MODELING
OF
CONFINED
CONCRETE
Esneyder
Montoya
A
Thesis
submitted
in
Confomiity
with
the
reqllirements
for
the
Degm
of
M88ter
of
Applied
Science
GradwPi,
Department
of
CM1
Engineering
University
of
Toronto
The
author
has
granted
a
non
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licence
dowing
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L i i
of
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owership
of
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copyright
in
this
thesis.
N e i k
the
thesis
naa
substantial
extracts
h m
it
reproâuced
without
the
audior's
permission.
L'auteur
a
accord
une
licence
non
exclusive
pennettant
B
la
Bibliothèque nationale du
Canada
de
nptoduin,
*,
didisiuer
ou
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de
cette
thèse
sous
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ou
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masenie
la
proprietd
du
droit
d'auteur
qui
protège
cette
thèse.
Ni
la
thhe
ni
des
exhits
substantiels
de
celleci
ne
doivent
&e
impnmCs
ou
autrement reproduits
sans
son
autorisation.
Modeling
of
Confined
Concra8
Master
of
Agplied
Sciem,
2000.
Graduata
Deportment
of
Civil
Engineering
University
of
Toronto
Abstract
Constitutive
modelr
for
confineci
conciete
were
invesügated
using
the
nonlineer
finite
dement
program
SPARCS,
in
which
the
influence
of
lateml
pressure in
tiad
reinforced
cancrete
(RC)
columns
subj8CSeâ
to
monotonie
axial
compression
is
fonnulated
in
ternis
of
threedimensional
stress
states.
A
combination
of
ascending
and
decrcending
kar t ch
of
axial
stress
axial
strein
relationships
for
confineci
concrete
proposeâ
by
various
authors
mre
evatuated.
The
influence
of
variable
Poirwn's
Mo,
cornpression
sdening
of
concnite,
and
concret8
ver
spalling
in
the
~88ponse
of
RC
columns
nnrs
alro
analyzed.
Data
from
cdumns
testeâ
by
various
remarchem
was
u W
to
establirh
the
validity
of
the
pro08dums
implemented
in
SPARCS.
In
g m l,
g m û
agreement
wîth
the
Acknowledgements
The
author
is
grateful
for
the
constant
support of
Professors
F.J.
Vecchio
and
S.
A
Sheikh
during
this
project;
and
the
financial
support
provided
by
the
Netunl
Sciences
and
Enginming
Research
Council
of Canada.
Contents
Ud
of
figures
List
of
tablas
1. Introduction
1.1.
Overview
of
Concrete
Response
1.2.
Objectives
1.3.
Summary
2.
Literrture
Review
2.1.
Plasticity
and
Fradure
Energy
based
models
2.1
.1
.
Liu
and
Foster
(1
998)
2.1
2.
Xi.;
MacGregor;
and
EM
(1
996)
2.1
3.
Karibanis
and
Kiousir
(1
993)
2.1
4.
Chen
and
Mau
(1
989)
2.2.
Linear
and
Nmlineat
Elastic
ModeCs
2.2.1.
Mau;
EM;
Md
ïhou
(1
998)
2.22.
Bamgar
cnd
Maipudi
(1997)
ii
iii
xiii
wxi
1
1
3
4
7
8
8
10
12
13
15
15
16
2.2.3.
Bortolotti
(1
994)
2.2.4.
Sdby
(1990)
2.2.5.
Vlcehio
(1992)
2.2.6.
Selby
and
Vecchio
(1
993)
2.2.7.
AbdelHalirn
and
AbuLebdeh
(1 989)
3.
SPARCS
24
3.1.
Program
Description 24
3.2.
Finite
Element
Library
26
3.2.1.
Hexahedron
26
3.2.2.
Pentahedra
(Wedge) 27
3.2.3.
Tniss
Bar
28
3.3.
Corwtitutive
Models
for
Concrete
29
3.3.1.
Base
StmssStrain
Cuwes
for
Concret8
in
Compmssion,
PrePeak
Behaviour
3.3.1
.l.
Hognestad
parabole
3.3.1
2,
Thorenfeld
et al.
3.31
.3,
Hoshikuma
el
al
3.3.2.
Base
ShbuStrain
CUIV~S
for
Concret8
in
Compnmion,
PosPeak
Behaviour
33
3.3*2.1.
Moditied
Kent
and
Park
(Scott
1982) 34
3.3.2.2.
Popovicr
(1
973)
35
3.3.2.3.
HoshiCaima
ebt
al.
(1
996)
36
3.3.3.
Base
StressStrain
Cuwe
fw
Concmte
in Tension
3.3.3.1.
V mi o
(1982)
3.3.3.2.
CollinbMitchell(1987)
3.3.3.3.
larmo,
Maekawa
et.
al.
3.3.4.
Feilue
Criteria
for
Conmete
3.3.4.1.
HsiehTingChen
Criterkm
(1 979)
3.3.4.2.
Cracking
Criteria
3.3.4.2.1
.
MohrCoulomb
Criterion
(stress formulation)
3.3.4.2.2.
MohtCoulomb
Criterion
( mi n
formulation)
3.3.4.2.3.
CEBFIP
Criterion
3.3.5.
Confined
Strength
tp
and
Stmin
at
Peak
Stress
sp
3.3.5.1.
Selby
(1 993)
3.3.5.2.
Veochio
(1
992)
3.3.6.
Compression
Sdtening
3.3.6.1.
V d i o
and
Collins (1982)
3.3.6.2.
Vecchio
and
Collins (1986)
3.3.6.3.
Vecchio
1992A
3.3.8.4.
Vecchio
19928
3.3.7.
VafiabCe
Poisson's
Ratio
3.4.
Constitutive
Modd
for
Steel
3.5.
SPARCS
Stnidun









4.1,
Introdudion
4.2.
Selection
of
Parametric
Variables
4.2.1.
PrePeak
Base
Curves
for
Conccete
4.2.2.
PostPeak
Base Curves for
Concrete
4.2.3.
Concmte
Cracking
4.2.4.
Confinement
Enhancement
4.2.5.
Variable
Poisson's
Ratio
4.2.6.
Compression
Softening
4.3.
Sheikh
and
Uzumeri
Tests
(1980)
4.3.1. Column
Geometry
4.3.2. Longitudinal
Bar
and
Tie
Setup
4.3.3. Test
Instrumentation
and
Procedure
4.3.4.
Selected
Sheikh
and
Uzumeri
Columns
60
4.4.
Finite
Elements
MocCels
4.4.1.
Geometry
4.4.2.
Material
Types
4.4.3.
Parameter
Combinations
4.4.4.
Anelyris
PrOCBCILIre
4.5.
Analysir
Rewlb
4.5.1
.
Column
2A11
4.5.1
1.
Cmss
Section
and
Profile
Results
4.5.1.2.
LodDefomiation
Curves
Combination 2
4.5.1.3.
Summary
of
LoadûehnWion
Cuwes
74
4.5.2.
Column
48349
4.5.3.
Column
2C517
4.5.3.1.
Cmss
Section
Results
4.5.3.2.
Summary
of
LoadDeformation
Cuwes
4.5.4.
Column
40624
4.6.
Effedr
of
Mode1
Combination
on
the
Response
of
the
Seleded
Columns
4.6.1.
Peak Load
4.6.2.
Strain
at
Peak Load
4.6.3.
PostPeak
Behavior
4.6.4.
Effect
of
Compression
Softening
4.7.
Cornparison
of
Analytical
Results with a
Previous
Version
of the
Program
4.8.
Study
of
Poisson's
Ratio
4.8.1
Procedure
to
Obtain
the
Experimental
Poisson's
Ratio
4.8.2
xperimental
Variable
Poisson's
Ratio
S.
ConObontion
with
ExpedmntJ
Studios
122
5.1.
Introduction
122
5.2.
Lui,
Forâr,
and
AttaICf
Tests
(1
998)
123
5.2.1.
Column
Gemetry
123
5.2.2.
Longitudinal
and
Lateral
Steel
Anangement8
124
5.2.3.
Tm
kidnmentsüon
end
P w
124
Con
t8
nt

5.2+4+
Seleded
Liu
at
al.
Columns
5.2.5.
Fhite
Element
ModeIr
.
5.2.5.1
.
Geometry
5.2.5.2.
Material
Types
5.2.5.3.
Material
ModeIr
5.2.6.
Analytical
and
Experimontal
Results
of Liu
el
ai.
Columns
5.2.6.1
.
Column
2C6410S5015
5.2.6.2.
Column
2C601
OS1
001
5
5.2.6.3.
Column
2C6û1
OS1
501
5
5.2.6.4. Column
2C8410S5015
5.2.6.5.
Column
2CMSSO15
5.3.
Mander et
al.
