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8/9/2019 PhD Thesis G a Chang
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SEISMIC
ENERGY
BASED
DAMAGE
ANALYSIS
OF
BRIDGE
COLUMNS
Gilberto
Axel
Chang
Advisor
Dr
John
B
Mander
A dissertation
submitted
o
the
Faculty
of
the Graduate
School
of
the
State
UniversitY
of
New
York
in
partial fulfillment of the requirementsor the degreeof
Doctor
of
PhilosoPhY
by
July
1993
8/9/2019 PhD Thesis G a Chang
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8/9/2019 PhD Thesis G a Chang
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Abstract
This study is concernedwith the computationalmodeling of energy absorption
(fatigue)
capacit),
of
reinforced
concretebridge
columnsby using a cyclic dynamic
Fiber
Element
computational
model.
The results are used
with
a smooth
hysteretic
rule
to
generate
eismic
energydemand.
By comparing
he ratio
of energy demand
o
capacity,
inferences f column
damageability
r
fatigue osistance re
made
The
complete
analysis
methodology
or
bridge columns
s
developedstarting
rom
basic
principles.
The hysteretic
behaviorof steel
einforcement s
dealt
with in
detailed:
stability, degradationand consistency f cyclic behavior s explained. An energy based
universally
applicable
ow
cycle
fatiguemodel for steel
s
proposed.
A hysteretic
model
for confined and
unconfined
concretesubjected
o
both
tension
or compressioncyclic
loading s developed,
which
is
also
capableof simulating
gradual
crack closure.
A
Cyclic
Inelastic Stnrt-Tie
(CIST)
model
is developed,
n
which
the
comprehensiveconcrete
model
proved to
be suitable.
The
CIST
model is
capableof assessing
nelastic
shear
deformations
with high accuracy,
within
the
context
of a
Fiber Element
(FE) program.
A
parabolic fiber
element
with
parabolic
stress unction element
for uniaxial flexure is
developed,as well as a rectangular iber element with a quadratic interpolation function
suitable
or
biaxial
flexure.
A
smooth ule-based
macro
model for
the
simulation
of
the
hystereticbehaviorof
reinforced
concrete
elements
s developed.
The model
s
capableof accurately imulating
cyclic behavior
when compared
with actual experimental data,
through
use of an
automated system
dentification
procedure
which
proved to
be
very effective
in
finding
the model
parameters o
best approximate
member
behavior.
The macro model was
oalibrated o simulate he behaviorof a full sizebridgepier and then implemented nto a
SDOF
non-lineardynamic analysis
rogram
o
generate
nelastic
esponse
pecffa.
In
addition
to
the
usual
ductility-based
nelasticspecffa,severaladditional
energy
spectra are also
generated
which
include: viscous
damping,
hysteretic energy, cyclic
(fatigue)
demand.
These
spectra
are used
as
part
of a
rational methodology n which
the
cyclic
demand on bridge
columns
is compared with
the capacity
predicted
by
Fiber-Elementanalvsis.
8/9/2019 PhD Thesis G a Chang
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Acknowledgments
My profoundgratitudes expressedo my advisorDr. JohnB. Mander,whosebrilliant
ideasand
helpful
comments
ade
possiblehe
completion
f
this work.
His
competent
guidance asstamped
great
mpressionn
my
academic
hought.
The
financialsupport
of
my home
nstitution
Universidad
ecnol6gica e
Panam6s
deeply
appreciated.
Special
hanks o Ing
H6ctor
Montemayor or
his
profound
commitment
and encouragement,
o Ing.
Luis Mufroz
or his sincere
riendshipand
to
Ing. Jorge
uis Rodriguez
or his spiritual
ellowship.
I am also thankful o LASPAU for its financialsupportand specially o Ms. Sonia
Wallenberg
or her
personal
nterest.
I
wish
to thank
my friends at
the
Civil
Engineering
Department
or their friendship,
specially
oy Lobo
My appreciation
oes o
my
family for
their
patience
nd support,specially
o my
parents-in-law hose
oveand
caro
werealways
elt.
This
researchwas conducted
t
the department
f Civil
Engineeringat
the
State
Universityof New York at Buffalounder he supervisionf Dr. JohnB. Mander. Drs.
Andrei
M., Reinhornand
Ian G.
Bucklealso served n
the
committee, nd
Dr Peter
Gergely
s
outside
eader.
The
NationalCenter
or Earthquake
ngineering
esearch t
the
State
Universityof
New
York at
Buffalo
provided
inancial
upport
or the work contained
n
Sections
and 6,
this
assistance
s
gratefully
cknowledged.
To my beloved
Marfa,
or
your ove.
8/9/2019 PhD Thesis G a Chang
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Table
of Gontents
1. ntroduction
1.1 Background. . . .
1.2 Integrationf Previous esearch ork
1.3
Seismic valuation ethodologies
1.4
Scope f Presentnvestigation
2.Hysteretic
nd
Damage
odeling
f
Reinforced
teel
Bars
2.1 ntroduction
2.2
Monotonic
tress-Strain
urve
2.2.1
The
Elastic ranch
2.2.2
TheYield
Plateau
2.2.3
Strain
Hardened
ranch
2.3The
Menegotto-Pinto
quation
2.3.1
Computat ion
f
Parameters
,
n
nd
R .. .
2.3.2
Menegotto-Pinto
quation
imiting
ase
2.4
CyclicProperties
f Reinforcing
teel
2.4.1
Envelope
ranches
Rules
and2)
2.4.2Reversal ranchesRules and4)
2.4.3
Returning
ranches
Rules
and
6) .
.
.
2.4.4
First
Transition
ranchesRules
and
8)
2.4.5
Second
ransition
ranches
Rules
and
10)
2.4.6
Strength egradation
2.5
Stress-Strain
odel
Verification
2.6Damage
Modeling
1-1
1-2
1-3
1-5
2-1
2-1
2-1
2-2
2-2
2-2
24
2-7
2-10
2-10
2-12
2-15
2-23
2-24
2-26
2-26
2-36
2.7
Damage
ModelmplementationndVerification . 241
2.8
Stra in ate
Ef fects
. . . . . .
2-ST
2.9
Conclusions
.
2-Sg
3. Modeling
tress-Strain
yclic
Behavior
f Concrete
3.1 ln t roduct ion. . . .
3-1
3.2
Review
f Previous
Work n
Stress-Strain
elations
or
iii
Concrete
3-2
8/9/2019 PhD Thesis G a Chang
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3.5
3.6
3.7
3 2.1
Monotonic
ompression
tress-Strain
quation
3-2
3.2.2
nitial
Modulus
f
Elasticity
. 3-12
3.2.3 Strain
t
PeakStress
or
Unconfined
oncrete
3'14
3.2.4 Characterist ic
f
heDescending
ranch
f heMonotonic
. . 3-16
Stress-strainurveor Unconfinedoncrete
3.3
Recommended
omplete
tress-Strain
urve
or
Unconfined
. . .
3-17
Concrete
3 .4Con f inemento f
oncre te
. . . . .3 -22
3.4.1Conf inement
odels
. . . . .
3-23
3.4.2Confinement
echanism
..
. . .
3-29
3.4.2.1 onf inementof
i rcu larSect ions
. . . . .3-29
3.4.2.2 onfinement
f Rectangularections
. 3-30
3.4.3Conf inementEf fectonStrength. . . . .3-32
3.4.4Conf inementEf fecton
uct i l i ty
. . . . . . 3-34
3.4.5 Confinement
ffect
n the
Descending
ranch
. . . 3-35
Concre te inTens ion
. . . .3 -35
Dynamic
f fects
nConcrete
ehavior
. . .
. . . . . . 3-37
Modeling
ysteretic
ehavior
. 3-39
3.7.1
Basic
Components
f a Hysteretic
odel
. . 3-39
3.7.2A General pproacho Assessing
egradation
ithin
. . . 3-40
Partial
ooping
n a
Rule-Basedysteretic
odel
3 .7 .2 .1
i rs tPar t ia l
eversa l
. . . . .341
3.7.2.2
Part ia l
eloading
,.
342
3.7.2.3
Partial nloading
roma
Partial eloading
. 345
3.7.3A Smooth
ransition urve
orMathematical
odeling
. 3-46
3.8 Cyclic
Properties
f Confined
nd
Unconfined
oncrete
. .
. . 3-49
3.8.1Compression
nvelopeurve
Rules
and5)
...
3-49
3,8.2Tension nvelope urveRules and6) .. 3-51
3.8.3
Pre-Cracking
nloading
ndReloading urves
.. 3-52
3.8.4
Post-Cracking
nloading
nd
Reloading urves
. 3-58
3.8.5
Pre-Cracking
ransit ion
urves
..
.. 3-59
3.8.6
Post-Cracking
ransi t ionurve
. . . . . . 3€1
3.9
Model erif ication
..
. .
3€4
3.10
DamageAnalys is . .
