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

Shear transmission on crack surface of ECC
T. Kanakubo
University of Tsukuba, Tsukuba, Japan

K. Shimizu
Railway Technical Research Institute, Tokyo, Japan

S. Nagai & T. Kanda
Kajima Technical Research Institute, Tokyo, Japan

ABSTRACT: In this paper, method of single plane shear test under tensile stress as normal stress on crack
surface is conducted to clarify shear transfer mechanism on crack surface of ECC (Engineered Cementitious
Composites). The PVA fiber is adopted for ECC. From the test results, shear strength on crack surface isabout half of tensile strength. This means that normal stress for crack surface is possible to keep crackstrength under shear. Predicting method for shear capacity of steel reinforced ECC beams is proposed basedon the arch-truss method with the summation of the tensile strength of ECC which is obtained by bending
test. The experimental values show good agreement with calculated values.
1 INTRODUCTION
High Performance Fiber-Reinforced Cementitious
Composites (HPFRCC), which show the strain hardening branch and multiple cracking under uniaxial
tensile stress, have been focused by lots of researchers because of its unique mechanical performance.
Engineered Cementitious Composites (ECC) exhibit
a maximum tensile strain of several percent owing to
the synergetic effect of high-performance fiber andspecifically designed mortar matrix. Unprecedented
high-performance structural members can be expected when ECC is applied to seismic components.
It has been cleared that shear capacity of steel reinforced members using ECC subjected to earthquake force (bending - shear force) increases rather
than ordinary steel reinforced concrete members(Shimizu 2004). It has been considered that tensilestress and shear stress can be transmitted throughmultiple cracks of ECC in the member as a result of
the bridging action of high-performance fiber. However, shear transmission capacity of ECC can not be
evaluated quantitatively because of difficulties oftesting method and phenomenon itself. Thoughmany uniaxial tension tests for ECC has been performed, the reports describing direct shear test for
ECC are very limited.
In this paper, method of single plane shear test
under tensile stress as normal stress on crack surface
is conducted to clarify shear transfer mechanism on
crack surface of ECC. The PVA fiber is adopted for
ECC. Predicting method for shear capacity of steel

reinforced beams is proposed based on multiplying
the reduction factor for tensile strength of ECC. The
tensile strength of ECC is obtained experimentallyfrom the evaluation of tensile properties using bending test result.
2 SINGLE PLANE SHEAR TEST
2.1 Loading method
Several testing methods for shear properties of ordinary concrete have been proposed. Some of thesemethods are shown in Figure 1. The shear test for
concrete is categorized in two types; single plane
shear test and double-plane shear test. Figure 1
shows the examples of single plane shear test. These
test methods are utilized to investigate the shear
properties of not only concrete but also reinforced
concrete. In ordinary reinforced concrete specimens,
crack width of crack surface is restricted by steel reinforcement and shear action is caused by bearing
action by coarse aggregate and Dowel action by reinforcement. The controlled normal stress can beapplied on crack surface when the testing method
shown in the left figure is used.
In case of ordinary concrete, concrete itself does
not carry tensile stress directly at crack surface after
crack opening. It is considered that normal stress canbe observed as the component force of bearing action of aggregate and Dowel action of reinforcement
to perpendicular direction of crack surface.

Shear
plane

P
Shear
plane

P

(1)

The proportionality coefficient D(h,T) is calledmoisture permeability and it is a nonlinear
Specimen function
Specimenof the relative
humidityhandtemperature
T (Bažant
SpecimenSlit
Shear& Najjar 1972). The moisture mass balance requires
planethat the variation in time of the waterP mass per
unit
volume Pof concrete (watercontent
w
)
be
equal
to
the
P
Figure
1. Examples
single plane
shear
divergenceof theofmoistureflux
J test.
Slit

In case of HPFRCC, however, HPFRCC can(2)− ∂w =tensile∇ • J stress after cracking. And it is considcarry∂tered that tensile stress to perpendicular direction ofcrack
and shearstressalong the as
crack
surThesurface
water content
w can
be expressed
the sum
faceby bridgingwater
actionfiber. Furthermore,water, wateritof theactevaporable
weof(capillaryisveryimportanttoinvestigate
shear
transmission
vapor, and adsorbed water) and the non-evaporable
behavior
for HPFRCC(MillsIt seems
1966,(chemicallybound) after
watercrackwnopening.
tobedifficulttoobservethese
actions
by using forPantazopoulo & Mills 1995). It is reasonabletomermentioned
method.water is a function ofassumethat thetestingevaporable
In thishumidity,
study, newly
developed
loading systemisrelativeh, degreeof hydration,
αc, andproposedtoinvestigate
directshear
properties
ofdegree of silica fume reaction, αs, i.e. we=we(h,αc,αs)
HPFRCC/ECC
under sorption/desorption
tensile stress as shownisothermin Fig= age-dependenture
2. Shearforce1997).
is applied
as thissameassumption
as that shown(Norling
Mjonell
Under
and
in
the
right
figureinFigure1.
Thetensileforce
isby substituting Equation 1 into Equation 2 one
subjected
tothespecimen
directly
using
two
oil
obtains
jacks, which is separately controlled from appliedshear
The screw∂bolts
at the both∂w
∂w force.
we embedded∂he α&surfacedends
arepulled
via
round
&
α
w&n load++) =+ ∇ • (D ∇− eof specimen
h
(3)
c
s
∂α radius∂αh ∂ting∂devices.
Theh roundedoftheloadingdecsvices is decided to not restrict the displacement ofthespecimento shearis theslopeforce.
of the sorption/desorption
where∂we/∂hdue
Beforetesting
for
ECC,moisture
the loading
is conducted
isotherm (also calledcapacity).
The
for
ordinary
reinforced
concreteandmortarspecigoverning equation (Equation 3) must be completedmen
in order toboundary
confirm theof this loadby appropriate
andadaptability
initial conditions.
ingThesystem.
It is considered
no dataofexists
previrelation
between thethatamountevaporable
ously
for
single
plane
shear
test
of
reinforced
water and relative humidity is called ‘‘adsorption
concrete
under controlledtensile stress.
isotherm”specimen
if measured
with increasing
relativity

humidityand ‘‘desorption isotherm” in the opposite
Side viewcase. Neglecting their difference (Xi et al. 1994), in
the following, ‘‘sorption isotherm” will be used with
Specimenreference to both sorption and desorption
conditions.
By the way, if the hysteresis of the Tensile
moistureforce
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 therelativity humidity. The shape ofUnit:thesorptionmm
isothermfor HPC is influenced by many parameters,
Top view
especially those that influence extent
rate of the
Slidingand
devicechemical reactions and, in turn, determine pore
structure and pore size distribution (water-to-cement
ratio, cement chemical Specimen
composition,
SF content,
Screw bolt
curing time and method, temperature,mix additives,etc.). In the literature various formulations can befound to describe the sorption30kNisothermof normal
Oil jack
concrete (Xi et al. 1994). However, in the present
paper the semi-empirical expression proposed by
Norling Mjornell (1997) is adopted because it

30kN
Oil jack
Figure 2. Loading system for single plane
shear
test.

