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STRUCTURAL BEHAVIOR OF FIBER
REINFORCED
MORTAR
RELATED
'IO
MATERIAL
FRACTURE
RESISTANCE
by
ROBERT
JAMES
WARD
B.E.
University
College
(1986)
Galway, Ireland
SUBMITTED
IN PARTIAL
FULFILLMENT
OF
THE REQUIREMENTS FOR THE DEGREE OF
MASTER
OF
SCIENCE
IN
CIVIL ENGINEERING
at
the
MASSACHUSETTS
INSTITUTE
June
1989
OF TECHNOLOGY
Massachusetts
Institute
of
Technology
1989
All
rights
reserved
Signature
of Author
Certified
by
Accepted
by
-------
Ch
Signature
redacted
Department
of Civil
Engineering
Signature redacted
Victor
C. Li
Associate
Professor
of Civil
Engineering
Thesis
Supervisor
Signature
redacted
J
Ole
S.
Madsen
airman,
Departmental
Committee
on Graduate
Students
Department
of Civil
Engineering
NSS.
JIST.
T d
JU '89
C
ARCHIVES
0
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STRUCTURAL BEHAVIOR
OF
FIBER REINFORCED
MORTAR
RELATED TO
MATERIAL FRACTURE
RESISTANCE
by
ROBERT
JAMES
WARD
Submitted to the Department of
Civil Engineering
on May 11, 1989, in partial fulfillment of the
requirements for the Degree
of Master
of
Science
in
Civil Engineering
ABSTRACT
An indirect J-integral
technique
for measuring
the tension
softening
curve of non-yielding
materials like
concrete
is
presented and is
applied
to
mortars
reinforced
with
various short fiber
types
(steel
and
synthetic). The tension
softening
curve serves
to
characterize and
measure material fracture
resistance. The dependence
of
structural
behavior
on
material
fracture
resistance is
investigated
through
third-point
loading tests on unreinforced fiber
mortar beams
which fail
in flexure and through
center-point loading tests on
longitudinally
reinforced
fiber
mortar
beams
without shear stirrups which
fail
in
shear.
Conclusive
evidence,
relating improved structural
performance directly
to
improvements in material
fracture
resistance
due to fiber reinforcement,
is
presented.
Semi-empirical
formulae are
proposed which
relate
flexural toughness
indices
directly to parameters
involving
just
the flexural strength,
the
splitting tensile strength and
the fiber
length. Also,
semi-empirical
design
formulae
are
proposed
which
relate
the
ultimate shear strength
of
longitudinally
reinforced beams
with
fibers as
shear reinforcement,
to
the
material fracture resistance
(represented by a
combination
of
flexural and
splitting tensile
strengths), the
shear span/effective
depth
ratio,
the
longitudinal
reinforcement ratio
and the
beam
depth. All
proposed formulae are suitable for
the purposes of design and quality
control.
Thesis Supervisor: Dr.
Victor
C. Li
Title:
Associate Professor of Civil Engineering
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A
I'DG ENT
I wish to
thank the
following
people:
Professor Victor
Li
for
introducing
me
to
the
subjects of fracture
mechanics
and
fiber
reinforced
concrete,
and for
overseeing
my
entire
research
program.
He
provided
me with
invaluable
guidance,
support
and
encouragement
and
above
all I
thank
him for
making
my
stay at M.I.T.
a
truly enjoyable
experience.
My
fellow graduate
students
who
so
often
offered
their
advice
and
help,
especially
in the
laboratory.
Professor
Stanley
Backer
who
always
ensured
that I
had
a large
and varied
supply
of synthetic
fibers
with
which
to work.
Mr.
Rick
Smith of
Ribbon
Technology
Corporation,
Ohio,
who
provided
me
with
all
the steel
fibers.
Mr.
Kevin
Grogan of
W.
R.
Grace and
Co., Cambridge,
Mass.,
who
provided
me
with
the
superplasticiser.
The
U.S.
National
Science
Foundation
and
the Shimizu
Corporation
of
Japan
who
both
provided
vital
funding
which
made
this
research
program
possible.
Miss
Irene Uesato,
who
took
time
out
of her
very
busy
schedule,
to
type
this
thesis.
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TABLE
OF
CONTENTS
Title Page
Abstract
Acknowledgements
Table
of
Contents
List
of
Figures
List of
Tables
Chapter
1:
INTRODUCTION
Chapter
2: TENSION
SOFTENING
CURVE
BY
J-INTEGRAL
TECHNIQUE
2.1
Introduction
2.2
Theoretical
Basis
of
J-Integral
Technique
2.3
Numerical
Verification
of
Test
Technique
2.4
Experimental
Procedure
2.4.1
Specimen
preparation
2.4.2
Testing
procedure
2.5
Results
and
Data
Analysis
2.6
Calculation
of
GF
2.7
Discussion
and
Recommendations
for
the
J-Integral
Test.
2.8
Current
Status
of
J-Integral
Test
Technique
Chapter
3:
FLEXURAL
BEHAVIOR
OF
FIBER
REINFORCED
MORTAR
3.1
Introduction
3.2
General
Behavior
of
Fiber
Concrete
in
Flexure
3.3
Experimental
Program
3.3.1
Specimen
preparation
3.3.2
Testing
procedure
3.4
Compressive
Strength
3.5
Splitting
Tensile
Strength
3.6
Flexural
Strength
3.6.1
Size
dependence
of
flexural
strength
3.6.2
Flexural
strength
related
to
fracture
resistance
3.7
Flexural
Load-Deflection
Curves
4
1
2
3
4
6
10
11
17
17
18
21
22
22
24
25
27
28
30
40
40
42
46
46
47
48
49
50
51
52
57
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3.7.1
Effect
of
fiber
type
and
volume
fraction
on
the
flexural
load-deflection
curve
57
3.7.2
Effect
of beam
size
on the
flexural
load-deflection
curve
60
3.8 Flexural
Toughness
Indices
62
3.8.1
15'
10'
3
and
150
indices
62
3.8.2
Tmaxp
T
50
and T
10
indices
65
3.9
Simple
Flexural
Toughness
Estimates
68
Appendix
3.1
Comparison
of Size-Effect
Predicted
by
the
Weibull
and
Fictitious
Crack Models
73
Chapter
4: FIBERS
AS SHEAR REINFORCEMENT
IN
LONGITUDINALLY
REINFORCED
BEAMS
102
4.1
Introduction
102
4.2
Advantages
of
Fibers
as
Shear
Reinforcement
104
4.3
Experimental
Program
107
4.3.1
Specimen
preparation
107
4.3.2 Testing
procedure
110
4.4 Observed
Failure
Modes
111
4.4.1
Beam-action (a/d
> 2.5)
111
4.4.2
Arch-action
(a/d
< 2.5)
113
4.5 Test
Results
and
Discussion
115
4.5.1
First crack
strength
115
4.5.2
Shear
span/effective
depth
ratio
118
4.5.3
Reinforcement
ratio
121
4.5.4
Beam depth
122
4.6 Simple
Design
Formulae
for
the
Shear Strength
of
Reinforced
Mortar
Beams
with
Fibers
124
Chapter
5: CONCLUSION
144
Chapter
6:
RECOMMENDED
FUTURE
RESEARCH
147
References
154
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List of
Figures
Fig.
