
0
0
0
MAX.
40,0eo
80,000
WALL
MOMENT,
120,000
MAX. WALL
kN-m
I
4.0
2.0
0
DUCTILITY
FACTOR
20
1
0
500
MAX.
8EAM
Figure 3. Response envelopes for moment
4
1
\
1500
1000
MOMENT,
kN-m
I
0
MAX.
t
\
10
0
3EAM
DUCTILITY
!
1
20
FACTCR
and ductility factor showing eITects of varying slope of inelastic portion
curve
of moment–rotation
~
A set of analyses was conducted to determine the significance of variations in post-yield slope of the
primary curve within the range observed in tests. The 20-storey coupled wall structure was analysed four
times with different percentages of effective elastic slope used for post-yield slope of the primary curve. Cases
covered are listed in Table II.
Response envelopes for these four cases are compared in Figures 3 and 4. Results indicate that for the
range of values assumed, maximum wall displacements and forces are not significantly affected by variations
in post-yield slope. On the other hand, maximum forces in beams appear to be affected by changes in
magnitude of the second slope. This can be attributed
to high levels of inelastic action and associated
ductilities in coupling beams.
Effects of varying unloading and reloading stiffnesses were investigated. As shown in Figure 5, parameters
‘u’ and ‘r’ define unloading and reloading stiffnesses, respectively. In a previous study3 of isolated walls, it was
concluded
that within the assumed range, variations
in unloading
and reloading stiffnesses do not
significantly affect dynamic response. In this investigation,
‘u’ and ‘r’ were held constant at 01 and 00,
respectively, for wall elements. For coupling beams, three analyses were carried out using unloading and
reloading parameters listed in Table III.
Maximum forces and displacements for the three cases considered were found to differ by no more than 5
per cent.
20
———-aw=gyo,
—.
—aw=5”/0,
aBM=
—aW=50/0,
..aW.
aBM= 10%
10%
aBM= 60/0
~0/0, aBM=3e10
o
I00
c1
MAX. HORIZONTAL
200
DISPLACEMENT,
3C0
-8000
MAX.
mm
8000
0
AXIAL
16000
FORCE IPI WALLS
24000
,kN
20
20
I
d
>
w
J
10
>
CC
o
iG9
r)
o
400
MAX.
BEAM
Figure 4. Response envelopes for displacement,
800
SHEAR , kN
I20G
o
2000
MAX.
WALL
4000
6000
SHEAR , kN
axial force and shear showing effects of varying dope of inelastic portion of momentrotation curve
5
Figure 5. Moment-rotation
curve showing parameters
defining unloading
and reloading
stiffnesses
Table III. Unloading and reloading parameters for coupling beams
Case
Unloading
parameter, u
Reloading
parameter, r
01
03
0
0
01
1.0
;
3
AXIAL
FORCE-FLEXURE
INTERACTION
MODEL
Coupled walls undergo significant changes in level of axial force during response to lateral forces. Variations
in axial forces directly influence force~eformation
characteristics
of members. Strength and stiffness
properties of coupled walls can be altered significantly by changes in the level of axial force.4 Therefore, the
reducing stiffness model was modified to include axial force–flexure interaction. The modified model is shown
in Figure 6. A set of hysteretic loops, corresponding
to different levels of axial force, is used as a guide in
predicting change in stiffness due to axial force effects. The basic concept in introducing the effect of changing
axial forces is to update stiffness for the subsequent time increment, based on axial force calculated for the
current time increment. Consequently,
if there is an increase in axial force then the hysteretic loop will be
directed towards the corresponding
moment–rotation
loop which has higher yield strength. This will
M
Figure 6. Moment–rotation
6
curve including
axial load effects
produce a higher slope of the hysteretic loop indicating an increase in stiffness. Details of the model are
explained with examples in Reference 5.
Figure 7 shows a plot of base moment versus hinge rotation for the 20-storey coupled wall structure when
the modified model was used. Also shown is the moment–rotation
curve associated with the primary curve
without interaction. Reduction in wall strength and stiffness when the wall is in tension and the reverse effect
when the wall is in compression can be observed when the interaction effect is included.
80,000
t
60,000
.——
40,000
z
z
/
1
20,000
t
/.
