Fracture Mechanics of Concrete and Concrete Structures High Performance, Fiber Reinforced Concrete, Special Loadings and Structural Applications- B. H. Oh, et al. (eds)
ⓒ 2010 Korea Concrete Institute, ISBN 978-89-5708-182-2
Fiber reinforced concrete characterization through round panel test - Part
II: analytical and numerical study
F. Minelli & G.A. Plizzari
University of Brescia, Italy
ABSTRACT: The round determinate panel according to ASTM was found to be a reliable, consistent and repeatable test method for the measurement of the energy absorption in Fiber Reinforced Concrete (FRC) composites. A smaller panel was proposed and experimentally investigated by the authors in the Part I of the present paper. The results of several tests on about 100 round panels (ASTM and small panels) evidenced a lower
scatter, as in the larger ASTM panels, with significant advantages. In fact, small round panels are easier to
place and handle and their weight is only 40 kg, if compared to the 91 kg of the ASTM panel. Given this broad
experimental database, an analytical approach is herein reported toward the definition of a suitable simplified
cohesive stress-crack width constitutive law for FRC, determined from round panel tests according to the requirements of the main structural codes. To this aim, crack development was measured during the tests. In addition, a kinematic approach was proposed to determine the constitutive laws from the load point displacement
of the slab.
1 INTRODUCTION
A standardized Round Determinate Panel (RDP) test
is nowadays available and published by ASTM
(2004) for the measurements of the energy absorption at 40 mm displacement, for characterizing Fiber
Reinforced Concrete (FRC) elements with special
emphasis on sprayed concrete in tunneling applications. It is a statically determinate test, with round
shape slab having a diameter (φ) of 800 mm and a
thickness of 75 mm, supported in three points at 120
degrees. The standard test is rather straightforward
and only requires load and displacement measurement. The specimen weight is 91 kg.
In the first part of this investigation, the authors
demonstrated that a smaller round panel with a diameter of 600 mm and a thickness of 60 mm is characterized by a similar scatter in results, which is much
lower compared to the classical tests on small
notched beams (Minelli & Plizzari 2007 and 2009).
Moreover, the three crack measurements could be
also included in the test, as the crack pattern is repeatable and predictable; therefore, the post-cracking
material properties can be adequately determined. In
addition, handling and placing such a specimen is
easier as compared to the classical panel since its
weight is only 40 kg. Finally, standard servocontrolled loading machines generally fit with the geometry of the small panel, allowing for a proper
crack controlled tests with a close-loop system.
In order to use this test procedure for characterizing FRC, which means determining fracture proper-
ties such as toughness indexes, residual post cracking
strengths and equivalent post-cracking stresses, one
should be able to find sufficient information from the
experiments. The procedures adopted in many standards for beam tests (UNI 2003, RILEM 2003 &
CEN 2005) are based on the determination of a
stress-crack width curve, in which:
-the stress can be derived from theory of elasticity,
i.e. the flexural stress is just the moment over the resistance modulus. This is a rather rough simplification but it allows for the definition of suitable postcracking strengths or toughness indexes without any
non-linear analysis, which is difficult to perform;
-the crack width is measured and monitored (Crack
Tip Opening Displacement – CTOD), especially
while dealing with notched beams.
Stresses causing cracks (i.e. normal stresses along
the crack line) and crack width records are therefore
needed if the procedure well recognized for beams is
to be transferred to round panels.
The virtual energy-based yield line method (Johansen 1972) in which the uniaxial flexural capacity
of the material upon cracking is used together with
an assumed pattern of failure to predict the point
load capacity, might be also used to determine a simple crack width as a function of the point load displacement (Bernard 2000, 2002 and 2006). However, the yield line method was originally developed
for nominally plastic materials; therefore, its application for the prediction of load capacity in structures
made of lightly reinforced concrete exhibiting brittle
behavior might be questionable.
=The
− D (present
h, T )∇h investigation focuses on the applica(1)
bility of round panel tests in defining a simplified and
easy-to-use
constitutive coefficient
law for FRC
material.
NuThe proportionality
D(h,T)
is called
merical
elastic
analyses
and
yield
line
method
will
be
moisture permeability and it is a nonlinear function
both
this aim.hIn
the Tlatter
apof theutilized
relativetohumidity
andparticular,
temperature
(Bažant
proach
will
be
examined
and
evaluated
as
far
as
the
& Najjar 1972). The moisture mass balance requires
prediction
of
cracking
is
concerned
and
compared
that the variation in time of the water mass per unit
against
experimental
results.
volumethe
of concrete
(water
content w) be equal to the
divergence of the moisture flux J
2 ∂ANALYTICAL
STUDY
w
−
∂
t
= ∇•J
we (h α c α s ) =
,
,
⎡
G1 (α c , α s )⎢⎢1 −
⎢
⎣
∞
(g α
c
e
∞ − α )h
⎡
(g α
c
c −
⎢
K (α c α s ) e
⎢
10
10
(2)
isotherm (also called moisture capacity). The
governing equation
(Equation 3) must be completed
P⋅D
by
appropriate
boundary
and initial conditions. (1)
σ t = 0.001816 2
t
The relation between the amount of evaporable
water and relative humidity is called ‘‘adsorption
isotherm” if measured
P ⋅ D 3 with increasing relativity
ηhumidity
0
.
000429
(2)
=
RPS
and ‘‘desorption
isotherm” in the opposite
E ⋅t3
case. Neglecting their difference (Xi et al. 1994), in
the following,
‘‘sorptionload
isotherm”
will be(aused
with
where
P is the external
in the center
unit load
reference
to
both
sorption
and
desorption
conditions.
of 10 kN was applied), D is the diameter of the panel
By the way,
if the
hysteresis
of inthethemoisture
measured
from the
supports
(550 mm
case of
isotherm
would
be taken
account,
two(60
different
small
round
panel),
t is theinto
panel
thickness
mm),
relation,
evaporable
vs relativemodulus
humidity,
E
and ν are
Young’swater
and Poisson’s
of must
conbe used
according to
signareofdefined
the variation
of the
crete,
respectively.
Allthe
units
according
to
relativity
humidity.
