Tension stiffening response of high-strength reinforced concrete ...

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ABSTRACT
The
purpose
of this
research
was to experimentally determine if any relationship
exists between the concrete strength and the post-cracking behaviour of a reinforced
concrete member.
A
total
of
nine
large-scale reinforced concrete tension
specimens,
using
either
high
or normal-strength concrete, were
constnicted
and tested in addition to a
series
of six small-scale reinforced
hi&
strength concrete specimens (pilot specimens).
High
strength concrete
refers
to
a
materiai
meeting the
requirements
of the
Canadian
specification
CSA
A23.1-94
(fcl
>
70
MPa).
The
extension response
with
respect to increasing tensile load of
al1
specimens
was
measured,
adjusted
to account for
shnnkage,
and subsequently presented as bond
performance
(Pc)
factor values.
Shrinkage
corrected response
c we s
more appropnately
evaluate the
post-cracking
behaviour, which wouid otherwise underestirnate the tension
stiffening
potential.
In
addition to concrete
strength,
other variables were tested for
influence on
post-cracking
behaviour and included: the steel reinforcing ratio, grade of
reinforcement
(deformed
bar or prestressing
strand),
and concrete section and bar
layout
(effective area).
A
series
of comparative based results regarding specimen variables were
then
produced
fiom
the
accumulated
research data. Observation of crack
width
and
spacing
developments were
also
made for the large scale specimens
during
testing.
It
was found that,
although
the high strength specimens were indeed stronger
(ie.
larger
cracking
load) and
stiffer
than
the normal strength members, concrete
strength
had
liale
or no effect on the relative post-cracking behaviour
(ie.
pc
was largely unaffected
by
concrete compressive strength). Perhaps the only
variable
influencing the
pc
values
obtained
was the steel reinforcing ratio.
A
pc
function
(for reinforced concrete
in
tension)
is suggested
which
better
fits
the experimental data
than
other published models do.
The
effective area of concrete in tension
given
in
the
CEB-FIP
code
is
found to reasonably
agree
with the information collected
under
this programme.
The
author
would
like
to
thank
many
people
who
were
able to
offer
assistance and
guidance
t h u g h
out
the
duration
of this research
project,
most
significantly
research
supervisor Dr.
P.
H.
Bischoff, whose
insight
and
advice
proved to
be
invaluable.
In addition,
without
the facilities
and
personnel of the University of New Brunswick
Structures Engineering Laboratory,
realisation
of this
project
could
not
have been
achieved.
Special
thanks
are
therefore extended to the
Department
of Civil
Engineering
technicians
Dan
Wheaton,
Don
Goodine,
and
Ken
Noftell.
Fellow
students
Augustin
Dukuze
and
Jonathan
Irving
also provided continued
advice
and
ready
hands
both
before and
during
testing.
Materials
donated by Mr. Don Eisnor at
StresCon
Ltd.
and
Dr.
T.W.
Brernner,
through
LaFarge
Canada
Ltd.,
were
dso
gratefuly
accepted
for
use in
this
research
project.
The
financial
assistance provided
by
both the National Sciences and Engineering
Research
Council
of Canada and the Brernner
Scholanhip
Fund
allowed
for
the
continued
development and
eventual
completion of
this
project.
Finally, the ongoing
interest
expressed
by
farnily
memben,
fiends,
and
CO-workers
at both Eastern Designers
and
Roy
Consultants
make
completion of this
project
most
fulfilling.
TitIe
Page
Abstract
Acknowledgements
Table
Of
Contents
List of Figures
List of Tables
Nomenclature
Chapter One: Introduction
and
Objectives
L
-1
Limit
States Design of
Reinforced
Concrete
L
.2
Direct Tension Members
1.3
High
S
trength
Concrete
1.4
Research
Objectives and Scope
L
-5
ïhesis
Outline
Chapter Two: Background Information
2.1
Coucrete
Properties
2.1.
L
Plain
Concrete
in
Direct Tension: S train
Sofiening
2.1.2
High
Siren,@
Concrete
in
Direct Tension
2.1.3
Shnnkage
of
PIain
Concrete
2.2
Steel Reinforcing
Properties
2.3
Bonded Composite Behaviour
in
Direct Tension
2
-3.1
Behaviour
Prior
to
Cracking
3.32
Post-Cracking Behaviour of
Reinforced
Concrete
2.3.3
Effective Embedment Zone for Concrete
in
Tension
2.3-4
Crack Development
and
Concrete
Strength
2.3.5
Resmined
Shrinkage
and
Creep
1
i
i
i
i
i
i
v
v
i
i
n i
i
s i v
1
3
4
5
6
6
9
9
11
13
14
16
17
19
21
25
26
29
Table of Contents
Chapter Three:
Srnall
Scale Study
3.1
Small
Scaie
Test Programme and Specimen
Details
3
-2
Experïmental
Procedure
3
2.1
Steel Bar Tension Tests
3
-2.2
Specimen
instrumentation
and
Testing
3
-3
Expenmental Results
3
-3.
i
Unbonded
Series
Sepcimens
3
-3.2
Bonded
Series
Specimens
3.4 Discussion of Results
3
-5
Summary
and
Limitations
Chapter
Four:
Large
Scale
Specimens
4.1
Specimen
Details
4.1.1
End
Region
Details
4.2
Experimental
Procedure
4.3.1
Experimentai
Instrumentation
4.2.2
Test Procedure and
Data
Collection
4.3
Large Scale Specimen Test Results
High
Strength
Concrete with Unbonded
1SM
Rebar
High
Strength Concrete with Bonded
15M
Rebar
(S250)
High
Strength Concrete
with
Bonded
1SM
Rebar
(L250)
High
Strength
Concrete with Bonded
15M
Rebar
(L
100)
High
Strength
Concrete with Bonded
20M
Rebar
High
Strength
Concrete with Bonded
Prestressing
S md
Normal
Strengtti
Concrete with Unbonded
1
5M
Rebar
Normal
Streqoth
Concrete with Bonded 15M Rebar
Normal
Strength
Concrete
with
Bonded
Prestressing
Strand
4.4
Demec
Point
Sbriakage
Data
Tabfe
of Contents
4.5
Development
of
Crack
Widths
and
Spacings
Chapter Five: Discussion of Large Scale
Results
5.1
Behaviour
Pnor
to
Cracking
5
-2
Behaviour
Me r
Initiai
Cracking
5.2.1
The
Effect
ofconcrete
Strength
5.2.2
The
Effect of
Reinforcing
Ratio
5.2.3
The
ERict
of
Bar Arrangement
5-3
The
Group
Envelope
Curve
5.3
Small
Scale
Specimen
Cornparison
Chapter Six:
Analysis
of
Composite Flexural
Members
6.1
The
General
Flexural
Response
6.1.1
Uncncked
Section Response
6.1.2
Cncked
Section Response
6.1
-3
Effective Member
Rigidity:
ET,
6.2 The Case
Study
Member
6.3
Calculation of
the
Case Study
M-4
C w e
6.4
Calculation of the Case Study Flexural Response
Chapter Seven: Conclusions
List of References
Appeadir:
Materia1
Properties
A.
1
Experimental
Concrete
A2
Reinforcing Steel
A.3
Additional
Large
Scde
Experimental
Plots
LIST
OF
FIGURES
Figure
Typicai
tensile
load
vernis
strerch
in
Iength
of
a
reinforced
mortar
prism
and
separate
bare
steel
wire,
as
tested
by
Considere.
1899.
Typicai
reiriforced
concrete
flexural
member and the
associated
M-6
diagram.
Direct
tensile
response of
an
axialiy
reinforced
concrete member.
Axial
stress
versus
displacement
of
plain
concrete
in
tension
aiter
Gopalarainam
and
Shaw.
Cornparison
between
matenal
propenies
for
plain
high
and normal
strength
concrete tension
specimens.
Stress-strain
regime
for
typicai
defomed
steel
bar
and
presûessing
stxand
reinforcement
rnaterials.
Typical
N-A
response for reinforced concrete specimen
under
uniaxial
tension.