Specimens
(1
984)
5.3.1
.Column
Geometry
5.3.2.
Longitudinal
and
Lateral
Steel Arrangements
5.3.3.
Test
Instrumentation
and
Procedure
5.3.4.
Selected
Mander
el
al.
Sp8dmens
5.3.5.
Finite
Element
Modds
5.3.5.1.
Gmetry
5.3.6.
Analyticil
and
Experimental
Reurlts
of
Mander
et
d.
Specimens
5.3.6.1.
Wall
11
5.3.6.2.
Scott
Column
5.4.
Sheikh
anâ
Uz u M
Colwnns
5.4.1,
Column
4
Content
5.4.2.
Column
48420
5.4.3.
Column
4D322
General
Reaultr
153
Analysir
of Short Column
Sedias
Oesigned
According
to
CSA23.394
and
AC191
M5R
Code
Provitsions
155
Cornparisons
with
SheikhUwmeri
and
RasviSaatdoglu
Confinement
Moâels
159
Finite
Elernent
Analysis
of
a
Shear
Wall
164
6.
Conclusion
6.1 Conclusions
6.1.1.
Summary
6.1.2.
Specific
Conclurions
6.2
Improvements,
Limitations and
Remmmendations
for
Future
Work
8.2.1.
Impmvemnts
in
SPARCS
6.2.2.
Limitations
of
the
Analysis
6.2.3.
Recommendaüons
for Future
Work
Appndices
178
A
ôesfgn
of
Short
Cdum
&dons
.ceodîng
to
CSA23=381
8nd
ACll.3188UR
178
X
Mo&Iing
of
cenfinedCOlrcna&
Al.
Section
Pmperties
A2.
Design
using
Canadian
Standard
CSA23.394
(Seismic
Provisions
Section
21)
A3.
Design
check
American
Standard
ACI31&95R
(Seismic
P
ravisions
Sedion
21
)
B.
Shaikh
and
Uwmrri,
and
Ruvi
and
Srritcioglu
Modal
Calculations
B.
1.
Data
for
Wall
1
1
8.2.
Sheikh and
Uzumeri
Model
8.2.1.
Parameters
6.2.2.
StressStrain
Cuwes
8.2.3.
Ma l
Load
Versus
Axial
Strain
Cuwe
8.2.4.
Parameten
d;r.
Q,
and
ki
for
All
Columns
8.3.
Rasvi
and
Saatcioglu
Moûel
8.3.1.
Parameters
8.3.2.
StressStrain
Cuwes
6.3.3.
Axial Load
versus
Axial
Strain
Curve
8.3.4.
Rarvi
and
Saatcioglu
Parameten
for
All
Columns
8.4.
Axial
Shortening
CUMS
8.4.1.
Sheikh
and
Uzumeri
Columns
84.2.
Liu el
al.
Columns
8.4.3.
Scott
Colunn
(Mander)

C.
Vuirble
P O ~ ~ ~ S
Ratio
in
Liu
et
al.
Columiis
206
List
of
Figures
3.1.
Eightnoded
brick
(hexahedron)
3.2.
Deformed
brick
3.3.
Pentahedra
(Wdge)
3.4.
Truss
Bar
3.5.
Hognestad
Parabola
3.6.
Generalized
Popovicr
Strsssstrain
Cuwe
3.7.
Ascending
Branch
of
Hoshikuma
et
al.
Mode1
3.8. Description of
Laterd
Prassure
f i
3.9.
Adapteâ
Version
of
the
Modified
Kent
and
Park
PostPeak
Model
3.1
O.
Popovicr
Portf
eak
Relationrhip
3.1
1.
Hoshikuma
et
al.
PosbPeak
Curve
3.12.
Reinforcd
Concrete
in
Tension
3.1
3.
MohrCoulomb
Criterion
3.14.
Comprsuion
Soflening
Ef f d
3.1
5.
Lateml
Expansion
3.16.
Poiuon's
Ratio
Relationrhip
3.17. Steel
Model
48
3.18.
Secant
Modulus
Definition
51
3.19.
Solution
Algorithi
for
SPARCS
(Selby
and
Vecchio
1993)
53
4.1. Column Dimensions
58
4.2.
ShdkhandUzumeri
Colurnn
Sections
59
4.3.
Section
Datail
for
2A11
62
4.5.
Material
Types
4.6.
Concrete
Axial
Stress
for
Combination
2
(2A11)
4.7. Variation
of
sd
/o,
Ratio
for
Combination
2
(2441
1
)
71
4.8.
Profile
Deformation,
Column
2A11
(Combination
2)
4.9.
Concrete
Lateral
Stress
States,
2A11,
Combination
2
4.1
0.
Rqxmse
of
2Al1,
Combination.
2
73
4.1
1.
Lateral
Reinforcement
Response
of
2A11,
Combination
2
74
4.12.
Axial
Response
of
2A11,
Combination 1
75
4.13. Axial
Respnam
of
2A14,
Combination. 3 75
4.1 4.
Axial
Responre
of
2A11,
Combination.
4
4.15.
Axial
Rmponse
d
2A11,
Combination.
5
4.16.
Axial
Response
of
?Al1,
Combination.
6
4.17.
Axisl
Response
of
2/41
1,
All
M e t
Combinations
4.1
8.
Lateml
Reinbmmmt
Respmse
of
2A11,
Ail
Combinations
78
4.1
9.
Axial
Respome
of
48319,
Combinatbn.
1
81
4.20.
Axial
Rerpocire
of
48319,
Combinafion.
2 82
4.21.
Axial
Response
of
4631
9,
Combination.
3
4.22.
Axkl
Responro
of
4831
9,
Combination.
4
4.23.
Axial
Responw
of
4B3=1@,
Combination.
5
r
4.24.
Axial
Response
of
4831
9,
Combination.
6
4.25.
Axial
Respnse
of
4831
9,
All
Model
Combinations
4.26.
Confinement
of
2C517,
Combination.
2
4.27. Sketch
of
Tie
Strain
Distribution,
Column
2C517
4.28.
Axial
Repense
of
2C517,
Combination.
1
4.29.
Axial
Responw
of
2C517,
Combination.
2
4.30.
Axial Response
of2C2C5
Combination.
3
4.31
.
Axial
Responro
of
2C517,
Combination.
4
4.32. Axial Response
of
2CS17,
Combination.
5
4.33. Axial
Response
of
2C517,
Combination.
6
4.34. Axial Response
of
2CS17,
Combination.
7
4.35. Axial Response
of
2C517,
All
Model
combinations
4.36.
Lateral
Reinfbmment
Response
of
2CS17,
All
combinations
4.37.
Axial
Responw
of
4W24,
All
Madel
Combinations
4.38.
Compeiirons
of
Strength
Increasa
4.39.
Ef f e
of
Portpak
Cuwes
(1:
W i e d
Kent
8
Parlc,
2:
Popovicr,
5:
Hoshikuma
et
ad.)
4.40.
Effed
of
Compnssion
Sdtening
441.
R.uilt
C O ~ ~ ~ O ~ S
f ~ r
2Al1
4.42.
Rouit
Comwms
Ik
2CH7

4.43.
Equilibrium
of
Forces
in
a
Circuler
Section
4.44.
LoadStrain
Curve
of Column
2C901
OSlûû25
4.45.
toadPoisson's
Ratio
Cunre
of
Column
2C9ûiOS10025
4.46.
Poisson's Ratio
Ma l
Strain
Curve
of
Column
2C9010S10025
4.47.
Volumetric
Strain Variation of
2C901
OS1
0025
4.48.
LoadStrain
Curve
of
Column
2CûMS5015
4.49.
LoadPoisson's
Ratio
Curw,
Colum
2C80bS5025
4.50.
Poisson's Ratio Axial Strain
Cuwa
of
Column
2C8WS5025
4.51.
Volumetric
Strain Variation
of
2Cüû6S5025
4.52.
Lateral
Expansion Variation
of
Series
I
4.53.
Laterd
Expansion Variation of
Series
II
4.54.
Lateral
Expansion
Variation
of
Series
III
5.1.
Detail
of
Columns
Tested
by
Liu et al.
5.2.
Cross
Section
Model
for
Liu Columns
5.3.
Material
Types
for
Liu et
al. Columns
5.4.
AxiPl
Shortening
of
2C6ûlOSSO15
S.S.
Lataml
Expansion
of
2C601
OS5W
5
5.6.
Aaal
Shortening
of
2C80iOSI
001
5
5.7.
Lateml
Expansion
of
2C601OS10015
5.8.
Mal
Shortening
d
2C801OS15015
5.9.
Lateml
Expansion
of
2C6û1OS1
SO15
5.10.
Axial
Shortming
of
2CBO1OS5015
5.1t.
Latemi
mnsiond2C80fOSSO15



5.12.