2.1
Cohesive
Zone
Ahead of
the
Crack
Tip
Fig.
2.2
Specimen Configurations
used
with J-Integral
Test
Fig.
2.3
Inside
of
Omni-mixer
Fig.
2.4
Specimens
covered
in
Plastic
just
after
Casting
Fig. 2.5
Notched
Beam
ready
for
Testing
Fig.
2.6
Loading
Machine
Used for
J-Integral
Test
Program
Fig.
2.7
Average
Load
versus
Load
Point
Displacement
Curves
Fig.
2.8
Average
Load
Point
Displacement
versus
Crack
Opening
Curves
Fig.
2.9
J-Integral
versus
Crack
Opening
Curve
Fig.
2.10 Deduced
Tension
Softening
Curve
Fig.
2.11
Comparison Between Deduced
and
Directly Measured Tension
Softening
Curves
Fig.
2.12
Flexural
Load
Deflection
Curve
Corrected
to Account
for
Energy
Supplied
by Beam
Self-Weight
Fig.
2.13
Tension
Softening
Curves
Deduced
by
Indirect
J-Integral
Technique
Fig.
3.1
Specimen
Geometry
and
Loading
Configuration
for
Flexural
Test
Fig.
3.2
Wooden
Mold for
228
mm Deep
Beam
Fig.
3.3
Beams
Covered with Plastic
Just
After Casting
Fig.
3.4
Cylinders
in
Plastic
Molds
and
Covered
with
Plastic
Just
After
Casting
Fig.
3.5
228
mm
Deep
Beam
ready for
Testing
Under
Third
Point
Loading
Fig.
3.6
Cylinder
with
LVDTs
on
either
side
ready
for
Compression
Test
Fig.
3.7
Cylinder
ready for
Splitting
Tension
Test
Fig.
3.8
Effect
of
Fiber
Reinforcement
on
Compressive
Strength
Fig.
3.9
Effect
of
Fiber
Reinforcement
on the
First
Crack
Splitting
Tensile
Strength
Fig.
3.10
Influence
of
Beam
Size on
Flexural
Strength
Fig.
3.11
Average
Normalized
Flexural
Strength
Values
as
a
Function
of
Beam
Depth
Fig. 3.12
Theoretical
Curves
Relating
the
ff/ft
Ratio
to
the d/lch
Ratio
Calculated
Using
the
Fictitious
Crack
Model
[101
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Fig. 3.13 Empirical Relationship
Between
the Flexural/Tensile
Strength
Ratio
and
the Compressive
Strength for Plain Concrete [401
Fig. 3.14 Empirical Relationship Between the
Flexural/Tensile
Strength
Ratio and
the
Material
Characteristic
Length
for
Plain
Concrete
[10]
Fig.
3.15 Effect of
Fiber
Reinforcement on the Ratio Between Flexural
and Splitting Tensile
Strengths
Fig. 3.16
Typical Tension Softening Curves
of Mortars
Reinforced
with
various
Fibers
Fig. 3.17
Flexural Load-Deflection
Curves for Mortars
Reinforced with
various Fiber Types
Fig. 3.18 Flexural Load-Deflection Curves
for
Kevlar Fiber
Reinforced
Mortar Beams
Fig.
3.19
Flexural Load-Deflection Curves
for Steel
Fiber (25 mm)
Reinforced
Mortar
Beams
Fig. 3.20
Flexural Stress versus Normalized Deflection
for
Different
Beam Sizes
of
Kevlar Fiber
Reinforced Mortar
Fig. 3.21 Flexural Stress versus Normalized Deflection
for
Different
Beam Sizes of Steel Fiber Reinforced Mortar
Fig. 3.22
Calculation
of Flexural
Toughness
Indices from
Flexural
Load-Deflection
Diagram
Fig.
3.23
Influence
of
Beam
Size
on
Flexural
Toughness
Indices
5
I10'
130 and
150
Fig.
3.24
Influence
of
Beam
size
on
Flexural
Toughness
Index
Tmax
Fig.
3.25
Influence
of
Beam
Size
on
Flexural
Toughness
Index
T
5m
Fig.
3.26
Influence
of
Beam
Size
on
Flexural
Toughness
Index
T10
Fig.
3.27
Various Flexural Toughness
Indices
for
Mortars Reinforced
with each Fiber Type
Fig.
3.28 Semi-Empirical
Relationship
Between
the
f /ft Ratio
and
the
Flexural
Toughness
Index
Tmax
Fig.
3.29 Semi-Empirical
Relationship Between the
Flexural Toughness
Index
T
10
and a
Parameter
involving
the Flexural and
Tensile
Strengths
and the
Fiber Length
Fig.
3.30
Theoretical
Curves
Relating
the
f
/ft
Ratio
to
the
d/lch
Ratio Calculated Using the
Fictitious Crack
Model
[38]
Fig.
3.31 Comparison of
Size
Dependence of Flexural
Strength
Predicted
by
the Weibull
and Fictitious Crack
Models
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Fig. 4.1
Specimen Geometry
and
Loading
Configuration for
Shear
Beam
Test
Fig. 4.2 Rebars
Fixed at
Proper Spacing
Using Short
Steel
Bars
Fig.
4.3 Rebars
Fixed
in
Mold
Ready
for
Casting
Fig.
4.4
Shear
Beam Ready for
Center
Point Loading
Fig.
4.5
Loading
System for
Large
Shear
Beams.
Bottom
Beam is just a
Support
Fig.
4.6
(a) Typical
Crack
Shape in
Plain Mortar
Beam with
a/d > 2.5
(b)
Typical
Critical
Crack
Shape
in
Fiber Reinforced
Beam
with a/d
> 2.5
Fig. 4.7
Force
System in
Reinforced Concrete
Beam at
a
Diagonal
Shear
Crack
Fig. 4.8
Failure Patterns for
Shear
Beams
with
a/d
=
3.0
and
d =
204
mm
(a)
=
Plain
Mortar
(c)
Kevlar
2%
(b) =
25
mm Steel 1% (d)
25 mm
Steel
2%
Fig.
4.9 Shear
Failure Patterns
in
Beams with a/d =
3.0, d = 204 mm
and 25 mm
Steel
Fiber Reinforcement
2%
Fig.
4.10
Typical
Shear Failure
Patterns
in
Beams with
a/d
<
2.5
(a) Splitting Failure
(b)
Shear
Compression
Failure
(c) Flexural
Tension Failure
Under Eccentric
Compression
Fig.
4.11
Typical Splitting Tension
Like Failures
of
Plain Mortar
Beams with
a Reinforcement
Ratio
of 2.2%
and
Loaded with
a/d
= 1.0
[18]
Fig.
4.12
Typical
Shear Compression
Like
Failures of
Beams Reinforced
with
2%
Acrylic Fibers.
The Reinforcement
Ratio
is
2.2% and
a/d
=
1.0
[18].
Fig.
4.13 Cracking
and Ultimate
Shear
Strengths of
various
Fiber
Reinforced
Mortar
Beams
Fig.
4.14 Influence
of the
Shear
Span/Effective
Depth
Ratio on
Ultimate
Shear
Strength
of
Fiber
Reinforced Mortar
Beams
Fig.
4.15
Influence
of
the
Shear
Span/Effective
Depth Ratio on
the
Maximum
Bending
Moment in
Fiber
Reinforced
Mortar
Beams with
Longitudinal
Steel
Fig.