The
shape
of
the
sorption
SI unit system.
isotherm
for HPC
is influenced
many
The elastic
analysis
was alsobyable
to parameters,
predict the
especially
those
that
influence
extent
and
of the
the
distribution of radial and tangential stressesrate
along
chemical
reactions
and,
in
turn,
determine
pore
line of crack formation (i.e., the radial bisector bestructure
andpair
poreofsize
distribution
tween
each
pivot
supports),(water-to-cement
as depicted in
ratio,
cement
chemical
composition,
content,
Figure 3. Note that the elastic solution isSF
an easy-tocuring
time
and
method,
temperature,
mix
additives,
use tool that allows a simplified calculation of the
etc.). In thepost-cracking
literature various
formulations
can for
be
equivalent
strength
being feasible
found
to
describe
the
sorption
isotherm
of
normal
practical and design applications, in the same way as
(Xi beam
et al.tests.
1994). However, in the present
itconcrete
is done for
paper
the
semi-empirical
by
The results show that theexpression
peak tensileproposed
stresses ocNorling
Mjornell
(1997)
is
adopted
because
cur near the loading point and a broad area ofit
⎤
⎥
− α c )h ⎥⎥
⎦
1
1
1
1
,
As diffusely reported in Part I of this paper, more
thanThe
100water
experiments
round
panels were
carried
content
woncan
beUniversity
expressed
as Brescia.
the
sum
out
in
the
last
4
years
at
the
of
of
the
evaporable
water
we (capillary water, water
Based
on
broad water)
experimentation,
an analytical
vapor, was
andthis
adsorbed
and
the of:
non-evaporable
model
then
developed
to
the
aim
(chemically local
bound)
water opening
wn (Mills 1966,
-determining
stress-crack
cohesive law
Pantazopoulo
&
Mills
1995).
It
isprocedure
reasonable
to
from
round
panels,
similarly
to
the
well
assume
that
the
evaporable
water
is
a
function
of
acknowledged
for beams;
relative humidity,
h, degree of hydration, αc, and
-defining
suitable
equivalent
post-cracking
strength
degree
of
silica
fume
reaction,
α
s, i.e. we=we(h,αc,αs)
directly
from panel tests,
in the same way as for
beam
= age-dependent
sorption/desorption
isotherm
tests.
(Norling
Mjonell
1997). Under
thisrelations,
assumption
and
First
of
all,
appropriate
general
accordby tosubstituting
Equation 1 into Equation 2 one
ing
the thin plate theory, were determined for deobtains
scribing the maximum tangential stress (as well as the
radial and tangential stress pattern, as shown in Fig. 1
∂w
∂w ∂h
and
Fig. 2)
and the displacement
the+ load
point.
e α& + ∂wate α&
w&the
+ ∇ • ( D ∇h ) =
− e
(3)
c
s
n
An ∂elastic
com∂αthrough
h ∂t finite helement∂αanalysis,
c
s
mercial software STRAUS7 (2004) with two dimensional elements (plates) was performed with the folwhere ∂w
e/∂h is the slope of the sorption/desorption
lowing
results:
Proceedings of FraMCoS-7, May 23-28, 2010
explicitly accounts for the evolution of hydration
reaction and SF content. This sorption isotherm
reads
⎣
+
(4)
⎤
1⎥
⎥
⎦
where 1.theradial
firststresses
term σ(gel
isotherm) represents the
Figure
r on a small round panel, elastic
physically
bound
(adsorbed)
water and the second
analysis
with FE
program
Straus7 (2004).
term (capillary isotherm) represents the capillary
water. This expression is valid only for low content
of SF. The coefficient G1 represents the amount of
water per unit volume held in the gel pores at 100%
relative humidity, and it can be expressed (Norling
Mjornell 1997) as
c α c+ ks α s
G (α c α s ) = k vg
c vg s
(5)
,
1
where kcvg and ksvg are material parameters. From the
maximum amount of water per unit volume that can
fill all pores (both capillary pores and gel pores), one
can calculate K1 as one obtains
g αc αc h
⎡
⎛
⎞ ⎤
∞
10⎜
−
⎟ ⎥
⎢
1
⎝
⎠ elastic
Figure 2. tangential
round panel,
− 0.188stresses
+ 0.22 t on− a small
⎥
⎢1 −
0
1
⎥
⎢
analysis with FE program Straus7 (2004).
⎦
⎣
( ,
)=
1
⎛
⎞
∞
1.2
10⎜
−
⎟
Comparison distribution
σ r(r)/σ
-σt(r)/σt,max,
⎝ 1
⎠ −r,max
1
w
σα s G
α s
c
s
K αc α s
e
e
(6)
g αc αc h
Small Round Panel
The material parameters kcvg and ksvg and g1 can
be calibrated by fitting experimental data relevant to
Serie1
s (r)
free0.8 (evaporable) water content
in concrete at
s (r)
Serie1
various ages (Di Luzio & Cusatis
2009b).
1.0
Stress σ r /σ r,max and σ t/σ t,max [MPa]
J
t
t
r
x
r
0.6
y
Support
Point
2.2 Temperature evolution
0.4 that, at early age, since the chemical reactions
Note
associated with cement hydration and SF reaction
are0.2exothermic, the temperature field is not uniform
for non-adiabatic systems even if the environmental
temperature is constant. Heat conduction can be
0.0
described
in50concrete,
at 150
least for200temperature
not
0
100
250
300
exceeding
100°C Radial
(Bažant
&
Kaplan
1996),
by
Coordinate r [mm]
Figure
3. Elastic
radial and
tangential stresses along the line
Fourier’s
law, which
reads
of crack formation (radial coordinate).
q = − λ ∇T
z
(7)
heightened tensile stress arose along the crack line.
While the radial and tangential stresses have the same
where q isat the
the center,
heat flux,
T is thetoward
absolute
magnitude
they diverge
the
temperature,
and
λ
is
the
heat
conductivity;
in this
edge of the panel along each radial bisector, the
ra-
dial stress dropping to zero for meeting the boundary
condition at the edge (see also Bernard 2006).
By using the above equations for the determination of the local tangential maximum tensile stress
along the yield line, it is possible to come up with a
σ-w cohesive law and compare it with those determined from beam tests, whereby w defines the experimental measured crack widths.
As an alternative, without direct measurement of
the crack width, a kinematic approach based on the
yield line theory (more properly defined as virtual energy-based yield line method, Johansen 1972) could
be adopted to calculate the crack width, as also done
by Bernard (2002), Tran et al. (2001) and Lambrechts (2004). In a yield line analysis of a round
panel, the governing mode of failure is taken to comprise three symmetrical yield lines-cracks emanating
from the center of the face opposite the point load
and running radially to the edge while bisecting each
sector between adjacent pivot supports.