,D
/
g
-40,000
1%
t
---
/
/
y
“
---
-60,000
-0001
60,000
wall
@
t
40,000
20,000
E
o
ii
‘L
~-
of
Hinging
I
z
20’000
-L+’”
-40,000
0.(331
0
Rototion
L
I
Region
----
-60,000
Tension
-80,00Q
—
,y
without
interachon
O.ooz
(rodions)
P
-—-
,/
Curve
Interoctiofl
Model w!th M-P
—
–0.002
;
Primary
M–P
Comp<es.mn
t
80,000
Tension
———
CompressIon
————
Primary
Curve
m– P
Interoct,on
without
Model with t4-P interaction
,,,
,
1
– C1.ooz
Rototion
Figure ‘7.Effect of moment–axial
0.001
o
–0.001
of
Hinging
Region
load interaction
0.002
(radians)
on hysteretic
loops
The significance of axial force–flexure interaction
on response envelopes was investigated.
For this
purpose, the 20-storey coupled wall structure was analysed twice. The modified model with axial force–
flexure interaction was used in the first analysis. In the second analysis, axial forc+lexure
interaction was
ignored. Response envelopes fur the two cases are compared in Figures 8 and 9.
The results indicate that maximum forces in the walls can be affected significantly by axial forces. When the
effect of axial force was ignored maximum shear and moment in the base wall were underestimated
by as
much as 50 per cent. Moreover, the sequence of yielding and the yielding pattern for the two structures were
different since the yield level is affected by concurrent
axial force. It should be noted that the difference
between the maximum wall ductilities of the two cases compared in Figure 8 due to the lower yield level of
7
20
——
‘i20
Without M-P
1-
With
M-P
Interaction
Interaction
i
z
J
10 –
>
a
o
+
u)
\
‘\
10
\\
\
o —
0
40,0CX2
c)
MAX.
WALL
00,000
MOMENT,
120,000
2.0
0
kN-m
MAX. WALL
4,0
DUCTILITY
FACTOR
20
20
\
I
I
\
I
ii
>
u
-1
\
ii
I
10
I
~,.
>
I
>
a
0
1m
a
Q
tm
/’
/
,),
,)[i,
/
0
0
0
500
MAX.
Figure 8. Response
/
BEAM
envelopes
1000
MOMENT,
for moment
1500
kN–m
andductility
10
0
MAX.
BEAM
20
DUCTILITY
factor showing effects ofmoment-axial
FACTOR
load interaction
tension walls. It does not necessarily imply increased horizontal displacements
of tension walls. Maximum
horizontal displacements
are shown in Figure 9.
It can be concluded from the above comparison that in performing dynamic inelastic analysis of coupled
walls, effects of axial forces should be considered. Inaccurate response quantities can result if these effects are
ignored.
STRENGTH
REDUCTION
UNDER
LOAD
REVERSALS
Reinforced concrete members generally show strength reduction or strength decay under repeated load
reversals. The term strength decay is used to signify a reduction in maximum force that can be carried by a
member under successive reversing loads. The degree of strength decay depends on structure geometry,
reinforcement detailing, confinement and history of loading. Experiments have shown that a reduction in
strength generally occurs under high levels of inelastic deformation. G–gIn the case of coupled wall Sttltctures,
wall ductilities are usually limited to values of approximately
3.0, Coupling beams can be expected to
dissipate most of the energy by developing ductilities in the range of 6.0 or more. Therefore strength decay in
coupled wall structures is generally associated with coupling beams.
20
0
0
I00
MAX. HORIZONTAL
200
300
DISPLACEMENT,
-6000
mm
MAX.
0
8000
AXIAL
16000
24000
FORCE IN WALLS
,kN
4(200
6000
20
0
400
MAX, BEAM
Figure 9. Response
envelopes
800
I200
SHEAR , kN
fordisplacement,
2000
MAX.
WALL
SHEAR,
kN
axial force and shear showing effects of moment–axial
load interaction
The reducing stiffness model was modified to include strength decay as shown in Figure 10. Strength
reduction is defined by the ‘strength decay guideline’, Examples of rapid and mild reduction are given in
Figures 11 and 12. The strength reduction guideline is defined by three parameters as follows:
a. ductility ratio on the primary curve at which strength reduction starts;
b. slope of the strength reduction guideline, KO;
c. minimum moment, M~in, belc)w which no further strength reduction occurs.