Proceedings of FraMCoS-7, May 23-28, 2010

explicitly
accounts
the evolution
2.2
Loading
test for for
reinforced
concreteof hydration

reaction and SF content. This sorption isotherm
reads
2.2.1 Specimen and materials
The reinforced concrete / mortar specimen for single⎡
plane shear test under tensilestress is shown⎤ in Fig1⎢⎥ure
) 1 − is 100mm square.
we (h3., α cThe, α ) sectional= G1 (α c , α ssize
+ The
⎢
⎥
s
∞
(g α
10position− αofc )hspecimen
notches are set at the central⎢⎥
1 ce plane⎦(4)and the dimension of ⎣shear
is 70mm
and
40mm for reinforced concreteand∞ − α )h (NC)⎡ 10(g αspecimen
⎤
c
1 c
⎢
⎥
mortar specimen(MT),respectively.
K1 (α c , α s ) e−1⎢ are embedded. The⎥ yield
Four D4 reinforcements⎣⎦
strength and tensile strength of D4 is 290MPa and
377MPa,respectively.
Maximum
size representsof coarse agwhere the
first
term
(gel
isotherm)thegregatefornormalconcreteis
15mm.
Compressive
physicallybound(adsorbed)
water
and
the
secondstrengthof normalisotherm)
concrete represents
and mortar theat thecapillary
testing
termis (capillaryage
30.0MPa
and
50.6MPa,
respectively.
water.
This
expression isis thevalidlevel
onlyoffortensile
low content
The
test
parameterstress.
the amountofof SF.tensile
The coefficient
G1 represents
The
stressis
kept
constant
through
shear
water
per
unit
volume
held
in
the
gel
pores
at
100%
loading.
The targetand
tensilestressis from 0 to
14kN
relative
humidity,
itspecimen
canbe expressed(Norlingasthetensileforce.
The
identifications
are
Mjornell 1997) as
indicated by the combination of the type of matrix(NC or MT) and the target tensile force (00 – 14).

c α c+ ks α sG (α c α s ) = k vg
c vg s(5),

1

Screw bolt

Notchwhere kcvg and ksvg are material parameters. From the
maximum amount of water per unit volume that canfill all pores (both capillary pores and gel pores), one
can calculate K1 as one obtains
NC:70mm
MT:40mm

J = − D ( hP, T )∇hw −0

K (α c α s ) =
,

⎡
⎢
−
⎢1 −
1
⎢
⎣

α s + 0.22α s G

0.188

c

s

⎛

10⎜

e

⎝

g α c∞ − α c ⎞⎟h ⎤⎥1

D4
deformed bar
D4鉄筋

⎠

⎥
⎥
⎦

(6)

g αc − αc h−eFigure 3. Reinforced concrete specimen and
mold.
1

⎛
10⎜
⎝

⎞
⎟
⎠

∞

1

1

TheTestmaterialparameters k vg and k vg and g1 can
2.2.2
results
becalibratedbyfittingdata relevant
toThe crack patterns
afterexperimentalloading are shownin Figure
freeIn case(evaporable)
waterlevelcontent
in stress,
concreteat
4.
of the lowerof tensile
comvarious
ages
(Di
Luzio
&
Cusatis2009b).
pressive strut of the matrix is observed and cracks
are formed diagonally between the loaded and supported
position (MT08,
NC08). In case of the higher
2.2 Temperature
evolutionlevel of tensile stress, on the other hand, one crack is
Note that,at early
since the(MT14,
chemical
reactionsformedat thecenterage,of specimen
NC14).
cs

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 notexceeding 100°C (Bažant
& Kaplan NC08
1996), by
MT08
Fourier’s law, which reads
q = − λ ∇T

(7)where q is the heat
flux, T is theNC14absolute
MT14temperature,
and
λ
is
the
heat conductivity; in thisFigure 4. Crack patterns after loading.

Shear stress is measured by shear force divided
by area of shear plane. Shear deformation and axial
deformation are measured by displacement transducers which are set as shown in Figure 2. Shearstress versus axial deformation and shear deformation relationships are shown in Figure 5 and Figure
6. Both for NC specimens and MT specimens, themaximum shear stress decreases as the target tensilestress increases. And the deformation at maximum
stress increases as the target tensile stress increases.
NC00
NC08
NC14

6
4
2
0

Shear stress (MPa)510Axial deformation (mm)

15

NC00
NC08
NC14

6
4
2
0

5
10
Shear deformation (mm)15Figure 5. Shear stress – deformation relationship for NC specimen.

Shear stress (MPa)

8
MT00
MT08
MT14

6
4

0

Shear stress (MPa)8510Axial deformation (mm)