Hinge Line
Zero Displacement
B
Yeld Line
Q
C
Central Load Point
P
Support Pivot
with r =275 mm. J = − D ( h, T )∇h
The rotation at the support αP is equal to:
The proportionality coefficient
δ Q = α P ⋅ PQ
(5) D(h,T)
moisture permeability and it is a nonlinea
theand
relative
h and temperature
δq onehumidity
can obtain:
By substitutingofPQ
& Najjar 1972). The moisture mass balanc
BQ
⎛ η RPS ⎞
water mas
BQ ⋅ ⎜
⋅ the variation in time of the(6)
⎟ = α Pthat
3 of concrete (water content w) be eq
volume
⎝ 2r ⎠
divergence of the moisture flux J
The rotation at the support P therefore yields:
3 η RPS − ∂w = ∇ • J
(7)
αP =
∂t
2 r
The water content w can be expressed a
The rotation in
the point water
of experimental
of Q,
theatbe
evaporable
we (capillary wa
crack measurement,
can
derived
as:
vapor, and adsorbed water) and the non-e
η
ϕ RPS = 2 ⋅ α P = (chemically
3 ⋅ RPS
bound) water (8)
wn (Mil
r
Pantazopoulo
& Mills 1995). It is reas
assume that the evaporable water is a fu
The corresponding
bottom
crack opening
relative
humidity,
h, degree(Fig.
of 5)
hydration
is therefore given degree
by:
of silica fume reaction, αs, i.e. we=w
⎛ 3 η RPS ⎞ sorption/desorption
⎛ ϕ =⎞ age-dependent
⎜
⎟
wm = 2t ⋅ tg ⎜ RPS(Norling
⎟ = 2t ⋅ tgMjonell
Under (9)
this assum
⎜ 2 r 1997).
⎟
⎝ 2 by
⎠ substituting
⎝
⎠
Equation 1 into Equati
obtains
where r is the radius
(r = 275 mm) and t is the panel
thickness (t = 60 mm).
It is worth noticing
the crack opening
ob∂w
∂w
∂w ∂that
h
e
e α& + w
e
&
α
+
(
)
+
∇
•
∇
=
−
D
h
tained using Equation 9 is constant along the radius,
c
s
h
∂α
∂α
∂h ∂t
for a given value of panel displacement. However,
ins
c
the actual case, the crack width will increase from the
external surface towhere
the load
/∂h is the slope of the sorption/
∂wepoint.
isotherm (also called moisture capac
governing equation (Equation 3) must be
by appropriate boundary and initial conditi
The relation between the amount of e
water and relative humidity is called ‘‘
isotherm” if measured with increasing
humidity and ‘‘desorption isotherm” in th
case. Neglecting their difference (Xi et al.
the following, ‘‘sorption isotherm” will be
reference to both sorption and desorption c
Figure 5. 3D representation
the failure
mode
of a roundof the
By the ofway,
if the
hysteresis
panel (a) (Bernard, isotherm
2006). Crack
width,
bottom
surface
of a
would be taken into account,
two
round panel (b).
relation, evaporable water vs relative humi
be used according to the sign of the varia
Figure 6 shows the comparison between experirelativity humidity. The shape of the
mental and analytical crack width: the experimental
isotherm for HPC is influenced by many p
curve was calculated from a linear regression of all
especially those that influence extent and
test results available. It is a bilinear curve, with a first
chemical reactions and, in turn, determ
constant line with zero crack representing the unstructure and pore size distribution (watercracked phase, in which the vertical displacement inratio, cement chemical composition, SF
creases prior to cracking (up to a value of around
curing time and method, temperature, mix
η0=0.54-0.63 mm, in which the lowest value is apetc.). In the literature various formulatio
propriate for normal strength concretes while the
found to describe the sorption isotherm
highest is more suitable for high strength concretes),
concrete (Xi et al. 1994). However, in th
and a second linear branch, showing a quite good fitpaper the semi-empirical expression pro
ting of the experiments (R2=0.99). The difference beNorling Mjornell (1997) is adopted b
tween the two curves (with the experimental showing
/2
Pivot Lines
t
Figure 4. Yield Line Approach for the determination of crack
width.
2t tg( /2)
The crack width was experimentally measured at a
distance from the point load of 120 mm (point Q in
Fig. 4). Based on geometry consideration, it can be
written that:
BC
η RPS
=
BQ
(3)
δQ
where CQ=120 mm (in the present experiments)
whereas B is the zero displacement point, ηRPS and δQ
are the vertical displacements at points C and Q, respectively. From trigonometry, one can derive that:
2r
η RPS
=
BQ
δQ
⇒ δ Q = BQ ⋅ ⎛⎜ η RPS ⎞⎟
⎝ 2r ⎠
(4)
Proceedings of FraMCoS-7, May 23-28, 2010
J = − Dcracks
( h, T )∇hfor the same vertical displacement) is
larger
(1)
probably due to the fact that a little elastic deformationThe
is always
measuredcoefficient
by the LVDTs
exproportionality
D(h,T)inisthecalled
periments
(150
mm
gauge
length)
and,
most
impormoisture permeability and it is a nonlinear function
tantly,
their location
120
mm from Tthe
load
of the relative
humidityis hatand
temperature
(Bažant
point,
while
the
yield
line
theory
would
match
better
& Najjar 1972). The moisture mass balance requires
with
a measurement
at r/2offrom
the load
point
that the
variation in time
the water
mass
per(i.e.,
unit
137.5
mm).
volume of concrete (water content w) be equal to the
divergence of the moisture flux J
explicitly
accounts
of ashydration
states
(crack
widths for
fromthe0.6evolution
to 3 mm),
follows
reaction
(Fig.
9): and SF content. This sorption isotherm
reads
we (h α c α s ) =
,
,
⎡
G1 (α c , α s )⎢⎢1 −
⎢
⎣
1
∞
(g α
c
e
∞ − α )h
⎡
(g α
c
c −
⎢
K (α c α s ) e
⎢
10
10
(2)
w
t
The water content w can be expressed as the sum
of the evaporable water we (capillary water, water
vapor, and adsorbed water) and the non-evaporable
(chemically bound) water wn (Mills 1966,
Pantazopoulo & Mills 1995). It is reasonable to
assume that the evaporable water is a function of
relative humidity, h, degree of hydration, αc, and
degree of silica fume reaction, αs, i.e. we=we(h,αc,αs)
= age-dependent sorption/desorption isotherm
(Norling Mjonell 1997). Under this assumption and
by substituting Equation 1 into Equation 2 one
obtains
s
&+ &
stitutive cohesive σ-w law based on round panel experiments.