Axial
Shortening
of
2CB06SSO15
5.13.
Lateral
Expansion
of
2C806SSO15
5.14.
Mander
et
al. Wall
5.15.
Square
Column of Scott
SWes
5.1
6.
Finite
Elernent
M W
for
WalI
1
1
5.1
7.
Prototype
of
a
Hollow
8ridge
Pier
5.1
8.
Axial
Shortening
of
Wall
11
5.1
9.
Ma l
Shortening
of
Scott
Colwnn
5.20.
Axial
Response
of
4C65
5.21.
Axial
Response
of
4 M
5.22.
Lateml
Expansion
of
4B420
5.23. Axial
Response
of
40322
5.24.
Maximum
Anal
ytical
to
Eqmimental
Load Ratio
(wo
J
Softening
)
5.25.
Maximum
Analyücal
to
Experimmtal
Load
Ratio
(W.
I
Softening)
5.26.
Localization
of
Cr
Sections
5.27.
Column
Section
and
FE
Mesh
(CSA,
ACI)
5.28.
Axial
Shortening
of
Section
in
Central
Zone
5.29.
Axial
Shortming
of
Section
In
Plastic
Hinge
ZOM)
5.W.
W l
Compwiron
for
Column
4831
9
5.31.
Model
Compwiwxi
of
Column
4û624
5.32.
Modd
Compatkon
of
Colunn
2 C 8 MH MS
5.33.
Modd
Compsr i
for
WalI
11
5.34.
Ldsr
et
al.
Wall
SW16
Content
5.35.
Finite
Element
Mesh
for Wall
SW16
5.36.
Material
Zone
distribution
for
SW?6
5.37.
Horizontal
Responm
of Wall
SW16
5.38.
Sketch
of
Horizontal
Displacement
of
Wall
SW16
A. 1. Design
Sedion
Using
CSA
and
AC1
Codes
B.1
Cross
Section of
Wall
1
t
8.2.
Strain
Cuwes for
Conwete,
Wall
11,
Sheikh
and
Uwmeri
Model
8.3.
Stmin
Curve
for Longitudinal Steel, Wall 1 1
8.4.
LoadStrriin
Curve
of
Wall 1 1,
Sheikh
and
Utuneri
Model
B.5.
Stmin
Cuwes
for
Concrete,
Wall 1
1,
Rasvi
end Saatcioglu Model
8.6.
LoadStrain
Cuve
of
Wall
1 1,
Rasvi and Saatcioglu Model
8.7.
Column
2A11
8.8.
Column
4665
8.9.
Column
2C54
7
B.10.
Column
4831
9
B.
1
1.
Column
46420
8.12. Column
4D322
6.1
3.
CoCumn
4D624
B.14.
Column
21OS5015
B15.
Column2C601OSi~~5
8.16.
Column
2C6015û15
817.
Column
2CSO1OS5û15
B.f8,
Column
2CM50 15
B. 1
9.
Scott Column
C.1,
LoadStrain
Curve
of
2CW1
OSHM
5
C.2.
LUPoimon's
Ratio
Cwe
Of
21
055015
C.3.
Posson's
RatioAxial
Strain
Cunre
of
2C60t
0SW15
C.4.
VolumeûieAxial
Strain
Curve
of
2C60..10S5015
C.5.
LoadStrain
Cuwe
of
2C603
OS1
001
5
C.6,
LoadPoisson's Ratio
Curve
of
2C601
OS1
001
5
C.7.
Poisson's RatioAxial Strain
Curve
of
2C1
OS1
001
5
C.8.
VolumetricAxial
Strain
Curve
of
2C1
OS1
001
5
C.8
LoadSîrain
Curve
of
2C601
OS1
501
5
C.10.
LoadPoisson's Ratio
Cuwe
of
2C6û1
OS1
501
5
C.11.
Poisson's
RatioAxial
Strain
Curve
of
2C60iOS15015
C.
1
2.
VolumetricAxial
Strain
Curve
of
2C1
OS
1
501
5
C. 13.
LoadStrain
Curve
of
2C801
OS504
5
C.14
LoadPoisson's
Ratio
Cuwe
of
2C80i
OS5û15
C.l
S.
Poisson's
RatioAwal
Strain
Curve
of
2C801
OSSO1 5
C.16.
Volumetri~~Axial
Strain
Curve
of
2C8û10S5015
C.17.
LoadSttain
Curve
of
2Cûû6S10015
C.18.
LoadPoisson's
Raüo
Cuwe
of
2C806Slûû15
C.19
Poi m's
RBtiOlAXial
Strain
Curve
of
2CûMSlûO15
C.20.
V o l u r n ~ ~ a l
Strain
Curve
of
2cBo6S10015
2
LoadStrain
Cuwe
d
2C906S10025
C.22.
LodPoisson's
Ratio
Cwe
of
2CQWS10025
C.23. Poisson's RatioAxial
Stroin
Cuve
d
2C908S10025
C.24.
VolumetricAxial
Strain
CUM
of
2CWS1
Oe2S
C.25.
LoadSWn
Cuwe
of
2C906S5ô25
C.26.
LordPoisson's
Ratio
Cunn
d
2C90665025
C.27.
Poisson's
RatioAxial
Stnin
Curw
of
2C906S5025
C.28.
Volumetri~~Axial
Stnin
Curve
of
2CgOSSiôU25
C.29.
LoadSWn
Curve
of
2 CS1
OS1
00Q
C.30.
LoadPoiuonJr
Ratio Cuwe
of
2C901
OS1
000
C.31.
Poisson's
Ratiodxial
Strain
Cuwe
of
2C901
OS1
000
C.32.
VolumetricAxial
Strain
Cuwe
of
2C901
OS1
000
List
of
Tables
4
1
SheikhandUzumei
Sel8Cfed
Column
Properties
4.2.
Finite
Element
Model
Gemmetry
4.3.
Element
Material
Types
4.4.
Parametric
Combinations
for
SPARCS
Analyses
4.5.
Analytical
to
Experimental
Ratios
for
2Ai
1
4.6.
Analytical to
Experimental
Ratios
for
4831
9
4.7.
Andytical
to
Expeimental
Ratios
for
2C517
4.8.
Analytical to
Experimental
Ratios
for
4û624
4.9,
Analytical
to
Expewirnentel
Peak
L o d
Ratios
4.10.
Strength
lncreare
4.1 1.
Anelytical
to
Experimental
Peak
Strain
Ratios
4.1
2.
Selby'r
anâ
Cumnt
Modela
4.1
3. Liu
et
al. Column
Pmpwtim
for
the
Study
of
Poisron'a
Ratio
4.14.
a
and
Y.
Values
fbr
Liu et
al.
Columns
5.1,
Liu
et
ai.
Columns,
Material
PmpWes
5.2.
Gmmby
of
Finite
Element
Modela
Con
               
5.3. Liu
et
al.
Material
Types
5.4.
Mander et al.
Material
Properties
5.5.
Material
Propwties
of
Additional Sheikh and
Uwmeri
Columns
5.6.
Maximum
Analyüaal
to
Gcprimental
Load Ratio
5.7.
Short
Colurnn
Sections
in
Seismic
tones,
Model
Comparisons
5.8.
Theordical
to
Gcperimentai
Maximum
Load
Ratios for Square and
5.9.
Theoretical to
Experimental
Maximum
Load Ratios
for
Cirwlar
Columns
5.1
0.
Material
Properties
of
Wall
S
W16
5.1
1.
Material
Zones
for
FE Model of
SW16
k
1.
Code
Provisions
for
designing
Compression
Members
B. 1. Wall 1
1
Properties
6.2,
Parameters
for
SheiWiUzumeti
Model
8.3.
Rasvi
and
Saatcioglu
Parameters
Chapter
1
Introduction
Threedimensional
nonlinear
finite
element
analysis
of
reinforced
cancrete
has
been
investigated
for
almost
four
decades.
Constitutive
material
models
that
take
into
account
the
influence
of
triaxial
states
of
strains and
stresses
have
been
developed
for
both
plain
conmete
and steel.
These
models
have
b e n
based
on
piinciples
founded in
theories
such
as
plasticity,
fracture
mechanics
and
elasticity
(linear and nonlinear).
In
classical
plasticity
theory,
both
materials
(i.e.,
conmete
and
steel)
behave
elastically
until
'yieldingn;
then,
materials
behave
plastically
and
follow
associative or
nonassociative
flow
rules.
The
conpt
of
Yfradure
energy"
is
used
in
ftacture
mechsnicr
to
establish
failum
criteria
that
depnd
on
the
state
of
stress
to
which
an
element
of
concmte
is
subjected.
Failuo
sucfaces
define
the
upper
boundery
of
conaute
ttrwigth.