Tensile
load
carried
by
concrete
done.
Idealised
differentiai section of
a
longitudinaliy
reinforced concrete
axial
tension member.
Uncracked
reinforced
concrete tension
element
as
represented
by
equivaient
springs
in
parallel.
Cracking
development and
dismbution
of forces
in
the concrete and steel for
a
typical
tension member
under
increasing
load.
Zone of effective embedrnent for reinforced concrete
in
tension.
Typical
primary
and
secondary
cracks
displayed
in
longitudinal
cross sections
of reinforced concrete axial tension members.
Reinforced
concrete tension member load
versus
extension response,
modified
to
account
for
restrained
shnnkage.
Relationship
between the
experirnental,
member, and
idealised
response
origins
as
a
result
of member
shrinkage.
Plan
of the small scaie test specimens. Copper sheathing
was
used to
constmct
the
unbonded
(U)
series
specimens.
Cross-sections of the six
small
scale specimens tested.
Load
versus
extension
results
fiom
the
srnaIl
scale
reinforcing
bar
samples
tested
in
tension.
Experimental
test
setup
and
instrumentation
configwation
for pilot
series
members.
Small
scale specimen P-20M-U.
Smail
scale
specimen
P-
1
SM-U.
Page
1
3
4
12
13
16
18
18
19
21
23
26
2'7
32
32
34
35
37
39
43
44
vii
List
of
Figures
Small scale specimen P-9M-U.
Small
scale specimen P-20M-B.
Small scale specimen
P-ISM-B.
S md
scale specimen P-9M-B.
The
bond
performance
factor.
for the
three
bonded
srna11
scale
specimens:
P-2OM-B,
P-
1
SM-B,
and
P-9M-B
Comparison
of results for the specimen P-LOM-B
using
equaI
to
180.200.220
GPa.
Cornparison
of
results
for
the
specimen
P-
1SM-B
using
equaI
to
180.
I9O.220
GPa.
Comparison of
renilts
for the specimen
P-9M-B
using
equal
to 180.200.220
GPa.
Exterior
details
and dimensions of the
large
scale
spechnens.
Specimen
member
designation
legend.
Cross-section
details
of the
nine
large
scaie
specimens.
CIear
cover spacing
within
the central test
lcngth.
Large
scale specimea
corbel
reinforcement
demik.
Test
fiame
and
specimen schematic.
Photogaph
of
typical
large scale specimen erected
in
test
frame
prior
to loading.
Instrument
output of Ioad
versus
extension for specimen
HS-
I5M-U-S250.
Instniment
ourput
of Ioad
versus
average extension for specimen
HS-15M-U-S250,
Specimen response for
HS-
15M-U-S250
with
the restrained shrinkage value offset applied.
Test results of specimen
HS-
15M-B-S250
with the restrained
sfirinkage
value
ofEset
applied.
Test results of specimen
HS-15M-B-L250
with
the restrained shrinkage value offset applied.
Test results of specimen
HS-
15M-B-L
100
with
the
restrained
shnnkage
value offset applied.
Test results of
specimen
HS-20M-B-S250
with
the
resmined
shrinkage
value
ofEset
appiied.
Test results of specimen
HS-
13P-B-S250 with the restrained shrinkage value offset applied.
Test results of specimen NS-
1SM-B-S250
with the
restrained
shrinkage value offset applied.
Test results of specirnen
NS-13P-B-S250
with the restrained
shrinkage
value offset applied,
Restrained
drying
shrinkage data collected for the
high
strength
specimens and
prisms.
Resmined
dryhg
shrinkage data
collected
for the
normal
strength
specimens and
prisms.
Typical
crack patterns
deveIoped
in
large
scaIe
specimens.
Idealised
Ioad
vernis
snain
for
specimens
constructed
with
l12-inch
prestressing
strand.
viii
List of Figures
Average concrete load vs. member extension and total average crack width for
specimens
constructed with
l12-inch
presaessiog
strand.
Bond performance factor,
PC.
for specimens constructed with
I12-inch
prestressing
süand.
ldealised Ioad
v e m
straul
for
specimens
constructed
with
20M
bar-
Average concrete load
vs.
member extension
and
total average crack width for
specimens
constructed
with
20M
bar.
Bond
performance
factor,
Pc.
for specimens
consmicted
with
20M
bar.
Idealised load
versus
strain
for specimens constructed with
15M
bar.
Average
concrete load
vs.
member extension
and
total average crack width for specimens
constructed
with
15M
bar.
Bond
performance
factor.
&,
for specimens constructed with
LSM
bar.
Average concrete load
vs.
member extension and total average crack
width
for
specimens
constructed fiom
hi&
suengih
concrete.
Bond
perfonriance
factor,
&,
for specimens constructed
fiom
high
strength concrete.
Average concrete load
vs.
member extension and to
ta1
merase
crack
width
for specimens
constructed
fiom
normal
strength
concrete.
Bond performance factor.
Pc.
for
specirnens
consmcted
fiorn
normal
strength concrete.
ldealised
load
versus
strain
for specimens having variation
in
cross-section
and bar arrangement.
Average concrete load
vs.
member extension
and
total average crack
width
for specimens
variation
in
cross-section and
bar
arrangement.
Bond
performance
factor,
P,,
for specimens
having
variation
in
cross-section
and bar arrangement.
Large
scale specirnen
group
&.data
with
model
equation and
prdiction
fùnctions.
Large
scale
specimen group
R,,data
with
model
equation.
SrnaII
scale specimen data and the
Pc,prediction
equations.
Typical
flexurai
mernber, or
beam,
subject to transverse load
Q
and the
resulting
shear
(V),
moment
(M),
rotation angle
(8),
and
deflection
(6)
functions.
Typical
flexucal
section
under
extemal
bending
moment,
Mx,
and a
differential
section
dr.
Typical
moment-curvahire
(M-4)
response,
both
with
and without tension
stflening.
Stress
and
strain
profile for
an
uncracked
section
within
a
reinforceci
concrete
member,
Cracked
section strain and
stress
profile.
The
reinforcing
carries
al1
of the load
necessary
to
establish
equilibrium.
List of
Figures
Cracked section
response
including
tension
stiffening
effect and the effective
area
of concrete in tension.
The
example
simply
supportecf
flexural
member
snidied
to evaluate
various
tension stiffenîng
rnodels.
Flow
chart
detailing
calculation
of the
M-#
relationship.
M-#
relationships
obtained
for the four tension
stifiess
rnodek
evaluated
using
the
example
flexural
member.
UDL
value,
W.
versus
centre-line
deflecaoa
6,
for the
exarnple
fiexural
member.
Effective flexural
rigidiry,
WIg,
versus
UDL
value.
W.
for the
example
flexurai
rnember.
Test
resuIts
from
the
defonned
bars
used
in
the
smale
scaie
experiments-
Load
versus
extension for the
15M
bars tested.
Load
versus
extension for the
20M
bars
tested.
Load
versus
extension
data
for
the
7-wire,
113-inch
prestressing
strand
used
in
the
large
scale
specimen
construction.
Total
of average crack width
versus
idealised
member
strain
for
specimens
constmcted
with
/?-inch
prestressing strand.
Idealised rnember load
versus
tom1
of average crack widths for specimens
comtmcted
of
l
iz-Înch
prestressing
strand.
Total of average crack width
versus
idealised member
strain
for
specimens
constructed
with
20M defomed bar.
Idealked
member load
versus
total of average crack
widths
for specimens
constmcted
of
20M
dehrmed
bar.
Total of average crack width
versus
ideaiised member
strain
for specimens constmcted
with
15M
deformed
bar.
Idealked member load
versus
total of average
crack
widths
for specimens constructed
of
15M
deformed
bar.
Total of average crack
width
versus
idedised
member
strain
for
specirnens
constmcted
with
high
strength concrete.
Idealised member load
versus
total of average crack
widths
for
specimens
constmcted
of high strength concrete.
Total of average crack
width
versus
ideaiised member
strain
for
specirnens
constructed
with
normal
strength concrete.