. . . .3€4
iv
8/9/2019 PhD Thesis G a Chang
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3.11
Conc lus ions
. . . . .3€6
4. Damage odeling
f
Reinforced
oncrete
olumns
usingFiber-Element
nalysis
4.1 ntroduction
4.2Moment-Curvature
nalysis
or
Uniaxial
ending
4.3Moment-Curvature
nalysis
or
Biaxial
ending
4.4 Force-Displacement
nalysis
. .
4.4.1Elastic
lexural
eformation
4
4.2
Plastic
lexural
eformation
4.4.3
Elastic
hearDeformation
. .
4.4.4 nelastic
hear
Deformation
4.4.4.1
Proposed
yclic
nelastic
trut-Tie
CIST)
Model
or
Shear
Deformations
.
4.4.4.2
Crack nclination
ngle
4-30
4.5 Validation
f Fiber-Element
odel
..
4-92
4.6
Conc lus ions
. . . . . .
4-3S
5. SmoothAsymmetric
egrading
ysteretic
odel
with
Para
meter
dentifi
ation
5.1 ntroduction 5-1
4-1
4-1
4-9
4-16
4-16
4-17
4-18
4-20
4-21
5.2A
Smooth
Curve
o FitTwo
Tangents
5.2.1The
Menegotto-Pinto
quation
5.2.2
Computationf Parameters
,
n
ndR
5.3
Description
f
Smooth
ysteretic
odel
5.3.1
Monotonic
nvelope
urves
5.3.2Reverse
urves
5.3.3Transition
urves
5.3.4
Model
Summary
5.4 Parameter
dentification
5.4.1
Optimization
ethod
5.4.2
Scaling
5.4.3
ConstrainingheParameters
..
5.4.4Init ial
st imate
5.4.5
Order
f Parameter
dentification
5-2
5-2
54
5-7
5-7
5-9
5-12
5-16
5-18
5-19
5-20
5-21
5-22
5-22
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5.5
Verification
f
S
5.6
Conclusions
mooth
Modeland
System
dentification
Method
6.
Assessmentf
Hysteretic
nergy EMAND
6.1 ntroduction
6-1
6.2
Elastic esponse
f a
SDOFSystem 6-1
6.3 nelastic
esponse
f a
SDOFSystem 6-4
6.4
Inelastic
esponse
pectra
6-7
6.4.1
Displacement
ucti l i ty
pectra 6-7
6.4.2
Energy ased pectra
6-8
6.5 mplementation
nd
Results . . . 6-11
6.6An ll lustrative
xample
. 6-14
6.7Conclusions . 6-15
7.
Summary,
onclusionsnd
Recommendations
7.1
Summary
.
7-1
7 2SomeSpecific onclusions
7-2
7.3Recommendationsfor
uture esearch...
74
Appendix
A. References
5-23
5-24
VI
8/9/2019 PhD Thesis G a Chang
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List
of
Symbols
a = sffess lock depth
A ,
:
atea
of
the
confined
core
concrete
measured
o the
centerline
of
the
perimeter
hoop
Aror,
:
area
of bound
concrete
nder
compression
A
:
effectively
confined
area
or rectangular ection
As
:
gross
section
area
Aq
:
shear
area
Au
:
total area
of
hoop steel
Au
:
areaof
the
shear
einforcement
A,
:
areaof
longitudinal
steel
Arn
:
hoop cross
sectional
rea
A,,
:
total longitudinal
steelarea
Ar,
:
total areaof
transverse
einforcement
parallel to the x
axis
lsr : total areaof transverseeinforcementparallel to the y axis
b
:
breadth
of
the
section
b
:
fatigue strength
exponent
b,
:
concrete ore
dimension
n
the x
direction
c
:
distance
rom where e-
is measured
o the neutral axis
c
:
damping
coefficient
c
:
fatigue ductility
exponent
D
:
total
damageaccumulated
d
:
column
diameter
dt
:
bar diameter
of steel
d
:
concrete
ore dimension
n the
y
direction
D
:
damage
or
one cycle
of a
given
amplitude
Ae
d,
:
diameter
of circular
or
spiral
hoops
v11
8/9/2019 PhD Thesis G a Chang
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f,ep
:
energy
absorbed
n
a elastic-perfectly
lastic
oop
Ey
:
final slope
E1,
:
hystereticenergy
absorbed
E1
:
absolute
inetic
energy
E
o
:
tangentmodulus
of elasticity
at
the nitial
point for
a softened
urve
e
:
average
strain
E2
:
viscousdamping
energy
8,,
:
tangent
modulus
at
the
returning
point
E,
:
sffain
energy
E,
:
elastic
modulus
of
elasticity
or
steel
,E.""
:
secant
modulus
Et
:
tangent Modulus
Et
:
total
energy
at
a
given time
8,"
:
tangential
Young's
Modulus of concrete
F
"
:
compressive
orce
in concrete
strut
Ft
:
tensile
orces
n concrete
ie
F,
:
forceon steelhoop
ft
:
strain
under
the confining
fluid
pressure
f
:
form factor
f,
:
stress
amplitude
f,
:
concretestress
f ,
:
confined
concrete
strength
f
:
unconfined
concrete
strength
fo
=
damping
force
fro
:
unbalanced
orce
ft
:
final
stress
or
a
softened
curve
f,
:
inertia force
fl
:
confining
sffess
fu
:
confining
sffess
n the x direction
vll l
8/9/2019 PhD Thesis G a Chang
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8/9/2019 PhD Thesis G a Chang
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M
:
moment about
he centroid
Mj
:
moment at
first
yielding
M^o
:
maximum
moment
Mr,
:
crackingmoment
m
=
total mass
N
:
effective
or
equivalent
numberof equi-amplitude ycles
Ny
:
number of cycles
o failure
P,
:
appliedaxial
oad
O
:
post
yielding
slope
atio
R
:
radius of curvafure
parameter
R
:
symmetry
parameter
R ,
:
critical value
of
R
Rmo
:
critical
value of
R for
the M-P
equation
R,,
-
force reduction
factor
s
:
center
o
center
hoop spacing
s/
:
clear
ongitudinalspacing
between pirals
n which archingaction of
the
concrete
develops
,So
:
spectralacceleration
S7
:
spectraldisplacement
Su
=
pseudo
velocity
T
:
period
of
the structure
/
:
time
To
:
period that
separates
long and
medium
period
behavior
tr
:
displacementof
the system
espect
o the
ground (deformation)
V
=
shear
orce
V,
:
shearstrength
of concrete
V,
:
shearstrengthofsteel
w
:
specificweight
W,n
:
modulus
of
toughness f
the hoop
steel
8/9/2019 PhD Thesis G a Chang
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.r
:
system
isplacement
xs
:
ground
displacement
Xp
:
plastic
displacement
t : system elocity
i-s
:
ground
isplacement
i
:
system cceleration
is
=
ground
cceleration
vsp
:
non-dimensional
palling
train
Xv
:
leld
displacement
X,
:
maximum
nelastic isplacement
esponse
y(x)
:
non-dimensionaltress
unction
z(x)
=
non-dimensional
angent
modulus
unction
o
:
fraction
of shear
orce which
is
added
o the
axial
load
y
:
sheardistortion
e
:
strain at any
fiber
eo
:
strain
at
the
centroid
A
:
total deformation
L,
:
elastic
flexure
deformation
Lp
:
plastic
flexure deformation
Lr
:
elasticshear
deformation
[sp
:
inelastic sheardeformation
A€o : strain amplitude
Ay
=
yield
displacement
2eo
:
total
strain
range amplirude
et,
:
strainat
peak
sffess
or
confinedconcrete
et
:
unconfined
concretestrain
at
peak
stress
eca
:
ultimate
compressionstrain
E1
:
strain at
final
point
for a softenedcurve
xi
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Etr^
:
location of
the tensionenvelope ranch
eo
=
strainat
initial
point for softened
urve
Ei^
:
location of
the compression
nvelope
ranch
ep
:
plasticstrainamplitude
Ept
:
Plastic
strain
Er,
:
strainat
the returning
point to the
envelope urve
es
=
steelstrain
tsfr
:
strain
hardening strain
€sro
:
standard eviationof
the
shain
history response
E,u
:
strain at ultimate sffess
Eun
:
unloading strain
from
an envelopecurve
etf
:
fatigue ductility coefficient
ey
:
leld
strain
Eo
=
average ongitudinal strain on
the
concretestruts
ev
:
sfain on
the
transverse oops
Q,
:
magnitude
of
plastic
curvature
0u : ultimate curvafure
Qy
:
leld
curvature
p
:
ductility
LLef
:
effectiveor equivalent
equi-amplitude
ycle ductility
€
:
damping ratio
9
:
volumeffic ratio
of
the ongitudinal
steel
n the
confined
core
p,
=
ratio
of
hoop reinforcement
o
volume of concrete
ore
measured o
outsideof
the hoops
P,n
=
volumetric ratio
of
transverse oops
P'
:
Ar'/sd
py
=
Arrlsb
6tf
:
fatigue
strengthcoefficient
0
:
angleof inclination of cracks
espect o the
ongitudinalaxis
xl l
8/9/2019 PhD Thesis G a Chang
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0p
@4
Cr) 1
:
plastic otation
:
damped
requency
=
naturalangular
requency
xl l l
8/9/2019 PhD Thesis G a Chang
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List
of
Figures
Section
1
Fig.