Finally,
in
linear
elasticity,
the
simplest
mode1
follmvs
a
Hdce's
Iaw
whefe
stresses
am
dindly
PIopOrtional
to
Wainr,
without
changes
in
matrn0al
pmpdesw
In
nonlinear
elasticily,
comate
a d
steel
bahve
material
properties
are
changed
as load
increaws
(or
deweases).
Secwit
and
tangentiel
stiffness
matri*s
have
ken
developed
to
account
for
the
change in
material
behaviour,
and a
number
of
mocklr
have
been
implemented
in
finite
element
programs
mat
use
eithew
formulation
(Chen
1982).
Behaviour
of
confined
c omt e
is
different
ftom
that
of
unconfined
conuete.
Concret8
cm
be
considered
mnfined
when
subjeded
to
triaxial
comprenions;
the
triaxial
compression
increases
the
conaete'r
capacity
to
sustain
largef
compressive
sbmgVis
and
deformations.
When
a
concrete
element is
laterally
rdnforced
(e.g.,
by
ties,
hoops
or
rpirals)
and
subjected
to
&al
compression,
lateral
expansion
of
aie
ekment
in
the
plane
perpendicular
to
the
axial
compression
activates
the
lateral
steel,
which
confines
th8
element
by
exerting
lateral
pressure.
Confined
concrete
generally
fails
in
a
ductile
manner,
Werear
unconfineâ
comtete
fails
in
a
bfittle
manner.
As
tensile
strains
develop
in
unconfined
concret8
rubjecteâ
to
compression,
concret8
Mens
and
strength
demases.
It
is
also
knuwn
that Poisson's ratio for
concrete
is not constant as
load
increases;
it
inereares
with
m*al
strain
increments.
This
phenornenon
is
beneficial
in activating
lateral
steel.
The
degm
of
confinement
ir
masund
analytically
by
the
increment
in
compressive
strength
and
compressive
itrriin
at
peak
stress
with
respect
to
the
unconljneâ
compn#rive
aC8ngVi
and
mi n
at
peak,
respecüvely.
1.2
Objectivas
The
general
purpose
of
this
work
was
to
detemine
the
capabilities
of
SPARCS
to
model
confimd
concrete.
SPARCS
is
a
nonlinear
elastic
finite
elemmt
pmgnm
developed
a
the
University
of
Toronto
(Sdby
1990,
Sdby
and
Vecchio
1993)
for
Vie
analysis
of
minforced
concret8
solids.
The
prognm
user
constitutive
models
bas&
on
the
Modifed
Compression
Field
Theory
(MCFT)
(Vwxhio
and Collins
1986)
and
has
incorpo,mted
simple
finite
elements
such
as:
trws
bars,
&iodeci
bricks
and
h o d d
wsdges.
Details
of
the
program
will
be
given
in
Chapter
3.
The
rpific
objectives
of
this
work
can
be
wmmarized
as
follawcr:
r
To
extend
the capabilities
of
SPARCS
to
analyze
large
models
unâer
imposed
displacements.
r
To
mak
a
parametric
study
on
the
influence
of
various
models
on
confined
concrete
behavior;
incîuding:
barn
stressstrain
curve
for
c o mt e
in
compression; compression
softening;
cracking
criteria;
strength
enhancement;
and Poisson's
ratio.
To
impiement
baae
stress$train
m e s
for
prepeak
and
postpeak
behavior.
r
To
corrObOTate
the
analytical
model
for
variable
Poisson's
ratio
impl8mentd
in
SPARCS
wHh
expwimentaI
wlb.
O
To
corroborate
the
COCtfimment
modrls
i m p l ~ t e d
in
SPARCS
with
cdumns
testad
under
mOclOtmic
i msi ng
axial
campnuion.
such
concealeâ
colwms
in
wllk
and
To
comment
on
ACI41û95
anâ
CWCAN
4423.3094
code provisions
for
the
design
of
tie
setups
of
short
columns
ôased
on
msults
obtained
fmm
mld
sections
that
satirfy
Wir
requiriwnts.
1.3
Summuy
A
set
of
ten
columns
testd
by Liu et
ai.
(1998)
was
analyzed
to
detemine
the
validity
of
the
variable Poisson's ratio
mode1
in
SPARCS
(Kupfer
1869). Plots
of
the
axial load
venus
Poisson's ratio, and axial
stmin
versus
Poisson
ratio
obtained
from
the
expwimental
msuk
will
be
given in
Chapter
4.
For
the
parametric
study,
a
set
of four
cdumns
tested
by
Sheikh
and
Uzumefi
(1978)
wem
modeled
with
SPARCS
and
analyzed
to
establirh
the
senribiiity
of
each
of
the
modela
rnentioned
above,
in
the
ôehaviour
of
confined
conmete.
Dwing
mis
study,
the
stress$train
m e
for
concrete
proposed
by
Hoshikums
et al.
(1996),
and
a
tentative
mode1
for
strength
enhancement
proposed
by
Vecchio
(1992)
wen
implemented in
SPARCS
and
used
in
the
parametric
study.
A
tt\Orough
mvision
of
the
mwilts
of
this
study will
al«,
k
given in Chapter
4.
Once
the
panmeûic
study
was
completed
and
the
influence
of
each
modd
haâ
be8n
eaiiblished,
sets
of
cdumnr
testecl
by
Liu
et
al. (1
QM),
Mander
et al.
(1988),
Scott
et
al.
(1982)'
and
thme
addi t i d
Sheikh
and
Uarmwi
colurnns
wsn
modokd
in
SPARCS
and
cuinpad
and
d i w d
with
the





axial
strain
arrve,
the
experimental
to
analytical
p a k
l a d
ratio,
the
stfength
gain in the
concrete,
and
the
lateral
expansion
(i.e.,
tie
or
spiral
strain history).
Two
short
cdumn
sections
digned
according
to
the
AC1
31û95,
and
CSAiCAN
23.3994
provisions
were
modeled
with
SPARCS.
The
results
are
discussed
almg
with
the analytical
cwobwation
in Chapter
5.
The
SheikhUzumeri
(1
982)
and
RezviSeatdoglu
(1
999)
analytical
models
for
confinai
cdumns
were
used
to
cornpute
the
m*al
response
of
the
specimens
modeled
with
SPARCS.
Camparisons
of
these
models
with
the
finite
element
solutions are given in Chapter 5 and
Appendix
B.
A
genenl
discussion
and conclusion
of
aie
results
is
pmsented
in
Chapter 6,
with
reaimmendations
for
Mure
work
Due to the
amount
of
msub
deriveû
from
SPARCS,
an
spcialpurpose
postprocessor
program
with
threedimensional
graphical
and analytical
capabilities
w u
developeâ
for
cheddng
and
plotüng:
geometry,
deformation
animation,
sûess
and strain
statea
in
cross
sections,
key
indiceton
of
damage,
secant
moduli
convergence.
Some
of
the
plots
prssented
in
this
wwk
have
b e n
taken
directly
fmm
thir
poapuor.
In
general,
al1
the
column
models
analyzed
with SPARCS
showed
excellent
correlation
with
the
experimental
result~.
The
strength
gain
avetrago
was
3%
higher
than
the
actuel
tests
wiOi
a
standard
deviation
of
11
%.
The
port
peak
behavior
of
romo
d
the
modeleâ
Sbikh
and
Uawneri
calumns
was
significantly
i mpved
w t m
compeâ
with
a
pnvious
version
of
the
pmgram
(Selby
anâ
V a i o
1993).
ExpwhentaI
1oaddal
mi n
wwm
for
Liu
et
al.
columns
wr e
adequately
simulateâ.
The
ovmll
behaviw
of
th
Scott and
Mander
specirn8ns
was
wll
capturd
by
SPARCS.
Finally,
the
cumnt
mdel i q
tooh
of
SPARCS
will
help
in
the
understanding
of
cornplex
elementr
or
sûudums
whem
thmedimensional
analysis
is
unavoidable.
Chapter
2
Literature
Review
Confined
mcmt e
can
be
defineci
as
ancrete
that is
restrained
latemlly
by
reinforcemant
wnsisting
of
steel
8timps
or
spimls.
This reinforcement
exerts
lateral
passive pressure
against
the
concret8
a8
it
expands
due to
the
Poisson's
efbct
when
subjected
to
compressive
load
in one
direction.
The
major
effect
of
the
confinement
is
to
enhance
the
stfength
and
dudility
of
reinforcd
oonaete.
Concrete
conlinernent
ha8
been
studied
sin
early
in
the
century
(Richart
et al.
1928),
and
with
smal
interest
in
column
behavior
for
the
last
Wree
decades
(from
Kent
and
Park (1972)
to
Rami
and
Saatcioglu
(1
998)).