Idealised member load
versus
total of
avcrage
crack
widths
for specimens
constmcted
of
normal
strength concrete-
Total of average crack width
versus
idealised member
strain
for specimens
consmicted
with
variation in cross-section and
bar
layout.
List
of
Figures
A-
lob
Idealised
member
ioad
versus
total
of
average
crack
widths
for
specimens
constructed
having
177
variation in
cross-section
and
bar
layout.
Table
Ti tl e Page
2-1
Tende
tests for plain concrete
specimens
and
their
failure
modes
including:
modulus
of
1
O
rupture, split
cylinder,
and direct tension.
3-
1
Small
scaie
study
specimen
cross-section
and
material
detaiIs.
34
3-2a
Property
values for the
sample
bar
lengths
tested
in
tension.
38
3-2b
Bar
stiffiiess
values
used
for
small
scale
specimens.
38
3-3
Response
data
co1Iected
fiom
bonded and
unbonded
series
srna11
scde
specimens
having
4
1
variation in the
reinforcing
steel.
3 4
Concrete and steel
reinforcing
property
values for
the
srna11
scate
specimens
tested. 51
3-5
Cornparisons
between
the
experimentally
determined
and
expected
results
for
uncracked
52
srnaII
scale specimens.
4-
1
Dimensional
details
of
large
scale
specimens.
62
4-2
Large
scale
specimen
concrete
mix
proportions.
63
4-3
Specimen match
cured
ultimate
compressive
snength
and
spIit-cylinder
data.
66
M a
Saxnple
ISM
deformed bars tested
in
tension.
68
U b
Sample
20M
deformed bars tested
in
tension.
68
4-5
Response
data
collected
for
hi&
and
normal
strength
reinforced
concrete
tension members
74
havuig
variable
reinforcing
conditions.
4-6
Data
regardiig
restrained
shnnkage
values
in
the
large
scaie
reinforced
concrete
tension members.
NS-2OM-U
I
and
U2.
4-7
Average
and
expected crack
widths
with
respect to Load Stage.
102
4-8
Average crack
spacing
values for
the
large
scale
specimens tested
as
compared
to
estimates from the CEB
and
Mitchell and Collins.
5-
1
Concrete and steel
reinforcing
property
values
for
specimens
tcsted.
1 06
5-2
Cornparisons
between
experimentaiIy
determined and expected
resdts
for the
uncncked
large scaie
specimens.
5-3
Comparison
groups
of
Iarge
scale
specirnens.
1
08
5-4
Determination of
the
avaerage offset between the idealised and member
orïgins
for
the
large
scale specimen
group.
6-
1
Cornparisons
of
axial
and
flexwal
members.
132
6-2
M-4
values for the four
dflerent
tension
stiffhess
models
evaluated
using
the
example
flexural
member.
xii
List
of
Tables
Maximum
moment, deflection,
and
average mernber
rigidity
values
for
the
exampfe
flexurai
member.
Concrete cylinder
data
coliecred
for
hi&
strength concrete
mixtures
"A
tbrough
"E".
Concrete cylinder
data
coiiected
for normal
streagth
concrete
m k m
"A"
and
"B".
Quantities
of
materials
and
admixtures
tbr
each
high
strength
mixture
produced.
Quantities
of
materials
used
by
Maio
et
al
for a
120
MPa
compressive strength
concrete
as
compared
to a
typical
concrete
mix-
Property
values for
the
bar
coupons tested in
the
small
scale
experünental
prognmme.
ResuIts
fiom
the
15M
bars
tested
in
the large scale programme.
Rcsults
from
the
20M
bars
tested in the large scale programme.
xiii
Ac.
Ag.
At
Net,
gross,
and
effective
tensile
areas
of concrete
normal
to
length
Lc
As
Net
area
of
reinforcing
steel
norrnai
to
length
Ls
b?
c.
d.
h
Dimensions of a composite member cross-section
including:
width.
depth to
HNA.
depth to
reinforcing
steel bar
group.
and
overall
depth
c c
Resdtant
compression force
in
concrete for
flexural
sections
C
1.
C?
...
Labels for
conseculatively
occuring
lateral
crack
withia
the
longitudinal
gauge
lenggh
Lo
dc
depth
of
reinforcing
bar
clear
cover
Ec,
E<
Measured
and
estimated
values of plain
concrete
elastic
rnodulus
(Secant
Method)
Es
-Ep
Elastic
modulus
of
reinforcing
steel and
presûessing
strand
h
'
Ultimate
compressive stress of concrete
cylinders
at
failure
J&,
,/,p.
ft
Flexural.
split-cylinder.
and direct tensile stress
limits
of plain concrete
memben
for tension (cracking)
faiiure
,
fP
Normal stress developed
in
concrete.
deformed
bar.
and
prestressing
srrand
under
axial
load
fcc.fcf
Concrete stress at the
exneme
fibres
of a
flexural
section
b./p u
Yield
and
ultirnate
stress values for deforemed
reinforcing
bar
and
prestressing
strand
H
Hydraulic
system force
used
ro
load
the
large
scale
specimens in tension
42
Effective Moment of
Inertia
for
cracked,
reiaforced
concrete
member
Ig.
It.
Ic,
Gross,
uncracked
ùansforrned,
and cracked moment
of
inertia
values
sf
reinforced
concrete flexural members
jd
Moment
arm
between
resultant
tension and compression force
vectors
of a
flexural
member
,
Axial
stifiess
values
(AE/L)
for concrete. steel. and composite member
kl.
k2
Bond
and
strain
gradient coefficients
used
for
the
CEB
estirnate
of
mean
crack
spacing
sm
Lc.
Ls
Length
of concrete and steel
with
respect to
the
Longitudinal
axis
Lot
Lu
Gauge
length
used
for bonded or unbonded
reinforced
axial
tension
mernbers
lb
Bond
Iength
required
to
transfer
force
between
concrete
and
steel
reinforcing
at
cracks
Mer,
MS,
Initial
cracking,
service level, factored
level,
and
ultimate
limit
state
(resistance) moment of
a
M ~,
Mu
reinforced
concrete
f l e d
member
Mt,
Mx
Interna1
moment
resistance
and external applied moment for
flemiral
member section
n
Steel-to-concrete
rnodular
ratio,
Es/&
xiv
Nomenclature
Number
of
lateral
cracks observed
in
IongitudinaI
section of reinforced concrete
Axial
t ende
force applied with respect to the member
and
idealised
ongins
initiai
cracking
and
reinforcement
yield
force for
a
reidorced
concrete member
Average
tende
load
canied
by the concrete,
defomed
bar,
and
prest~ssing
s m d
within
a length of reinforced concrete
under
the applied
teusile
load
Ni
Average
tensile
load
canied
by
conmete
at
initial
cracking
and
reinforcing
steel
at
yielding
Force
in
reinforcing
steel
due
to
(rcstraiued)
shrinkage
Constants used
in
the
modified
Ramberg-Osgoode
equation (to
mode1
strand
response)
Lateral
forcing
function
(with respect to length) of a
Fiexural
member
Average of the four
individual
large
scale
specimen
restraining
md
reaction
forces
Absolute
residual
value
allowed
for
equations
of
equilibrium
using
numerical
analyses
Length
between the extreme crack locations within gauge
Iength
Lo
Experimentally
measured
crack
spacings
of
a
reinforced concrete member
Reinforcing
bar
spacing
in
composite member cross-section
Mean
spacing
of
lateral
cracks
within
gauge length
Lo
of a reinforced concrete member
Resuitant
tensile
force
in
concrete before cracking,
in
concrete after cracking. and steel
reinforcing
for a
fiexural
section
Shearing
force
in
a
flexural
member
with
respect to length
Values of
dead
live.
service.
and
factored
uniformly
distributed
load applied to
flexural
members
Average
measured
Iatenl
crack width at crack location within gauge length
Lo
Characteristic crack
width.