4.16
Influence
of the
Longitudinal
Reinforcement
Ratio
on
the
Ultimate
Shear
Strength
of
Fiber
Reinforced
Mortar
Beams
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Fig.
4.17
Influence
of
Beam
Depth
on the
Ultimate
Shear
Strength
of
Fiber
Reinforced
Mortar
Beams
Fig.
4.18
Semi-Empirical
Relationship
Between
Ultimate
Shear
Strength
and the
Material
and
Geometrical
Properties
for
a/d
>
2.5
Fig.
4.19
Semi-Empirical Relationship Between Ultimate
Shear
Strength
and
the
Material
and
Geometrical
Properties
for
a/d < 2.5
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List of
Tables
2.1 Summary of J-Integral Test Results
3.1
Fiber
Properties
3.2
Flexural,
Splitting Tensile
and
Compressive Strengths
of
Mortars Reinforced
with various Fiber
Types
3.3 Flexural Toughness Indices of Mortars Reinforced
with
various
Fiber
Types
3.4 Size Dependence
of
Flexural Strength Predicted
by the
Weibull
and
Fictitious
Crack Models
4.1
First Crack and
Ultimate
Shear
Strengths
4.2
Flexural,
Splitting
Tensile
and Compressive
Strengths
10
Table
Table
Table
Table
Table
Table
Table
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1 INTRODUCTION
Concrete
is used as
a structural
material
in
vast
quantities
worldwide.
Due to its low tensile strength
and
susceptibility
to cracking
designers
have conventionally just considered
its compressive strength and
have used steel
reinforcement to resist tensile
stresses in
structures.
However,
in
recent years
the
inadequacy
of just
considering
compressive
strength has
been
realised. Reinforced
concrete structures
usually crack
before the full service load is applied
due
to
the
low
tensile strain
capacity
of the concrete.
These
cracks
allow penetration
of
chloride
ions
and
other
environmental
agents
into
the reinforcing
bars,
causing
corrosion.
Subsequent expansion
of the
reinforcing bar
is followed by
cracking and spalling
of the concrete cover. Shrinkage and thermal
stresses
can lead to
significant cracking even before any load is applied
to
a structure. Anchorage failure
of
reinforcing
bars
is due
to
cracks
propagating
along
the
steel-concrete interface.
Prestressed
concrete beams
exhibit web cracking
and
surface
spalling
due to transverse tensile
stresses. Reinforced
concrete
beams
crack
diagonally under principal
tensile
stresses
caused
by combined
moment and shear. Even
a concrete
cylinder in uniaxial
compression can
fail
due to local
transverse
tensile
forces.
The
use
of
low volume
fractions short fibers
to reinforce plain
concrete matrices
has led
to
only small changes
in
pre-peak stress-strain
behavior but
very significant
changes in the composite's
resistance to
crack propagation
under tensile
forces.
The development of
high strength
concrete has produced
a material
which has
very
high
compressive
strength
but
which
is perceived
as being
even more brittle
than
ordinary concrete
due
to a less than
proportional
increase in its crack
resistance.
All the
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above
phenomena
illustrate
the
important
role
which
tensile
properties
may
play
in
determining
the
structural
performance
of
concrete
in
numerous
applications.
This
realization
has
led
to
dramatically
increased
research
in
this field
in
recent
years
(eg.
NATO
Advanced
Research
Workshop
on
Application
of
Fracture
Mechanics
to
Cementitious
Composites,
Evanston,
1984;
International
Conference
on
Fracture
of
Concrete
and
Rock
sponsored
by
RILEM
&
SEM,
Houston,
U.S.A.,
1987;
American
Concrete
Institute
Symposium
on
Applications
of
Fracture
Mechanics
to
Concrete
Structures,
Seattle,
U.S.A.,
1987;
MRS
International
Meeting
on
Advanced
Materials:
Symposium
on
Fracture
of
Rock
and
Concrete,
Tokyo,
Japan,
1988;
International
Conference
on Fracture
and
Damage
of
Concrete
and
Rock,
Vienna,
Austria,
1988;
International
Workshop
on Fracture
Toughness
and
Fracture
Energy
--
Test
Methods
for
Concrete
and
Rock,
Sendai,
Japan,
1988;
International
Workshop
on
Applications
of
Fracture
Mechanics
to
Concrete
Structures,
Lulea
University
of
Technology,
Lulea,
Sweden,
June
28-30
1989;
International
Conference
on
Recent
Developments
in
Fiber
Reinforced
Cements
and
Concretes
and
International
Conference
on
Recent
Developments
in
the
Fracture
of
Concrete
and
Rock,
both
Cardiff,
Wales,
1989).
Since
concrete
is
a
non-yielding
brittle
material
which
fractures
by
crack
propagation,
many
researchers
have
adopted
a fracture
mechanics
approach
to the
study
of
its
tensile
behavior.
A
general
consensus
has
been
reached
that
whilst
linear
elastic
fracture
mechanics
is not
directly
applicable
to
concrete
structures
due
to
the
relatively
large
non-elastic
zone
in
front of
a growing crack
[1],
it is
possible
to
use
a non-linear
fracture
mechanics
theory
which
accounts
for
the
growth
of
a
fracture
process
zone
ahead
of
the
crack
tip
[1,2,3,4,5].
Carpinteri
[6]
related
such
structural
phenomena
as
the
size-effect
and
the
transition
from
brittle
to ductile
behavior
to an
energy
brittleness
number
sE
defined
by:
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(1.1)
sE
= GF
ft
d
where
GF
is the
fracture
energy,
f t
is the
tensile
strength
and d
is the
beam
depth.
Hillerborg
[1]
defined
the
material characteristic length
1
ch
as:
1ch
= GF
E
/ f
t
(1.2)
where
E
is the
modulus
of
elasticity,
for
similar
purposes.
Li and
Liang
[4]
related
the
process
zone
size
to
the
characteristic
length.
They
showed
the
importance
of
1
ch in
determining
the
transition
from
brittle
to
ductile
behavior
and
the
limitation
of
LEFM
(linear
elastic
fracture
mechanics)
and
strength
concepts
in
their
application
to
grain
brittle
materials
such as
concrete
and
FRC
(fiber
reinforced
concrete).
However
two
major
problems
blocking
the
widespread
use
of
fracture
mechanics
ideas
in
concrete
design
codes
is
first
the
lack
of
a standard
test
method
to
characterize
and
measure
fracture
resistance
and
secondly
the
lack
of
adequate
experimental
data
relating
changes
in structural
behavior
directly
to
changes
in
material
fracture
resistance.
It
is
the
purpose
of this
thesis
to
address
both
of
these
problems.
Concrete
structural
members,
unlike
their
counterparts
in
steel,
are
more
often
than
not
made in
situ,
and
their
quality
is
almost
exclusively
dependent
on the
workmanship
of
concrete
making
and
placing
and
on
the
curing conditions which
exist
while
the
concrete gradually
gains
strength.
The
exact
influence
of
each
of
these
factors
on
the
quality
of
the
final
product
is
usually
not
very
well
understood.
Also,
in
many
cases,
when
we
measure
some
property
of
concrete
we
find
it
very
difficult
to
determine
exactly
how
this
property
affects
the
behavior
of
concrete
in
various
structural
applications.