Reloading stiffness is defined by the point on the reduction guideline corresponding
to the maximum rotation
in the preceding cycle.
Two sets of analyses were conducted using the E–W component of the 1940 El Centro record to investigate
the effect of beam strength reduction on dynamic response. The first set involved investigation of decay rate.
TWO different rates of strength decay were used as shown in Figures 11 and 12. The second set of analyses
involved investigation
of the deformation
level at which strength reduction starts. For this purpose, the
‘Rapid’ rate of decay was modelled, starting either at rotational ductility of 3.5 or 6.0. Response envelopes for
each set are compared in Figures ‘13 and 14 with cases where no strength reduction was considered.
The results indicate that if beam ductility demands reach the range where strength loss occurs, significant
effects can be expected in structure response. Effects of strength decay in coupling beams are most noticeable
in increased horizontal displacements of the structure and coupling beam ductility requirements. Therefore,
9
M,V
Strength
Decay
Guide Line
J,
MY ,Vy .
\
\
Figure 10. Force deformation hysteretic loop including strength decay
M
‘\
My
~---
---
---
---
-----
-man.
-
— ——
‘-=i
—
o
RAPID DECAY
STARTING
KO= 20
AT ,u
’10
= 3,5
K,
K2= 6 “A K,
I
MY=339
kN-m
rdmm= 34
kN–m
Figure 11. Rapid strength decay
M
WY
-~---
----
.--—
----
A
-
\
-a
#--
‘min.
--e
MILD DECAY
STARTING AT A = 3.5
KO=IO % K,
K2 = 6 O/. K,
MY= 339
I
Figure
10
12. Mild strength
M ~,n=
decay
203
kN-m
kN–m
20
20
d
ii
z
BEAM
-J
STRENGTH
—
10
>
a
o
t(I’I
~
0
0
No Decoy
‘—
Mild
. .
Ropid
Decoy
starts
DECAY;
A
F
a
Decay
Decay
at
““..
40,000
MAX.
WALL
I
80,000
MOMENT,
10
o
b
(J3
o
120,000
kN-m
0
MAX.
2.0
WALL
4,0
DUCTILITY
FACTOR
20
ii
>
U
-1
+
10
a
0
+
m
o~
0
o
I 00
MAX. HORIZONTAL
Figure
’200
DISPLACEMENT,
13. Rf!sponse envelopes
10
o
300
mm
MAX.
BEAM
showing eflects ofrateof
20
DUCTILITY
strength
FACTOR
decay
in modelling hysteretic loops for dynamic inelastic analysis, care should be taken
reduction of beams appropriately
depending on the expected maximum deformation
SHEAR
n
to simulate
level.
strength
YIELDING
Tests of isolated walls conducted under slow load reversals at the Construction
Technology Laboratories
indicate that flexural yielding is usually accompanied by shear yielding. b. 7 To investigate the effect of shear
yielding on dynamic response, the shear force–shear distortion hysteretic loop was modelled on the basis of
Takeda’s2 rules as shown in Figure 2.
The 20-storey coupled wall structure was analysed twice, first with shear yielding and then without shear
yielding. Effective elastic flexural and shear stiffnesses were used to define the primary force–deformation
relationship.
Table IV gives the stiffness parameters
used in the analyses. The shear deformations
were
computed using shear stiffnesses based on the hysteretic model shown in Figure 2. The shear yield level in the
model was governed by flexural yielding, a phenomenon observed in laboratory tests.6’ 7 Accordingly, shear
yielding was allowed following flexural yielding even if the previously specified shear yield level was not
achieved. Because the structure under consideration
was designed to avoid premature shear failure, shear
yielding could only occur due tcl the flexural yielding and the subsequent
change in shear resisting
mechanism. Response envelopes for the two cases are compared in Figures 15 and 16.
11
i
20
id
BEAM
STRENGTH
—
DECAY?
-1
No Decay
10
?