15

MT00
MT08
MT14

6
4
2
0

0 .5
⎡
⎛ m − vu ⎞ ⎤
w
∂
⎟ ⎥
⎢
− = ∇ •0.5J ⎜⎜(2)∂tvu ⎟⎠ ⎥⎢π
−1 ⎛ m − vu ⎞⎝⎟⎟ −= m ⋅ ⎢ − tan ⎜⎜
⎥
v
2
vu ⎠
⎝ The
+ u w ⎥can be expressed a⎢water 1content
vu ⎥⎢⎣of the evaporablem −water⎦ we (capillary wa

vapor, and adsorbed water) and the non-e(chemically bound) water wn (Milm = 3.826 f c
Pantazopoulo & Mills 1995).(3)It is reas
assume that the evaporable water is a fu
where vu = shear relative
strength;humidity,p = reinforcement
fyh, degreeratio;of hydration= yield strength ofdegree
reinforcement;
σN =reaction,
normal stressof silica fumeαs, i.e. we=w
(compression is =positive);and fc = sorption/desorption
compressive
age-dependent
strength of concrete.
(Norling Mjonell 1997). Under this assum
Figure 7 shows
relationships
betweenob- Equatiby thesubstituting
Equation
1 intoserved test resultsfor NC and MT specimens andobtains

calculated shear strength. Though specimens tested
in this study are subjectedcalcu∂w∂w∂w ∂h under tensile stress,e α&lation value by Equation(2)
evaluates
test
re-e α&s + w(D ∇) =+ ∇ •well
− e
h
c +
h
∂
α
∂
α
∂
∂
h
t
sults. The calculation value underestimatesc test res
sult in the region of higher tensile stress (lower p
fy+σN). It is considered
that
Dowel
which
is
is theaction
slope ofthe sorption/
where ∂we/∂hignored in Equation(2)
relatively
becomeslargerisotherm (also called moisture capac
rather than contactgoverning
action in equation
this region.
(Equation 3) must be

by appropriate boundary and initial conditi
The relation between the amount of eNC
water
and relative humidity is called ‘‘
6isotherm” if measured with increasing
4
humidity and ‘‘desorption isotherm” in thcase. Neglecting
their difference (Xi et al.
Test
2
the following,Mattock-Hawkins‘‘sorption
isotherm” will be
Li-Maekawa
0referencetobothsorptionand desorption c
0
1
2
3
4
5
By thep f +σway,(MPa) if the hysteresis of theisotherm would be taken into account, two
8MTrelation, evaporable water vs relative humi
6
be used according to the sign of the varia
relativity humidity. The shape of the4
isotherm for HPC
is influenced by many p
Test
2
Mattock-Hawkins
especially thosethat
influence extent and
Li-Maekawa
chemical
reactionsand,in turn, determ
0
0
1
2
3
4
5
structure
and
pore
size
distribution(waterp f +σ (MPa)cementshearchemicalcomposition, SF
Figure 7. Comparisonratio,of maximum
stress.
curing time and method, temperature, mixetc.). In the literature various formulatio
2.3 Loading test foundfor ECC
to describe the sorption isotherm
concrete (Xi et al. 1994). However, in thpaper
the semi-empirical expression pro
materials
2.3.1 Specimen and
Norling Mjornell (1997) is adopted b
820p⋅ fy +σ N

1
3

8

0

volume of concrete (water content w) be eq
divergence of the moisture flux J

05
10Shear deformation (mm)15Figure 6. Shear stress – deformation relationship for MT
specimen.

The measured maximum shear stress is comparedwith evaluation method proposed previously. Equation (1) (Mattock & Hawkins 1972) and Equation(2) and (3) (Li & Maekawa 1990) have been proposed by single plane shear test results. However,these tests were done under compression normal
stress. Equation (1) has been proposed by regressionvu (MPa)0& Najjar 1972). The moisture mass balanc)vu = 1.38 + 0.8( p ⋅thatf y +theσ N variationin time of the(1)water mas

y

N

y

N

vu (MPa)

Shear stress (MPa)8= − D (h, TEquation) ∇hanalysis from testJ results.
(2) and (3) havebeen introduced by considering contact behavior ofcrack surface and theof roughness
The probability
proportionality
coefficientof D(h,T)
crack surface. The
Dowel
actionhas
notmoisture permeability and been
it is ain-nonlinea
volved in Equationof(2)(3). humidity h and temperature
theandrelative

Proceedings of FraMCoS-7, May 23-28, 2010

J = −ECC
D (h, Tspecimen) ∇hThefor single plane shear test under(1)
tensile stress is shown in Figure 8. The sectional sizeis 100mm
square. The notches
are setat theis central
The proportionality
coefficientD(h,T)
calledpositionofspecimen
andthedimension
shear
moisture permeability and it is a nonlinear offunction
plane
70mm.humidity h and temperature T (Bažant
of theisrelative& Najjar 1972). TheNotch
moisture mass balance requires
Screw boltthat the variation in time of the watermass per unit
volume of concrete (water content w) be equal to thedivergence of the moisture flux J− ∂ =8.∇ECC• J specimen.
Figure∂wt

isotherm for HPC is influenced by many parameters,
especially those that influence extent and rate of theTable
1. Propertiesof PVAfiber.in turn, determine pore
chemical
reactionsand,Fiberpore size distribution
Tensile(water-to-cement
Elastic
structure
andFiber
Diameterlengthstrengthmodulusratio, cement
chemicalType(mm) composition, SF content,
(mm)
(MPa)
(GPa)
curingtime
andmethod,temperature,mix
additives,
PVA
12
0.04
40
etc.). In theliterature
various1600
formulations
can befound
todescribethesorption
isothermofnormalTable 2. Mix proportion of PVA-ECC.
concrete
etbyal. 1994).
However,in the present
Water
by (XiSand
Air content
Fiber vol.
binderratio (%) expression
fraction proposed(%)
paperratio
the binder
semi-empirical
by
0.42
0.77
10
1.5, 2.0 because it
Norling Mjornell(1997)
is adopted

e

⎢
⎣

xn

Kφ(αu

(2)