-1Summaryof
esearch
igni f icanceofh isStudy
ntheContextfa . . . . . .1-4
Seismic valuation ethooologY
Section
2
Fig.2-1The
enegot to-Pinto
quat ion
. . . . . . .
2-3
Fig.2-2Different
urves
avingheSame tart ing
ndEnding ropert ies
.... . . . .
2-4
Fig.2-3
Tension
ndCompression
nvelope urves
2-1'l
Fig.2-4
Effect
f heStrain
mplitudef
heReversaln he Equation
. . . .
2-15
Parameters
Fig.2-5Reversal
rom
YieldPlateau
2-16
Fig.2-6Definition
f
heReversal nloading
ranch
2-16
Fig.2-7Effect f heStrain mplitudef Loop n he nitialModulus ndR
parameter
or
Reinforcingars
fy
=
Sgksi)
Loading)
. .
Fig.2-8Effect f
heStrain
mplitudef Loop
n he nitialModulus ndR
parameter
or
Reinforcing ars
fy
=
53 ksi)
Unloading)
Fig.2-9Effect f he
Strain
mplitudef Loop n he
nitialModulus ndR 2-19
parameter
or
HighStrength ars
fy
=
123ksi)
Loading)
Fig.2-10Effect f
heStrain
mplitudef Loop n he nitial
Modulus ndR 2-20
parameterfor
igh
Strength ars
fy
=
123ksi)
Unloading)
Fig.2-11 itting
f M-P
Equationo
a Loading oop f Reinforcing
teelBars
. . . .
2-21
(fY
=
53 ksi)
..
Fig.2-12 itting f M-PEquationo an Unloadingoop f Reinforcingteel 2-21
Bars
fy
=
53 ksi)
Fig.2-13Fitting
f M-PEquation
o a Loading
oop f HighStrength teel 2-22
Bars
fY
=
123
ksi)
Fig.2-14
itting f M-P
Equationo
an Unloadingoop f HighStrength 2-22
Steel
Bars
tY
=
'tZg
ksi)
.
.
Fig.2-15Sequence
f
Partial eversals
2-23
Fig.2-16Flow
of Rulesat
EveryReversal
ndTargetStrain 2-25
Fig.2-17Degradation
f Reinforcing
ndHighStrength teelBars
2-27
Fig. -18Comparison
f
Degradingodelwith
xperimentalesults .. .
2-27
Fig.2-19StressDegradation
imulation nd
Fracture rediction n Steel
2-28
Bars
Fig.2-20
tress-Strain
xperimenty
Kent ndPark
1973),
pecimen
.... . . . .
2-29
Fig.2-21Stress-Strain
xperimenty
Kent ndPark
1973),
pecimen
.... . . . .
2-29
Fig.2-22 tress-strain
xperiment
yKent nd
Park(1973),pecimen
.... . . . .
2-30
Fig.2-23 tress-Strain
xperimentbyKentand
ark(1973),pecimen
5 ... . . . . 2-30
Fig.2-24Stress-Strain
xperiment
y Kent
andPark
1973),
pecimen 1 2-31
Fig.2-25 tress-Strain
xperiment
y Kent
ndPark
1973),
pecimen 7 .
.
..
. . .
2'31
2-17
2-18
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Fig.
2-26
Stress-Strain
xperimenty Ma,Bertero ndPopov
1976),
2-32
Specimen
Fig.2-27Stress-Strain
xperiment
y Ma,Bertero ndPopov
1976),
2-32
Spec imen
. . . .
Fig.2-28Stress-Strain
xperiment
y Panthaki
1991),
pecimen 2
Fig.2-29Stress-StrainxperimentyPanthaki1991), pecimen 3
Fig.
2-30
Stress-Strain
xperimenty Panthaki
1991),
pecimen ' 6
Fig.2-31Stress-Strain
xperimenty Panthaki
1991),
pecimen 19
Fig.2-32Stress-Strainxperiment
y Panthaki
1991),
pecimen l
Fig.2-33Stress-Strain
xperimenty Panthaki
1991),
pecimen 4
2-33
2-33
2-34
2-34
2-35
2-35
2-36
2-40
2-42
2-43
2-44
2-45
2-46
2-47
2-48
2-49
2-50
2-51
2-52
2-53
2-54
2-55
2-56
2-57
Fig.2-34Stress-Strainxperimenty Panthaki
1991),
pecimen 5 . . .
Fig.2-35Determination
f Equivalenttrain mplitude
Fig.2-36
HighStrength ar,Specimen 18
Panthaki,
991)
Fig.2-37
High
Strength ar,Specimen 10 .
Fig.2-38 ighStrengthar,Specimen13 .
Fig.2-39HighStrength
ar,Specimen 12 .
Fig.
2-40
HighStrength ar,
Specimen
4
Fig.2-4'lHighStrength
ar,Specimen 7
Fig.2-42
High
Strength ar,Specimen 14 .
Fig.2-43
High
Strength ar,Specimen 9
Fig.2-44HighStrength ar,Specimens11,P2andP3
Fig.2-45
Reinforcing
ar,Specimen l
Fig.2-46Reinforcingar,Specimen 9
Fig.2-47Reinforcingar,
Specimen
5
Fig.2-48 einforcingar,
Specimens11,R7and
R10
Fig.2-49HighStrength ar,
Specimen 20,
Low-High tepTest
Fig.2-50HighStrength ar,Specimen
21,
High-Low tepTest
Fig.2-51 ncipient ailure
rediction
Section
3
Fig.3-1Characteristics
f
the
Stress-Strain
elationor Concrete
. . . . 3-2
Fig.3-2Comparisonf Different tress-Strainquationsor Concrete
.. . ... 3-6
Fig. -3Equat ionuggestedyYoung1960) . . . " . , . . . 3-6
Fig. -4Equat ionuggestedySaenz
1964)
. . . . . . . . .
3-7
Fig.
3-5Equationroposed
yPopovics
1973)
... . . . ^
3-7
Fig.3-6Equation uggestedySaenz
1964)
3-10
Fig.3-7Equation uggestedy
Sargin
1968)
3-10
Fig.
3-8Equation roposed
yTsai
1988)
3-11
Fig.3-9Comparisonf Different quationsor heSecantModulus
f 3-20
Concrete
Fig.3-10Comparisonf Different quationsor he Strainat Peak
Stress 3-20
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Fig.3-11
Proposed
heoreticaltress-Strainurvesor Unconfined
3-21
Concrete
Fig.
3-12Theoretical
tress-Strainurves uggestedyColl ins ndMitchell
. . . 3-21
Fig.3-13
Tsai sEquation
arametersor Unconfined
oncrete
3-22
FiE. -14Some
Proposed
tress-Strainurvesor
Confined oncrete
3-28
Fig.3-15ConfinementechanismorCircularndRectangularross 3-31
Sections
Fig.3-16Confined oncrete
trength atio .
Fig.3-17Comparison
f Different odelsor
Triaxial onfinement.
. . .
Fig.3-18Characteristicf
heFalling
ranchorConfined oncrete
Fig.3-19Definitionf
Falling
ranchorConfined oncrete
Fig.3-20Relationshipetween urves
n
a Rule-Basedodel
Fig.3-21 arget oint nd
Reloadingoint
n a Complete eversal
Fig.3-22
Reloading
roma Partial nloading
Fig.3-23Unloadingroma Partial eloading
Fig.3-24 Smooth ransitionurve
Fig.
3-25Tension
ndCompressionnvelope urves
Fig.3-26
Cyclic
Compressionharacteristicsf Concrete
Fig.3-27Complete nloading
ranch
Fig.3-28Complete oading
ranch
Fig.3-29
Loading
ndUnloadingurves fterCracking . . .
Fig.3-30Partial nloading
urves
orTension ndCompression
Fig.3-31Transition urves
Before
racking)
Fig.3-32Transition urves
After
Cracking).
Fig.3-33RelationshipmongheModelRules
Fig.3-34UnconfinedyclicCompressionestby
Sinha,Gerstle ndTulin
(1
e64)
Fig.
3-35UnconfinedyclicCompressionestby Karsan
ndJirsa
1969)
Fig.3-36
Unconfined
yclicCompressionestby
Okamoto
1976)
Fig.3-37
Unconfined
yclicCompressionest
by Okamoto
1976)
Fig.3-38
Unconfined
yclicCompressionestby
Tanigawa
1979)
Fig.3-39Cyclic ension estbyYankelevskyndReinhardt1987)
Fig.