Exceeded
by
oniy
5%
of crack
widths
Expected
mean
and
maximum
lateral
crack
widths
of a reinforced concrete member
Water to cementicous materials ratio for a given concrete
miu
Bond
and
load factors used
in
Mitchell
and Collins post-cracking response equation
Strain
gradient coefficient used
in
the
Gergely-Lutz
maximum
crack width
calcuiation,
w-
Bond performance factor,
P ~ ~ o
representing
tension
stiffening
in
reinforced
concrete
with
respect to
ideaiised
strain
Ei
Mass
density
of concrete,
approximately
2400
kg/m3
Average extension
in
a
longitudindly
reinforced concrete tension member
Nomenclature
Ac,
As,
Am
Average extension of concrete,
reinforcing
steel,
and composite rnember
under
axial
load
A
A
Shrinkage
of
reidorced
(remahed)
or
piain
(h)
concrete
8x,
Amax
Point
and
maximum
deflections of a
flexural
member
h m
the
longimdinal
axis
EcT
ES,
Em
Average
strain
of concrete.
reinforcing
steel, and composite member
under
axial
load
&CC.
E C ~
Concrete
strain
at
the
exueme fibres
of
a
flexurd
section
&cf.
%-cr
Average
appiied
stress
in
concrete
berneen
cracks
and
steel
reiaforcing
at a crack
EP
fi,
EU
Reinforced
(restrained),
plain
(k),
and
ultimate
shtinkage
strain
Ei,
EW
Idealised
and
smeared
crack
saains
of
a
composite member
(fiom
idealised
origin)
ECC.
&y
Cracking
and
yielding
strains
of a composite
axial
member
P
Reinforcing
ratio
(net steel-to-net
concrete),
or
As
/
Ac
@b
Reinforcing
steel
bar
diameter
4-
(Px
Angdar,
or
rotational.
gradient
of
a
flexural
member.
ex
Anguiar.
or rotational, displacement of
a
flexural
member
from
the longitudinal
axis
act
Stress
hct i on
of concrete
in
tension,
&ft
r
Bond
force
rransfer
stress
at the
steel/concrete
interface
flc
Bond performance factor,
representing
tension
stiffening
in
reinforced
concrete
with
respect to
smeared
crack
strain
E~
[
r +n~l
Composition factor of
reiaforced
member,
L
+
( EA) ~
/
(EA)c
xvi
CHAPTER
ONE:
INTRODUCTION
AND
OBJECTIVES
Tests of reinforced tension members were carried out as early as
1899
by
Considerel.
The extension response of mortar prisms
axially
reinforced with steel wire was measured, as
shown
by the typical result
given in Figure
1-1.
The response
slope
of the reinforced mortar
prism,
following
initial cracking load
Ncn
was found to both lie
above
and
be
parallel to the response of an equivalent bare steel wire sample. This fact
led to the conclusion that the mortar,
although
cracked, continued to add
/
/
i
.
/
-
.
I
/
i
-
1
Yield
Point
Average
load in
Reinforcement
Stretch
in Prism
Length
I
Figure
1-1.
Typical
tensile
load
versus
stretch
in length of
a
reinforced
mortar prism and
separate
bare
steel wire, as tested
by
Considere,
1899'.
stiffhess
to the encased steel. Both samples were ultimately
lirnited
by
the
steel wire yield
capacity,
Ny.
Similarly,
Morsch?
in
1908,
obsewed
that concrete
remaining
intact
between
two
adjacent cracks was able
to
decrease the
extent
to which steel
reinforcement stretched as
compared
to bare steel samples. This effect was
Chapter
I
:
Intmduc
tion
and
Objectives
referred to
as
tension stiffening and was
attributed
to surface fiction, or
bond,
between
the two
component
materials.
Bond force transfer allows
sections of uncracked concrete to assume some portion of the total applied
axial load,
N.
Load
sharing
correspondingly
reduces the average steel
stress, force, and
strain
to result in
a
lowered
overall
member extension as
compared to bare steel. This
thesis
is concemed pnmarily with the
phenornenon of tension stiffening and quantifies the average tensile capacity
of certain cracked high strength reinforced concrete
lengths.
1.1
Limit
States
Design
of Reinforced
Concrete
The strength of concrete in direct tension is
known
to be about
10
to
20 times lower
than
in direct compression. For this reason
reinforcement
is
provided in
the
tensile
regions
of concrete members to
cany
or resist
tensile forces at crack locations. This results in composite action which
increases the strength and
ductility
of the concrete member,
allowing
for
greater
load and span combinations. Reinforcement may include either
deformed
steel bar or prestressing strand. Figure
1-2
shows a
typical
flexural
member
(beam)
subjected to applied end-moments of increasing
magnitude,
M,
and the resulting
deflection,
6,
fiom
the longitudinal
axis.
In
this case, the upper fibres of the mernber are in compression, while the
Iower
fires
are
in tension where cracks develop perpendicular to direct
tensile stresses.
Limit States Design
methodology
defines
conditions, or
States,
which
will
produce
a
satisfactory member response to
various
load conditions.
Firstly, the
ultimate
limit
resistance of the rnember
factored load case
(Mf).
state is
defined
as the maximum possible load
(Mu)
and
cannot
be exceeded
At ultimate load
levels,
it
is
by
the governing
expected that the
Chapter
1
:
introduction and
Objectives
critical
member section
will
be at
a
crack location where tensile force
can
only be
transferred
by the given steel reinforcement. The
amount
of steel
employed is proportioned so as to
produce
a ductile member
failure,
rather
than
the
undesirable
bnttle
mode. Secondly, the
sewiceability
Iimit
state
must
also
be checked to
ensure
that any applicable
limiting
tolerances are
not exceeded by
the
governing
unfactored, or
service,
load case
(Ms).
Continued
serviceability
of reinforced concrete members, as
determined
by
maximum allowable deflection or crack width, preserves both aesthetic
,
Ultirnate
Load
i
t
-
Member
Member
Deflection
at
Centre-Line
(6,
)
Figure
1-2.
Typical
reinforced
concrete flexural
member
and the associated
M-6
diagram.
appeal
and
exposure to
aggressive
environments.
While
tension stiffening
is
not considered at ultimate load levels, this phenornenon
can
significantly
affect service load deflections,
making
the effect
worthy
of consideration.
As
seen
in Figure
1-2,
the service moment
MS
would
produce
deflection
61
using
the average member or "stiffened" response
line,
or deflection
62
Chapter
1:
introduction
and
Objectives
using the cracked section response. Deflection
61
is
noticeably smaller
than
62-
1.2
Direct Tension
Members
Axial tension members, generally avoided in reinforced concrete
structure design,
can
be used to
experimentally
isolate
and
study
direct
tensile behaviour
for
application
with
other
member
sections which
experience combined or reversing stress
States.
A typical reinforced
concrete rnember
under
direct tensile force,
N,
and the resulting
longitudinal
extension,
A,
can
be
seen
in Figure
1-3.
Even
after
cracking
Y
Reinforced
Member
I
I
Bare
Steel
I
i
1
t
I
L+ A
l
l
I
!
Member Extension
(A)
Figure
1-3.
Direct
tensile
response
of
an
axially
reinforced
concrete member.
has
occurred,
it
is noted that
the
member still maintains
a
stiffkess
greater
than
a bare steel
sarnple,
i.e.
tension stiffening effect
shown
by the shaded
area,
and
passes
through
four distinct phases
under
increasing axial tension.
Chapter
1
:
Introduction and
Objectives
These
are: uncracked linear
elastic
behaviour, line
OC;
initial and
continued cracking, line
CS;
stabilised crack development, line SY; and
finally,
post-yielding.
The values
Pc
and
Ps
represent the average load carried by the
concrete and the steel over the gauge length
of
the member. From the
point of initial cracking (point
C)
onward, the value
Pc
continues to
decline
until yielding of the steel reinforcement (point
Y).
At a crack location,
only
the steel reinforcement
can
carry
the
total applied force N, but is
lirnited to the value
Ny.