Rc!pid
Decoy
;
Starting
cd :
——
Ductility
of
6.0
,.....,.
Ductility
of
3.5
CO
1
40,000
0
MAX.
80,000
WALL
MOMENT,
2C)
$
/
...
z
..”
-1
:0
&
p
)
,~,,...””’”
U7
/
.“
I
I
I
I 00
200
300
MAX. HORIZONTAL
Figure 14. Response
C)lSPLACEMENT,
envelopes
FACTOR
1:
1’
1:
);
, .,..”’
w
,..
o
DUCTILITY
\’
.,
c1
4.0
\!
J
.,.”’
.
WALL
20
,...”
/..””
...
Ic)
1
2.0
.,.””
/
w
J
I
MAX.
,,,...”’
/
>
lx
o
tL9
kN-m
/
/’
n
“o
120,000
I
o
mm
I
1
10
0
showing effect ofvarying
..””
MAX.
BEAM
ductility
I
DUCT!LITY
atonset
FACTOR
ofstrength
Table IV. Stiffness parameters
Walls
Floor level
Effective elastic ‘Ef’
(million kNm2)
Post-yield ‘EI’
(primary curve,
million kN.m2)
Effective elastic ‘GA’
(million kN)
Post-yield ‘GA’
(primary
curve,
million kN)
Effective elastic ‘EA’
(million kN)
12
lst–
5th
238
6th–
12th
190
Beams
13th–
20th
154
All
floors
00651
9“5
7“8
00040
454
3.60
2.94
0.289
0.271
0.218
0.178
0.00173
11.8
632
507
41.4
4.10
I
20
decay
20
.
Elastic
Shear
Inelastic
Shear
with
‘—
Inelastic
without
o
40,0C0
MAX,
WALL
Pinching
80,000
MOMENT,
10
Shear
Pinching
120,000
0
kN–m
2.0
MAX, WALL
DUCTILITY
40
FACTOR
20
d
>
!&!
-1
10
>
&
o
tUY
o
500
MAX.
BEAM
1000
MOMENT,
1500
kN-m
0
MAX.
10
BEAM
DUCTILITY
20
FACTOR
Figure 15. Response envelopes for moment and ductility factor showing effects of inelastic shear and pinching
Results of the analyses indicate. that for the structure and the ground motion under consideration,
shear
yielding has little effect on total displacement, shear force and moment envelopes. Maximum beam rotational
ductilities, on the other hand, show considerable
increase when shear yielding is included. This can be
explained by a reduction in the flexural component of deformations due to increased shear distortions while
total deformations
remain essentially unchanged.
In addition to the comparison of response envelopes, behaviour of the wall hinging region was examined.
The hinging region was taken as the lower 63 m portion of the walls. Results indicate that for the structure
under consideration, shear displacement constitutes about 50 per cent of the total horizontal displacement of
the top of the hinging region prior to yielding.5
During dynamic response, shear force and moment response are not always in phase. This means that,
contrary to static test conditions,
maximum moment and maximum
shear do not necessarily occur
simultaneously.
Examination
of the hinging region shows that during the inelastic range, horizontal
displacement of the hinging region due to shear yielding can be as high as 65 per cent of total hinging region
displacement when the shear force is at its maximum. 5 Because maximum horizontal displacement does not
necessarily occur when the shear is maximum, this behaviour is not reflected in the response envelopes shown
in Figure 16.
13
.,..
/“
.-.’
...””
...
,..
..
..”
—
Elastic
...
...
.,.
...
...
‘—
lnelostic
Shear
without
Pinching
,’
,.,
..”’
o
Sheer
Inelastic
Shear
with
Pinching
100
MAX, HORIZONTAL
200
300
DISPLACEMENT,
-8000
mm
0
MAX,
20
16000
8000
AXIAL
FORCE IN WALLS
24000
,kN
r
“’”J
“j
10
:1
i,\
‘!
,
L~
o
MAX. BEAM
Figure
16. Response envelopes
I
0
4Q0
%00
SHEAR , kN
fordisplacement,
I200
\...
o
I
I
4000
2000
MAX,
.,
WALL
axial force and shear showing
SiiEAR,
6000
i(N
effects ofinelastic
shear and pinching
The same set of analyses was repeated ufider a different earthquake motion. Input motion used for this case
was the E–W component of the 1940 El Centro record. Comparison
of response envelopes indicates that
shear yielding has very little effect on response.