3-40Confined oncrete yclic
est
by Mander t al.
1984)
Fig.
3-41Confined oncrete yclic estby Mander t al.
(1984)
Fig.3-42Comparisonf theProposedension ranch
quation ithother
Analytical quations
3-33
3-33
3-38
3-38
3-40
3-43
3-43
3-44
3-48
3-50
3-54
3-55
3-56
3-58
3-60
3-62
3-62
3-63
3-68
3-68
3-69
3-69
3-70
3-70
3-71
3-71
3-72
Chapter
Fig.4-1Definition
f
Global ndLocalCoordinates
. . . . 4-5
Fig.4-2Definitionf Variablesna FiberElement
. . . . 4-7
Fig.4-3
Definitionf
Variablesor Biaxial ending
4-10
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Fig.4-4Element
ode
Numbering 4-13
Fig.
4-5Flexural
eformation
n a
Column
4-'16
Fig.
4-6Shear
Deformation
na Column 4-22
Fig.4-7
Equivalent
trut-Tie
odelor ShearDeformations. . . .
4-22
Fig.4-8EquilibriumndStrainDeformationn heCyclicneiastic trut-Tie 4-23
Shear
Model
Fig.4-9Definition
f Average ongitudinaltrain n ShearConcrete trut
4-24
Fig.4-10Comparison
f heAnalytical tress-Strainelationshipith
he 4-33
Experimental
ehavior f PlainConcreteromAycardi t al.
(1992)
or
Specimens
and4
Fig.4-11Comparisonf
Propossediber
Element
odelwith
Experimental 4-37
ResultsromAycardi
t al.
1992)
pecimen, P
=
0.10 c Ag . .
Fig.4-12Comparison
f Proposed
iberElement odelwith
Experimental 4-38
Resultsrom
Aycardi t al.
1992)
pecimen,
P
=
0.30 c Ag .
Fig.4-13Comparisonf ProposediberElement nalysis ithExperimental 4-39
andAnalytical
esultsromMander t al.
(1984)
olumn
Fig.4-14
Predictionf LowCycle atigue
racture
f Longitudinalars or 4-40
Co lumn . . . .
Fig.4-15Comparison
f ProposediberElement nalysis ithExperimental
4-41
andAnalytical
esultsromMander t al.
(1984)
olumn
. . . .
Fig.4-16Comparisonf
ProposediberElement
nalysis
ith
Experimental 4-42
andAnalytical esults
romMander t
al.
1984)
olumn
Fig.4-17 nalytical imulationf a
Full
SizeShearCritical ridge ier 4-43
Tested y Mander t al.
1993)
Chapter5
F ig .5 -1 heMenegot to -P in toEquat i on
. . . . . . . . 5 -3
Fig.5-2Computationf Parametersor
he
M-PEquation ., . .
5-4
Fig.5-3Monotonicnvelopeurves .. . . 5-8
Fig.5-4Reverse oading
urve
5-10
Fig.5-5Reverse nloading
urve
5-13
Fig.5-6Transition urves
5-14
Fig.
5-7
Logical ranching iagram
5-17
Fig.5-8Comparisonf MacroModelSimulationseneratedhrougha) 5-28
Experimentalata,
c)
FiberElement xperimentimulation
Fig.5-9MacroModelSimulationf a
Full
SizeBridge ierBased nActual 5-29
Experimentalata
Fig.
5-10Simulationf
heCyclic ehaviorf a FullSizeBridge ierBased
5-30
on a FiberElement imulated xperiment
Fig.
5-11Macro
ModelSimulationf a 113 caleColumn ased n 5-31
Experimentalata
Fig.
5-12Macro
ModelSimulationf a 113 caleColumn ased n Fiber 5-32
Model
Simulated
xperiment
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Fig.5-13Macro
Model
imulationf a Bridge ol low olumn ased n Fiber
5-33
Element
imulated
xPeriment
Chapter
6
Fig. -1
Proposedhree
evel
eismic valuationethodology
.
.. . . . . .
6-2
Fig.
-2Equivalent
ingle-Degree-Of-Freedom
ystem
.
...
^
6-6
Fig.6-3aForce
Correction
actor . ...
. . . 6-6
Fig.6-3bStepBy-Step
ntegration . . . ..
6-6
Fig.6-4Symmetry
arameter 6-10
Fig.6-5
nputGround
Motions sed
or
SpectralAnalysis
6-12
Fig.6-6Energy,
uctility
nd
Low
CycleFatigue emand pectraor El
6-16
Centro
1940)
-S,
with5olo iscous amping atio ndPGA
=
0.348
Fig.6-7Energy, uclility
ndLowCycle
atigue
emand pectraor 6-18
Pacoima
1971) ,
i th %Viscousampingat io ndPGA
=
1.17 . . . . .
Fig.6-8Energy, uctility
ndLowCycleFatigue emand pectraor San 6-20
Salvador1986), i th olo iscous amping atio ndPGA =0.695 ... .
Fig.6-9Energy,
uctilityndLowCycle
Fatigue
emand pectraor Taft 6-22
(1952)
21E,with5%
Viscous amping atio ndPGA
=
0.156
Fig.
6-10Energy,
uctility
ndLowCycleFatigue emand pectraor
6-24
MexicoCity
1985),
ith5%Viscous amping atio ndPGA
=
0.171
Fig.6-11Energy, uctility
ndLowCycle atigue emand pectraor
6-26
Sinusoidalnput ,wi th
olo iscousDampingat ioand GA
=
1.09 . . . . . .
Fig.6-12Energy, uctility
ndLowCycle
atigue
emand pectra
or
El 6-28
Centro
1940)
-S,with
5%
Viscous
amping
atio nd
PGA
=
0.348
.
(Elasto-Perfectly
lastic
Model)
Fig.6-13Energy,
uctilityndLowCycle
atigue
emand pectraor 6-30
Pacoima
1971),
i th
olo iscous amping atio ndPGA
=
1.17
.
(Elasto-Perfectly
lasticModel)
Fig.6-14Energy, uctility
ndLowCycleFatigue
emand
pectraor
San 6-32
Salvador
1986),
ith
5%Viscous amping atio ndPGA
=0.695
.
(Elasto-Perfectly
lastic
Model)
Fig.6-15Energy, uctility
ndLowCycle
atigue
emand pectraorTaft 6-34
(1952)
21E,
with5olo iscous amping atio ndPGA
=
0.156
.
(Elasto-Perfectly
lastic
Model)
Fig.
6-16Energy,
uctility ndLowCycle
atigue
emand pectraor 6-36
Mexico ity 1985), ith5olo iscous amping atio ndPGA = 0.171 .
(Elasto-Perfectly
lasticModel)
Fig.6-17Energy, uctility
ndLowCycle atigue emand pectraor 6-38
Sinusoidal
nput, ith5%Viscous amping atio ndPGA
=
1.0
g
(Elasto-Perfectly
lasticModel)
Section
7
Fig.7-1 ummaryf Research
ignif icancef hisStudyn heContext f a . . . . . . 7-6
Seismic valuation
ethodology
XVII I
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Section
Introduction
1.1 Background
In order to designor analyzehe behaviorof bridge substructurespiles and
columns f
piers)
hatmay
be either
einforced,
r
fully or
partially
prestressed
oncrete,
it is
essential
hat
analyticalmodels
be developed
hat
accurately eflect
the true
non-linear
dynamic cyclic
loading
behavior
of
those
members.
Current
analytical
modeling echniques
f structural lement se eithera macromodeling
approach
e.g.
DRAIN, Kanaan
nd
Powell,
1973;Allahabadi
nd
Powell,
1988)
or micro inite
element
approache.g.ANSYS,Kohnke,1983). t is consideredhata coarsemacroapproachn
which
lumped
plasticity
within elements
s
used o
predict
esponse
ehavior, n
many
instances,s too
crudewhen
ooking
at detailed
ehavior f
joints
and
plastic
hinges.
On
the
other
hand,
ophisticated
inite
elementmodels
may equire
mesh epresentation
hat
is too fine, thus
prohibiting
he
analysis f large
or evenmoderate
ize bridges. It is
considered
hat the most
appropriate
ompromises to
use a combination
f the two.
Fiber
elements
anbe used
or this
purpose.
Fiber
elements
an be incorporated
nto
a
non-linear time-history
structural analysis
computer
program
using two
different
approaches:
irect iber modeling,
r
indirect
iber modeling.
The first
has ecently
een
incorporated
nto the atest ersion
f DRAIN-2DX,
but
s in
a relatively
rude orm
and
still
may require
some urther
refinement,
ut
the
approach
hows
great
promise.
The
second pproachs the
subject f
this
study
or
the
purpose
f use
with
programs
uchas
IDARC
Park
et al., 1987)
or
DRAIN-2DX).