Polyvinyl
fiber
shown in
The water alcohol
content (PVA)
w can beexpressed
as Tablethe sum1
isfor ECC.water
The volume
fractionwater,of PVAfiwaterof utilized
the evaporable
we (capillary
ber
is
set
to
1.5%
(PVA15)and
2.0%(PVA20).
Mix
vapor, and adsorbed water) and the non-evaporable
proportionis shown(Mills2. Mod1966,(chemicallyof PVA-ECC
bound) water
wnin Table
erate-heatPortlandcementand Itflyisash(type II by
Pantazopoulo& Mills
1995).
reasonabletoJIS
A 6201)
as binder.
limestone
assumethat arethe used
evaporable
waterCrushed
is 2a functionofwhich
surfacearea is of2500cm/g is usedasrelativespecifichumidity,h, degree
hydration,
αc, andthefineofaggregate.
PVA-ECC
mixed
using
we=wbye(h,αc,αsa)degreesilica fumereaction, was
αs, i.e.
practical
mixer with volumeof 1m3. The specimens= age-dependent
sorption/desorption
isotherm
wereuniformly
continuously
castassumption
into the mold.
(Norling
Mjonelland1997).
Under this
and
After
steam
curing
for
8
hours
with
35
Celsiusdeby substituting Equation 1 into Equation 2 one
gree,
the specimens were cured in atmospheric enviobtains
ronment.
properties
are shown in∂wof PVA-ECC∂Mechanicalw ∂he α& + ∂wwith
e α&100&φ-200mm
Table) = test pieces+ ∇ • ( D ∇type− e 3. Cylinderh(3)cs + wn
h
∂
α
∂
α
∂
∂
h
t
size are used for compression
c test. sTo determine thetensile characteristics of HPFRCC, lots of direct tension(Kanakubo/∂h is thehave
slopebeenof theproposed
sorption/desorption
wheretest∂wemethods
2006).
Tensilecharacteristics
of
HPFRCCare conisotherm (also called moisture capacity).
Thesideredto equation
be sensitiveby boundary
andgoverning(Equation3) mustcondition
be completedsize
and shape of
specimen.
Instituteby appropriate
boundary
andJapaninitialConcreteconditions.
(JCI)
gives
of the
TheStandard
relation (JCI-S-003-2007)
between the amountof oneevaporable
simplemethods
ofreversecalculation
frombending
water and relative humidity is called ‘‘adsorption
characteristics
to tensile withones. increasing
The assumptionofisotherm” if measured
relativity
stressdistribution
for pure isotherm”
bending region
humidityand ‘‘desorption
in theindicatedoppositein
JCINeglectingStandard istheirshown
in Figureultimatecase.
difference(Xi9.etTheal. 1994),
instrain(ε)
andtensilestrength
(f
)
can
be
obtainedtu,bthe following,‘‘sorption isotherm”t,b will be used withfromthe tomaximum
momentcurvatureby thereferenceboth sorption
and anddesorption
conditions.
bendingeasy calculation.
In thisBy the test
way,results
if thebyhysteresisof the moisture
study,
strainPVA20
andtwoPVA15
isisothermultimate
would betakenofinto
account,
different
1.07%
and0.42%,
respectively.
However,itisconrelation, evaporable water vs relative humidity, mustsidered
PVA15todoes
not showbe usedthat
accordingthe signof thestrain-hardening
variation of the
behavior
uniaxial tension
relativityinhumidity.
The test.
shape of the sorption

Proceedings of FraMCoS-7, May 23-28, 2010

Table
3. Mechanicalpropertiesof PVA-ECC.
explicitly
accounts
for theevolution of hydration
Compression
Tension*reaction and SF content. This
sorption
isotherm
Elastic
Comp.
TensileUltimate
reads
ID
Modulus strength
strengthstrain
(GPa)
(MPa)
(MPa)
(%)
PVA20
16.7
42.5 ⎡
4.79
1.07⎤
PVA15
40.1 ⎢
2.41 10.42⎥ +
) = G (α , α ) 1 −we (h, α c , α s16.2
c bendings ⎢ test10results
1* Calculated
value from
(JCI-S-003-2007)∞(g α− α )h ⎥

1

1

c

c

⎥
⎦

⎡ 10(g α ∞ − α )h
⎤
c
1 c
⎢
−M1⎥
c , α s )⎢e
⎥max
⎣
⎦

(4)

where the first term (gel isotherm) represents thephysically bound (adsorbed) water and the second
term (capillary isotherm) represents the capillary
ε tu,bt,bwater. This
expression is fvalid
only for low content
Figure9.
Stressdistribution
assumption
in JCI-S-003-2007.
the amount ofof SF. The coefficient G1 represents
water per unit volume held in the gel pores at 100%
2.3.2
methodrelativeLoading
humidity,
and it can be expressed (NorlingIncase ofreinforced
concrete specimen, tensile
Mjornell
1997)
asstress is kept constant as the target level. In case of
ECC, however,stress which is carried by fic αtensile
G1 (αafter,α ) = kc+ k s αdiffers
(5)
ber
first
cracking
by each specimen.
c s vg c vg s sMoreover, axial deformation (crack opening) at thesame tensilestressis different. For loading of ECC
ksvg arematerial parameters. From thewhere kcvg andspecimens,
the target
is set to axial deformation. The
maximum amount of water per unit volume that cantensile stress is kept constant when the axial deforfill all pores (both capillary pores and gel pores), one
mation reaches the target deformation (wexp) after
can calculate K as one obtains
first cracking 1in the pre-loading only by tensile
force. After that, shear load is ⎡applied to the speci⎛⎞ ⎤∞− α ⎟h ⎥level⎜g α
men under constant tensile stress.
Asa cresult,⎢c⎝⎠w − differs
α s + α s −G ⎢ −e⎥cof tensile stressbys each⎢ specimen. It can
be
⎥
⎦ (6)
⎣ constant is similar
said
that
the
tensilestresskept
K (α α ) =levelc ofs crack strength. ⎛⎜ g α ∞ − α ⎞⎟he ⎝ c c⎠ −
10

0

0.188

0.22

1

1

1

,

1

10

1

1

2.3.3 Test results
ksvg andg1 canmaterialofparameters
kcvg andafter
TheTheexamplescrack patterns
loadingarebecalibratedbyfittingexperimental
data
relevant
toshown in Figure 10. Only single crack is observed
freePVA15(evaporable)
waterIn content
concreteatfor
specimens.
most of inPVA20
specivarious
ages
(Di
Luzio
&
Cusatis2009b).
mens, single crack between the notches and multiple
cracks towards the loading points can be observed. It
isthat evolution
crack surfaces contact at final and2.2considered
Temperaturecompressive strut is formed.

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 notexceeding 100°C (Bažant & Kaplan 1996), by
Fourier’s law, which reads
PVA20 No.1

PVA15 No.2

wexp = 0.0245mm
wexp = 0.108mm
q = − λ ∇T
Figure 10. Examples of crack patterns after loading.