Constitutive
material
models
for
concrete
in
triaxial
stress
states
have
been
proposeci
and
adopted
for use in
numeical
analyses,
uring
fradure
mechanicl,
plaaticity
or
nonlinear
elasüc
analysis
approaches.
Stressmin
cuwes
for
confined
conmete
have
bmn
deriveci
fmm
@ally
loaded
cdumns
tested
undet
different
load rates.
arrangements;
tie
setup;
lateml
steel
spadng;
and
cover
dimensions
(mg.,
Ksnt
anâ
Park
1972,
Sheikh
and
Uuimeri
1978,
Scott
et al.
1982,
Mander
et al. 1988,
Rami
and
Saatcioglu
1992,
Curwn
and
Paultre
1995).
The
analytical
stress
mi n
wwes
were
fit
with
great
accuracy
to
mts
of
colwnns
teaed
by
their
awn
a
authors,
but
they
Iack
general
applicability.
The
search
for
a
general
constiMive
mode1
that
can
be
applied
not
only
to
reinforcd
concrete
columns, but to
confined
concrete
in
other
structure
elements
where
threedimensional
behavior
can
be
expeded
(i.e.,
concealeâ
columns
in
walls,
beamcolumn
joints), is
undennay.
This
chapter
describes
b M y
sorne
of
the
recent
publisheâ
workr
in
rnodeling
of
confineci
concrete,
its applicability and
success,
as
well
as
a
brief
discussion
of
each
one.
2.1
Plaatidty
and
Fracture
Enrrgy
Basad
Moâels
2.1.1
Uu
and
Fostrr
(1
888)
The
authors
revised
the
material
mode1
proposed
by
Caml
et
al.
(1992),
and
calibrateâ
the
parameten
of
the
mode1
for
highstrength
concmte
(HSC).
The
'mkmprane
modal"of
Carol
et al.
is
bared
on
the
microplane
conpt
introdud
by
Bazânt
et
al.
(1984). At
the
micro
Ievel,
an
arbitrary
plane
that
passe8
through
a
point
maiin
the
concret8
is
used
to
define
nonnal
and
shear
rtnins
acting
on
the
plane.
The
fomicK
Pcb
perpmdiailarly,
and
the
latter
acts
parailel
to
the
pl am
The
normal
strain
is
divided
into
its
volumetic
and deviatoric
components,
and
rtressrtnain
nlatiwhips
for
the
volumetric,
deviatoric,
and
tangentiel
stresses
(i.e.,
shear stresses)
are
detemineâ
using
empirical
parametem.
A
mlationship
ir
thm
established
ôetween
the
microplane
stresses
and
macro
stress
tenrors
using
the
principle
d
virtual
work.
The
change
in the
stress
tensor
dm,
is
given
by
where
D~~
is
the
tangential
material
stiiess
matrix,
and
d e
is
the
change in
the
strain
tansor.
It
was
assumed
that
conesponding
8tresses
and
strains
have
the
same
direction
(i.e.,
shear
stress
and
shear
mi n,
normal
stress
and
normal
strain.).
The
empirical
parameters
needed
to
calibrate
the
mode1
showeâ
a
wide
range
of
variation;
as
pointed
out
by
various
reseanhers.
The
authon
developed
an
wigsymmetric
m&l,
using
the
rnodel
of
Card
et al. to
analyze
confird
and
unconfined
HSC,
with
sûengths
up
to
100
MPa.
A
set
of
five
parameters
wvm
calibrated
for
diffatent
cancrete
strengths
and
different
valwr
of
the
aJas
ratio,
Wre
trf
ir
the
major principal
stress,
and
is
the
rninor
principal
stress.
confined
cylinders
with
dt
confining
pressures
tested
by
Dahl
(1992)
were
used to
match
the
expewimental
data
with
the
analyticd
results,
showing
good
agreement in
both
pre
and
postpeak
behavior.
An
analytical
cirarlsr
cdumn
was
alro
modeleci
using
either
four
or
eight
node
cwisymmetric
dementa,
tnirs
bars
for
the longitudinal steel, and
a
point
type
dement
for
the
lateral
confining
steel.
It
was
shown
thot
the
analyses
using
fow
OF
e1gM
d e
elemento
are
similar
for
concrete
strength
up to
60
MPa
and
diier
for
90
MPa
comt e.
Radial expansion
in
the
cirarlar
section
seems
to
be
concentrateci
in
a
hinge
region,
after
the
peak
stress
is
reached.
The
stimips
did
not
yield
at
peak
load and
the
mode1
was
able
to
represent
convete
softening
and
cowr
spalling,
as
well
as
dudile
or
brittle
postpeak
behavior.
2.1.2
Xie;
MacGmgo~
and
Elwi
(19B6)
ln
remnt
yearr,
various
researchen
have
testeci
high
strength
concrete
(HSC)
columns,
and have
k n
investigated
finite
element
column
models
of
normal
strength
concrete
(NSC).
Numerical
analyris
plays
an
important
rob
in
investigating
parameten
af
column
failum
wid
stretches
#a
study
of
column
behaviour
beyond
expetimental
tests.
The
oôjective
of
thir
paper
was
to
evaluate
analyücally,
the
behavior
of
four
columnr
tested
at
the
University
of
Alberta.
Thom
analyses
were
canjed
out
using
a
finite
e l m t
d l
in
which
th8
material
mockl
proposed
by
Pramono
and
Willam
(1
989)
wrrt
implemnteô.
The
constitutive
d l
is
a
f ndun
8nergybad
plasticity
fomiulatian
thrit
use8
the
same
fomiulaüon
for
c o m~ r#r h
and
tamion
in
CO(Nnefe
and
considen
rtroin
Menin9
for
triaxial
states
(tensile,
compressive
and
triaxial
compmrive
8).
At
initial
load
stages,
the
modd
bhaves
elastically
within
a
yied
s u m.
As
load
increases,
plastic
flow
ocairs
within
a
failure
surface
accofding
to
a
nonassociative
ruk.
if plastic
flow
continues
beyond
failure
surface,
the
material
will
follow
a
suftening
path.
The
failure
criterion
usd
war
proposd
&y
Leon
(Romano
1969)
and
was
rnodified
to
account
for
the
triaxial
stress
states.
Parameters
needed
in the
material
mode1
wem
adjusteci
for
HSC
and
wem
taken
from data
obtained
from
cylinders
testecl
at
the
Univemity
of
Alberta.
Specitically,
a
n w
definition for
'cmck
spaUg"
as
a
function
of principal stress
ef
accounts
for
l eu
stMened
HSC
cdumns
with
moderate
confinement.
Finally,
the
program
ABAQUS
was
ured
to
implement
the
new
mockl.
Four
~olumns
testeci
by
Ibrahim and
MacGregof
(1994)
were
chosen
to
pmvo
the
analytical
rnodel.
These
columnt
had
equal
dimensions and steel
properties,
but
different
hwk
spacing
and
concrete
strength,
with
the
latter
ranging
from
59.3
to
124.8
MPa.
Eccentric
loading
was
applied,
cawing
a zero
stress
condition on
one
of
the
dges
of
each
column.
One
quarter
of
each
d
the
colunnr
was
mudeled
wing
the
ABAQUS
20
node
brick
fw
concret8
elements
in
which
longitudinal
and
pamlleltoeâge
reinforcement
wre
mbedded
(i.e.,
meamd
out
into
concret@).
Oiamondtype
internai
sümp8
were
Wi e d
as
ûuu
bars,
anâ
ab
aswmptions
wem
made
for
di r pl ~ c ont r oI I ed
loading
and
mrtnintr.
Litemmehiew
12
The
msub
agnw,
well
with
8xperimental
values
ancl
the
materiil
mode1
has
enough
capscity
to
describe
failum
modes
for
columns
ranging
from
poorly
confinai
to
well
conflnd.
Poorly
confined
columns
prssented
brittle
behavior
after
pek
load.
Alüiough
triaxhal
compressive
stress
sûates
developed
at
tk
levels,
local failure
occurred
at
midlevel
betwwn
stimipr
due to
tenrile
strains
in
the
c omt e
wre.
Cover
spalling
almg
the
compretssive
fa
was
typical
of
wellconfined
colmn
failure. High moment
and
load
capacities
after
cover
spalling
were
the
result
of
highet
lateral
pressures
due
to
yielding
of
stimipr.
ABAQUS
a l l m
the
uwr
to input
material
models
and failure
criteria.
It
seems
that
some
difficulties
arose
h m
the
lack
of
automation in load
increments
within the
program.
Use
of
ûuss
ban
attachexi
only
to
the
ends
could
not
adequately
mpresent
the
&ual
confining
behwior.
2.1.3
Karibanis
and
Kiousis
(1893)
A
DruckerPrager
plasticitytype
moâel
was
developed
to
analyze
aie
behaviout
of
columns
confined
with
either
tier
or
spiralt.