A fiber model
epresentation
an
eapture
details of features uchas the critical concrete nd steel strainsas part of the analysis
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process hrough
he direct
ntegrationof stress-strainesponse.Most
existing ime-history
computer
programs
ocus on
determiningthe nelasticdemands
aused
y a
given
seismic
excitation.
As
part
of a
fiber
element
analysisof components he
inelastic
capacity
of
members can also be determinedas part of a preprocessing post-processinganalysis.
Further, as
part
of a
post-processing nalysis,
he
damage
sustained
y components
and
subassemblages an
be determined as
the ratio
of demand versus
capacity. This
dissertation
ocuses
on
this
damageability
concept
as
part
of
the
modeling for
bridge
substructures.
1.2 Integrationf Previous esearch ork
Considerable
work has beenundertaken
y
Mander,
Priestley
and Park
(198a)
in
developing
moment curvatureand
force-deformation
models
based
on a
fiber
approach,
directly
integrating
stress-strain
elations or reinforced
concretemembers
Mander
et
al.,
1988a,
1988b).
Dynamic
reversed
cyclic
loading
of
members s
accounted or and
inelastic
buckling of
longitudinal reinforcement, ransverse
oop fracture,
and
concrete
crushingmodesof failure are determined rom energy considerations.Good agreement
has been demonstratedwhen
tested
against a variety
of
physical
model
experimental
results. This
fundamentalwork was
followed
by Zahn
et al.
(1990)
who developed
energy-based
esigncharts or bridge
pier
with
ductile detailing.
The
need for sophisticated
ools to
analyze
sffuctures
subjected
to
earthquake
loadingshas
produced
a
great
deal of research. Much
of
this
research s the
coordinated
effort of
many researchers hat
share a common
purpose,
to
gain
insight into this very
complex
problem.
The
complexity of
the
problem
underlies
both
the randomness
of
earthquakemotions
and
the nonlinearhysteretic
behavior
of structuralcomponents. At
the
end,
the
goal
is to
develop
rational
methods
of design,
hat
will consider
both
the
demand
that the ground
motion will impose
on
the
structure and
the
capacity of
the
sffucture
o
meet
those
equirements.
The
demand on a sffucture can be of
two types:
displacementductility demand and
energy demand. The
former
dictates bearing set width requirements
and secondaryP-A
|
-^/.
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load
effects,
while the
latter leads
to
failure
of the
constituent
materials.
steel
and
concrete, hrough
ow
cycle fatigue.
It
will
subsequently
e
shown
that
the
two
are
also
interrelated.
Much
of the research
ffort had
been
concentrated
n
the
ductility
demand,
although energy demand esearch s gaining popularity among researchers The
capacity
of structural
elementss,
of course,
a fundamental
roblem.
A
computer program
to
simulate
he
cyclic
behavior
of
reinforced
concrete
s
presented
n this
study. Every
major
aspect
of
its
development
s
presented.
Advanced
models
for
concrete
and
steel are
proposed,
with
improvements
over previous
models.
Mathematical
models or
the
description
of
damage
n
steel
elements
re ncorporated.
A
uniaxial
moment-curvature
nd force-deformation
micro model is presented s well as a
biaxial
moment-curvature
iber
element
model.
A general
pulpose
macro
model
with
system
identification
for
uniaxial
moment-curvature
or force
deformation
was
implemented.
Theseprograms
can
be inte$ated
as
part
of
an analysis
methodology
outlined
in
F ig
1 -1 .
1.3
Seismic
valuation
ethodologies
Herein
a three
evel
seismic
valuation
methodology
s proposed.
The
first
is
based
n well-known
oncepts
f ductility
and
uses
imit
analysis
echniques
rom
which
capacity/demand
C/D)
ratios
are calculated
or
structural
trength
nd
ductility.
This
is
called irst-order
pproach
s t
doesnot
concern
tself
with
cyclic
oading
effects
and s
similar
to the
procedures
iven
n
ATC
6-2.
The
second
s new
approach
dvanced
herein,
s
based
on fatigue
or damage
oncepts
nd s
concerned
ith
comparing
energy
absorption
apacities
with
seismic
energy
demands.
This
is
called
a
second-order
approach,
s t is
more
refined
aking
nto
account he
earthquake
uration
and
would
be
used
when the
results
rom
a first-order
analysis
re n
doubt.
A third
and
more
refined
analysis evel
concern
multi-degree
f freedom
system
analysis,
n
which
rationally
implemented
ysteretic
erformance
s
used.
l-3
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Step
r.
Strength
emand,
(d)
Step2" Strength apacity (c) (Limit Analysis)
C ( c t
I
Step.
r*
=
ffi
=
t
lIf
r,,
>
1.5
STOPI
Step
.I Ductility
Demand
(d)
Step
.1
DuctilityCapacity
(c)
per
ATC
6-2
Step . rt = g? [If ru 2 1.5 STOP]
$la1
Step
.2
Rotational
emand
0
p(A,
N(A
=
f
(R
$,
E
Q,
HYst.moe[)
Step
.2
Rotational apacity
o(c),N(c)
M/r)
Step.2 ,r
=ff i
[If r^'21.5 STOP]
Step
7. Generate
emberSpecific
Hysteretic
Models
From
Steps
.2and
5.2)
Step
8.
Perform
imeHistoryAnalysis
(IDARC
or DRAIN-2DX)
Step9.
ExamineCritical
Members erformance.
Use
Fiber-Element
o
predict
detailed
behavior
ased n
memberime-history.
SEISNtrC
EVALUATION
METHODOLOGY
Fig.
1-1
Summary
of
Research ignificance f
this
Study
n the
Contextof a Seismic valuationMethodology.
F a
; v
:''l d
a t r
<
v E
t s €
€ a r
- a
p
.=
al
0
, >,
. = E
- E
b<
13
q,)
L b D
o
EF
6 t t
>F
O A
= a
= ?
2 2 2
F F
b i
: ; o
v c t
a E
;A
Sections & 5
Sections
-
4
Section 5
Section
l-4
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1.4 Scope
of
Present
nvestigation
Firstly.
this
investigation
deals with
the modeling of
the
hysteretic and fracture
characteristics
f
reinforcing
steel.
The low
cycle
fatigue behavior of
steel
s modeled
based
on experimental
data.
The importance of
this modeling is that it allows the
prediction of
the
fatigue
ife of
longitudinal bars
n
the
context of a
reinforced concrete
member
subjected
o
cyclic
loading.
Thus,
his modeling will allow
to predict the failure
of
a
member due
to
low
cycle
fatigue,which
is
predominant
on
well
detailed
beams
and
columns
with low
levels of axial
load. Numerousexamples
are
presented
o
show
the
capacity
of
the model
to simulate
both
the
sffess-strain
yclic behavior
and
the
fatigue
fracture.
Secondly,
his investigation
egards
with the modeling of
the
behavior of both
confined
and unconfined
concretesubjected
o
cyclic
compression nd
tension
Section
2).
This
is the first
time
any
model
have
attempted
o model
cyclic behavior of concrete
in both
tension and compression.
The need
for
such
model is more
obvious
when
considering
shear
deformations
where
the tension
capacity
of reinforced steel
plays
an
important
role,
as
in
the Modified Compression
Field Theory
Collins
and Mitchell.
1992),and
the
Softened
Truss
Model
Hsu,
1993).
Section
4
deals
with
the Fiber Elements
modeling
of
the
moment-curvature
behavior
of a concrete ection
and
with
the
assessmentf deformations.
A cyclic strut-tie
model
is
developed
o
assess hear
deformations.
This
cyclic
sffut-tie
model for
shear
deformation,
which
makes
good
use of
the
comprehensive
onstitutive
models
developed
in
sections
and3, allows
to
simulate
he
behaviorof shear
dominated
members.
Section5 presents smoothmacro model that can be incorporatednto non-linear
dynamic
analysis
programs
o accurately
epresent he hysteretic
behavior
of concrete
elements.
The advantage f
this model,
over
previous
models,
ies
upon
the
capability of
the model
to represent ealistically
the hysteretic
behavior of concrete elements.
A
parameter dentification
procedure s also
presented,
which
is use for
the
automatic
identification of
the model
parameterso representa
provided
hysteretic
behavior.
This
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automation
makes
the
use
of
a more comprehensive
macro model more
appealing
han
unrealistic
simpler
macro
models.
Section
6
presents he development
of spectralcharts
for both
elasto-perfectly
plastic structuresand typical bridge
pier
sffuctures.
The hysteretic
behavior of bridge
piers
is simulated
by
the
macro
model developed
n
section 5
which was calibrated o
simulate
he
behavior
of an
actual ull
size bridge
pier
testedcyclically.
The model was
also calibrated
o
simulate
he
hysteretic
behavior
of an analytically
produced
srmulated
experiment
using
actual
material and
section
properties
nto
the
fiber
model
column
analysis
program
developed
n
section
4.