(7)

where
is theheataxial
flux,deformation
T is theandabsolute
Shearq stressversusshear
temperature,
and
λ
is
the
heat
conductivity;in thisdeformation relationships of all specimensareShear stressTensile stress

Shear stress /Target tensile stress

Shear stress /Target tensile stress

Shear stress /Target tensile stress

J = shear− D (h, Tstrength) ∇horder to express theunshown in Figure 11. Shear stress is normalized1by
PVA20
der
tensilestressfield,‘tensileshear
the target tensile stress for each specimen. Shear
0.8
strength’ is newly definedas that the coefficient D(h,T)
stress is also revised to include P-δ effect due to axThe proportionality
0.6
shear stress shows
almostconstant and it is a nonlinea
ial tensile force. In the PVA20 specimens, shear
moisture permeability
0.4with increment of
deformation,
stress increases with a certain stiffness at first. Whenof shearthe relative
humidity h andNo.1
temperaturein other words ‘shear
yielding’.
Thethe ratio of shear stress to the target tensile stress
No.2
0.2& Najjar 1972). The moistureNo.3
mass balanctensile shear strengthis 1.63MPa
andtime of the water mas
reaches to around 0.4, stiffness clearly reduces andthat
thevariation
in
0
0.5
1
0 and PVA15
0.5
1
for PVA20
axial and shear deformation increase with almost0 1.29MPa
volumeof concrete(water contentw) be eq
Axial deformation (mm)Shear deformation (mm)specimens as thedivergence
average. These
valconstant shear stress. After that, in case of smallofthemoistureflux
J1
ues are 41% and 54% of target tensile
level of target axial deformation (No.1
PVA20
0.8
– No.4), shear stress starts to increase− ∂w = ∇ • Jagain at around 0.4mm of shear de0.6∂tformation. And then, shear stress
0.4
shows the maximum and specimen
The water content w can beNo.4
expressed a
No.5
0.2
fails. In case of large level of target axwaof the evaporable water we (capillaryNo.6
ial deformation (No.5 – No.6), re0
vapor,
andadsorbed
the non-e
00.5
100.5 water) and1
increment of shear stress after 0.4mm
Axial deformation (mm) (chemicallyShear
deformationwater(mm)bound)
wn (Milshear deformation can not be observed
Pantazopoulo & Mills 1995). It is reas1and specimen fails. In the PVA15
PVA15
assume that the evaporable water is a fu
0.8
specimens, specimen fails at smaller
relative humidity, h, degree of hydration
deformation without re-increment of
0.6
degree of silica fume reaction, αs, i.e. we=w
shear stress. The difference of fiber
0.4= age-dependent sorption/desorption
No.1
volume fraction causes clear difference
No.2(Norling
Mjonell1997).
Under
0.2
No.3this assumof shear behavior.
by substituting Equation 1 No.4
into Equati0
Failure progress for ECC specimen
0
0.5
1
0
0.5
1
obtains
can be explained as follows as shown
Axial deformation (mm)Shear deformation (mm)in Figure 12;
Figure 11. Shear stress – axial and shear deformation relationship.
∂w∂w ∂he α& + ∂we α& + w
Stage I:+ ∇ • ( D ∇h ) =
− e
c
s
h
∂α
∂α
∂h ∂t
Tensile loading starts. Tensile stress
c
s
I.
II.
III, IV.
V.
increases as an elastic manner.
Tensile loadingFirst crack
Shear loading
Surface contact
Stage II:
where ∂we/∂h is the slope of the sorption/
First crack occurs. Tensile stress indiisotherm(also called moisture capac
cates first crack strength. Axial deforgoverning
equation(Equation 3) must be
mation increases with almost constantbyappropriate
boundary
and initial conditi
or slight increment of tensile stress.
Fiber
Fiber
The relation between the amount of e
After reaching the target axial deforIII IV
V
I
II
water
τ and relative humidity is called ‘‘
mation (wexp), tensile stress keeps conif measured
with increasing
Shearisotherm”
τcstant and shear loading starts.
loading
humidity
and‘‘desorption
isotherm” in thbstartsStage III:acase. Neglecting
their difference (Xi et al.
w
Shear stress increases as almost elastic
00‘‘sorption
isotherm” will be
Axial deformation the following,
Shear deformationmanner. If capacity for shear is small,referencetobothsorptionand desorption c
specimen fails. (Branch a)
Figure 12. Failure progress of ECC specimen.
By
the
way,
if
the
hysteresisof theStage IV:
isotherm would be taken into account, two
Shear deformation increases with alTable 4. Test results of ECC specimen.
relation, Tensile
evaporable water vs relative humi
most constant shear stress and tensile
TargetTargetbeusedaccordingto Shearthe sign of the varia
axial
shear
tensilestress. If capacity for shear is notIDstrengthrelativity
humidity.
The shape of thedeformation strength
stressenough, specimen fails. (Branch b)τ(MPa) by many p*
maxisothermτffor(MPa)
HPC is influenced
σt,cons (MPa) wexp (mm)Stage V:especially1.87those that influenceextent and
No.1
3.58
0.0245
2.62
Crack surfaces contact and shear stress
No.2
3.86
0.0495
1.79
2.52 in turn, determ
chemical
reactionsand,starts increasing again. Compressive
No.3
3.84
0.0598
1.84
2.70
structure1.73
and pore size2.89
distribution (waterstrut via crack surface formed. After
No.4
4.25
0.121
ratio, cement
chemicalcomposition, SF
No.5
3.72
0.124
1.43
1.43
maximum shear stress, specimen fails
curingtime
andmethod,temperature,
mix
No.6
4.28
0.161
1.13
1.13
by compression. (Branch c)etc.).
In
theliterature
variousformulatio
No.1
2.14
0.09181.15
1.78
The test results are shown in Table
No.2
2.47
0.108found to1.56
describe the1.56sorption isotherm
4. As a result, target axial deformation
No.3
2.44
0.122concrete1.25
1.57 However, in th(Xi et al. 1994).
ranges from 0.02mm to 0.16mm and
No.4
2.59
0.162paper the
1.19semi-empirical1.19 expression pro
stress,respectively. And these are 34% and 54% of
the target tensile stress (constant) is * Shear stress at ‘shear
yielding’
Norling Mjornell (1997) is adopted b
tensile strength shown in Table 3. These results also
almost same as first crack strength. In
max

f

PVA15

PVA20

exp

Proceedings of FraMCoS-7, May 23-28, 2010

J = − D ( hthat, T )∇normalhindicatestress for crack surface is possi(1)ble to keep crack strength under shear.