From
a linear
of experimental
is the
sloperegression
of the sorption/desorption
where ∂we/∂h
results,
an
analytical
relation
between
the crackThe
tip
isotherm (also called moisture capacity).
opening
displacement
(CTOD)
and
the
vertical
disgoverning equation (Equation 3) must be completed
placement
at midspan
of and
a beam
7), according
by appropriate
boundary
initial(Fig.
conditions.
to UNI,
was
derived
as
follows:
The relation between the amount of evaporable
water and relative humidity is called ‘‘adsorption
CTOD m = 1if.03measured
⋅η B
(10)
isotherm”
with increasing relativity
humidity and ‘‘desorption isotherm” in the opposite
case.
Neglecting
theirmodel
difference
et al. 1994),
in
From
a kinematic
shown(Xi
in Figure
8, it was
the following,
obtained
that: ‘‘sorption isotherm” will be used with
reference
4 ⋅ ηtoB both sorption and desorption conditions.
ϕ
ByB =the way,
if the hysteresis of the moisture
(11)
l
isotherm would be taken into account, two different
relation,
evaporable⎛ ϕwater
⎞ vs relative humidity,
⎛ 2 ⋅η B ⎞ must
wm = 2 ⋅ (h − a 0 ) ⋅ tg ⎜ B ⎟ = 2 ⋅ (h − a 0 ) ⋅ tg ⎜
⎟ (12)
be
used according⎝to2 the
sign
of
the
variation
⎠
⎝ l ⎠ of the
relativity humidity. The shape of the sorption
isotherm
for HPC
is influenced
many
parameters,
Note that
the same
approachbyfor
calculating
the
especially
those
that
influence
extent
and
the
rotation and crack width can be followedrate
for of
beam
chemical
reactions
and,
in
turn,
determine
pore
according to CEN (2005), provided that a notch
structure
andmm,
porerather
size distribution
(water-to-cement
than 45 mm,
is used. Figure
depth
a0=25
chemical
content,
8ratio,
wouldcement
also apply
for the composition,
CEN beam testSF
(2005).
curing
time
and
method,
temperature,
mix
additives,
The Italian Standard UNI 11039 (2003) defines
etc.).
In the equivalent
literature various
formulations
canfeq(0be
two
different
post-cracking
strengths,
found
to
describe
the
sorption
isotherm
of
normal
0.6) and f eq(0.6-3), which represent the toughness given
concrete
(Xi by
et fibers
al. 1994).
However, inserviceability
the present
to
the matrix
in a conventional
paper
the
semi-empirical
expression
proposed
by
(crack width from 0 to 0.6 mm) and ultimate limit
Norling Mjornell (1997) is adopted because it
Proceedings of FraMCoS-7, May 23-28, 2010
(4)
⎤
1⎥
⎥
⎦
where the first term (gel isotherm) represents the
physically bound (adsorbed) water and the second
term (capillary isotherm) represents the capillary
water. This expression is valid only for low content
the amount
of
of SF.7.The
G1 represents
Figure
Crackcoefficient
width at bottom
surface of a small
round panwater per unit volume held in the gel pores at 100%
el.
relative humidity, and it can be expressed (Norling
Load Points
Mjornell 1997) as
c α c+ ks α s
G (α c α s ) = k vg
c vg s
(5)
,
1
where kcvg and ksvg are material parameters. From the
maximum amount of water per unit volume that can
fill all pores (both capillary pores and gel pores), one
can calculate K1 as one obtains
B
c
&+
⎣
+
B
Figure 6. Crack width at bottom surface of a small round
panel.
∂w
∂w ∂h
e α ∂we α w
+ ∇ • ( D ∇h ) =
− e
(3)
n
h allows
∂αfor cthe ∂definition
α s
∂h ∂tprocedure
This
of a con-
1
1
1
,
− ∂ = ∇•J
∂
⎤
⎥
− α c )h ⎥⎥
⎦
Figure 8. Kinematic model for 4 point bending tests UNI
⎡
⎛
⎞ ⎤
∞
beams.
10⎜ g α
− α ⎟h ⎥
⎢
w
0
K (α c α s ) =
,
1
α s + 0.22α s − G
− 0.188
c
s
⎛
10⎜
⎝
e
1
⎢1 −
⎢
⎣
e
g αc − αc h
−
∞
1
⎞
⎟
⎠
⎝
1
c
c⎠
⎥
⎥
⎦
(6)
1
The material parameters kcvg and ksvg and g1 can
be calibrated by fitting experimental data relevant to
free (evaporable) water content in concrete at
various ages (Di Luzio & Cusatis 2009b).
2.2 Temperature evolution
Note that,
at earlyofage,
since the
chemicalstresses
reactions
Figure
9. Definition
equivalent
post-cracking
according
to UNI
Standard
(2003).
associated
with
cement
hydration and SF reaction
are exothermic, the temperature field is not uniform
l
U1
for
systems
if the environmental
(13)
f non-adiabatic
=
⋅ even
eq, (0 − 0.6) isb(constant.
2
temperature
Heat
conduction can be
0
.
6
h − a0 )
described in concrete, at least for temperature not
exceeding 100°C l(BažantU 2& Kaplan 1996), by
(14)
f
=
Fourier’s
reads⋅
eq, (0.6 −law,
3) which
b(h − a0 )2 2.4
q = − λ ∇T
(7)
By using Equations 11 and 12, it is possible to
correlate
ranges
vertical
ηB
where qcrack
is the
heatwithflux,
T displacements
is the absolute
ϕ
of
beams.
and
rotations
B
temperature, and λ is the heat conductivity; in this
For round panels, it was experimentally found that
(Fig. 6):
wm = 0.67 ⋅ η RPS − 0.32
= − D (the
h, T )P-w
∇h experimental graph
U2 is the area Junder
calculated in the crack range from w0 + w1 to w0 + w2
(Fig. 10);
The proportionality coefficient D(h,T)
z k2 = 0.001816 moisture
mm-1 is the
constant already
permeability
and it calcuis a nonlinea
lated throughoutof the
elastic humidity
analysis hpreviously
the relative
and temperature
shown.
& Najjar 1972). The moisture mass balanc
z
that the variation in time of the water mas
volume of concrete (water content w) be eq
divergence of the moisture flux J
Load
C aric
Table 1. Midspan displacements and rotations corresponding
to the CTODm defined by the UNI standard 11039 (2003).