The
development
of
reliable inelastic spectra
is
an
important aspect when
assessing
he
energy
and
ductility demands n
ductile structures.
A methodology or the
consideration
f energy-based
nelasticdamage ssessment
s
given
in
this
section.
This
approach
of
simulating stnrctural
behavior by
means
of a
Fiber Elements
model analysis
and
then
calibrate
a
macro model to represent
he
actual behavior
is
presented n this investigation
as
the most rational way
of
simulating
the
behavior of a
reinforcedconcrete trucnlre
without
the
costly
mplementation
more refined
procedure
asFinite Elements.
Finally
some
conclusions
nd recommendations
or
further
research
re
presented
in
the
last section.
This
investigation
has
shed some
ight into the need for some well
designed
experiments
o
look
into
the
behaviorof somespecific
variables.
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Section
Hystereticnd
DamageModeling
f Steel
Reinforcingars
2.1 lntroduction
The hystereticbehavior
of
the reinforcing
and
prestressing
teel bars
influences
the hysteretic
behavior
of a
sffucturalconcrete
member. Fracture
of a
reinforcing
bar may
also
be defined as
ailure of
the member tself. It is
very
important
o
thus model
both
the
hysteretic
and
the
fatigue
properties
of
the
reinforcing
bars
accurately. Tests
performed
by
Kent and Park
(1973),
Ma
et
al.
(1976)
and Panthaki
1991)
were
used
o calibrate he
stress-strain
model advanced
herein. The degradingcharacteristicof
steels
with
leid
stresses
anging from 50
ksi to 120 ksi were
studied, and
damage relationships
were
incorporated
nto the model. The Menegotto-Pinto quation
1973)
used
by
Mander
et al
(1984)
s
used
herein o represent
he
oadingand unloading
stress-sfrain
elations.
2.2
Monotonic
tress-Strain
urve
Numerous estshave shown
hat the monotonic
stress-strain urve for reinforcing
steel
can be describedby
three
well defined branches. The corresponding elations for
stress
f,)
and
angentmodulus
E,)
after
Mander
et. al.
(1984)
are
given
below:
2.2.1
The
Elastic
Branch
0
(
e,
(
eu
f,
=
E,E,
P T
_
D S
(2-r)
(2-2)
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wher
f
e:
er=f i
in which,e_u
yield
strain,
r:yield
stress,E,
:
Elastic
Modulus
of Elasticity.
2.2.2 TheYieldPlateau y( r-r r,1
f'
=fy
Et=0
in
which,
€r1
:
sffain hardening
train.
2.2.3
Strain
Hardened
ranch
s 2
r,n
f,=f*+(fy-f,,)
l##l'
r r -,
(
e :--*-\l
*
-f, l( '-;,)
Lt
=
Lsh
tgn[€ ,
€ ,
) l
1,
_1ry
where:
=
E,r13
- J
f r u - f ,
in
which,
,r, i, the
stress
t ultimate
tress nd
,u
:ultimate
(maximum)
tress.
These
relations
an
be
represented
y a single
quation
s
given
n
Eq.
(2-45)
2.3 TheMenegotto-Pinto
quation
The Menegotto-Pinto
1973)
(M-P
hereafter)
s
useful for
describing
curve
connectingwo tangents
ith
a
variable
adius
of curvature
t
the
intersection
oint
of
thosewo
angents,
sshownn Fig.2-1.
The
M-P
equations
expressed
s:
f ,
=f,
+
Eo(e,
e )
1Q+
I -Q
(2-7)
The angent
modulus
t any
point
s
given
by:
(2-3)
(2-4)
(2-s)
{2-6}
.,
.,
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E*,
_
QEO
(2-8)
r*lr,9'-t:|-^
I Jrn-J
with a
secant
modulus connecting
he origin coordinates
eo,
fo)
and
the
coordinatesof
the
point underconsideration
e,,/) defined s:
t ,
=*=
E* .
dtr
{ _ {
/ s l o
h 1
l S € C - a
a
c s
-
L o
in which
€,
:
steel
strain,
f,
:
steel
stress,eo
=
sffain at
initial
point,
/,
:
sffessat
initial
point, E
o
:
tangent
modulusof elasticity
at
initial
point,
Q,
R
and
fs11
rc equation
parameterso control
the
shape
of
the curve.
It
should
be
noted
that as t is
presented,
q.
(2-7)
has he following
properties:
(1)
a slope
Eo at
the starting
coordinate
(eo,
fo),
(2)
it
approaches
he
slope
QEo
as
€5
oo.
For
computational
tractability
R needs
to
be
limited
to
about
25. This
essentially
epresents
bilinear
curve
given
by a single
equation.
To
use
this
equation
t
is necessary
o
develop
an algorithm
to
compute
the
parameters
Q,
*
andR.
A
procedure o
compute
hese
parameters
s
presented
n the next
section.
0.8
f _ f
':
':
0.6
t . - l
C N J
O
0.4
(2-e)
R
F
J
)
I
Fig.2-1
TheMenegotto-Pinto
quation
Eo@,
eo)
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2 3.1Computation
f
Parameterc
,
f ^
and
R
Let
the
denominator
n
the M-P
equationbe such
hat,
-
,Oa l
I l - €,-e, l l^
A = l l - / - l p n = ' - - - - ' : t
t
L
I
l ,n- l l
)
The
derivative
of
I is therefore:
dA
_A( l -A -R)
de'
t '
- to
Eq.
(2-7)
canbe expressed
n terms
of
,,4
as:
f , =fo Eo(e,-d(e+y)
\ , ' r )
and
then the
derivativeof
f,
respect o
e,
gives
a
tangentmodulus
which is:
(2-10)
E,=#=s (e*+)-' #(+#)
(2-1
)
(2-12)
(2-13)
Fig.2-2 Different
Curves
Having he
SameStarting
and Ending Properties
A A
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By substituting
q.
(2-11)
nto
(2-13)
and
earranging.
L
=
o
-'-.?
E
'
trR+l
By evaluating his equation&t 0, = Ey, andsolving for Q,
tf
-
o^^u'
(2-t4)
Q=
(2-1s)
(2-r6)
|
-A-8+r)
Solving
or
Q
in Eq.
(2-12),
Er..
A
l
t r
-n
Q =
o -
t -A- ,
Eq.
(2-15)
wasobtained
rom an equation
elated
o
the final
slope
{),
thus
his
equation
guarantees
hatat
the target
point the
slope
condition
s met Elef)
=
E.f
Eq.
(2-16)
was derived
rom
the
ordinate
quation o,
by satisfying
his
equation,
he
ordinate
ondition
s
met
f,(e)
=fJ.
To satisfy
othconditions,
t is necessary
o
equate
both
Eqs.
2-15)
and
2-16).
(2-r7)
I
where :
A- ' .
The
solution
rocedures as ollows:
fr-f
(1)
E. .
=;--
- 'f
-€ o
(2) R*o
=
t#
, the derivation f this expressrons
given
in the next
subsection
.3.2.
It is not
possible
o reach
he
point
(ef,ff)
with
the
slope
E7
with
a
value of
R
<
R6o.
Evaluation
f
the M-P equation
or
the
caseof
R
=
R,,;o s only
possible
y
taking he
imit of
the expression,
o a
value
of
R slightly
greater
hen
Rn;o
has
o
be
used,
n order
o apply
he
expression
s
t is
shown
n Eq.
(2'7)'
(3)
If R, io 0,
it means
hat
the three
points
are
aligned,
hus
take
Q:
I and
f n
=fr.
Thevalueof
R
need
ot o
be
modified.
1 _ n R + r
_
a ( l - a R )
n
Et-
E*#-
+Ei
'
--
'
-
0
'
r -d t -a
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(4)
If R
<
R-o
then
akeR
:
R*n
+
0.01
(5)
Solve
or the
valueof a
in the ollowing
expression:
E
-
E, , "4
+
E
"a( , -
aR
-
o
'
t - a I - a
To find
the
value
of a
the ollowing
procedure
s
used:
(a)
Define afunctionf(a) as:
f(a)=r.,-t,,#*t,ff
@)
Evaluate
(1-e)
andf
e),
where is
a small
(c)
If
f
(I-e)
*
f
(e)
>
0,
no
solutions found,
so
ofe and epeat tep b) .
(e)
Takeasan nitial
estimate:
R-o
ao=_f
(2-1e)
value
=0.01).
decrease
he
vaiue
(2-18)
(2-20)
(d)
If
f
(l-e)*f
(e)
<
0
then
a solution s found
n
this
interval"
The
quadratically
onverging
ewton-Raphson
rocedure
an
hen
be used
o
find
the
solution.
(f)
If
/(a")"/(l
-e)
<
0
then
replace
ao by
J,r.
untii
the
inequality s false o
ensure
roper
convergence. f this
condition s not met the
algorithm
will find a solutionoutside he meanineful anee.