For
these
reasons
much
of
the
concrete
design
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code
is
based
purely
on
empirical
data.
We
make
a
specimen
and test
it and
we
hope that
if
we build
a
somewhat
similar
structure
in the
field
its
behavior
under
load
will
not
differ
greatly.
The
availability
of
a
huge
base of
experimental
data
together
with
the
use
of
conservative
safety
factors
allows
us
to
design.
However
modern
trends
towards
structures
which
are
novel
and
more
complicated
and
also
subjected
to
more
severe
loading
conditions
and
environments
dictate
the
need
for
a
design
code
which
has
a
greater
basis
in
understanding
how
and
why
concrete
structures
behave
as
they
do
and
how
individual
material
characteristics
influence
structural
behavior.
Concrete structural members
are
increasing
in
size due
to
advances
in
materials
and
improvements
in
design
and
construction
techniques
[7].
It
is
well
known
that
larger
size
structures
fail
in
a more
brittle
and
'fracture'
like
mode.
Presently
the
ACI
code
does
not
account
for
size-effect
in
the
ultimate
shear
strength
of
longitudinally
reinforced
concrete
beams.
This
may mean
that
the
code
is
unconservative
for
large
beam
sizes.
Another
trend
in
the
construction
industry
is
the
gradual
adoption
of
high
strength
concrete
as
a
construction
material.
LeMessurier
[8]
suggested
that
very
tall
buildings
(say,
height
=
800
m)
could
be
economically
constructed
by
transferring
all
the
loads
to
very
large
reinforced
concrete
columns
around
the
perimeter
of
the
building.
Sidesway
due
to
wind
loading
governs
the
design
of
such
tall
buildings
and this
would
be
inversely
proportional
to
the
concrete
modulus.
Thus
high
strength concrete with
high
modulus
is
ideally
suited for
this
application.
However,
there
is
relatively
little
experimental
data
available
relating
to
behavior
in
large
structures
of
a material
which
is probably
much
more
brittle
than
plain
concrete.
The
development
of
fiber
reinforced
concrete
over
the
past
two
decades
or
so
has
ignited
the
dream
of
a non-brittle
cement
based
material.
It
has
also
presented
a
great
challange
to
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researchers
and designers
alike. Due
to
the huge variety
of fibers
available,
steel
and synthetic,
long
and short, straight,
crimped,
hooked,
deformed,
etc., and the option
of using various volume fractions
up to
about
2 ,
or of
combining different
fiber types
in
a single
mix,
a
whole
new dimension
of
cement based
materials
have been
created. The
possibility
of
overcoming
the extra brittleness
of high strength concrete
due to
aggregate fracture
by providing
stiff
fibers which slip
out of
the matrix
may allow us to use
similar design rules for high strength
concrete as
ordinary strength concrete
if we
first agree
to reduce
the
brittleness
with
some fiber
reinforcement.
It will
not be possible to
have
experimental data for all possible
material and structural variations
which we will want to
use.
We
do not
want
our ability to creatively
use new
materials and
new
structural
forms
to be
drastically limited
by our lack of
understanding
of
basic
structural
behavior. Fracture
Mechanics ideas
are very
useful for
explaining
certain
trends in
the
structural
behavior
of
a
brittle material such as
concrete.
It predicts the
size-effect
in
structural
strength. It can
also
explain
why
the
strength
of
members,
made
with
high strength concrete,
in which
the
concrete
is subjected to
tensile stresses such as the diagonal tensile
stresses
in
a reinforced concrete beam,
may increase at
a lesser rate
than
the compressive strength. Hawkins [9] argued in favor of
fracture
mechanics
playing
an
important role in
a revised reinforced concrete design
code.
Fracture mechanics
ideas
may allow
us
to understand the behavior
of
a wide
range of new
materials
and
structural
forms
with
only
a
limited
number
of
experimental
tests.
Economic considerations
often play a powerful role
in shaping our
design
procedure. Designers
will
not use
a
complicated
design
procedure
which can
save the client
say
1%
due
to reduced material
costs
but adds 2%
to
his
costs
through greater
design effort.
Many designers
and concrete
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producers do
not believe
that the
advantages
of
considering
a fracture
resistance
material
property
outweigh the disadvantages
associated
with
more troublesome design
procedures.
At the outset it was the hope of
the
work
presented
in
this
thesis
to find
a
way
of
producing conclusive
evidence
of a link between material fracture
resistance and structural
performance
and also to develop semi-empirical
design formulae
which
have
a
rational
basis
in material
properties and at the
same
time
represent
experimental data reasonably well.
It was hoped
that
these formulae could
take
account of
material
fracture resistance
in
a simple and
efficient
manner which
would be
acceptable
to both researchers, designers and
builders alike.
Initially
the
tension
softening curve
is presented as
a
material
property
which characterizes fracture
resistance.
A
relatively simple
experimental procedure
is then proposed
to measure
this
curve.
Using
this
procedure the
tension softening
curves
of
mortars
reinforced with
various
fibers
is measured. In
chapters 3
and
4
the
fracture resistance
is
manipulated
through fiber reinforcement
and
the
effects
on
structural
performance
are
examined.
Tests
on unreinforced
beams failing in
flexure
and longitudinally
reinforced
beams
failing
due to diagonal
tension cracks
establish
conclusive
experimental
evidence
of
the
link between
greater
fracture
resistance
and better
structural
performance.
Simple
semi-empirical
formulae
are developed
which relate fracture
resistance
to
structural
behavior
in a manner
which
is suitable for the
purposes
of
design
and
quality
control.
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2
TENSION
SOFTENING
CURVE
BY
J-INTEGRAL TECHNIQUE
2.1
Introduction
The
fracture
toughness
KIc,
used in linear elastic
fracture mechanics,
cannot
be
used
for
analysis of
most
concrete
structures. This is because
of
the size-dependence
of
toughness values measured using
laboratory
size
specimens, caused by the
large
process
zone
size in front
of the
crack tip
relative
to
the
specimen dimensions. Hillerborg
[1] showed that 3-point
bend concrete specimens would
need to
be
at least
1
- 2 m in depth to
give
a valid
KIc measure. Thus
it
is
necessary
to
look outside
LEFM
to
obtain a
fracture
resistance
parameter
which
is
a
true
material property. To
do
this
it is necessary
to
appreciate
the physical processes
which lead to
fracture in
a non-yielding
material like
concrete.
When
loaded
in
tension
the
material
follows a stress-strain curve
which is
almost
linear
initially and then
becomes increasingly
non-linear
due to
distributed microcracking
throughout
the
material volume. When the
maximum
load
is
reached cracking
concentrates on
a
narrow
fracture zone
across the
material leading
to the development of
a macrocrack. If the
specimen
is
loaded
in a displacement
controlled
machine
the load gradually
drops
as the
macrocrack
width increases.
Energy is absorbed
across
the
fracture
plane by
microcracking in
the cement
paste and the
cement-aggregate
interface
as
well as by
fiber
pull-out
or
fiber breakage
where
applicable. The
constitutive relationship
between the
tensile
stress
transferred
across
a
crack plane and the
separation
distance of
the
crack
faces
is a material property
and
is
generally
referred to
as the
tension
softening
(a-6)
curve. A
number
of material
parameters
may be calculated
from this
curve.