Reinforced concrete tension members
can
be
studied
and evaluated in
terms
of the basic physical properties of the concrete and steel, which
include: the
material
elastic
modulus, and the respective limiting
material
stress values
under
both tension and compression. In addition
to
the
basic
physical properties, other considerations are important when
studying
reinforced concrete,
including
shrinkage
and
creep
strains, the reinforcing
bond characteristics, effective area of concrete in tension,
and
the
specific
curing and loading histories
of
the member.
1.3
High
Strength
Concrete
The
Canadian
Standards
Associationi,
in specification
CANKSA
A23.1-94,
has recently defined high strength concrete as being a
rnaterial
meeting or exceeding 70
MPa
compressive strength at
28
days of moist
curing. Normally, structural concretes are mixed in the
20
to
40
MPa
range,
but
possible strengths have
sharply
increased over the past few
decades~.
For special projects, where economics and engineering feasibility
indicate, concretes
above
60
MPa
and up to
150
MPa
have been
satisfactorily placed,
while
other experimental concretes have attained
Chapter
1:
Introduction
and
Objectives
strengths
of
200
MPa
and
hi ghe~.
Recently the
Canadian
reinforced concrete design
specification6
(CANKSA
A23.3-94)
has been updated to
reflect
the current information
regarding high strength concrete properties derived
fkom
the
literature.
Higher compressive strength improves the tensile capacity of plain concrete
but
also reduces the ductility of the
material
at
failure7.
It
is
therefore not
specifically
known
how plain concrete strength
can
affect the tension
stiffening process in reinforced concrete, although some
authors
indicate
that an increased strength would
reduce
tension stiffening
effectss-10.
1.4
Research
Objectives and
Scope
The
purpose
of this research was to experimentally determine if any
relationship
between
the concrete strength
and
the post-cracking
behaviour
of
a reinforced concrete member existed.
A
total of
nine
reinforced
concrete tension
specimens
were
constnicted
and tested, using either
high
or normal-strength concrete.
The extension response of each
individual
specimen
was
measured, and
subsequently adjusted
to
account for shrinkage, with respect to increasing
tensile load.
Observation
of crack width and spacing developments were
also made.
A
series
of
comparative based results regarding specirnen
variables were
then
produced
fkom
the data.
1.5
Thesis
Outline
Background
information
conceming this research project
is
given
in
Chapter Two, and is
derived
fiom
both
the
current
topic
literature
and
basic engineering
theory.
Chapter
Three
contains
the testing results
from
a
series
of small
scale
Chapter
1 :
Introduction
and
Objectives
specimens
(0.9
metre gauge length) which also provided
information
to
implement
the eventual large scale
program
and develop an appropriate
high strength concrete
mix.
Six pilot specimens, having a target concrete
strength of
70
MPa,
were constructed using either bonded or unbonded
rebar in three different reinforcing ratios. Results
fiom
these
tests are
presented and discussed at the conclusion of Chapter
Three.
Chapter Four
contains
the information
pertaining
to the large scale
research
program;
including the specimen design, investigative procedure,
and
experimental results for the
nine
reinforced concrete tension specimens
(1.8 metre gauge length). Six specimens were cast using high strength
concrete, while the remaining three were normal strength. Of the
nine,
two were cast with unbonded rebar, while the remaining seven were
normally bonded. Two specimens incorporated prestressing strand (not
stressed)
and
seven others used normally deformed steel bar. The results
of
these
tests, along with crack width and spacing data, are presented in
Chapter Four and discussed in Chapter Five.
Comparison
groups
are created in Chapter Five to better identify
which of
the
large scale specimen variables most affect tension stiffening
and
cracking.
In
particular,
attention was paid to the effects
of
concrete
strength, reinforcing ratio, and bar placement in terms of a tension
stiffening
function,
pc,
and load
versus
crack width and extension data.
Chapter Six
provides
estimates of maximum defiection for a simply
supported reinforced concrete flexural member experiencing different
levels of tension stiffening as a result of applied transverse load.
The
beam, having typical reinforcing
details
and span length was evaluated
using the
p,
curve
experimentally
determined
from this work, models
from
the literature, and the case
of
zero tension
stiffness.
Analysis
of
the
Chapter
1:
introduction
and
Objectives
rnember is camed out using a simplified representation
of
the cross-section
and
subsequent
numerical
integration
of
the
M-t$
relationship to determine
the maximum span
deflection,
6.
From this information, an effective
member
flexural
rigidity
(Ere)
can
be determined and cornparisons
between the different models are
made.
Chapter Seven
provides
the
conclusions to discussions presented
individually
in Chapters
Three
through
Six.
An
appendix
completes this
thesis
and
contains
information regarding
the experimental
material
properties for the concrete and reinforcing steel.
Additional specimen data, which
is
similar
to
that already presented in
Chapter Five,
are
also provided for interested readers.
Reinforced concrete
is
an
internally
indeterminate structure of two
materials which
can
be solved
using
the principals of force equilibnum and
strain compatibility via individual
material
behaviour characteristics.
Material
properties, in reference to the stress
versus
strain
(o
vs.
E )
characteristics,
are discussed for plain concrete and reinforcing steel in
sections
2.1
and
2.2
respectively. The load
versus
deformation response
of
an axial specimen
(N
vs.
A)
is
determined experimentally and used to
obtain the
O-E
response, although it is more convenient to express
N
vs.
E
for composite
specimens.
Composite behaviour, along with a pertinent
research literature review,
is
presented in section 2.3. Composite
behaviour in tension includes both the initial
linear
elastic
response
and
the
more complex post-cracking phase.
2.1
Concrete
Properties
Concrete strength is in reference to
the
ultimate compressive stress
Cfc')
of
representative
cylinders
and
can
be
used to predict other concrete
properties including the
elastic
modulus
(Ec)
and the direct tensile strength
(fi).
The secant modulus
of
elasticity,
at
0.4fcr,
of plain concrete in
compression
can
be approximated using one of the
two
fomulae
given
by
the
Canadian
reinforced concrete design
specification6,
depending on the
strength of the material. The
first,
which
is
valid
for
fc'
values between
20
and
40
MPa,
is
given as:
Chapter
2:
Background
Information
For higher strength concretes, another
empirical
equation is used,
which is a
function
of the hardened concrete density. This formula is
valid
for concretes having a
mass
density between 1500
and
2500
kg/&
(typically
above
40
MPa)
where
y,
is concrete
density typically between
2300
and
2500
kg/d
The
tensile strength of plain concrete
is
typically also expressed
as
a
factor of
the
square-root
of
fc'.
The
measured tensile strength
depends
upon the type of tensile test: either direct tension, split cylinder, or
modulus
of
mpture,
as shown
in
Table 2-1. While the compressive
response of plain concrete in
compression
is
parabolic,
the response of
Table
2-1.
Tensile tests for plain
concrete
specimens
and
their
failure
modes including: modulus of rupture, split cylinder, and direct tension.
Modulus
of
Rupture.
1
Tensile Load Procedure
Test
I
Third-point
ioading
in
flexure.
Failing
i
Accepted
!
Stress
1
Value
1
Split Cylinder
I
Test.
Cylinder
tested on
edge.
Direct
Tesion
Prismatic
member
in
axial
tension.
l
i
I
l
*
-iTT
i
8P
1
l
1
I
1
FL-I
=-
f,~
n202
ft
=-
P
A c
l
0.5
@'
1
0.33
I
Chapter
2:
Background
Information
uncracked concrete loaded
in
direct tension behaves as an
elastic
material
having a linear stress-strain relationship,
Ec,
until
60
to 75 percent of the
ultimate tensile stress
~alueu.~*.
It
is
often
assumed however that
linearity
continues until tensile
failure.
Shrinkage,
or the
volumetric
reduction of
a
concrete mass over time,
is usually measured as a strain value
(i.e.
iength change in relation to
original length) and
can
be termed either "free"
or
"restrained". Free
shrinkage is the
naturally
occumng value in plain, or unreinforced sections
of concrete. Lengths of steel reinforcing work to reduce the
amount
of
free
shrinkage that
would
otherwise
be produced in a
similar
plain concrete
section. Restrained shrinkage
is
therefore measured in reinforced sections
of concrete. The relationship between free and restrained shrinkage is a
fûnction
of the axial rigidity
(EA)
ratio between the steel and concrete, as
given in section
2.3.4.