ηB
CTODm
ϕB
[mm]
[mm]
0.6
0.58
0.052
3
2.91
0.0259
(15)
k2 ⋅ D U1
⋅
w1
t2
(16)
f eq 2 =
k2 ⋅ D
U2
⋅
2
w2 − w1
t
(17)
where:
z f eq,1 is the equivalent post-cracking strength relevant for serviceability limit states, corresponding to
feq(0-0.6) of UNI beams;
z f eq,2 is the equivalent post-cracking strength relevant for serviceability limit states, corresponding to
feq(0.6-3) of UNI beams;
z U1 is the area under the P-w experimental graph
calculated in the crack range from w0 to w0 + w1 (Fig.
10);
The water content w can be expressed a
of
the evaporable
water we (capillary wa
U vapor,
U
and adsorbed water) and the non-e
(chemically bound) water wn (Mil
Pantazopoulo & Mills 1995). It is reas
w
wassume
+w
w +w
w is a fu
that the evaporable
water
Figure 10. Load-crack
width
curve
for
small
round
panels
relative humidity, h, degree of dehydration
picting areas U1 and degree
U2.
of silica fume reaction, αs, i.e. we=w
= age-dependent sorption/desorption
This approach(Norling
can be used
to determine
average
Mjonell
1997). Under
this assum
values, standard deviations
and
characteristic
values
by substituting Equation 1 into Equati
of the toughness parameters
obtains defined, taking a signifi1
2
1
m
f eq1 =
t
0
Therefore one can define two crack width ranges
and determine the corresponding equivalent postcracking strengths, which are:
z f eq,1 from w0 to w0 + w1;
z f eq,2 from w0 + w1 to w0 + w2.
being w0 the crack width at first cracking,
w1=0.55 mm and w2=2.75 mm (Table 2). The two
equivalent post-cracking strengths can therefore be
defined as:
w
0
Table 2. Vertical displacement and crack width values corresponding to the same yield line rotation between panels and
beams.
ηRPS
wm
ϕRPS=ϕB
[mm]
[mm]
0.0052
0.82
0.55
0.0259
4.11
2.75
− ∂ = ∇•J
∂
0
By imposing the rotation of round panels to be
equal to the one of beams, according to the yield line
theory, using Equation 8 one can first determine the
vertical displacement at the point load ηRPS, and then
the crack width corresponding to the rotation given
(Equation 15). The crack width values then define
two ranges that allow the definition of two equivalent
post-cracking strengths, relevant for serviceability
and ultimate limit states, respectively. The results of
this procedure are reported in Table 2.
2
cant advantage from the lower scatter of panel tests
over beams (see Conti
and Flelli 2009). ∂w
∂w
∂w
−
e ∂h + ∇ • ( D ∇h) = e
h
∂α
∂h ∂t
α&c + e α&s + w
∂α
c
s
where ∂we/∂h is the slope of the sorption/
isotherm (also called moisture capac
governing equation (Equation 3) must be
by appropriate boundary and initial conditi
The relation between the amount of e
water and relative humidity is called ‘‘
isotherm” if measured with increasing
humidity and ‘‘desorption isotherm” in th
case. Neglecting their difference (Xi et al.
the following, ‘‘sorption isotherm” will be
reference to both sorption and desorption c
Figure 11. Typical Load-w curve for a small round panel dethe way, if the hysteresis of the
fining areas E1 and EBy
2.
isotherm would be taken into account, two
relation,
vs relative
A similar approach
canevaporable
be derivedwater
starting
from humi
be
used
according
to
the
sign
of
the varia
the load-displacement curve of panels (Fig. 11), nerelativity
humidity.
The
shape
glecting the crack width measurements (often not of the
isotherm
for HPC isone
influenced
by many p
provided). In such
circumstances,
can define
especially
those
that
influence
extent
suitable displacement ranges for the definition of feq1 and
chemical reactions
and, in turn,
ranges, calculated
from determ
and feq2. The displacement
structure
and
pore
size
distribution
Equation 14, are incorporated into the definition of (watercement chemical
composition, SF
the two equivalentratio,
post-cracking
strengths as:
curing time and method, temperature, mix
k ⋅ D E etc.). In the literature various formulatio
f eq1 = 2 2 ⋅ 1 found to describe the sorption(18)
isotherm
η1 concrete (Xi et al. 1994). However, in th
t
k ⋅D
Epaper
the semi-empirical expression
pro
2
f eq 2 = 2 2 ⋅ Norling
(19)
Mjornell
(1997)
is
adopted
b
t
η 2 − η1
Proceedings of FraMCoS-7, May 23-28, 2010
f
w
t
(20)
(2)
⎡
⎤
wu
f FtuThe
= k ⋅water
⋅ ( f Ftsw−can
0.5 ⋅be
f eq 2expressed
+ 0.2 ⋅ f eq1 )as
0 (21)
sum
⎢ f Fts −content
⎥ ≥ the
wi 2
⎣
⎦
of the evaporable water we (capillary water, water
vapor, and adsorbed water) and the non-evaporable
where:
(chemically bound) water wn (Mills 1966,
z k is a coefficient equal to 0.7 for uncracked secPantazopoulo & Mills 1995). It is reasonable to
tions
under
and equal water
to 1 inisthe
other cases
assume
thattension
the evaporable
a function
of
k=1
in
this
case;
relative humidity, h, degree of hydration, αc, and
z wi2 is the average of the crack width values for
degree of silica fume reaction, αs, i.e. we=we(h,αc,αs)
In this case wi2=1.8 mm
(avwhich
feq2 is calculated.sorption/desorption
= age-dependent
isotherm
erage
value
between
0.6
and
3
mm
according
to
UNI
(Norling Mjonell 1997). Under this assumption and
11039);
by substituting Equation 1 into Equation 2 one
z wu is the greatest value of the crack range for
obtains
which feq2 is calculated. In this case wu=3 mm (according
to UNI 11039);∂w
∂w
∂w ∂h
e α& linear
e constitutive
e
a typical
law
α&s + w&n
( D ∇h ) =
∇ •depicts
− Figure +12
(3)
c +
h
∂
α
∂
α
∂
∂
h
t
obtained from the equivalent
post-cracking
stresses
c
s
calculated from beam test.
where ∂we/∂h is the slope of the sorption/desorption
isotherm
σ t (also called moisture capacity). The
governing equation (Equation 3) must be completed
by appropriate boundary and initial conditions.