The
mode1
indudes
a
nonassociative
flow
ruleI
rtfain
hatûening,
and
a
limitai
tensile
strength
for the
concmte.
Steel
plasticity
was
modeled
with
a
Rambergûsggood
sûessrtnin
wrve.
ïhe
elastic
behaviow
of
ancrate
f dl ws
a
simple
H W s
IawI
and
in
h
plastic
range
follows
a
loaâing
function
which
ir
based
on
a
min
hardening
funcüon
of
pkticrtnins.
(Le.,
i mv( WQbi e
rtninr)
.
ElartopMic
stress
incrementa
mm,
computeà
wing
e
rather
complicated
constitutive
mabrix
ttmt
8
waluated
numerically.
In
this
model,
different
confining
zoner
betwwn
ties
or
spirals
weie
defined.
The
least
confined
zone
was
midway
between
lies.
While
plastic
behavior
was
obsmed
in
the
wsakest
zone,
other
zoner
could
experien
elastic
unloading
during
large
deformations
(i.e.,
postpeak
behavior)
The
mode1
sucwssfUlly
predided
the
behaviour
of
some
of
the
columns
tested
by Mander et al.
(1998),
and
lyengar
et
al.
(1970).
H
is
to
be
noted
that
the
mode1
was
developed
only
for
circular
columns,
but
can
be
usd
in
columns
of
different
size
because
it
is
not
bssed
on
Mistical
regressions
of
a
determineci
set
of
specimens.
2.1
A
Chan
and
Mau
(1
888)
The
authors
recalikated
the
mode1
proposed
by
Bezant
and
Kim
(1979)
with
ercperimental
data
from
uniaxial
biaxial
and
triaxial
tests
from
other
nwearctiers.
Regions
of high
axial compressive
stress
and
low
lateral
stresses
in
those
tests
were
of
sWal
importance in
Mdivvely
simulating
the
behaviour
of
concmte
columns.
The
original
mdel
of
Bazânt
and
Kim
is
an
incmmental
stressmin
relationrhip
of
the
type
where
d#
and
de
am
the
incremental
stress
and
incremental
m i n
matric~s,
nspedively;
and
the
material
maMx
is
mlwlated
as
wtiere
the
right
hand
ride
ternis
stand for
elastic,
plastic and
fredure
material
matrioes,
respedively.
nie
material
rnatrix
CI
is
mitten
in
tems
of
invariants
that
make
it
suitabk
for
finite
element
anallyis
(FEA).
The
recalibrated
mode1
was
irnplemented
in a
FEA
progrorn
that
uses
twodimensional
(2D)
axisymmetric
triangular
ring
elements
for plain
concrete,
and
elastoplastic
springs for
sted
spinlr.
A
NewtonRaphson
rdieme
was
used
for
the
loaddisplacemcmt
history,
and
the
cracking
of
conmete
was
modeled
using
an
iterative
produre
that
accounted
for stress
nlease
(i.e.,
stress
softening
).
Circular
cylinden
with
spird
confinement
testeâ
by
Ahmad and Shah
(1982)
and
by
Mander et
al.
(1984)
wbn
geometrically
madeleci
in
the
numetical
analyses. Only
slices
of
experirnental
columns
were
modeled
due
to
their
syrnmetry
and
the
assumption
that
the
behavior
of
aie
fest
of
the
column
is
similar.
The
analytical
rmults
showad
vwy
good
agre«nent
with
those
obbined
fm
the
tests.
The
nuwc8l
mwlts
prov8d
10
be
an
8xcellrnt
methOd
for
analyzing
stress
dirtrikrüonr
within
the
caicnite
c m
of
columnr.
The
authon
compafed
their
recalibrated
mode1
with
cylinders
with
mdmt e
to
hioh
volumetric
nüos
p
@
r
1.57%

3.1%)
and
wÏth
a
wdlmfined
cduM
tested
by
Mander
et al.
The
moâel,
houmver,
was
not
compared
with
pooriy
confiruad
columns to
demonstrate
its
ability
to
pmdict
a
wide
range
of
column
khavion.
2.2
Linear
and
Nonlineu
lutlc
Models
2.2.1
Mau;
hi;
and
Zhou
(189û)
An
analytical
study
of
confinement
in
cirailar
columnr
is
presented
in
thir
pawr.
The
geneml
solution
folIowa
a
Iinear
elastic
appmach,
whereby
the
radial, tangential and axial
stresses
are obtained from
a
solution of the
displacements
of
a cylinder
subjected
to
a
ring
load
(i.e.,
radial
compmssion).
The
Wt en
defined
an
'8
&ning
stressN
as
aie
average of the
tangential and
d i a l
Messes
in
the
cylinder.
The
eMi w
confinihg
dsss
ir
a
fundion
of
the
distance
from
the
axis
of
the
cylinder.
Several
cornparisons
were
made for
âiienmt
spiral
pitcht~radius
ratios, and for
different
transverse
sections
dong
the
axh
of
the
Sinder.
From
these
compafirons,
an
ua~rage
confinement
hctof"
was
detined
for
sections at
miblevel
bekwn
spimlr
as
s
fundion
of
the
spiral
pi tcMmdi us
ratio.
It
was
found
that
thir
factor
cm
k
u s d
in
the
interpretation
of
nonlinear
tests
as
wa8
estrblirhed
from
analyses
of
~ ~ r n m r
testeâ
by
other
authors.
Linear
elastic
analysam
~ l s
~~
in
rectangular
columns
cordineci
with
welded
wire
faMc
(WWF),
wbm
shilar
confinement
fadm
w m
obtained.
222
Buzegar
.nd
Maddipuâi
(t887)
The
authon
developed
a
thmedimcmrional
(30)
nonlineat
finite
element
analysis
(NLFEA)
pmgmm
for
modeling
of
reinforcd
concrete
structurer.
The
concret8
is
modeled
as
an
otthotropic
matefial,
with
rmeared
cracks
in
nonorthogonal
directions.
Concrete
failure
stresses
are
calailated
wing
the
fiveparameter
ultimate
stiength
criterion
proposeci
by
Willam
and
Wamnke
(1975).
The
hpl ast i c
constitutive
rnodel
of
Stankwki
and
Gentle
(1985)
was
adopted
with
some
modifications. In
mis
model,
increments
of
principal
strains
are
tnindomeâ
into
oûahedral
normal
and shear
Wain
increments.
The
latter
a n
multiplid
by
a
material
rnatrix
to
calculate
increments
in
octahedral
normal and shear stresses,
The
material
matrix
is
a
fundion
of
the
tangential
shear and bulk
rnoduli,
and
coupling
rnoduli.
The
postpeak
behaviour
of
concrete
was
also
taking
into
acewnt
A
stress
decrement
L
computed
using
a
reduction
factor that affects
the
tangent
stiflhess
material
mat e
Smeared
cracks
in
multiple
directions are
based
on
a
fradure
energy
mode1
developed
by
De
Bont
and
Nauta
(1985).
Finally,
the
authors
also
implementrd
models
for
embeddeâ
minforcement
and bond slip.
To
conoborate
the
poapeaak
model,
the
program
wu
uwd
to
analyze
om
of
the
columnr
tested
by
Scott
et
al. (1982);
the
rsurltr
dmmâ
good
agreement
with
2m2.3
~0l ot t l ( 18@4)
The
author
proposeâ
a
#lum
criterion
for
concrete
in
ternis
d
the
tensile
strengai
of
ancrete,
and
the
intemal
hidion
angle.
B a d
on
this
criterion,
a
constitutive
law
for
c omt e
in axial tension
was
deriveci,
The
failure
criterion
was
developâ
to
be
applied
to
mially
loeded
confined
m e t e
colums.
It
was
shown
mat
the
confinement
strength
is
a
fundion
of
the
tenrile
strain
of
the
cover
concret8
at
failure.
ln
the
case
of
cirarlar
cdumnr,
the
cover
shell
is
idealized
as
a
tube
subjeded
to
intemal
radial tensile pressure due to
expansion
of
the
conmete
m.
The cover
starts
its tension
softening
behaviour
after
it
reaches
its
peak
tensile
strain.
A
formula for
the
minimum transverse steel ratio
was
deduceâ
from
the
failum
criterion.
K
the
quantity
of
transverse
steel
exceeds
the
minimum, cover
concrete
wuld
mach
the
ultimate
tensile
strain,
A
set
of
tolumns
testeâ
by
0th
was
compred
with
the
formulae
deducd
for
peak
lord
and peak
strain.
Analytical
mults
varied
fmrn
about
Ml%
to
16û%
of
the
experimental
mults.
2m2.4
Sdby
(1
980)
The
author
develop8â
program
SPARCS,
the
fint
University
of
Toronto
FEA
pmgm
for
nonlinear
elastic
anilysir
d
r e i n f'
ammte
rdidr.