It is obviously important to use reasonably accurate estimates of flexural and shear stiffnesses of members
in defining primary force–deformation
relationships.
Tests indicate that flexural and shear cracking of
members reduce their stiffnesses substantially. Effective flexural stiffness prior to yielding can be as low as 30
per cent to 50 per cent of untracked stiffness. Effective shear stiffness can be as low as 10 per cent to 30 per
cent of the stiffness associated with gross (untracked) area. In the above two sets of analyses, 50 per cent and
10 per cent of untracked stiffnesses were used for flexure and shear, respectively.
It should be noted that the
relatively high reduction in shear stiffness was intentionally used to stimulate the inelastic shear deformation
effect.
A third set of analyses included higher shear stiffness. In this set of analyses, flexural stiffness remained the
same while the shear stifiness was increased by a factor of 5 to 50 per cent of untracked
stiffness. Results
indicate that shear yielciing effects on response envelopes are further reduced showing a maximum of 5 per
cent difference in response between the structure with elastic shear and the structure in which shear yielding
was considered.
14
PINCHING
IN HYSTERESIS
LOOPS
Force–deformation
relationships
of concrete members generally show a ‘pinching’ action under cyclic
loading. Pinching occurs on reloading after cracks open significantly during previous loading in the opposite
direction. When the load reverses, initial stiffness is very low until cracks close. An increase in stiffness occurs
in the later part of the reloading branch as the contribution
of concrete to stiffness increases. This behaviour
was modelled as shown in Figure 17.
M,V
LJ’ pinching
‘ffec’
Figure 17. Force–deformation
hysteretic
loop including
pinching effect
Tests have shown that pinching action is most apparent in the shear force–shear distortion relationship.d’
7
Therefore,
the inelastic shear model was used to investigate
pinching effect on dynamic response. The degree
of pinching used was determined after examining available test data.
The previously
hysteretic loops.
shown in Figures
However, results
characteristics
of
selected 20-storey coupled wall structure was analysed with pinching in shear-distortion
Response envelopes are compared with the case in which pinching was not allowed, as
15 and 16. Results indicate that pinching action has little effect on response force envelopes.
also indicate that a shift in the axis of oscillation can occur under specific frequency
exciting force.
CONCLUSIONS
Based on results of this investigation, the following conclusions can be made.
1. Axial force–moment
interaction effects due to coupling should be considered in dynamic analysis of
coupled walls, Stiffness and strength of walls depend on the concurrent level of axial force.
2. Rotational ductility requirements for coupling beams can be significantly increased by early and rapid
strength reduction in coupling beams.
3. Maximum forces and displacements
do not appear to be significantly affected by shear yielding or
pinching in hysteretic loops.
4. Variations in post-yield loading, unloading and reloading branches of the hysteretic loop, within the
range observed in tests, do not significantly affect dynamic response.
ACKNOWLEDGEMENT
The
investigation
under
Grant
Construction
features
conclusions
National
reported
No.
ENV77-
in this
15333.
Technology
were
Science
in this
into
program
paper
is sponsored
project
Laboratories,
implemented
expressed
paper
The
are
in major
was conducted
a Division
of
DRAIN-2D
those
of the
by
authors
part
in the
Portland
Dr.
by the
Engineering
Cement
T. Talcayanagi.
and
do
not
National
Science
Development
Association.
Any
necessarily
opinions,
reflect
Foundation,
Division
of the
Some
modelling
findings
the
views
and
of the
Foundation.
15
REFERENCES
1. A. E. Kanaanand
G.H. Powell, `General purpose computer program forinelastic dynamic response ofplane structures', Report No.
EERC73+,
Earthquake Engineering Research Center, University of California, Berkeley, CA, Apr. 1973
2. T. Takeda. M. A. Sozenarld N. N. Nielsen. `Reinforced concrete res~onse tosimulated
earthquakes', J.struct. die. ASCE96,25572573 (1970).
3.
4.
5.
6.
7.
8.