The proportionality coefficient D(h,T) is calledmoisture permeability and it is a nonlinear function
METHODSHEAR T (Bažant
3of PREDICTING
the relative humidity
h andFOR
temperature
CAPACITY
OF
ECC
BEAMS
& Najjar 1972). The moisture mass balance requiresthat the variation in time of the water mass per unit
3.1
Specimensfor evaluation
volumeof concrete(water content w) be equal to thedivergenceofthemoisture
fluxcapacity
J
Predicting method for shearof steel reinforced (longitudinal bars and stirrups) ECC (R/ECC)beams
A small number of previous stud(2)− ∂w = is∇ introduced.
•J
ies∂thave been reported concerning shear testing for
R/ECC beams. In this study, test results reported by the
The (Shimizu
water content
be areexpressed
sum
authors
2004 w&can
2005)
utilized as
to theconfirm(capillarywater,
wateroftheevaporable
water
wpredicting method for shear ecapacity. All beam specivapor,were
and adsorbed
and the non-evaporable
mens
tested bywater)
bending-shear
loading (anti(Mills
1966,(chemically moment)bound)as shownwater in wFigure
symmetrical13.
nPantazopoulo
Mills 1995).
It is reasonableto
The list of&specimens
is shown
in Table 5.
assumethattheevaporable
water
is
a
functionofSpecimens have the 180 x 280mm size rectangular
relative humidity,
degreeof (M/QD)
hydration,is α1.5section,and shearh,spanratio(Lc, andwe=we(h,αc,αofs)
degree of silica
fume(Sreaction,
αs, i.e.
specimen)and 1.25
specimen).
Arrangements= age-dependent
main
bars and stirrupssorption/desorption
are D13 and D4 or isotherm
D6. Pa(Norling
Mjonell1997).
Under
thisassumption
andrameters are fiber volume fraction (Vf), shear spanby substituting
Equation(pw1) andinto yield
Equation
2 one
ration,ratio of stirrup
strengthofobtainsbar (σy). Fiber volume fraction of PVA fibermain
was set to 1.5 or 2.0%. Beam specimens named by
the∂wlast∂halphabet of F are∂w designed∂w to have flexural

explicitlybefore
accounts
for theevolutionof hydration
yielding
failure.
Other
specimens
are dereactionto show
and SF
content.
This sorption
isotherm
signedshear
failure before
flexural yielding.
reads

P

⎡
⎤
⎢ Loading beam 1⎥
we (h, α c , α s ) = G1 (α c , α s )⎢1 −+⎥∞10(g α− α c )h ⎥
⎢
e 1 c
⎣
⎦
Stub

SpecimenStub⎡ 10(g α ∞ − α )h
⎤
c
1 c
⎢
(α , α for− 1⎥
Figure 13. LoadingKmethodc s )⎢eR/ECC beam specimen.
1
⎥
⎣
⎦

(4)

Mechanical properties of PVA-ECC are shown in
Table
Cylindertype (geltest pieces
withrepresents
100φ-200mm
where6.thefirst term
isotherm)
the
size
are
used
for
compression
test.
To
determinethephysically bound (adsorbed) water and the second
tensilecharacteristics
of PVA-ECC,
term (capillary
isotherm)
represents JCI
the Standard
capillary(JCI-S-003-2007)
is adopted
as only
described
in Section
water. This expression
is valid
for low
content
2.3.1.
of SF. The coefficient G1 represents the amount ofwater per unit volume held in the gel pores at 100%
relative
humidity,
it can be expressed (Norling
3.2
Outlineof test andMjornell1997)
as results

The examples of crack patterns after loading areshown in Figureand shear crack are obc 14. Bendings(α , α ) = k α c + k α sG(5)servedcracks and restrain
c ats 0.0025rad.
1
vg c Thevg multipleseffect of crack opening could be observed. In shearcsfailurespecimens,
when load becomes aroundwhere ktype
vg and k vg are material parameters. From thethemaximum
value,
deformation
concentrated
maximum amount of water per unitwas
volumethat canona certain
shear
crack.pores
Theandwidthof other
fill allpores (bothcapillarygel pores),oneeαe + ∇ • ( D ∇h ) = e α
(3)
c
s
n
h
∂α
∂α
∂h ∂t
can calculate K1 as one obtainscsTable 5. List of R/ECC beam specimen.
Vf
L
bxD
Main Bar
Stirrup
ID
M/QD
⎡
⎛
⎞ ⎤
∞
(%) slope of the sorption/desorption
(mm)
(mm)
where
∂we/∂h is the
Arrangement σy (MPa) Arrangement ⎢ pw10(%)
⎜ g α σ−⎟h ⎥wyα(MPa)1 cc⎝⎠w0 − 0.188α- c s + 0.22α s s − G1 ⎢1 − 0.00
e⎥
PVA15-00L(also called moisture capacity). The
isotherm⎥⎢
719
PVA15-15Lequation (Equation 3) must be completed
2-D4@93(6)
governing⎣ 0.15295 ⎦(α , α ) =K
2-D4@47c s
1⎛⎞ 0.30byPVA15-30L
appropriate boundary1.50
and initial840conditions.
∞10⎜ g α− α ⎟hPVA15-60L
2-D6@59
The relation between
the amount of evaporable−1
e ⎝ 1 c c ⎠ 0.60
1.5
711
334
PVA15-89L
water
andrelative
humidityiscalled‘‘adsorption
PVA15-89LF
438
2-D6@400.89 sc
isotherm”
The material parameters k vg and k vg and g1 can
PVA15-89S if measured with increasing relativity
1.25isotherm”
700 in the180
691
427
humidity
and‘‘desorption
oppositebecalibrated
by fitting
experimental
8-D13
PVA15-120S
4-D6@59
1.20data relevant toxcase.
Neglecting their difference (Xi et al. 1994), pint=2.43%
free (evaporable)- water content0.00in concreteat
PVA20-00L
thefollowing, ‘‘sorption isotherm” will be 280
used with various719
ages (Di Luzio& Cusatis 0.15
2009b). 295
PVA20-15L
2-D4@93reference
to both sorption
and desorption
conditions.
PVA20-30L
2-D4@47
0.30
1.50
840
PVA20-60L
2-D6@59
0.60
By
the way, if2.0the hysteresis of the moisture 2.2 Temperature711
evolution
334
PVA20-89L
isotherm
would be taken into account, two different
PVA20-89LF
438
2-D6@40
0.89
relation,
evaporable water vs relative humidity, must Note that, at early age, since the chemical reactions
700variation of the
691 with cement hydration and SF
427 reaction
bePVA20-89S
usedaccordingto the1.25
sign of theassociated
PVA20-120S
4-D6@59
1.20−&+& + w&