2.1.1
Plain
Concrete
in
Direct Tension:
Strain
Softening
Data collected by Gopalaratnam and
Shaw16
(1985)
shows a
typical
plain concrete tensile response
curve
in
Figure
2-1.
Plain concrete loaded
in tension is characterised
by
an ascending (line
OC)
and descending (line
CF)
portion of the stress
versus
extension
ciirve,
where point
C
represents
crack initiation. Initially, the concrete has a near-linear
elastic
response
until the specimen cracks. Once cracking occurs, the two halves of the
specimen
then
undergo
elastic
shortening while the crack width, w,
continues to increase. The post cracking response of plain concrete is
referred to as "strain
softening".
The descending portion of the tensile response is not always properly
measured and
Guo
and
Zhangl7
(1987)
indicate that the tensile strain at
Chapter
2:
Background
information
maximum stress has been cited
between
60 and 800 microstrains
( p ~ )
by
vanous
authors.
The
reason for this, as Reinhardt et
ails
point out,
is
that
the softening response is discrete (or localised) behaviour, related
specifically
to
crack
widening.
It is therefore necessary to measure applied
tensile stress with respect to overall specimen extension, rather
than
a
strain value which would
Vary
with the
gauge
length
chosen.
Concrete
Prism
L
F
i
O
10
20
30
40
50
Specirnen Displacement,
A
(pm)
Figure 2-1. Axial
stress
versus
displacement
of
plain
concrete
in tension, after
~opalaratnam
and
Shaw?
In the past, reinforced concrete structures have been
modeled
using
finite
element procedures which incorporate strain softening data to
simulate
the tension stiffening response of cracked reinforced concrete as
"smeared" (or non-discrete) homogeneous elements. Many
authors
indicate
that this procedure is feasible once a
suitable
modification factor, applied to
the descending portion of the tensile stress-strain curve,
can
be
determinedi9.
Difficulties
anse in this method due to the wide discrepancy
in expenmental data and non-standardised testing practices.
Chapter
2:
Background
Information
2.1.2 High
Strength
Concrete
in
Direct Tension
It is
known
that increased concrete compressive strength results
in
both an increased failing tensile strength and brittleness
after
failure'o.
Typical
results from plain high and normal-strength concrete samples
tested in tension are shown in
Figure
2-2.
It
can
be
seen
that both the
failing tensile
stress,fr,
and the initial
elastic
slope value,
Ec,
of the
high
O
1
O
20
30
40
50
Specimen
Displacement,
A
(pm)
Figure
2-2.
Cornparison between
material
properties for plain
high and
normal strength concrete tension
spe~i rnens~~.
strength sarnple are greater
than
the
normal
strength sample. Precisely
what effect this change would have on the post cracking behaviour of
reinforced concrete (tension stiffening effect)
is
still
not clear in the
literature.
Based
on
observations of centre-point deflections
of
high strength
reinforced concrete slabs
(90
to 150 mm thick), Marzouk
and
Chen21
in
Chapter
2: Background
Monnation
1993 concluded that high strength reinforced concrete slabs behaved
differently
than
similar
normal strength
specimens.
This was
attributed
to
differences in the concrete
material
properties,
particularly
the increase in
post-cracking brittleness found
in
unreinforced high strength
sampIes.
Increased
brittleness
is
observed as an increase in slope of the descending
branch of the
a v.
A
response following initial cracking.
Finite
element
analysis output, using modified strain
softening
data
collected
fiom
plain
concrete cylinders, was compared to the experimental results. The
load
versus
deflection response of
these
high
strength reinforced concrete slabs
was indeed different, but
it
is
not clear if the relative tension stiffening
response
was
affected by concrete strength.
2.1.3
Shrinkage
of
Plain Concrete
The
shrïnkage
of plain concrete, or
free
shrinkage,
is
dependent upon
many variables, some of which are also linked to concrete strength.
Shrinkage
occurs
in
three
basic
ways:
(1)
plastic shrinkage prior to final
concrete set,
(2)
autogenous or basic shrinkage resulting from self-
desiccation, and
(3)
drying
shrinkage.
Plastic shrinkage occurs
pnor
to the final set of the hydrated cernent
paste.
During
the
first
few hours following placement
of
fresh concrete,
intemal
moisture
migrates
to the surface of the concrete as bleed water.
When
surface bleed water evaporates faster
than
can
be replaced
From
lower layers, pore water pressure is lost in the small voids
near
the surface
and allows for increased shrinkage of the matrix. Plastic
shrinkage
is
generally expected to be a small fraction
of
the ultimate shrinkage value
obtained,
The physio-chernical
process
of cernent hydration
produces
a volume
Chapter
2:
Back-mund
Information
of
paste
smaller
than
that assumed by the
anhydrous
cernent and water
separately. That is, water and cernent react to
produce
a
paste
which
continues to
shrink
as it hardens,
ultimately
resulting in a
rigid
matrix
under
internal contraction stresses.
Anhydrous
cores
of
unreacted cernent
are present in the hardened skeleton of any concrete mass, which in the
presence of internal moisture, will also eventually hydrate. This self-
desiccation process leads not only to additional strength gain through repair
of
intemal
micro-cracks, but increased shrinkage as well. Autogenous
shrinkage values are not
usually
measured in normal strength concrete
samples
as
the
concrete is still hardening while
it
is
occurring.
In general,
high strength concretes
use
more
unhydrated
cement
in the mix (with the
same
or
lesser
volumes of water) and should therefore
produce
larger
autogenous shrinkage values.
Drying
shrinkage occurs when the concrete mass
is
unable
to retain
internal moisture due to the arnbient environmental conditions (ie.
temperature
and relative
humidity).
Drying shrinkage
is
expected to
be
less in higher strength concretes
due
to the increased density and reduced
porosity of the matrix.
Primary
factors
of
the concrete
rnix
which affect the rate of shrinkage
include: water-to-cementious materials ratio,
(W/=rn),
relative humidity
(RH),
strength
and
elasticity
of
both
the aggregate and hardened cernent
paste,
and the volume of
paste
and
admixtures. It
has
been suggested that
high strength concretes should experience shrinkage values which are
approximately the
sarne,
or less,
than
normal strength
concretes??.
However, autogenous shrinkage values (rneasured between
72
and
5000
hours age) of ten high strength concrete mixtures
(44
to
100
MPa)
were
made by de
Larrard
et
al2,
(1994)
which varied
fiom
40
to
1
10
ps
and
Chapter
2:
Background Information
were proportional to the compressive strength values.
The Portland
Cernent Association,
in
publication
EB
I
14.0
1T?o,
States
that high strength
concrete batched with admixtures (including superplasticisers) will
generally increase
shnnkage
by
IO%,
as compared to NSC, but will be
correspondingly offset by the reduction in
mix
water used to obtain the
increased strength.
The ultimate free shrinkage value,
EU,
of plain concrete
c m
be
expected to range
fiom
200 to
1000
p ~,
while values
between 400
to 800
p~
are more
likelyi4.
In addition to the environmental conditions, the
shape, size, and surface area-to-volume ratio of placed concrete will also
impact
the
final
shrinkage
value
realised for
a
given
project.
2.2
Steel Reinforcing Properties
Reinforced concrete
is
typically constructed using either normally
deformed
(ribbed)
steel
bars
or prestressing steel. The primary difference
between the two are the respective yield
and
ultimate
strength values, as
shown
in Figure
2-3.
Deformed steel reinforcing bars
are
expected to
t
1
2000
.- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
Prestressing
Strand
(LOW
Relaxation)
i
Deformed Steel Bar
f
O
5
1
O
15
20
25
Steel
TensiIe
Strain
(microstrains)
Figure 2
4.
Stress-strain regime for
typical
deformed steel
and prestressing strand reinforcement materials.
bar
Chapter
2: Background
Information
follow the
classic
elastic-plastic stress
versus
strain
curve,
showing a
definite
plateau
after
the
yield
stress
value,&
Prestressing
steels
gradually
develop an ultimate
stress,fpu,
following a
separate
yield point.