16
A. T. Derecho, S. K. Ghosh, M. Iqbal, G. N. Freskakis and M. Fintel, `Structural walls inearthquake-resistant
buildings, dynamic
analysis of isolated structural walls, parametric studies’, Report to the National Science Foundation, Portland Cement Association,
Mar: 1978.
M. Saatcioglu and A. T. Derecho, `Dynamic inelastic response ofcoupled walls asaffected byaxial forces', Proc. CSCE-ASCE-ACICEBint. symp. nonlinear ales.concrete struct., SM Study No. 14, University of Waterloo Press, Waterloo, Ontario, 1980.
M. Saatcioglu, A. T. Derecho and W. G. Corley, `Coupled walls inearthquake
resistant buildings, modelling techniques and dynamic
analysis’, Report to the National Science Foundation,
Portland Cement Association, June 1980, National Technical Information
Service, 5285 Port Royal Road, Springfield, VA (NTIS Accession No. PB81- 132698).
R. G. Oesterlq J. D. Aristizabal-Ochoa,
A. E. Fiorato, H. G. Russell and W. G. Corley, ’Earthquake resistant structural walls—tests
of isolated walls—phase Ii’, Report to the National Science Foundation, Portland Cement Association, Oct. 1979 (NTIS Accession
No. PB80-t32418).
R. G. Oesterle, A. E. Fiorato, L. S. Johal, J. E. Carpenter, H. G. Russell and W. G. Corley, `Earthquake resistant structural wallstests of isolated walls’, Report to the National Science Foundation, Portland Cement Association, Nov. 1976 (NTIS Accession No.
PB271467/AS).
G. B. Barney, K. N. Shiu, El. G. Rabbatand
A. E. Fiorato, `Earthquake resistant structural walls-tests
ofcoupling beams', Report to
the National Science Foundation, Portland Cement Association, Jan. 1978 (NTIS Accession No. PB281733).
This publication is based on the facts, tests, and authorities stated herein. It is
intended for the use of professional personnel competent to evaluate the significance and limitations of the reported findings and who will accept responsibility for the application of the material it contains. Obviously, the
Portland Cement Association disclaims any and all responsibility for application of the stated principles or for the accuracy of any of the sources other
than work performed or information developed by the Association.
-------------
----------------------------------
--------------------
------------
1i
KEYWORDS: axial force, coupling beams, elastic slope, flexure interaction, force
deformation, moment interaction, pinching, post-yield slope, response envelopes,
rotational ductility, shear yielding, stiffness, strength decay, yielding level.
Modeling techniques for dynamic inelastic response analysis of
ABSTRAC~
coupled wall structures are investigated. Emphasis is placed on effects of parameters
defining the forcedisplacement hysteresis loop. Specifically, effects of axial forcemoment interaction, strength reduction, shear yielding, pinching, reloading, and
unloading branches of hysteresis loops are considered. Effects of modeling
parameters on selected response quantities are investigated and discussed in detail.
A 20-story coupled wall structure was selected for dynamic analysis. Ranges of
parameters characterizing forcedisplacement
hysteresis loops were obtained from
laboratory tests under slowly reversed static loading. Results indicate that wall
axial forces and beam strength reduction can have significant effects on response
envelopes. Moderate variation in unloading and reloading branches of hysteresis
loops and pinching appear to have little effect on dynamic response.
REFERENCE:
eling
Hysteretic
Portland
.Wwctural
----.-_------
Corley, W. Gene; Derecho, Arnaldo T.; and Saatcioglu,
Behavior
of Coupled
Cement Association,
Dynamics,
---------
Walls for
Dynamic
1984. Reprinted from
Earthquake
M. A40d-
(RD087.O 1D),
Analysis
Engineering
and
Vol. II, 1983.
-------------------------
---------------------------------
PCA
R/D
Ser.
No.
1684
?
PORTLAND
An organization
of cement
manufacturers
to improve
and extend
‘5420 Old
Printed
in U.S.A.
CEMENT
the uses of portland
Orchard
ml
cement
Road,
and
Skokie,
I ASSOCIATION
concrete
through
Illinois
scientific
research,
engineering
fieldwork,
and
market
development.
60077-4321
RD087.01
D