The
maximum
stress
is
the
tensile
strength,
the
crack
separation
at
which
the stress
transferred
drops
to
zero
is
the
critical
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crack
separation
Sc,
and
the
total
area
under
the
curve
represents
the
energy
required
to
form
a
unit
area
traction
free
crack
and
is called
the
fracture
energy.
The a-6
relationship
is
a basic
material
property
which
characterizes fracture resistance
in
a
material
which
exhibits
tension
softening
behavior
and
can
be
used
for
numerical
simulations
of
crack
formation
and
propagation
in
structures
constructed
from
such
materials
[1,4].
The
a-S
curve
can
be
obtained
from
a displacement
controlled
direct
tension
test.
However
an
inherent
difficulty
with
this
test
is the
stability
of
loading
the
specimen
during
the
softening
process.
Usually
a
very
stiff
testing
machine,
which
is
not
available
in
most laboratories,
is
required
to obtain
a
completely
stable
load-deformation
curve.
Successful
tests
have
been
performed,
using
mechanisms
such
as
parallel
steel
bars
in
the
direction
of
loading
and
closed
loop
feedback
systems,
by
Petersson
[10],
Gopalaratnam
and
Shah
[11]
and
Reinhardt
[5].
However
it
is unlikely
that
this
will
become
a standard
test
method
due
to the
need for
these
intricate
modifications
of
the
loading
machine.
This
chapter
focuses
on
an
indirect
J-integral
technique,
first
proposed
by
Li [12],
to experimentally
determine
the
a-S
curve.
This
test
procedure
is
simple
and
requires
only
a simple
testing
machine.
It
is
believed
that
any
standard
test
procedure
must
possess
these
characteristics.
2.2
Theoretical
Basis
of
J-Integral Technique
The
path
independent
J-integral
is
defined
as:
j
r (Wdy
-
T @u/ax
ds)
2.1)
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where
r
is
a
curve
surrounding
the
notch
tip,
W
is the
strain
energy
density,
T is
the
traction
vector
in the
direction
of
the
outward
normal
along
r
u
is the
displacement
vector
and
ds
is
an arc
along
r.
From
Eq.
(2.1)
Rice
[13,14]
produced
two
alternative
definitions
of
J.
He
used
the
Barrenblatt
approach
which
considers
a cohesive
zone
ahead
of
the
crack
tip
in
which
the
restraining
stress
a(S)
is
viewed
as
a
function
of
separation
S.
If
the
J-integral
is
evaluated
along
a contour
rl,
shown
in
Fig.
2.1,
which
runs
along
beside
the
cohesive
zone,
then
we get:
J
= f
a(S)
(dS/dx)dx
(2.2)
cohesive
zone
This
definition
may
be
interpreted
as
follows.
If
the
crack
opening
at
each
point
in the
cohesive
zone
increases
by
an
amount
dS
then
the
profile
of
the
cohesive
zone
boundary
extends
a
distance
dx.
The
quantity
a(S)dx
is
the
force
over
each
infinitesimal
area
and
a(S)
dx
dS
is the
energy
absorbed
during
increased
separation
dS.
Thus
Eq. (2.2)
defines
J
as
the
rate
of
energy
absorption
with
respect
to
cohesive
zone
propagation.
Eq.
(2.2)
may
also be
expressed
as:
J
=
fS
a(S)
dS
(2.3)
0
where
St
is
the
separation
distance
at
the crack
tip.
When
St reaches
c
the
real
crack
propatates
and
a
critical
J-integral
value,
J
is reached:
j
=
f
cr(8)
dS
(2.4)
S0
Jc
is
equal
to
the
total
area
under
the
a-8
curve
and
may
be
interpreted
as
the
rate
of
energy
absorption
in
the
cohesive
zone
with
respect
to
crack
tip
propagation.
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The
second
interpretation
of
J
may be
given
as:
J =
- [a(PE)/aa]
(2.5)
where
PE
is the
potential
energy
of
a body
with crack
length
a.
Thus
J
is
equal
to the
rate
at
which
the
potential
energy
of
a
cracked specimen
decreases
as the
crack
propagates.
The
basis of
the
indirect
J-integral
technique
of
finding
the
a-S
relationship
is
to find
J
experimentally
using
Eq.
(2.5)
and
then
to
substitute
into Eq.
(2.3)
and
find
a(S).
Potential
energy
may be
calculated simply
from
a load-displacement
curve. However, since
the
crack
tip
position
is difficult
to
locate
accurately,
it would
be
almost
impossible
to
directly
evaluate
Eq.
(2.5)
by
propagating
a
crack
in
a
single
specimen.
One
approximate
procedure
for
getting
around
this
problem
is
to
use two
cracked
specimens
identical
in every
respect
except
that
there
is
a
slight
difference
in
their
initial
crack
lengths.
If
the
load-load
point
displacement
(P-a)
curves
are
measured
for
each
specimen,
then
the
area
A(A)
between
the two
curves
up
to a load
point
displacement
A
represents
the
difference
in
energy
and Eq.
(2.5)
may be
interpreted
as:
J(A)
= A(A)
/
bia
(2.6)
where
Aa
is
the
difference
in
crack
lengths
and
b
is the
ligament
width.
If,
during
the experiment,
the
crack
tip
separation
8,
is
also
measured
then
it
is possible
using
the
A-8
relationship
to convert
J(A)
to
J(8).
Differentiation
of
Eq.
(2.3)
then
gives:
a 8)=
J(8)/98
2.7)
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and
the tension
softening
curve
may
be
determined from
the slope
of
the
J(8)
curve.
2.3
Numerical
Verification
of Test Technique.
This
method has
been
verified numerically
for
both
beam bending and
compact tension configurations.
A. Hillerborg (private communications,
1985) provided verification by employing his
fictitious crack
model in a
finite element
scheme to simulate
the load-load
point
displacement curves
and load-crack tip separation curves
of
a pair
of
three-point
bend
specimens
of
slightly different
crack lengths. He used an
artificial
bi-linear curve
as input for the
tension
softening behavior
in the
material
ahead
of
the crack
tips.
The objective
of
the
exercise was to
extract
the
same curve using
the
indirect
J-integral
technique
with his numerically
derived
test
results. The extracted
curve essentially
overlapped the
initial
assumed curve,
thus
verifying the theoretical
basis.
Reyes
[15]
used
a boundary
element method to
carry
out
a similar procedure with a
compact tension
configuration. Again the input
curve and
the
extracted
curve showed
excellent
agreement.
The theoretical basis
and numerical verification
confirm that this
test
technique
is independent of specimen
geometry
and
should
also
be
independent of
specimen
size. The
only
restriction on specimen
size
is
that the smallest
specimen dimensions
should be a number of times
larger
(maybe
four or five times) than
the
largest
single particles
in the
material.
Thus
minimum
dimensions depend
on
material properties
such as
aggregate size
and fiber length.
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2.4 Experimental
Procedure
This
J-integral
technique
has
been
used
by
Li and
co-workers
[12,16,17,18,19]
with
both
the
compact
tension
and
four-point
bend
beam
loading
configurations
illustrated
in
Fig.
2.2.
Hashida
[20]
also
used
a
compact
tension
configuration
and
Chong
et al
[21]
used this
technique
on
semi-circular
specimens
from
rock
cores.