(g)
With
o" u,lninitial-estimate the following
recursive
expression
shouldbe applied until convergence
s
met. It is important o note that
the
functionf
(a)
hasa singularityat d= l, so the valueof Aa shouldbe the smallerof 0.5(1
-
a") and
0.00.
2f(a) A,a
(2-2r)
i + l
=
a i -
f
(a,
+
A,a)
f
(ai
-
La)
(8)
After
the value
of a has
beendefined
hen,
I
P . ;
^ -
( l
- 4 " ) "
u
-
--v-
(9) The valuesof fo andQ are hencalculated s:
2-6
(2-22)
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J"n
=
fo
+
f@
-
e")
(2-23)
Er..
t r
-u
Q=ft Q-21
2.3.2 Menegotto-Pinto
quation imiting
ase
In step
2
of
the
procedure
outlined above, a
factor
R^o was introduced. The
derivation
of
that factor and
the relation
of
the Menegotto-Pinto
equation
to
a
power
equation
s the
subject
of
this
subsection.
The Menegotto-Pinto quation an be expressed y:
(2-2s)
where:
If the
curve
s to
pass
hrough
xr,yr),
it canbe
rewritten
as:
1
l f
- f
o
-
8 r . .
-
A t l -Q
Eo
x1-xo
-;=
a+;
Q'26)
and
ts
derivativeas:
=n*0-Q)
E o
Y '
A R + l
(
t Y - v
I
,q=lr+
r"---=- lR)F
\
|
, v c h l o l
. /
rf"
e
=
A-t
(2-29)
then
by solving
or
Q
in Eqs.
2-26)
and2-27),he ollowing
expression
s
obtained:
n r
|
-aR* t
-
a ( l
- cR )
n
Er-E*"ft+E"-ft/=o
(2-30
(2-27)
(2-28)
,''
1
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By solving
for
a
in this equation,
he
parameters
"1,
dnd
Q
are
given
by:
/ \
l ch= lo *Eo (x1 -xo ) l
J
, I
l ' ( t
o^)8,
and.
?
-
o^*'
Q =
" o
|
-
AR+l
Eq.
(2-30)
cannot
e
evaluated
s t
is
written
or
Q:1, but
i t
The limit
value of
the fraction
n the
second
erm s:
(2-31)
(2-32)
presents
limit.
(2-33)
(2-34)
(2-3s)
(2-36)
l i rn lo** ' =R+ I
a--+ l- I
-
A
while
the other
imit is:
, .
a ( l
-
a R )
h
lllil
---:-
=.t(
a - + l
l - A
So
the limit for
the
equation
when a
-+
I is:
Ey-
E,""(R+ )
+E,R
=
0
Solving
for R,
the following equation
or
the
critical
value
of
R can be derived:
^"r=ffi
This value, as
can be shown
numerically, epresents
he minimum value that R is
to have, so that a solution o meetthe conditionsof both slope and ordinatevalue at the
ending
point.
What is of interest
now, is
to know what the limit
for the
original equation
would
be.
Both
y"o
and
Q
tend to
infinity
as a
tends o one.
Eq.
(2-25)
can
be expressed
in
terms
of
a as:
=
lo
+ Eo(x xo)fm+
Q(l
-
m)l
where,
2-8
(2-37)
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[ , , I
x-xol^t -o^ ' l i
L'-
|
xrx"
,^
I
(2-38)
(2-3e)
(2-40)
(2-4r)
(2-42.)
(2-43)
(2-44)
When + l,
l im
m=
a- )1-
The imi t of
-
m),
s
a
complicated xpression:
Er
)yort-m)=jjl.#
E,
J 7
r
I t K
L o
l _ T - r , l "
=
- t - l
R+l
l x1 -xo l
l -
[ ' .
So,
Eq.
2-37)
anbe expressed
s:
The inal form of
the
imitingcase f
the
Menegotto-Pinto
quation
s:
=
o
+
E
o(x
x
)
+
A(x
-
x
")lx
-
x
olR
with,
I
x-xo
l ^
-o^ lF
ry4"1 "- )
and,
, -E - f -E* "
I \
-
F
Ln"
-
Lo
, - E n " - E o
" - ,
t R
l x t -
xol
Eq.
Q-aD
is
dealt with
in more detail n
section3.6.3.
It is worth noting here hat
this
equation
epresentshe most
"relaxed"
of all
the
curves
given
by
the M-P
equation,
but at
the
same
ime, the M-P equationcannotbe evaluated
or this
case,as it is
a
limit
expression.
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2 4 Cyclic
Properties
f
Reinforcing
teel
In
this
section
a
universally applicable
cyclic
stress-stralnmociel
s
advanced
or
ordinary
reinforcing
and
high strength
presffessing
ars. The model is
composed
of
ten
different rules, five for the tensionside and five for the compression rde. Each of the
rules
s
described eparately
n the following
sections.
2.4.1Envelope
ranches
Rules
and2)
The
envelopebranches re definedby the monotonic
sffess-strain
elation
which is relocated and scaled
to
simulate strength
degradation. The
shape
of
the
monotonic branch is kept
intact,
except hat
at
the
points
of
reversal
a scale factor
is
calculated. This combinedmodel ensures
egradationwithin local
cyclic,
a
phenomenon
not been modeled
before.
The model was
calibratedusing
experimental esults
given
by
Panthaki
(1991).
The
sffess-sffain
elation
for
the
tension
envelope
curve
can be
expressed s a single expression y:
Rule I
(Tension
Envelope Branch)
r=
;ff *
.=P rr;r;tl,|ffi l-]
L'-(.
.
.]
l
Q-asa)
-T
'
(2-4sb)
(2-45c)
(2-4sd)
where: t r r = t r - € j ,
o.=t:,ffi
in
which €t^
=
location
of
the tension
nvelope ranch. Eq.
(2-a5)
s
shown
plotted
n
Fig. 2-3. Also
shown
n this figure
s
the
compression
nvelope ranch
defined n
an
analogous
orm
as
ollows:
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Rule
2
(Compressionnvelope
ranch)
^
=
;ffr 1-
s'9
i'
'
-r'-'l'
|
fi
l'
L'-(.t J ]
(2-46a\
(2-46b)
(2-46d)
(2-46e)
p--
t
p-
t r -
u I
-
E,
sign(e,,
ess)
+
I
sign(e,, e;)E,n
Fig. 2-3 Tensionand Compression
nvelope
Curves
f;
-f'
f;
-f;
['.[?) ]
where:
€ = tt
- t l t
n-
-
F-
g;
-
etl,
y
-
'h
-f;
_f;
in which Ei^
:
locationof
the compression nvelope ranch.
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2.4.2 Reversal
ranches
Rules
and
4)
When a
reversal
akes
place
on an envelopebranch
a
reversal
cun'e connects
his
point
of
reversalwith a
target
point
on
the
opposite
envelope.
The curve
that
connects
these wo points will be referred o as a reversalbranch. In general, eversalbranchesare
completely
defined
by
the
extremum
points:
maximum excursion into
the tenslon
envelope
branch
€*o
,
and
maximum excursion nto
the
compressionenvelopebranch
€*n,
(Fig.
2-6). If a
reversal
akes
place
rom within the
yield
plateau
on
the tension
envelope
curve at
a coordinate
e;,
l;),
with
f;
=fi,
then
e'" .,
is
definedas:
€max
=
e;
-e]o^
The
target
strain
on
the compressive
nvelope urve
s
calculated s:
(2-47)
(2-48)
(2-4e)
(2-s0)
(2-s1)
(2-s2)
(2-s4)
(2-ss)
(2-s6)
where,
and
with,
While the targetslope s givenby:
with:
E n = E i ^ * t . l n
t-io
=
e;
+pr(e-rh-er)
f"-
e;^=ei-E
t 'no
-€J
n - -
' '
-
eIo-ei
L r -
-
- u
t
(
'
l )
**P'lr;-
E,
Ei^
=
€i
k;,,
+
euQ
-
k;e")
f+
eI=eI^+e n-?
L S
et=eX^*rmax
I*
E,
and
the target
sftess
f
the
leld
stress n
the
compressive nvelopebranch
(Fig.2-5).
In
the
case when
the reversal
akes
place from the
strain
hardened
curve
of
the tension
envelopebranch,
hen Eqs.
(2-49)
through
2-52)
aremodified as follows.
The
straine6o
is taken
as
he actual
maximum excursion
within
the
compressive nvelopebranchbut,
l r -" l t
l r ; l
(2-s3)
The
shifted
riginabscissa
or
thecompressionnvelope ranch
s
calculated s:
2-12
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in
which
k; , is a
factor
o locate
he
compression
nvelope
ranch
between
he
ej and
ei
as shown
n
Fie. 2-5. and
was
ound
o
be:
/ \
le-* i
Freu
=
exp
I
--;
1
[
5000
il- )
Finally
he
arget
tress
ndslope
[
and
E;
are
calculated sing
Eq.