relativity humidity. The shape of the sorption are exothermic, the temperature field is not uniform
isothermfor HPC properties
is influenced
by many parameters, for non-adiabatic systems even if the environmental
Table 6. Mechanicalof PVA-ECC.
*
especially thosethat
influenceextentand rate of the Tension
temperature
is constant. Heat conduction can be
Compression
chemical
reactions
and,
in
turn,
determine
pore
described
in
concrete, at Specimen
least forfortemperature not
Elastic
Compressive
Strain at
TensileUltimateTypestructureandmodulus
pore size distribution(water-to-cement
strength
strength
strengthexceeding strain
100°C (Bažant
& Kaplan 1996), by
(GPa)
(MPa)
(%)
(MPa)
(%)
ratio, cement chemical composition, SF content, Fourier’s law, which reads
0.36additives, 4.75
1.53
pw=0.00 - 0.30% L
curing time 19.5
and method, 39.1
temperature, mix
PVA20
19.5
45.8
0.39
4.33
0.67
pw=0.60 - 0.89% Letc.). In the19.9
literature various
formulations
can be 4.79q = − λ∇T 1.07
(7)
44.3
0.48
pw=0.89 - 1.20% S
found to describe
thesorption
isothermofnormalpw=0.00 - 0.30% L
16.3
1.02
concrete
et al. 1994).35.7
However, in0.35
the present 3.83
PVA15 (Xi
pw=0.60T- 0.89%
L absolute
19.4
50.3
0.39
3.66where q 0.41
is
the
heat
flux,
is thepaper the semi-empiricalexpression proposedby 2.41temperature,=0.89conductivity;- 1.20% S
17.4
42.0
0.42
0.43and λ is thepwheatin thisNorling Mjornell (1997) is adopted because it* Calculated value from bending test results (JCI-S-003-2007)

Proceedings of FraMCoS-7, May 23-28, 2010

J = − D ( h , T ) ∇h

PVA15-00L

The proportionality coefficient D(h,T)
moisture permeability and it is a nonlineaof the relative humidity h and temperature& Najjar 1972).
The moisture mass balanc
PVA15-89S
that the variation in time of the water mas
volume of concrete (water content w) be eq
divergence of the moisture flux J

PVA15-60L− ∂ = ∇•J∂
PVA20-89S
w

PVA20-00L
PVA20-60L
Figure 14. Crack pattern of R/ECC beam specimen after loading.

t

The water content w can be expressed a

Table 7. Test results of R/ECC beam specimen.
of the evaporable water we (capillary wa
Translational angle (x10-3rad) at
At maximum load
vapor, andUltimate
adsorbed water)
Failure and the non-eIDangle*1*2
Main bar
Shear force
Trans. anglewater
wn (MilStirrup yielding-3 (chemically -3 bound) Mode(x10 rad)yielding(kN)(x10 rad)Pantazopoulo&Mills1995).
It
is reasPVA15-00L
143
8.7
10.4
S
assumethattheevaporable
water
is a fu
PVA15-15L
2.60
170
15.0
17.9
S
h, degreeof hydration
PVA15-30L
3.27
183
12.0 relative humidity,
14.3
S
silica fume reaction,
αs, i.e. we=w
PVA15-60L
8.34
296
16.9 degree of 31.1
S
PVA15-89L
11.8
22.7
344
23.3 = age-dependent>50
F -> S
sorption/desorption>50*3 (Norling Mjonell>50
F Under this assum
PVA15-89LF
13.1
10.8
270*3
1997).
PVA15-89S
8.36
296
17.5 by substituting
46.0
S
Equation
1 into Equati
PVA15-120S
7.34
344
22.0
56.1
S
obtains
PVA20-00L
183
10.6
12.6
S
PVA20-15L
206
12.8
15.2
S
∂w
∂w ∂h 22.6
PVA20-30L
5.50
209
19.0
S∂w
eα
e +23.9∇ • ( D ∇h ) = S e α c
PVA20-60L
11.1
310
18.8 −
s
h
∂
α
∂
α
∂
∂
h
t
PVA20-89L
11.5
19.2
341
19.2
43.1
F ->c Ss
PVA20-89LF
11.6
10.8
272*3
>50*3
>50
F
PVA20-89S
9.43
337
16.6
38.6
S
where ∂w44.9
e/∂h is the slope of the sorption/
PVA20-120S
9.23
19.7
406
23.9
F -> S
isotherm (also called moisture capac*1 : Translational angle when load decreases to 80% of maximum. *2 : S=Shear failure, F=Flexural yielding
governing equation (Equation 3) must be*3 : At the last point for deloading (1/20rad.)&+cracks decreased due to localized deformation. The
large difference between PVA15 and PVA20 specimen could not be recognized. In PVA15-89LF and
PVA20-89LF specimens, load is in process of increasing when translational angle becomes 0.05rad.
Main test results are listed in Table 7.
3.3 Evaluation of shear capacity
It is considered that principal tensile stress of ECC
keeps tensile strength (σt) at shear failure based on
the knowledge from single plane shear test under
tensile stress. The average of tensile stress on beam
is expressed by σt with a reduction factor (νt). It is
assumed that beam specimen exhibits the maximum
load when compressive strut is failed by principal
compressive stress. As a matter of fact, compressive
failure at crack zone of beam specimen was recognized as shown in Figure 15. Because the compressive strut has some angles with main shear crack, local compressive failure takes place at the crack
surface. This approach has same way to ordinary RC&+w

by appropriate boundary and initial conditi
The
relation
betweenforce
the atamountbeam. In case of RC,
shear
transmitting
the of ewater
andrelative
humidityiscalled ‘‘
crack surface is carried by mainly bearing action ofisotherm”
ifmeasuredwithincreasingcoarse aggregate. In case of R/ECC, the force is carhumidityand ‘‘desorption isotherm” in th
ried by bridging actionof fiber.
case. Neglecting
their difference(Xi et al.
Based on the methodof Architectural
Institute ofthefollowing,‘‘sorption
isotherm”
Japan for ordinary reinforced concrete beam (AIJ will bereference
to bothbeam
sorptionand desorption c1990), shear capacityof R/ECC
is expressed
By
the
way,
if
the
hysteresis
as follows. This method is built up from the summa-of theisotherm
would
be taken (Vinto), account,twotion of the capacityof truss
mechanism
arch
relation, evaporable water vst relative humi
be used according to the sign of the varia
relativity humidity. The shape of theisotherm for HPC is influenced by many pespecially those that influence extent andchemical reactions and, in turn, determ
After loadingstructure and pore size distribution (waterratio, cement chemical composition, SF
curing time and method, temperature, mixetc.). In the literature various formulatio
max.sorption
loadfound to describe Atthe
isotherm
concrete failure(Xi etofal.R/ECC
1994).beam
However,
Figure 15. Local compression
speci- in thmen.
paper the semi-empirical expression pro
Norling Mjornell (1997) is adopted b
加力終了後