Th.
constitutive
nldionrhipr
fiom
th0
Modifid
Cmpnrrion
Field
ï hmy
(MCFT)
Conmte
wa8
moâeleâ
as
en
irotropic
mmal
bebm
craddng,
and
with
orthotropic
properties
dter
devdoping
cracks.
Steel
wru
srneamd
out in
the
comt e.
Principal
stress
directions
coind*ded
with
principal
rtrains.
The
nonlinear
elastic
analyses
wwe
carried
out
updating
the
8ecant
rüffness
matefial
matis
for
both
muet e
and
steel.
An
iroparametric
rightnoded
brick,
with
thme
degrees
of
freedom
(DOF)
per
node
and
a
clocedfm
stiffness
mate
was
implemented.
Material
behaviour
included:
a
Hognestad
parabols
for
stressstrain
uwe
for
concrete
in
compression;
canprersion
80ftening
due
to
transverse
tensile
strsins;
linear
elastic
khaviour
of
uncracked
ancrete
in
tension;
tension
stwening;
and stress checks in
cracked
concret@.
Steel
could
be
wienteâ
in any
diredion,
and
was
modeled
following
an
elasticperfedlyglastic
curve,
which
also
included
$train
hardrning.
Pef l H
bond
war
also
assumed.
First
checks
of
the
program
wre
made
cornparino
analytical
to
exparimental
mults
of
beams
subjected
to
torsion
and
testeâ
by
other
researdier.
2.26
Vecchio
(1
882)
In
19û6,
Vecchio
and
Collins
proposeci
The
MOdified
Compression
Field
Thtwy
(MCFT).
This
theoiy
ha8
been
used
to
pprrdict
and
to
analyze
the
behviw
of
planedmss
end
planemin
elements
wbjected
to
rhear
and
n o ml
stresses.
T b
original
MCFT
war
extendeci
b
aanunt
for
the
eîf8ds
of
tow
t nri k
stresses,
~~
mhr
due
to
confinam&
nd
variable
Poisson's
ratio.
ModMcationr
to
concret8
material
matruc
derived
from
the
MCFT,
and to
dnust rai n
cuwes
wwo
proposed.
In
the
finite
elment
fornulotion
of
the
MCFT,
orthobopic
materials
have
symmetrical
material
matrices.
As
compression in
conaete
increases,
the
secant
stiffness
for
different
values
of
compressive
stress
varies.
The
Poisson's ratio
is
expected
to
change as
a
result,
thus
making
aie
materiel
matrlx
for
ancrete
The concept
of
a
'prein
metM
is
i nt mdud
to
account
for
non
stress
related
strains
in
concrete
elements.
Prestrains
are
transfomed
into
Yorcesn
in this
formulati~.
The
expansion
eff8d
is
included
in
the
apmSfrein
mat&"
as
where
e&
is
the
prestrain
vector
in
the
principal directions
1
and
2,
vu
is
the
Poisson's
ratio
in
the
direction
i
when
subjecteâ
to
a
stress in the direction
j,
and
cf
ir
the
recant
stiffness
of
concmte
in the
direction
'
To
avoid
numerical
pmblems
in
the
solution
d
the
FE4
the
Poisson%
ratio
ve
wer
dividd
into an
"elatic
portionu
vr'
and a
msidual
component
vr
*

v~

ve
.The
dastic
porüon
vu
%as
d i d y
induded
into
the
material
matrix,
and
the
msiducil
portion
v~
'into
the
presûain
matiix,
keeping
the
symmetry
and
the
odhotmpical
condition8
of
the
material
ma


.

To
account
for
sûtmgth
enhancement,
a
formula
proposed
by
Kupfer
(196Q)
for
biaxial
stressstates
is
8180
included
in
the
formulation.
ï he
increment
in
the
concret8
strength
is
where
K.
is
aie
~&8ngth
enhanament
fador
(Kupfer),
P.
is
the
unconfined
strength
of
concrete
and
1,
is
aie
confind
strength. And
the
main
at peak stress
e,
=
Ke.e,
where
g
is
the
strain
at
peak
unconfined
stress.
In the
case
of
ûWal
sess
states,
the
author
suggested
a
tentative
formula
to
compute
the
strength
enhancement,
which
is
a
combination
of
the
relationships
proposeci
by
Richart
(1
928)
and
Kupfer
(1
969).
The
stressatrain
curve
for
confinecl
ancrete
is
a
liberal
modification
of
the
Modified
Kent
and
Park
mode1
(Scott et al.
1982)
which
consirts
of
a
parabola
for
ththe
ascending
branch
of
the
curve,
and
a
straight
line for
the
pst 
peak
behavior.
A
total
of
1
1
pamlr
tested
by
Vio
a d
Collin8
(1
986)
and
13
sbar
4 1 s
terted
by
Ldas
et
al.
(1990)
wsm
a~l yzeâ
using
the
FEA
progrpm
TRUC
The
panels
wem
subj8deâ
to
combinations
of
bim*al
and
$Mar
sûeases.
The
set
of
panels
chown
wws
those
which
apwiemcâ
awhing
d
Coclcrete,
and
walls
were
subjected
to
monotonic
inmments
of
lateml
load. and constant
compression
at
the
top.
The
walls
were
constmtd
with
conaled
columnr
at
each
end,
thur
providing
an
excellent test to
check
triaxial
compressive
stresses
under
the
load
conditions
menti
above.
The
effect
of
concret@
expansion
was
well
modeled
in
the
finb
ekment
approadi,
the
strength
enhancement
was
also
well
captwed
for
all
tests.
However;'
the
analytical
nrwlts
shwed
a
stiffer
responw,
and
underedimated
the
defledion
of
the
walls.
The
author
extendeci
the
s q e
of
the
MCFT
to
accwnt
for
strength
enhancement
anâ variable
expension
d
concret0
in
twodimenrional
strudural
elements.
2.2.6
8rlby
and
Vecchio
(1883)
SPARCS
was
furüw
updated
and
imptoved
to
include
hno
additional
finite
elements:
sixnoded
isopacametric
wedges
(rsgular
pentahedm),
and
ûuss
ban.
The
effect
of
expansion due to Poisson's
ratio
was
implemented
using
the
concept
of
a
@prestmin
metnbC(Vecchio
1992).
New
constitutive
models
were
added
to
SFARCS.
Base stress$train
curves
for
concret8
in
compression
now
indude
Popovicr
(1973)
Wednrwtnin
curve
for
high
Wmgth
o0c)ciete
(HSC). The
eff
of
confinernt
dm
to
triaxial
rtnu
states
was
also
implmmted;
a
failum
wwe
propomd
by
Hsieh
et
al.
(1970)
is
used
to
ampute
the
ultimate
coinpcwriue
dnngüi.
Finally,
a
mviwd
version
d
the
modilkd
lbnt
wid
Pwk
p o o t m
ampmssiva
wnm
(Scott
et
al.
1982),
and
the
variable
Poisson's
ratio
proposd
by
Kupfer
et
al.
(1969),
were
added
to
account
for
dudility
and
variable
lateml
expansion
in
conamte.
The
first
wthw
modeleâ
six
of
of
columns
tested
by
Sheikh
and
Uzumeri
(1978).
Eightnoded
kicks
wrs
used
in
two
diffmnt
types
of
meshw;
one
coniidered
boai
are
and
cover
m t e,
and
the
other
considered
only
the
conuete
are.
lmpooed
loado
were
applied
to
a
stiff
plate at
the
top of
each
model,
and
a
set
of
springs
was
used
to
mode1
the
postgeak
behaviour.
Column
peak
strengths
obtained
with
SPARCS
agfeed
fairly
well
with
the
experimental
mm~lts
from
Sheikh
and
Uzumeri.
However,
the
postpeak
responses
were
not
rstitfactory.
2.2.7
AbdelHalim
and
Abulebdeh
(1
988)
A
set
of
8
redangular
cdumna,
hno
of
them
originally
tested
by
Scott
et
al.
(1982)
wws
examined
analytidly.
The
linear
elastic
FEA
program
SAP
N
was
useâ
in
a
stepbyatep
lord
increment
approach
to
account
for nonlinear
material
behaviour
of
rei nf wd
concrete.
At
each
loaâ
s t a ~,
the
finite
elaments
(Bnoded
bricks)
were
checked
to
detrrmine
if
t h y
aither
cnckeâ
or
re8chad
a
previourly
defined
failun
rufla.
Al=,
material
plopcwti*e$
(i.e.,
mateda1
stiffnes8
matrices)
were
dionged
accarding
to
the
cumt
$train
state.