Because
the
experimental
procedure
is very
similar
irrespective
of
the
specimen
configuration
or
the
material
type,
a
thorough
description
of
specimen
preparation,
testing
procedure,
data
analysis
and
results
will
be
given
for
just
one
geometry
and
one
material.
The
geometry
used
is
the
beam shown
in
Fig 2.2 and
the
material
is
a Kevlar
fiber
reinforced
mortar.
2.4.1 Specimen
preparation
The
following
materials
were
used:
(i)
Type
III
rapid
hardening
portland
cement
(ii)
Sand
passed
through
a #8
sieve
(iii)
Kevlar
fibers
with
length
=
6.4
mm,
diameter
=
12
pm,
modulus
=
130
GPa,
tensile
strength
=
2.8
GPa,
and
density
=
1.45
g/cc.
(iv)
A
high
range
water
reducing
admixture
named
Daracem-100
and
classi-
fied
as ASTM
C-494
Type A.
The
cement:sand:water
ratio
was
1:1:0.5
by
weight.
The volume
fraction
of
fibers
was
2%
with
a superplasticiser
volume
fraction
of
0.75%.
An
Omni-mixer
in which
random
movement
of
the
particles
is
induced
to
occur
by
a
wobbling
flexible
drum
bottom
was
used.
A
photograph
of the
inside
of
the
mixer
is
shown
in
Fig.
2.3.
The
cement,
fibers
and
sand
were
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added
to
the mixer in that order. The plasticiser was mixed with
the water
and
added
last.
Mixing was
then
carried
out
for
approximately
3
minutes.
Some preliminary trial mixing suggested that there was
no
advantage to be
gained
by
dry mixing the cement, fibers and sand
before
adding water.
Plexiglass
molds
(63.5
x
114
x
432
mm)
were used. Casting
was done in a
direction such that
the
63.5
mm side
was vertical and
thus
all
specimens
were rotated
through
900 for testing.
Compaction was achieved using
a
tamping
rod and
a table vibrator.
For synthetic fiber
mortar tamping is
probably
much
more effective than
vibration.
It
proved relatively
difficult
to achieve a low
air content
comparable
to
that
which
could be
achieved with
just plain mortar.
After casting
the
specimens were covered
with
plastic and left
in the
molds
for
approximately 16
hours. Fig. 2.4
shows
some
specimens
just
after
casting.
They
were then
removed
from the
molds and
placed in water
at 22 0C. After 13
days
the specimens
were
removed from
the water and
all the
notches
were
cut
using a 1.5 mm thick
diamond blade.
Two beams had a through the
thickness
notch,
57.15 mm deep
on the
tension
side and two beams had a similar notch, 63.5
mm
deep. Crack
guides,
6.35 mm
deep
were
also
cut on
each vertical side of the
specimen,
extending from
the
tip of the notch
to
the top of
the
beam.
This left a
ligament
50.8
mm wide and either 56.85
or 50.5
mm
deep,
depending on
the
beam, at
the center of the span.
All the
specimens were
then
left in
air
until testing the following day
at
14 days
of
age.
Some preliminary
experimental work
showed
that
cast-in-notches
could
be used instead
of
saw-cut notches
if
a suitable
saw
is
not
available.
Thin
aluminium plates (0.8 mm
thick)
were placed in
the mold before
casting
and
coated with mold
release. They
were easily
removed from the specimen
about 10
hours after casting.
This
technique
assures
very
accurate
notch
length.
No
evidence of
fiber bundling
at the
plate tip was found.
Either
notching
technique
can
be used
depending on
preference
and available
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facilities.
2.4.2
Testing
procedure
Figs. 2.5 and 2.6 are
photographs
of the
experimental
set
up.
The
beam was
supported on a roller
at one end and a ball
at the other. The
load
was
applied via a universal
joint onto
a
steel plate
which was
supported
on the
top of the
beam
by a
roller
at one
side
and
a
ball at the
other. A displacement controlled
Instron loading
machine
was used with
load being
applied
by
a
screw
mechanism.
LVDTs (Linear
Variable
Displacement Transducers) with a displacement measuring
capacity
of
5.1
mm
were used to
measure the load point displacement and
the
crack opening.
One
LVDT,
placed at the front of
the beam measured
the
average load point
displacement. A
second LVDT was placed
on the front of the
beam
at the
notch tip
level to measure the crack tip opening.
The third LVDT was
placed
on the
back
of the beam,
63.5 mm from
the
bottom, corresponding to
the
position
of the
notch tip in beams with
a long
notch and
being 6.35 mm
(equal to Aa) above the notch tip in beams with a short notch.
The LVDT
holder
and
its target
were
placed
14 mm apart (inside to
inside)
on
opposite
sides
of the notched plane. Fig.
2.5 shows the positions
of
the
two
LVDTs
on
the front of the
beam.
The horizontally
placed
LVDTs have
the
shaft spring loaded
against
the target to ensure
permanent contact when
the
crack opens.
The vertical LVDT
shaft stayed
in
contact
with the target
under gravity.
The
cross-head
speed was
initially set
at
0.0254 mm/min. and
was
changed
to
0.0508 mm/min.
after
the load
dropped
to
30% of its
peak
value.
It
was later changed
to
0.127
mm/min. to
ensure
a reasonable test time.
Tests
were continued
until
the
beams failed
under self-weight
with
total
test time
being
about 50 minutes.
The data
was
passed
through a
fluke
data
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acquisition
system
and
stored
on
an
IBM
PC.
A
Basic
Program
was
used
to
convert
the
output
voltage
readings
into
load
and
displacement
values.
2.5
Results
and
Data
Analysis
Fig.
2.7
shows
the
average
load
versus
load
point
displacement
curves
for
each notch
length.
Each
average
curve
was calculated
by
taking
a
displacement
value
and
calculating
the
average
load
corresponding
to this
displacement
for
a set
of
beams
with
the
same
notch
length.
Fig.
2.8
shows
average
crack
opening
values
61,
62
and
63
expressed
as
a function
of
the
load point
displacement.
S
is
the
average crack opening
at
the
notch
tip
in the
short
notched
beams,
62
is
the
average
crack
opening
at the
notch
tip
in
the
long
notched
beams
and
63
is
the
average
crack
opening
in
the
short
notched
beams at
a position
6.35
mm
directly
above
the
notch
tip.
The
average
crack
opening
value
6 was
defined
as:
6
=
(6
1
+
63
)/2
(2.8)
Some
previous
experimental
work [12,16,17,18]
had
used 6
=
(8i
+ 62)
/2
as
the
definition
of
the
average
crack
opening.
However,
since
the
uncracked
ligament
Aa,
representing
the
difference
between
the
short
and
long
notched
beam,
exists
in
the
short
notched
beam,
it
is
considered
more
accurate
to
define
the
average
crack
opening
in
this
ligament
by
measuring
the
crack opening
at
each end (61 and
63)
and
taking
the
average
of
these
two values.
From
Fig.
2.8
it can be deduced
that using
63 instead
of 62
produces
an
average
6
value
which
initially
increases
slower
and
later
increases
faster
with
respect
to
the
6 values.
Eqn.
(2.7)
shows
that
this
produces
a
tension
softening
curve
which
is
higher
initially
and
becomes
lower
after
some
crack
opening.
It
will
be
seen
later
that
this
produces
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better
correlation
between the
deduced
tension softening
curve
and
the
softening
curve measured
by a direct
tension
test.