The relation between the amount of evaporable
water and relative humidity is called ‘‘adsorption
isotherm” if measured with increasing relativity
fFts
humidity
and ‘‘desorption isotherm” in the opposite
case. Neglecting their difference
(Xieq1et al. 1994), in
0.5f eq2-0.2f
the following, ‘‘sorption isotherm”
will be used with
reference to both sorption and desorption
f Ftuconditions.
By the way, if the hysteresis of the moisture
isotherm would be taken into account,
different
w i2 wtwo
w
u
relation,
evaporable
water
vs
relative
humidity,
must
Figure 12. Typical σ-w simplified cohesive constitutive law
be
used
according
to
the
sign
of
the
variation
of
the
according to CNR DT 204/2006.
relativity humidity. The shape of the sorption
isotherm
fornotice
HPC that,
is influenced
byofmany
parameters,
One can
to the aim
coming
up with
especially
those
that
influence
extent
and
rate
the
a simplified and easy-to-use constitutive law,of the
chemical
reactions
and,
in
turn,
determine
pore
standard neglects the first branch from the tensile
structure toandthepore
sizefFts
distribution
(water-to-cement
. Moreover,
the σ-w indistrength
value
ratio, incement
chemical
SF content,
cated
the picture
refers composition,
to a strain-softening
matecuring
time
and
method,
temperature,
mix
additives,
rial under tension.
etc.).
In the
literature
formulations
can the
be
Figure
13 and
Figure various
14 report,
as an example,
found
to
describe
the
sorption
isotherm
of
normal
characteristic values of equivalent post-cracking
concretecalculated
(Xi et al.by1994).
in theproposed
present
stresses
using However,
the procedure
paper the
expression
proposed
by
herein
for semi-empirical
small round panels
and the
classical
Norling
Mjornell
(1997)
is
adopted
because
method included in UNI 11039 (2003) for beams.it
Proceedings of FraMCoS-7, May 23-28, 2010
c α Comparison
s s feq(0-0.6),k - feq1,k
G (α4 c α s ) = k vg
+ kSteel
NSC
c cProperties
vg α sFibres;
determined at SLS
(5)
,
1
3.31
cfeq,RPS / feq,B s
vg= 1.22
vg2.91
where
k and k are material parameters. From the
3
maximum amount of water per unit volume that can
fill all pores (both capillary pores and gel pores), one
can2calculate K1 as one obtains
feq,RPS / feq,B
= 1.58
2.38
2.09
[MPa]
= 0.45 ⋅ f eq1
−Fts∂ = ∇ • J
∂
explicitly
accounts refers
for theto evolution
of hydration
The
exemplification
a normal strength
conreaction
content.
This of
sorption
isotherm
crete
withand
twoSF
different
contents
steel fibers,
30
reads60 kg/m3, according to Conti and Flelli (2009).
and
The advantage concerns the significantly smaller
dispersion of experimental
⎡ results that means
⎤ higher
1
characteristic values and,
an improved
⎢ therefore,
⎥
we (h, α c , α s ) =width
G1 (α c ,cohesive
α s )⎢1 −
+ in
⎥
stress-crack
law,
as
depicted
∞
α )h ⎥
10(g α
⎢
c −area
c under
Figure 15. The fracture energy,
the
e as1 the
⎣
⎦
(4)
σ-w diagram, is around 25% larger
in the two round
⎡ 10(g α ∞ − α )h
⎤
panel diagrams plotted, which
refer
to cthe character1 c
⎢
K1 (αin
− 1⎥14. The
istic values shown
c , αFigure
s )⎢e 13 and Figure
⎥
⎣ Beam 1, in the plot,
⎦
constitutive law related to
has
the only advantage to show the best performance for
wherecrack
the widths,
first term
(gel isotherm)
represents
the
high
exceeding
all other laws
depicted.
physically
bound
(adsorbed)
water
second
This
evidence
comes
from the
factand
thatthethe
two
term (capillary
isotherm)post-cracking
represents the
capillary
equivalent
characteristic
stresses
are
water. similar
This expression
only for low
rather
and then, is
in valid
the calculation
the content
subtrathe would
amountalso
of
of SF.would
The be
coefficient
G1 represents
hend
lower (Equation
21). This
water perthat
unitthere
volume
gel pores
at 100%
suggest
is aheld
needintothefurther
corroborate
relative
humidity,
and it can
be expressed
and
validate
the proposal
included
in CNR(Norling
(2006)
Mjornella broader
1997) asset of experimental data.
against
w
1
0
K (α c α s ) =
⎡
⎛
10⎜
⎢
⎝
−
⎢1 −
1
UNI
⎢
⎣
α s + 0.22α s G
− 0.188
UNI
c
RPS
,
s
⎛
10⎜
⎝
1
0
30 kg/m3
e
e
g αc − αc h
−
⎞
⎟
⎠
∞
1
g α c∞ − α c ⎞⎟h ⎤⎥
⎠
1
RPS
⎥
⎥
⎦
(6)
1
60 kg/m3
c equivalent
s
Figure
Analytical
calculationkof
post-cracking
The13.material
parameters
vg and k vg and g1 can
stresses from experiments. NSC beams
and panels from identibe concrete
calibrated
by(Conti
fitting
cal
batch
andexperimental
Flelli, 2009). data relevant to
free (evaporable) water content in concrete at
various
ages (Di Luzio
& Cusatis 2009b).
4
Comparison f
-f
eq(0.6-3),k
eq2,k
Steel Fibres; NSC
Properties determined at ULS
2.2 Temperature evolution
3
Note that, at early age, sincef the/ f chemical
reactions
=
associatedf with
cement
hydration
and
SF
reaction
1.32
/f =
1.14 the temperature field is not uniform
are exothermic,
2
for non-adiabatic
systems even if the environmental
temperature is constant. Heat conduction can be
described in concrete, at least for temperature not
1
exceeding
100°C (Bažant & Kaplan 1996), by
Fourier’s law, which reads
eq,RPS
eq,RPS
= −0 λ∇T
1.93
1.62
UNI
q
2.56
eq,B
eq,B
1.86
[MPa]
J = −D
, T )∇h mm and η =4.11 mm (Table 2).