Deformed bars typically have a minimum yield stress
of
300 to 400
MPa,
while prestressing steel is
much
higher,
typically
between
1000
and
2000
MPa.
Deformed bars and prestressing strand are accepted to have
initial
elastic
modulus values of
200
and 190
GPa
respectivelylz.
The
o-E
relationship of steel reinforcement
can
be
approximated by
formulae. In
the
case of
normally
deformed bars, a bilinear
cuve
bâsed on
Hook's
Law"
is
used (Equation
2-2a);
while
the modified Ramberg-Osgood
expression (Equation 2-2b)
can
be used for prestresssing
strandzs
f p
=
where
Q,
R,
and
S
represent constants used to
fit
the
curve to
the
particular
characteristics of the strand tested.
2.3
Bonded
Composite
Behaviour
in Direct
Tension
Composite behaviour results when two individual
materials,
such
as
concrete and steel, share
an
applied load.
A
typical
load
versus
extension
response for reinforced concrete
under
uniaxial
tension, Figure 2-4a, is
similar
to the approximate response in Figure
1-3.
As before the member
experiences four distinct phases of behaviour with increasing tensile load
value. They are:
initially
uncracked, continued crack development,
stabilisation, and yielding. The average load in
the
concrete,
Pc,
can
be
Chapter
2:
Background Information
determined
through
each
of
these
four
phases
as the value
Ps
subtracted
fiom
N
and
is
s how
in Figure
2-4b.
Linear
elastic
behaviour
can
be
assumed
in the composite section
up
until the initial cracking load
Ncr
is
achieved and
is
descnbed
fbrther
in
section
2.3.1.
The next
two
phases of composite behaviour (crack
Member
Extension
(A)
a)
1
1
I
Bond
Performance
Member
Strain
(E)
b)
Figure 2-4.
a)
Typical
N-A
response for reinforced concrete
specimen
under
uniaxial
tension, b)
Teosile
load carried
by
concrete
alone.
Chapter
2: Background
information
development and stabilisation)
can
be grouped together as the
post-cracking
response and are discussed
fûrther
in
section
2.3.2.
Ultimately
yielding of
the steel reinforcernent
occurs
and represents, for
al1
practical purposes,
the maximum value of
tensile
load which
can
be applied to a cracked
reinforced concrete section. Prestressing strand, having a gradua1 yield
development
curve
as opposed to a
sharp
transition, may in fact
allow
for
continued tension stiffening behaviour at
very
large
strain
values. Crack
development
under
axial load and restrained
shrinkage
are discussed in
sections
2.3.3
and
2.3.4
respectively.
2.3.1
Behaviour
Prior
to
Cracking
A
direct tension member
cm
be conveniently envisioned
as
a
series
of
sections
of
differential length
dx
connected
along a
common
axis
(shown
in
Figure
2-5),
each of which must
carry
the total applied load
N.
Through
bond force
transfer
(t),
the
individual
interna1
forces
Ps
and
Pc
are
,-
Concrete
Figure
2-5.
Idealised differential section of
a
longitudinally
reinforced concrete
axial
tension member.
x
differential
length:
dx
4
x+dx
Chapter
2:
Background
Idormation
developed
such
that, at
al1
sections, the
sum
of the
intemal
component
forces
is
equal to the extemally applied
load.
This condition is
referred
to
as force equilibrium and expressed as:
Sections experiencing tensile force are subject to an extension in
length, as
defined
by the individual
material
properties. At sections where
both the overall
length
(L)
and
change in length
(A)
of the steel and
concrete are equal, strain compatibility exists
and
is
written
as:
cm
=
EC
=
ES
where
member
strain
=
A/L
therefore
if
compatibility exists, the individual extensions become:
Am
= A c =
AS
(2-3)
Strain compatibility results
fiom
the bond developed between the
composite materials. Force equilibrium and strain compatibility
can
be
used to solve the
intemal
indeterminacy
of
a composite section via
the
elastic
modulus expression:
E
=
o
/
E
=
PL
/
AA.
Rearrangernent of this
expression
cari
then
be
substituted
into
Eq.
2-3,
producing:
where
Kc
and
Ks
represent the axial
stifhess
values
(EA/L)
of
the
concrete
and steel respectively.
Applying
the
strain compatibility condition
(Eq.
2-
4)
then
results in an expression relating the individual
internal
axial
stiffhess values to the
extemal,
or global,
st i aess
of
the
rnember
(bar-K):
Chapter
2:
Background
Infornation
where np is the axial
rigidity
ratio
between the steel and
the
concrete
( EA)/( EA).
Equation
2-6
represents the overall stiffness of
a
bonded
composite
system
and is analogous to a system
of
equivalent springs
connected in parallel, as shown in Figure
2-6.
Linear
elastic
behaviour
is
then
observed until the initial cracking load
Ncr
is
produced at
a
concrete
stress value equal to the direct tensile
Iimit,fi.
Figure
2-6.
Uncracked
reinforced concrete tension element
as
represented by equivalent springs in
parallel.
2.3.2
Post-Cracking
Behaviour of Reinforced
Concrete
An
exact
expression describing post-cracking behaviour is
difficu
determine
as
it
would
involve
many inter-related variables
including:
t
transfer length, member length, number of cracks,
cracking
load,
and
total
applied
load. In lieu of an analytical approach to understanding
pose-
cracking behaviour
in
reinforced concrete tension members, empirical
Chapter
2:
Background
information
expressions for expected
post-cracking
response have been suggested in the
literature.
These
functions
represent the tension stiffening effect in cracked
reinforced concrete and are referred to as the
P,
function, which
can
essentially be viewed as a bond performance factor. Bond performance
can
be evaluated as the
ability
of the concrete to assume load, in between
cracks,
fiom
the steel reinforcing through bond force transfer
(r).
Thus
pc
=
Pc
/
Pcr
=A&),
where
Pcr
is
the maximal
Pc
value occumng at initial
cracking. As shown previously in Figure
2-4b,
the concrete load with
respect to member strain
is
determined as:
Pc
=
N
-
Ps.
Member response
is given in Figure 2-4a and indicates that,
following
initial cracking,
continued load carrying capacity
is
possible until
N
=
Ny.
The
interval
between member cracking and member yielding is characterised by
additional crack development followed by a period of stabilisation,
referring to the fact that, although existing cracks continue to
grow
wider
under
an
increasing axial load, no new cracks will
form.
Figure
2-7
presents the member given in Figure
2-4
in
terms
of the
interna1
force values camed separately by the steel
and
concrete with respect to
member length at
five
different stages of load application.
Diagram
i)
shows the member just
pnor
to initial cracking and, with the exception of
the zones of bond force transfer
(lb),
the forces camed by the steel and
concrete are
taken
to
be
constant. The maximum load
carried
by the
concrete
at
any point
is
Pcr,
but the average load
Pc
is slightly smaller due
to the effect of bond-force transfer at each end.
Diagrams
ii)
through
v)
display
the
development of four cracks
(C
LC4)
as the
result
of
increasing
load
N.
Again,
the maximum
permissible
force within any section of
concrete
is
limited
to the value
Pcr.
As each successive crack
is
created,
bond transfer zones on either
side
of the formation are also produced.
Chapter
2:
Background
Information
i
i
i )
iii)
i v)
v)
Figure
2-7.
Cracking development and distribution of forces
in the concrete
and
steel for
a
typical
tension member
under
increasing load.
This has the effect
of
continuously reducing the average concrete value
(Pc)
in
relation to
Pcr
(Le.
decreasing
P,
value
)
during
crack development
and stabilisation (see also Figure
2-4b).
At the
same
time,
Ps
(average
force
carried
by
the reinforcing steel) must become larger
due
to the
increasing
extemal
load value.
Because
the allowable bond-stress
(r)
is greater in high strength
concrete
than
normal
strength
concrete, the total length
of
bonding (per
unit length)
will
be
smaller.