Numerical integration was
used to calculate
A(A)
which
was
then used
with Eqn. (2.6) to calculate
J(A).
Using
the S(a) relationship
obtained
from Fig. 2.8
and Eqn. (2.8), the J(6)
curve was calculated and
is
illustrated
in
Fig.
2.9.
Numerical
differentiation
of
the
J-6
curve
in
accordance
with
Eqn.
(2.7) was
achieved using
Taylor expansions at
five
consecutive
points J(S-g), J(S-h),
J(6), J(S+j), J(S+k)
and solving for
J'().
Fig. 2.10
shows the
deduced tension
softening
curve.
It shows an
initial rise from a =0 to a = ft
before descending back down
to
a 0.
This
initial
ascending
part may be
interpreted
as
the sum of
recoverable
elastic
deformation,
and distributed
inelastic
deformation in
the form of
microcracking,
which occurs between the
LVDT holder and
its
target,
prior
to the localization
of
inelastic
deformation
onto the fracture
process
zone.
The dashed line
in Fig. 2.10
shows
the
unloading
path
of
the
material
in this zone.
It is clear that
the area
between
the ascending
path and the
dashed line
represents
inelastic
energy absorbed
outside
the
fracture
plane and thus
should not
be counted
as
part of
the
tension
softening
curve.
Fig.
2.11
shows
the
corrected
a-8
curve,
accounting
for
this
energy loss
away from
the
fracture zone,
together with a
tension
softening
curve
for
the
same material
measured by
Wang
[22]
using
a direct
tension
test.
Considering
the
limited number
of
tests
carried out
(just two beams
for each
notch
length)
the
agreement
between
the
deduced softening
curve
and
the
directly measured
curve is
reasonably good.
The deduced
tensile
strength
is 3.35
MPa
compared
to
a
directly
measured
value of
3.95
MPa.
This
underestimation
is
consistent
with
previous
test
results
and some
explanation for
this
trend
will be
presented
in
Section 2.7.
The critical
J-integral
value
Jc
, defined as
the
total
area
under
the
deduced a-S
curve,
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is 1485
N/m
compared
to
a
directly
measured
value of
1250 N/m.
This
error
can
be
attributed
to the
small
number
of tests
and
would probably
be
eliminated
by
carrying
out
four or
five
tests
for
each
notch
length.
2.6
Calculation
of
GF
Fracture
energy
was
also calculated
using
a method
similar
to
that
recommended
by
RILEM
Technical
Committee
50
[23].
The RILEM method
is
based
on
the total
area under
the
load-load
point
displacement
curve
for
a
three-point
bend
test
on
a
notched beam.
However
the
results
should
be
similar for
the
four-point
bend
geometry.
The fracture
energy GF
is
given
as:
GF
= A
/
b(d-a)
(2.9)
where
A
is
the
total
area
under the
load
versus
load
point
displacement
curve,
including
a
correction
for
beam
self-weight.
The method
of
estimating
energy
supplied
by
beam
self-weight
was
as proposed
by
Petersson
[10]
and
is
illustrated
in
Fig.
2.12.
The calculated
GF
values
were
1181
N/m
and
1143
N/m
for
the
short
notched
and
long notched
beams
respectively.
These
values
are
less than
that
calculated
from
the
direct
tension
test
and
this
finding
is
the
opposite
of
test
results
obtained
by
Horvath
and
Persson
[241
which
showed
GF
values
to be
about
20% greater
than
direct
tension results.
Their
tests were
on
plain
concrete
specimens whose
load-deflection
curves
have
a
relatively
short
tail.
The self-weight
correction
may
not
be
very
accurate
for
fiber
reinforced
concrete
curves
which
have
a
much
longer
tail.
This
may
account
for
some
of
the
above
discrepancy.
It
should
be
remembered
that
GF
values
are
usually
expected
to
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overestimate
the true
fracture
energy because of
inelastic
energy absorbed
in
the
material
which lies
close to the
fracture
zone and
also in the
material
close to the
supports
and loading
points
where
concentrated
loads
are applied.
Using
a relatively
long notch
length
helps to
reduce this
inelastic
energy
absorption.
Another
interesting
point
concerning
the
GF
calculation
method
is
that as
the
beam
size is
increased,
while the
geometry
is kept constant,
the
loads on
the
beam before the
maximum load
is
reached
increase
approximately
in proportion
to the
beam
volume,
and thus
inelastic
energy absorbed
away
from the
fracture
zone
increases at a
similar rate.
However
the fracture
energy
absorbed, only
increases in
proportion
to
the cross-sectional
size. Thus
as
beam
size increases,
the
ratio
of
energy
absorbed outside
the fracture
zone to that
absorbed
within,
increases,
leading
to greater
overestimation
by
GF
of
the true
fracture
energy.
Horvath and
Persson
[24]
found calculated GF
values to increase
by
as much
as 50%
when
1455
x 300
x
100
mm beams
were
used
instead of
840
x
100
x 100
mm beams.
Thus the
GF fracture
energy
value
cannot
be considered
a
material
property,
but
is dependent
on
specimen
geometry.
GF should
be
closer
to
the
true
fracture
energy
value when
small size specimens
are
used.
However
specimen
dimensions
should be
at least
3 or
4 times
larger
than the
largest
particle
size in
the material
such as
aggregate
diameter
or
fiber length.
2.7 Discussion
and
Recommendations
for the
J-Integral
Test
A
summary
of all
test
results
obtained
by Li
and
co-workers
[12,16,17,18,19]
is
given
in
Table
2.1 and
the
deduced
tension
softening
curves
are
illustrated
in
Fig.
2.13.
There
is close
agreement
between
Jc
and GF*
However
the
deduced
tensile
strength
is consistently
less
than
the
actual
strength.
A
possible
explanation
for
this
tendency
and
a possible
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means
of
avoiding
it
is
now
presented.
The J-integral
method examines a portion
of
the
crack
plane
which
extends a distance Aa
between the positions of the
short and
long
notch
tips.
The actual
ligament
exists
in the
beam with
the
short notch. The
deduced a-S
curve
represents the average stress
transferred across this
area
(b
Aa),
as a
function
of the average crack
separation. The
loading
configuration
ensures that at
any
given time
during
the test the actual
crack opening
at
any point within the specified zone gets smaller as the
point moves from
the
initial
notch
tip
towards the neutral
axis. When the
Aa
zone is
transferring
maximum
load,
the
stress
along
some
line
within the
zone is equal to
the
tensile
strength.
At
all
other
points
the
stress
transferred must
be less than
ft. Thus when
the
average stress throughout
the zone is
calculated, it
must be less than ft.
Theoretically
this
problem is solved by letting Aa approach
zero.
In practice, however, there
is a lower
limit on Aa
so that
errors
in
the
measurement of
initial crack
lengths and in the loading machine
and instrumentation
are
not
significant.
Experience suggests
a lower limit of about
6
mm
for
the beam bending
configuration
in Fig.
2.2.
A possible extension of
the approach
to date
may
be to
use
different
Aa values
in a given test
program
and to
extrapolate the J-S or
the a-S
curve
to
obtain a curve as a-0. A
variation
of
this possibility
may be to
use similar Aa
values, to
vary
the
ratio Aa/(d-a),
and to extrapolate the
resulting
curves to find a solution
as this
ratio
approaches
zero.