The
columns
wm
brought
to
failus
and
the
analyücsl
rewlts
compared
against
üw
experint#rtrl
values,
and
against
The
threedimensional
rtnssstrain
nlationrhipr
and
ultimate
strength
surface
proposeâ
by
Celodin
et
al.
(1977)
wwe
adopted
in
these
analyses.
Steel
was
modeled
using
expecimental
stressmin
wwer
with
stnrin
harâening.
Good
agreement
with
the
experimental
nsultr
d
Scott
et
al.
wru
obtained.
This
is
one
of
the
fint
nonlimer
elastic
finite
elment
analyses
of
confineâ
concrete
repoted
in
the
Iiterature.
Voiumetric
ratio
of
confining
steel,
arrangement
of
longitudinal
steml,
and the
iwement
in compressive
strength
due
to
confinement
were
wme
of
the
aspds
investigated.
Chapter
3
SPARCS
In this
chapter,
a
brief
description
of
the
pmgmm
SPARCS
is
presenteâ.
The
finite
elements
developed
for the
pmgram
are
de&bed,
f o l l d
by
a
review
of
each
of
the
constitutive
modelo
for
concrete
and steel
that
have
been
implemented
to date. Finally, the
SPARCS
stnicture
and
solution
algorithm
is
desctibed.
3.1
Pmgnm
Description
Program
SPARCS
(i.e.,
Selby's
Program
for
the
Analysis of
Reinforcecl
Comt e
Solids)
is
a
nonlinear
finite
dement
program
that
has
been
developeâ
at
the
University of Toronto (Selby
1990,
Selby
and
Vecchio
1993
and1
997)
for
the
analysit
of
reinforced
concret8
rdids.
The
üifeedimensional
(3D)
state
of
stresses
in
reinforcBd
m t e
rdidr
is
taken
into
account
by
extrapdating
the
stress$train
cwws
derived
from
the
Modified
Compfession
Field
Theory
(MCFT)
(Vécchio
and
Collins
1986)
b
its
thfeedimensionrl
fomiulrüon.
The
30
dnss
te
is
mlated
to
the
30
mi n
state
through
a
comtiMive
where
(e)
is
the
stmu
vector,
[Dl
ir
the
material
stifhess
matfix,
and
{c)
is
the
sttain
vector.
The
material
maio<
[Dl
is
given
in
ternir
of
the
secant
stiffness
moduli,
Poisson's
ratio
v,
and
Shear
moduli
G
in
three
diredions
(Le.,
local, global
or
principal
directions).
Secant
moduli
Vary
at
each
losd
state
as a
fundion
of
the
stress
atate.
Stresses
are
cornputeci
accordhg
to
base
stress$train
curves
derived
for
both
conmte
and steel
subj8ded
to
either
compresrive
or tensile
strains.
Material
behwior
dations
have
been
adopted
from
the MCFT
and
theory
of
plardicity;
inciuding:
strength
Meni na
due to tensile
stmins,
strength
enhancernent
due to confinement, variable
leteml
expansion,
cancrete
cracking,
It
is
amurneâ
in
SPARCS
that
concret8
behaves
isotropically
befm
cracking, and
orthotropically
afterwards.
Cracks
are
assumed
to
be
smeareâ
within
concretet
thus
allm*ng
the
user
to
maintain
the
same
finite
elment
mesh
during
the
analysis
proceu,
and
not
having
to
change
it
due to
llocsized
cracks.
Although
cracks
am
arrumed
smewed,
shss
checks
at
crack
ruHaces
are
perfomd
to
reüw
compatibility
and
equilikiun.
St1
can
k
modeled
u
rmeared
within
the
conamte
elemmb,
or
repmmnteâ
as
trum
bprr
attached
to
rdi d
elements.
In
any
caset
p wk t
bond
A
description
of
the
finite
dement
library,
and
the
material
constituüve
laws
and
failure
criteria
followsi.
3.2
Fî nk
Elment
Ukuy
SPARCS
ha8
aime
finite
elements
in itr
libraw
an
&nodeci
brick
(hexahedon),
a
6noded
brick
(pentahedra
or
wedge),
and
a
tniss
bar.
3.2.1
Hexahedron
The
811oded
brick
ir
shomi
in
Flg.
3.1.
It
is
an
isoparametric
elemnt
with
orthogonal
rides and 24
degrees
of
freedom
(DOF),
thme
at
each
node.
The
relative
displacement
between
two
adjacent
nodes
is
assumeâ
linsar,
so
that
Wges
mai n
stfaight,
as
ahanin
in
F~Q.
3.2.
Infinitesiml
rotations
and
small
defOnnationr
are
assumed
in
the
cmputatim
of
the
elment
süffness
matrix
k:
whem
[BI
is
the
mi n
displament
ma*
mat
depends
on
linear
displament
fundions,
anâ
[a
is
as
defined
above.
The
dosedîbm
solution
of
Eq.
(3.2)
ir
obtained
by
direct
integration,
as
illcorporated
in
SPARCS
by
Selby
(1990).
Figura
3.2
Womnd
Brick
It
ir
noted
that
nonorthogonal
hexahedranr
are
not
allowed
in
SPARCS;
the
program
has
a
subrouthe
to
check
element
geometry.
Numkring
must
be
counterclese
as
s h m
in
Fig.
3.1.
3.2.2
Pentatwâtl
(Wedge)
The
6noded
kick
is
rhown
in
Fig.
3.3.
lt
is
also
an
iropammetric
element
with
18
âegmes
of
freedom
(d.0.f.);
thme
at
emch
nade.
It
must
be
prismatic
(Le.,
mintaining
th.
same
b9nsverse
wdim
throughout),
and
the
bottom
and
top
fiaces
(fsce
123
and
face
456
in
Fig.3.3)
mu&
k
of
wual
%ma
and
must
be
Iocated
in
paralid
planes.
Dista0111
am
not
al l wd
in
its
genenüon.
Linear
displacements
and
infinitesimal
d&ormationa
are
also
assumed.
The element
stiffness
matrix
k
(Eq.
3.2)
is
obtaineâ
from
numerical
gauss
integration.
3.2.3
Trum
Bar
This
element has
two
nodes,
and
three
d.0.f.
at
each
end, as
show
in
Fig. 3.4.
The
element
deformation
ir
computed
as the relative displacement
between
the
two
nodes
divided
by
the
Irngth
of
the
element.
A
simple direct
amputation
of the
element
stiffness
rnaûix
b
given
in
the
program
(Selby and
Vecchio
1993).
Although
the
element
formulation
ir
baseâ
rolely
on
axial
defomatians,
kidding
is
rot
taken
into
eccount
whm
the
bar
is
subjected
to
compression.
Bending
ir
alur
ignomd
in
its
stifhar
mrtrix.
This
section
describes
the
modelr
for
conmete
implementeâ
in
SPARCS
and
used
in
the
analysir
of
confined
concrete.
It
begins
with
the
base
stress
strain
cunres
for
concnte
in
compression
and tension, and continues
with
the
failure
criteria
for
concrete
under
M a l
state
of
stresses,
and
cracking. The
suggested
models
for
compreuive
strength
enhancement
are
also
ieviewed.
Finally,
comprersionsoftening
models
derived
ftom
the
MCFT
are
presented.
The
section
ends
with
o
description
of
the
variable
Poison's
ratio.
3.3.1
Bru
StmsStnki
Cuwes
for
Concmk
in
Compnuion,
Pnpeak
6ehaviour.
The
followhg
exprewionr
are
intendeci
for
the
modeling
of the
arcending
branch
of
the
8tress$train
curve.
The
parabola
proposed
by
Hognestad
is
a
widely
useâ
stressstrain
cuwe
far
the
behuviwr
of
n o ml
stnngth.
anâ
is
calculated
as:
whem
C
and
tr
are the
compressive
stress
and
strain
in
the
principal
Miredion
respectivdy,
and
f,
and
Q
an,
the
peak
stress
and
$train
et
peak
stress.
rerpctively.
Eq.
(3.3)
is
depided
in
Fig.
3.5.
3.3.1.2
Thonnfeldt
et
al.
(see
Collin8
et
al.)
This
relationship
is
a
generalized
model
of
the
baw
shssstrain
cuwe
pmposed
by
Popovics
(1973).
The
mlationship
represents
well
the
stiffer
ascending
branch
and
st eqw
falling
branch
of high
strength
concrete
(HSC).
The
stre#anin
cwve
ir
computed
as
follawr:
and
k
=
1
.O
for
the
ascending
bmnch,
or
for
the
destanding
bmnch
The
generelized
Popovics
arve
is
gmphed
in
Fig.
3.6.
A
variation in
the
descendhg
sl op
can
be
seen
as the
ancrete
strengai
i mases.
C
.
Figure
3.6
Gewalizrd
Popovfcs
StwmStraln
Cuwe
3.3.1.3
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