being
η(1h=0.82
2
(1)
According to the Italian Guidelines for the Design,
Construction
and Production
Control
of Fibre
ReinThe proportionality
coefficient
D(h,T)
is called
forced
Concrete
Structures
(CNR,
2006),
once
suitmoisture permeability and it is a nonlinear function
able
testshumidity
are performed
the corresponding
of thebeam
relative
h and and
temperature
T (Bažant
and
f
(also
indicated
as
f
and
values
of
f
eq1
eq2
eq(0-0.6)
& Najjar 1972). The moisture mass balance requires
if
the
aforementioned
UNI
test
is
considered),
fthat
eq(0.6-3)
the variation in time of the water mass per unit
itvolume
is possible
to define
a simplified
of concrete
(water
content stress-crack
w) be equal width
to the
constitutive
cohesive
law asflux
follows:
divergence of
the moisture
J
RPS
30 kg/m3
UNI
RPS
60 kg/m 3
(7)
Figure 14. Analytical calculation of equivalent post-cracking
where from
q experiments.
is the heatNSCflux,
thefrom
absolute
stresses
beams T
andis
panels
identitemperature,
and
λ isandtheFlelli,
heat2009).
conductivity; in this
cal
concrete batch
(Conti
3.00
= −the
D (h,pivot
T )∇h support might affect
stress disturbanceJof
the record.
- The 60 mm thickness
of the small round
panel D(h,T)
The proportionality
coefficient
should be well evaluated
especially
with
longer
fibers
moisture permeability and it is a nonlinea
(longer than 50 mm).
be
of theSuitable
relative limitations
humidity h should
and temperature
defined concerning
maximum
allowable
fiber
length
& Najjar 1972). The moisture mass balanc
for which the small
panel test
thatround
the variation
in can
timebeof considthe water mas
ered significant. Similar
limitation,
however,
shouldw) be eq
volume of concrete (water content
be defined also for
classical round
panels, where
divergence
of the moisture
flux Jthe
thickness is only slightly higher (75 mm).
- The thin plate geometry
cause a 2D orienta∂w = ∇ •might
Jevaluated and compared
tion, which should− ∂be
well
t
with that of beam tests and samples taken directly
from the field structural
Theapplication.
water content w can be expressed a
Cohesive σ-w constitutive Law
Beams vs. Round Panels
2.50
BEAM 1
Stress [Mpa]
2.00
ROUND PANEL 1
BEAM 2
ROUND PANEL 2
1.50
1.00
0.50
0.00
0
0.5
1
1.5
2
Crack Width [mm]
2.5
3
Figure 15. σ-w laws according to CNR DT 204/2006. Beam
test v. Panel test. NSC beams and panels from identical concrete batch (Conti & Flelli, 2009).
3 CONCLUDING REMARKS
In the present paper, numerical elastic and the yield
line approach were adopted to come up with simplified cohesive constitutive laws from round panel
small tests.
Panel test is quite appropriate for representing the
actual behaviour of FRC materials. It can be considered as a complete test for the characterization of
FRC: suitable range of crack widths were defined
and the corresponding equivalent (or residual) postcracking strengths were derived (from σ-w plots) following the same procedure as done for beam tests.
The kinematic approach for calculating crack
width turned out to be simple, effective and rather
accurate if compared against experiments.
Panel test can become a very promising test
method for the characterization of FRC composites,
as it is relatively easy to perform and it is not expensive, it might require only one vertical displacement
transducer and, last but not least, it is characterized
by a very low scatter.
Further studies are actually going on at the University of Brescia for supplementary refine and validate the proposed calculation.
Among the open issues for further studies, the following key-points should be kept in mind:
- the position of the crack measurements: the crack
width varies along the crack line, and changes in a
different way before and after cracking. Moreover,
we should also recall that tangential stresses have to
be considered with radial stresses, which are non
zero for a length of 2/3 of the crack line. One should
think at measuring crack close to the pivot support,
where radial stresses are negligible, but then the
crack width would be lower and, more notably, the
of the evaporable water we (capillary wa
vapor, and adsorbed water) and the non-e
4 ACKNOWLEDGMENTS
(chemically bound) water wn (Mil
Pantazopoulo
& Mills
1995). It is reas
The authors would
like tothat
sincerely
thank Engineers
assume
the
evaporable
water
is a fu
Moira Conti, Giovanni
Flelli
&
Nicola
Vezzoli
for
relative
humidity,and
h, degree of hydration
their assistance indegree
data processing
in performing
of
silica
fume
reaction,
αs, i.e. we=w
numerical and analytical
studies.
= age-dependent
sorption/desorption
(Norling Mjonell 1997). Under this assum
by substituting Equation 1 into Equati
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∂w
∂w ∂concrete
ness of fiber reinforced
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e α loaded
eα w
e h
round panel), pp. −9, 2004. + ∇ • ( Dh ∇h) =
c
s
∂α
∂α
∂h ∂t
Bernard, E.S. 2000 Behaviour of round steel fibrec reinforceds
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(also
called ofmoisture
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the behaviour
fibre rein- capac
governing
equation
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35,initial
April conditi
by appropriate
boundary
2002, pp 156-164 The relation between the amount of e
Bernard, E.S. 2006 water
Influence
on the Apparent
andof Toughness
relative humidity
is called ‘‘
Cracking Load of Fiber-Reinforced Concrete Slabs, Journal
isotherm”
if
measured
with
increasing
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1,
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Products – their
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– Measuring
the flexural
tensilewill be
the following,
‘‘sorption
isotherm”
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pp.18. and desorption c
to 14651,
both sorption
CNR DT 204/2006. By
2006.the
“Guidelines
for
the hysteresis
Design, Con-of the
way, if the
struction and Production Control of Fibre Reinforced Conwould be
takenof into
crete Structures",isotherm
National Research
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Italy. account, two
relation,
water vs relative
Conti, M. & Flelli,
G. evaporable
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be used according
the sign
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su piastre
sottili”, M.Sc. Thesis,
Department
DICATA;
University
relativity
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especially those that influence extent and
Association, U.K.
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and,
in turn,
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concrete
char- determ
structure
and
pore
size
distribution
acteristics. Study on toughness results 2002-2003”, Pro- (waterceeding of the International
Workshop
“Fiber composition,
Reinforced
ratio, cement
chemical
SF
Concrete. From curing
theory totime
practice”,
Bergamo,temperature,
September mix
and method,
24-25, 2004, pp. etc.).
135-148.