The
problem
with these
approaches
is
that
the
number
of
test
specimens
required
is
considerably
increased.
Another
source of
error
in
the
final result
is from the raw
data
curves. Because this
technique
relies
on
measuring the
difference
between
two load
versus
load point
displacement curves
obtained from notched
specimens
with
only a small difference
in
initial notch lengths, any
error
in
the
measured
curves results
in
a
magnified
error in the
difference
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between the curves.
Concrete, and
especially
fiber
reinforced
concrete,
is
a
heterogeneous
material
and
so
there
is
always some
variation in
test
results
obtained
from
supposedly exactly
similar
specimens. In
order
to
guard
against the
possiblity
that
a single exceptional
test result may
significantly
affect
the
final
deduced
tension softening
curve,
a
standardized
approach
is
suggested
for
examining
individual
raw
data
curves
and combining
them
to produce
average
curves which
are likely
to give
a
good
result.
Using
each load
versus load
point
displacement
curve, two
values
are
calculated,
namely GF
and ff(net)
where
ff(net)
is
defined
as:
f
f(net) =
6 M
max
b(d
- a)
(2.10)
were
Mmax is the
maximum
bending
moment
resisted
by the
notched
section.
It is reasonable
to
expect that
these
values
are
similar
for the
short
notched
and
long
notched beams.
Thus
it is
also reasonable
to assume
that
if
the
raw data
curves for
each
notch
length
are chosen
in
a
manner
which
ensures
that
the two
average
curves
have similar
GF and
ff(net)
values,
then
they
are likely
to produce
an
accurate
tension
softening
curve.
The
ff(net)
values
significantly
affect the
deduced
tensile
strength
and
there
is
a direct
relationship
between
the
two GF
values
and the J c
value.
This
approach
is especially
useful when
the
number of
tests performed
is small.
2.8
Current
Status
of J-Integral
Test
Technique
The J-integral
test
technique
just outlined
has
undergone
extensive
development
and
refinement
in the
last
few
years
both
here at
M.I.T.
and
elsewhere.
The
testing
details
outlined
here
represent
the
most recent
procedure
used
at
M.I.T.
for the
beam
bending
configuration.
Successful
application
of this
test method
to an
increasingly
wide
range
of
materials
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including
mortar
and
fiber
reinforced
mortar,
granite
[20]
and basaltic
rocks
[211
together
with
the possibility
of
using
various
specimen
geometries
such as
beams,
compact
tension
specimens
and semi-circular
specimens,
suggests
its
usefulness
as
a standard
test for
the
fracture
properties
of many
non-yielding
materials.
This test
is relatively
simple
and
can
be
performed
using
any standard
testing
machine.
Through
the
tension
softening
curve
it
provides
us with
a
basic
measure
of
the
fracture
resistance
properties
of
any non-yielding
material
such as
concrete.
Given
that
the
fracture
resistance
can
be
quantified
by
this
test
it
is now
essential
to examine
why we
need
to
measure
this
material
property
and
how
we can use
it
to
achieve
improved
and
more
efficient
material
performance
in
structural
applications.
It
is
imperative
that
not
only
does
a standard
test
provide
a good
measure
of
some
material
property
but
also
that
that
measure
is
capable
of
contributing
to
our
understanding
of
how
the
material
will
behave
in
practice.
The
aim of
chapters
3
and
4
is
to establish
fracture
resistance
as
a
material
property
which
can
dramatically
affect
structural
performance
and
which
therefore
must
be
considered
in any
accurate
design
code.
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Table
2.1 - Summary
of J-Integral
Test
Results
Matial
Deduced
Jc
8C
Dict
GF
#
ft
ft
(MPa)
(N/m)
( Im)
(M~a) (N/m)
2
3
4
5
6
7
8
2.09
2.80
1.78
2.09
2.08
2.01
2.11
3.35
83.8
1200
187
81
205
404
543
1485
140
3700
700
190
1100
1520
1340
2300
3.0
2.6
2.6
2.6
2.6
3.95
78
209
414
607
1162
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Fig.
2.1
1 a
J
J
127
r
ds
5t
S
=
separation
distance
:)
=
stress
transferred
Cohesive
Zone
Ahead of
the Crack
Tip
-4
-4
ii
a
127 5
Specimen width
=
63.5
19.1
82.5
Specimen
width
= 19.1
Dimensions
in
mm
Fig.
2.2
Specimen
Configurations
used
with
J-Integral
Test
33
j
i
25
127
I
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Fig.
2.3 Inside of Omni-Mixer
Fig.
2.4
Specimens Covered in
Plastic just after Casting
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Fig
2.5 Notched Beam
ready
for
Testing
Fig. 2.6 Loading Machine
Used
for J-Integral Test Program
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3000 r
2000
Crack
length
= 57.15mm
1000-
Crack
length
=
63.5mm
1000
2000
3000
4000
5000
Load
point
displacement
(pm)
Fig.
2.7
Average
Load
versus
Load Point
Displacement
Curves
3000.
M
82
2000-
83
0
o -
S
1000-
1000
2000
3000
4000
Load
point
displacement
(pm)
Fig.
2.8
Average
Load
Point
Displacement
versus
Crack
Opening
Curves
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I
- I ~
1000
2
Crack
opening
(pm)
Fig. 2.9 J-Integral
versus Crack Opening
Curve
correct
8
500 1000
1500
Crack opening (pm)
Fig.
2.10
Deduced
Tension
Softening
Curve
37
2000
r
1500
[
E
S..-
w
4.
C
1000
500
3000
3.0
(U
4.-
(I,
2.01
1.0
2000
2500
-
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4.0-
3.0-
2.0-
Deduced
curve
1.0.
Directly
measured
curve
500
1000 1500 2000
2500
Crack
opening
(pm)
Fig. 2.11
Comparison
Between
Deduced
and
Directly
Measured
Tension
Softening
Curves
Shaded
area
=
energy
absorption
recorded
in test
A
+ A2 =
energy
supplied by
beam
self
weight
0
A A
(Petersson,
10]
F0
=
applied load
which
gives
same
moment
at
mid-span
F
0
A
A as
beam
self-weight
] . 2
d
0
Load
Point Displacement
Fig. 2.12
Flexural
Load
Deflection
Curve
Corrected
to
Account for
Energy
Supplied
by
Beam Self-Weight
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4.0
3.0.
#8,
Kevlar
FRM, 6.4 mm,
2%
7, Acrylic
FRM,
6.4 mm, 3%
2 0
#6, Acrylic
FRM,
6.4 mm,
2%
#2,
Steel FRM, 9.53
mm, 1%
#4, Plain Mortar
1,
Plain
Mortar
1.0-
#5,
Acrylic
FRM,
6.4
mm, 1%
3 ,
Acrylic
FRM,
12.7
mm,
1%
C
= 3700Pm
500
1000
1500
2
25 0
Crack
opening
(m)
Tension
Softening Curves
Deduced by Indirect
J-Integral
Technique
ig.
2.13
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3
FLEXURAL
BEHAVIOR
OF FIBER
REINFORCED MORTAR
3.1 Introduction
Is
there
any
need
to use
fracture resistance
as
a material property
in
the design of concrete
structures? For many years the majority of
Engineers,
if
asked, would
answer
a definite no
to
this question. The
design code