Q-aQ.
for the
oading
reversal ranch,
he
shifted
ension
riginstrain
s
grven
by:
ei^
=
e;(l
-
kl,,)
+
e
b
kle
r-
t . .
E i = E i ^ * E - r n -
E r :
f
/ m ' ^
E i = E i ^ f € - i o -
, ,
( , , \
I le-i ' | |
Nreu=expl-------
t I
[
5000
il-
)
with:
where:
(2-s7)
Similarly,
(2-s8)
(2-se)
(2-60)
(2-6r)
(2-62)
envelope
ranchs
the initial Young's
(unloading)
can be
Then
the target
strain
on
the
tension
envelope
ranch
s
given
by:
eL
=et^*t.r*
In a
similarway,
he
arget
tress
,i
andslope
EI on
the ension
calculated
sing
Eq.
Q-
5) .
Experiments
performed
by
Panthaki
(1991)
have
shown
that
modulus
at
the
point
of reversal
rom
the tension
envelope
branch
expressed
s:
E
=
(1
-3
Leo)E,
(2-63)
While,
or a
reversalrom
he compressionnvelope ranch
loading),
he nitial Young's
modulus anbesivenby:
EI
=
(l
-A€,)E,
(2-64)
The
M-P
parameter was also
ound
expressed
s:
R-
=rc(
\
to
be
f i \ t
- l
E,)
a functionof
the
vield
sness.
hat
can be
(1-
l0Ae, )
2-13
(2-6s)
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for
the unloadins
branch.
and
(2-661
where
Leo
:
sffain
amplitude
for
the
cycle
and
E,
:
initiai Youn-e's
modulus
for the
reversalbranch,as
shown
n Fig.
2-4. Analytical
calibrationof
these
variables
are
shown
in
Figs.2-7
to 2-10 rom experiments
y
Panthaki
1991),
and
Figs.2-l l to 2-i4
show
someof
the
actualexperimental
oops hat
rvere
used
o fit the M-P equation.
The
unloading
and
unloading
branchare define
as:
Rule
3
(Unloading
Reversal
Branch)
Eol=Etrm*tln*
fot
=f*
Eot
=
Ei
En
=Elo
r _ f -
J
b3
-J
a
En=EL
The
nitial
slope
E; and
he
Menegotto-Pintoquation
arameter
-
are unctions
f the
strainamplitude
Aeo
of
the
oop,Eqs.
2-63)
and(2-65),
hich s defined s:
R.
=
20(*)
,
,
-
2oAe,
c . - _ E a 3
Aeo
=:--
L
Rule 4
(Loading
Reversal
ranch)
€ a + = E i n * t ^o
fo+
=fr
o
Eoq
=
El
E M = E L
fiq
=f i
Etc
=
EL
Leo=lryl
whereEI andR+ are calculated
sing
Eqs.
(2-64)
and
2-66),
respectively, y
havrng.
(2-67)
(2-68)
(2-6e)
2- t4
d2-10)
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2Leo-------2
Fig.2-4 Effectof the StrainAmplitude of the Reversal
on
the Equation
Parameters
2.4.3
Returning
ranches
Rules
and6)
When
partial
unloading
n
the reversal nloading
ranch
rule
3)
takes
place,
he
reloading
ranch
will be
called
oading eturning
ranch
rule
5).
An
analogous
ranch
will exist
when a
reversal
akes
place
on
the
loadingreversalbranch
(rule
4),
and
unloading
s done
hrough
he
unloading
eturning ranch
rule
6), asshown
n Fig. 2-15.
At the occurrence
f
a
reversal
on rule 3, rule
5 will start
and
the target strain €as
s
calculated s:
with,
€ns=EI^*€ro+Lele
Ae "=€a3ta5 L
t .2E,
f.:
0
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f A
,,a'
.o
f
-
,
G,n'Jt
\ t ;
a j ' a
(
E;.
f;
)
te;,fr-
)
Fig.
2-5
Reversal rom Yield Plateau
€max
---------)
Fig.
2-6 Definitionof
the ReversalUnloading
Branch
€s
T
I
f
/
max
f^o
I
I
J
./.- o
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+
Eo
E;
1 2
1
N R
0.6
0.4
v . z
0
1 A
1 . 6
1 . 4
1 . 2
I
N R
Fig.2-7
Effect
of
the
Strain
Amplitude
of
Loop on
the
Initial
Modulus
and
R Parameter
or
Reinforcing
Bars
(fu:
53
ksi)
(Loading)
R
N A
0 . 4
v . 1
0
2-17
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+
Eo
E"
1 . 2
1
0.8
0.4
u.z
0
? E
4
0 . 5
t . c
4 . 5
5
e"
(%)
(a)
R
2.s
0 .5
n
z
I
z.?
. 5
A
4.5
5
e"
(%)
o)
Fig.
2-9 Effect
of
the Strain
Amplitudeof
Loop
on
the Initial
Modulus
and
R Parameter
or High
Strength
Bars
({
=
123ksi)
(Loading)
2-t9
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t . z
0 . 8
f
L O
L a
( )
4 4 . 5
€a
3 . 5
. 5
6
(a)
2 . 5
l 4
R
? q
2 . 5
. 5
(b)
4 . 5 5
eo
( )
Fig.
2-10 Effect of
the Strain
Amplitudeof
Loop
on
t-he nitial Modulus
and R
Parameter
or High Strength
Bars
(1,
:123
ksi)
(Unloading)
2-24
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Fig.2-f
l Fitting
of
M-P Equation
to
a
Loading
Loop of
Reinforcing
Steel
Bars
(
{,:53
ksi)
8 o i
6 0*
l
l
2 0*
Experiment
Curve itting
-0 .015
-0.005
0.01
0.005
0.0 1
Fig.2-12
Fitting
of M-P
Equation
o
an Unloading
Loop of
Reinforcing teel
Bars
{:
53
ksi)
2-21
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1 5 0
Experiment
Curve
itting
-0.02
-n nn6 fr
Y
- h t t
+
- 1 0 0 -
- 1 5 0
n n l t r
u .uz
Experiment
Curve
itting
/
/:
/,
Fig. 2-13
Fitting of
M-P Equation
o
a
LoadingLoop
of
High StrengthSteel
Bars
{:
123ksi)
1 4 n
. -
l
100
-l-
I
-0.005
q
-50
-U .UZ
- 0 .015 -0.01
0.005
/
n n l - / n n l (
0.02
Fig. 2-14
Fitting of
M-P Equation o
an Unloading
oop of
High StrengthSteel
Bars
({
=
123
ksi)
. , , , . ,
L L L
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In a similar
way, a
partial
oading
rom the oading
reversalbranch
rule
4), which defines
ruie
6.
is calculated
s:
Eua=Ei^ t t ' , ,
+Le,e
(2-74)
(2-75a)
(2-7sb)
with,
Le r
= €a4 - Ea6
*
{-
0
>
Ae,. :+
3 L s
2.4.4 First
Transition
ranches
Rules
and
8)
The curve
followed
after
a reversal
from an envelopebranch
curve has been
named
eversalbranch,
he one
followed by a reversal
rom
a
reversalbranch
s
called he
returning
branch
The curve
then followed after
a reversal
from
a
returning branch is
called
the
irst
transition
branch
and a
reversal rom
this
will
lead
to
a second
ransition
branch.
These ive
typesof curvesare
llustrated
n Fig. 2-15.
It
shouldbe
noted
hat the
reversaland
the returning branches
orm
a
closed
oop and
the
first and second
ransition
branches
ycle
nside
his loop.
{
1 , 2
3 ,4
5 ,6
7 , 8
9 ,10
Envelope ranches
Reversal ranches
Returning
ranches
First ransitionranches
Second
ransitionranches
Fig.
2-15
Sequence
f
PartialReversals
2-23
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The targetsffain
of rule 5
eas s
given
n Eq.
(2-71),
his
equation s
different rom
the
starting
sffain of
rule 3
€.ot,but
if rule
5
would have reached
he
end
the
a
reversal
from
this
point
would
have
been
he
starting
point
for rule
3 agaln.
It means hat
in the
case of
a
reversal from rule 5
(incomplete
oading),
a
redefined
rule 3 needs to
be
calculated.
The
starting
strain
for this redefined
ule ought
to
be between
he previous
startingsffain and
the
targetstrainof rule 5.
By
using a
linear
proportion,
c *
_ . . - E a | - E a 5
r . - t b 5 - € o 7
cai
:
Yb5
ebs
-
%
-
eol€;;4;
(2-76)
It can be
noted that if the reversal
happens
when rule 5
has
ust
started Eat
=
Ea5,
hen
from
Eq.
(2-76)
Elt
=Eoz,
what
means hat
an
insinuation
of reversal
occurredat rule
3,
so
the
path
followed shouldbe on
the
unchanged
ule
3.
While if the reversal
occurredat
the end of rule 5 when Eai=Ebs, hat means t is a