In the literature various formulatio
Marinoni, Y. et al.: “Prove di frattura su calcestruzzi
found
to describe
the sorption
isotherm
fibrorinforzati in accordo
con le principali
normative
concrete Report,
(Xi et University
al. 1994).ofHowever,
internazionali”, Technical
Brescia, in th
paper2007.
the semi-empirical expression pro
Department DICATA,
Minelli, F. & Plizzari,
G.A. (2007),
“Fibre(1997)
Reinforced
Norling
Mjornell
is Conadopted b
crete characterization: Round Panel vs. beam test toward a
&+
Proceedings of FraMCoS-7, May 23-28, 2010
&+
J =harmonization”,
− D (h, T )∇h Proceedings of the 3rd Central European
(1)
Congress on Concrete Engineering CCC 2007 – Visegrád,
Hungary, 17-18 September 2007. G.L. Balázs & S.G.
The proportionality
D(h,T) University
is called
Nehme
(eds), Publishing coefficient
Company of Budapest
moisture
permeability
and it pp.
is a213-220.
nonlinear function
of Technology
and Economics,
Minelli,
F. & Plizzari,
G. A. 2009
panel vs.Tbeam
tests
of the relative
humidity
h and“Round
temperature
(Bažant
toward
a
comprehensive
and
harmonic
characterization
of
& Najjar
1972). The moisture mass balance requires
FRC materials”, Proceedings of the BMC-9 Conference,
thatWarsaw,
the variation
in time of the water mass per unit
Poland, 25-28 October 2009, A.M. Brandt et al.
volume
of
concrete
(eds.), pp. 23-32. (water content w) be equal to the
divergence
the moisture flux
J
RILEM Final of
Recommendation
TC-162-TDF,
“Test and Design Methods for Steel Fibre Reinforced Concrete; σ−ε Design
Method”, Materials and Structures, Vol 36, October
w
(2)
− ∂2003,
= ∇pp.
• J560-567.
∂t
Straus7.
2004.
Finite Element Analysis System,
www.strand7.com www.straus7.com.
The
water
w applica-tion
can be expressed
theTheory
sum
Tran,
V.G.
et al.content
2001. The
of Yield as
Line
Determinate
Panels,
De(capillaryEngineering
water, water
of totheRound
evaporable
water
we Shotcrete:
velopments,
Bernard
(ed.),
pp
245-254,
Swets
&
Zeitlinvapor, and adsorbed water) and the non-evaporable
ger, Lisse.
1966,
(chemically
bound) water wn (Mills
UNI 11039, “Steel Fiber Reinforced Concrete
- Part I: DefiniPantazopoulo
&
Mills
1995).
It
is
reasonable
to
tions, Classification Specification and Conformity - Part II:
assume
that
the
evaporable
water
is
a
function
of
Test Method for Measuring First Crack Strength and Ducrelative
humidity,
h, Board
degree
of hydration, 2003.
αc, and
tility Indexes”,
Italian
for Standardization,
degree of silica fume reaction, αs, i.e. we=we(h,αc,αs)
= age-dependent sorption/desorption isotherm
(Norling Mjonell 1997). Under this assumption and
by substituting Equation 1 into Equation 2 one
obtains
−
∂w ∂h
e + ∇ • ( D ∇h) = ∂we
h
∂α
∂h ∂t
∂w
α&c + e α&s + w&n
∂α
c
s
(3)
where ∂we/∂h is the slope of the sorption/desorption
isotherm (also called moisture capacity). The
governing equation (Equation 3) must be completed
by appropriate boundary and initial conditions.
The relation between the amount of evaporable
water and relative humidity is called ‘‘adsorption
isotherm” if measured with increasing relativity
humidity and ‘‘desorption isotherm” in the opposite
case. Neglecting their difference (Xi et al. 1994), in
the following, ‘‘sorption isotherm” will be used with
reference to both sorption and desorption conditions.
By the way, if the hysteresis of the moisture
isotherm would be taken into account, two different
relation, evaporable water vs relative humidity, must
be used according to the sign of the variation of the
relativity humidity. The shape of the sorption
isotherm for HPC is influenced by many parameters,
especially those that influence extent and rate of the
chemical reactions and, in turn, determine pore
structure and pore size distribution (water-to-cement
ratio, cement chemical composition, SF content,
curing time and method, temperature, mix additives,
etc.). In the literature various formulations can be
found to describe the sorption isotherm of normal
concrete (Xi et al. 1994). However, in the present
paper the semi-empirical expression proposed by
Norling Mjornell (1997) is adopted because it
Proceedings of FraMCoS-7, May 23-28, 2010
explicitly accounts for the evolution of hydration
reaction and SF content. This sorption isotherm
reads
we (h α c α s ) =
,
,
⎡
G1 (α c , α s )⎢⎢1 −
⎢
⎣
1
⎤
⎥
− α c )h ⎥⎥
⎦
∞
(g α
c
e
∞ − α )h
⎡
(g α
c
c −
⎢
K (α c α s ) e
⎢
10
10
,
1
1
1
⎣
+
(4)
⎤
1⎥
⎥
⎦
where the first term (gel isotherm) represents the
physically bound (adsorbed) water and the second
term (capillary isotherm) represents the capillary
water. This expression is valid only for low content
of SF. The coefficient G1 represents the amount of
water per unit volume held in the gel pores at 100%
relative humidity, and it can be expressed (Norling
Mjornell 1997) as
c α c+ ks α s
G (α c α s ) = k vg
c vg s
(5)
,
1
where kcvg and ksvg are material parameters. From the
maximum amount of water per unit volume that can
fill all pores (both capillary pores and gel pores), one
can calculate K1 as one obtains
w
0
K (α c α s ) =
,
1
⎡
⎢
−
⎢1 −
1
⎢
⎣
α s + 0.22α s G
− 0.188
c
s
⎛
10⎜
⎝
e
⎛
10⎜
e
g αc − αc h
−
∞
1
⎞
⎟
⎠
⎝
g α c∞ − α c ⎞⎟h ⎤⎥
1
⎠
⎥
⎥
⎦
(6)
1
The material parameters kcvg and ksvg and g1 can
be calibrated by fitting experimental data relevant to
free (evaporable) water content in concrete at
various ages (Di Luzio & Cusatis 2009b).
2.2 Temperature evolution
Note that, at early age, since the chemical reactions
associated with cement hydration and SF reaction
are exothermic, the temperature field is not uniform
for non-adiabatic systems even if the environmental
temperature is constant. Heat conduction can be
described in concrete, at least for temperature not
exceeding 100°C (Bažant & Kaplan 1996), by
Fourier’s law, which reads
q
= − λ ∇T
(7)
where q is the heat flux, T is the absolute
temperature, and λ is the heat conductivity; in this