This may
allow
more
cracks
to
form in
higher strength concretes, but the relative value
Pc
/
Pcr
will
tend
to a
Chapter
3:
Background
Information
constant value based on the
geometric
configuration of the bond stress-
tram
fer curve.
Mitchell and
Collinsll
(with earlier work
done
by Vecchio
and
col lin^)
use
a
collection of data
fiom
various
studies
(having variation in
section, size, and reinforcing ratio) to calculate the concrete contribution to
the post-cracking capacity
of
reinforced tension rnember with respect to the
concrete strain due to applied stress,
scf,
given in Equation
2-7a.
The
expression
can
also be
written
such
that the average concrete tensile stress,
fc,
is normalised to the direct tensile stress limit,
h,
performance factor
pc.
where
a
1
and
a
7
represent bonding and loading
producing the bond
(2-7a)
adjustment factors
respectively and
is
input in strains
( E=
1
x
10-6).
NO
high-strength
specimens were included
in
this
group,
but
it
can
be
seen
that Equation
2-
7a
excludes
any relationship to concrete compressive strength. Previously
Vecchio and
Collins
suggested
an
equation constant of
200,
rather
than
500,
emphasising the variable nature of reinforced concrete in tension.
The Comite
Euro-International
du Beton (CEB-FIP) in
1990
released
the
CEB
Design
Manual:
Cracking
and
Deformations
which included the
expected post cracking behaviour of reinforced
concrete?
In this
manual,
the relative post-cracking strength of concrete,
pc
,
is simply
taken
to be
the constant
This
mode1
represents
a
departwe
from
previous editions of the
manual,
Cbapter
2:
Background
Information
which required
lengthy
concrete
strength
and
deformation calculations.
Belarbi and
Hsu29(1994)
present
a
proportional
mode1
based on
concrete stress or strain following initial cracking. They suggest using an
initial cracking
strain
of
approximately
80
pe
and the following
relationship:
Reinforced concrete in direct tension maintains
load
carrying
capacity
after cracking through transfer of force using the steel reinforcement.
Behaviour
c m
also be expressed in
t ems
of the average mernber response,
including both cracked and
uncracked
sections,
through
bond
efficiency.
In contrast to this, the post-cracking behaviour
of
plain concrete in tension
is
defined
simply
by
the localised effect
of
crack widening. As described
previously,
strain softening data
is
commonly
modified
to
imitate
tension
stiffening behaviour for
use
with some sophisticated
analytical
procedures,
such
as
finite
elements. While strain
softening
and tension stiffening
phenomena are seemingly similar, it must
be
realised that they are in fact
very
different rnechanically. Strain
softening
describes
behaviour
taking
place
at
the crack (local effect), while tension stiffening includes member
response between the cracks (global effect).
2.3.3
Effective Embedrnent
Zone
for
Concrete in
Tension
The
transfer
of
force between concrete and steel occurs at the
interface of the two materials through bond. The stress developed in the
concrete at a section
VC)
is
then
nonuniforrnly
distributed over the net
concrete section area
(A,).
Point stress values in the concrete are
a
function
of the distance to the reinforcing
bar
and maximum values
occur
Chapter
2:
Background
Information
in the
region
adjacent to the steel,
tapering
off
further
away
fiom
the
bar.
The
CEB-FIP28
code
recognises
this fact
and
describes
a zone of
effective concrete which
may
cary
tensile force. As shown in Figure 2-
ga,
the effective zone of tension
(At)
surrounding
each
longitudinal
reinforcing bar is
taken
to be a maximum of
(15%)2,
where
is
the bar
diameter.
The tensile stress camed
in
this effective concrete area
is
assumed
to
be
uniformly
distributed.
If the net concrete area of the section
is
less
than
this
value,
then
the
effective area equals the net area, as shown
in
Figure
2-8b,
or
At
=
A,.
Overlapping zones of tension, resulting fiom adjacent longitudinal
bars,
are effectively shared
between
the
two.
irea
of
Concrete
,,,:,,.
Tension
-
in
i
w
ISIUI
1.
At
Specimen
!
Effectiv
Concrete
.
Steel
'
-
-
,:
\-
Reinforcing
Bar
@b
'-
Reinforcing
Bar
a b
I
!
a)
Effective area
b)
Net
area
larger
than
larger
than
net
area.
effective
area.
Figure
2-8.
Zone of
effective
embedment
for
reinforced
concrete in
tension.
2.3.4
Crack
Development
and Concrete
Strength
Cracks
form
in concrete
perpendicular
to the
axis
of load application
when tensile stress exceeds tensile
strength.
To study crack development,
Gotoi*
tested a
series
of
notched reinforced
prisms
loaded in tension and
Chapter
3:
Background
Information
reinforced with either
19
or
32
mm
diameter
deformed bar. At the
conclusion
of
the tests, the
pnsms
were bisected
longitudinally,
revealing
the
interna1
crack patterns
shown
in Figure
2-9.
It
was found that
two
types of cracks had formed,
both
of which were
present at the surface of the steel bars.
Primary
cracks were
found
to be
continuous through the section,
creating
two
halves of concrete connected
by
the steel.
These
formations
resulted
frorn
a
principal tensile stress
approaching the tensile
strength
(Le.
Pc
=
Pcr).
Secondary cracks formed
as
a
result
of
cmshing
or bearing failure
at
the
bond interface of the
steel
rib
and
the adjacent concrete, but
did
not
travel
to
the surface except at
high
stress
levels.
!
,.
lnternally
Cracked Concrete
I
Force components
1
/
,--
Uncracked
Concrete
,/-
on
concrete
and steel
Teightening force on bar.
(Due
to
wedge action
and deformation
of
teeth
of
corn
b-like concrete).
-
Crack
l,
',
1,
'-
Deformed Bar
,
(with
lateral
lugs)
\
Prirnary
Crack
Figure
2-9.
Typical primary and secondary cracks
displayed
in longitudinal
cross sections of reinforced concrete
axial
tension members.
Crack spacing in reinforced concrete
varies
but the expected average
spacing
between
cracks,
sm,
can
be
determined using
empirical
formulae
Chapter 2:
Background
Idormation
supported
by
Arnerican
and
European
specifications.
The
CEB-FIP
code
provides
a
value for average crack spacing
as28
where
dc
is the clear cover of the longitudinal reinforcing bars,
sb
is
the
maximum spacing between adjacent longitudinal reinforcing bars (but not
to exceed 15-bar diameters),
kr
and
k2
are bond and strain gradient
adjustment coefficients, is the bar diameter, and
p
is
the net
steel-to-
concrete cross-sectional area ratio. Transverse
reinforcement
may also
affect the nature of the longitudinal crack pattern developed in a tensile
member. Mitchell and
Collinsl2,
however, suggest a simple
estimate
of the
average crack
spacing
as
s m=3
dc
The minimum and maximum crack spacing values are expected to range
between
2
and
4
cic
and the expected average crack width,
wm,
is
then
based
on either
Sm
value
Wm
=Ecf
Sm
(2-9)
where
is
the
average strain in the concrete due to applied stress, or
Pc
divided
by
Ac
Ec.
Further, the characteristic crack width,
wk,
is
defined
as
a width that
is
exceeded
by
only
5%
of cracks
and
given
as the value
1
.7wm
by
the CEB-FIP code.
The American Concrete
Institute
(ACI)
bases the maximum crack
width
(wmax)
calculations on the
Gergely-Lutz
equation31
comprising
three
member variables, including: steel stress at the crack location, concrete
cover,
and
the area of concrete in tension
surrounding
each bar. The
expression is
written
Cbapter
2:
Background
information
(2-
1
O)
where
p
is
a
strain gradient coefficient (1.0 for
uniform
strain),
Es-cr
is
the
strain in the steel at the crack location
(N
/
EsAs),
dc
is the rebar cover to
the centroid of the reinforcing bar
group,
and
At
is the effective area of
concrete in tension divided
by
the total number of reinforcing bars
(At
=
2dcb
for flexure).
Concrete strength is