最大荷重直後Proceedings of FraMCoS-7, May 23-28, 2010

of the relative humidity h and temperature T (Bažant

V
Vt + V1972).
(4)& =Najjara + VECCThe moisture mass balance requires

thatthe variation in time of the water mass per unitV(5)t = b ⋅ jt ⋅ pw ⋅ σ wy ⋅ cot φvolume of concrete (water content w) be equal to theV
1 −theβ ) ⋅moistureν ⋅ σ B ⋅ b ⋅flux
D 2J
a = tan θ ⋅ (ofdivergencetan θ =

(L D ) + 1 − ( L D )2∂ = ∇ • J2
β− =∂ (1 + cot φ ) ⋅ pw ⋅ σ wy ν ⋅ σ B
w
t

(6)

⎡
⎤
1
⎢
⎥
we (h, α c , α s ) 0.5= G1 (α c , α s )⎢1 −+⎥∞10(g α− α c )h ⎥
⎢
e 1 c
⎣
⎦

(2)
(8)

(9)
The water content w can be expressed as the sum⎧2.0 water we (capillary water, waterof the evaporable⎪
vapor,θ ) and the non-evaporable(10)
cot φ = andmin ⎨adsorbedj ( D ⋅ tanwater)(chemically⎪ t bound) water wn (Mills 1966,
σ wy ) − 1It is reasonable toB ( p w ⋅1995).
Pantazopoulo⎩ ν&⋅ σMillsassumethat the evaporable water is a functionV(11)of
ECC = b ⋅ jt ⋅ν t ⋅ σ t ⋅ cot φrelative humidity, h, degree of hydration, αc, anddegreeof silica fume reaction, αs, i.e. we=we(h,αc,αs)
where,=
age-dependent
sorption/desorption isotherm
V = shear capacity(Norling
Mjonell1997).
Under this assumption and
Vt = shear capacity by truss mechanism
by substituting Equation 1 into Equation 2 one
Va= shear capacity by arch mechanism
obtainsb = width of member
jt = distance between compression and∂w∂tensionw ∂h bars
e α& + ∂we α& + w&
+ ∇ • ( D ∇h ) =
− e
(3)
c
s
n
pw=∂hstirrup∂α∂α∂t ratio hcs
σwy = yield strength of stirrup
σB = compressive strength of ECC/∂hcompressiveis the slopestrutof the sorption/desorption∂weof
φwhere= angleisotherm(alsocalledmoisture
capacity). The
θ = angle of arch mechanism
governing
equation(Equation3)mustν = effective coefficient of compressivebe completed
by strength
appropriate
boundary and initial conditions.
of ECC
relation
between the amount of evaporable
D =Thedepth
of memberwater
andrelative
L = clear span lengthhumidity is called ‘‘adsorption
if measured with increasing relativity
νisotherm”
t = reduction factor for tensile strength of ECC
humidity
and‘‘desorption
σt = tensile strengthof ECCisotherm” in the opposite

case. Neglecting their difference (Xi et al. 1994), in
the following,‘‘sorption
isotherm”
willofbeECC
used with
Reduction
factorfor tensile
strength(νt),
referencetobothsorptionanddesorption
conditions.
which may express the uniformless of tensile stress
By thethecrack
way,surface,
if theis hysteresisof thisthe study,
moistureonunknown. In
0.5
isotherm
wouldtaken into account,
twomaximum
different
isadopted.
Thebeexperimental
value ofrelation,
evaporable
relative humidity,
shear
force
by R/ECCwater
beamvsspecimen
is plotted must
with
be
used
according
to
the
sign
of
the
variationof thecalculated shear capacity in Figure 16. Both maxirelativity
shape shearof thecapacity
sorptionmum
shearhumidity.
force andThecalculatedis
isotherm for HPC
is influenced
by many
parameters,
standardized
by calculated
bending
strengthin shear
especiallythat influence
rate agreeof theforce.
The those
experimental
valuesextent
showandgoodchemical
reactions
and,
in
turn,
determinement with calculated values. The average of thepore
rastructure
and
pore
size
distribution(water-to-cementtios of experimental values to calculated shear caratio, forcementpacityshear chemical
failure typecomposition,
specimens is SF
1.09.content,
curing time and method, temperature, mix additives,etc.). In the literature various formulations can be
to describe the sorption isotherm of normal
4foundCONCLUSIONS
concrete (Xi et al. 1994). However, in the present
papermethod
the semi-empirical
proposedbyTheof single planeexpression
shear test under
tensileNorling
Mjornell(1997)
isadoptedbecausestress as normal stress on crack surface is newly de it

,
,

(4)

Shear failure
Flexural yielding

g c∞ − c h−c s e⎡ 10( αα ) 1.5 ⎤0.5⎢
1 1(α , α )1⎥
Shearcapacity1⎢ / Bending strength ⎥

K

0

(7)

ν = 1.70σ B −0.333

Proceedings of FraMCoS-7, May 23-28, 2010

explicitly accounts, forPVA20
the evolution of hydration
reaction and SF content.
,
PVA15 This sorption isotherm
1
reads
Maximum shear force /Bending strength

J = − D ( h, T )∇(Vh a) and the effect of ECC (VECC). The
mechanism(1)effective coefficient of compressive strength of ECC
is treatedas same ascoefficient
the case ofordinarycon(ν) The
proportionality
D(h,T)
is calledcrete
proposedbyfib.
moisture permeability and it is a nonlinear function⎣ calculated shear capacity⎦
Figure 16. Comparison between
andexperimental one.

where the first term (gel isotherm) represents thephysicallybound shear(adsorbed)
and theonsecond
velopedto clarify
transferwater
mechanism
crack
term
(capillary
isotherm)
represents
the capillary
surface of ECC. From the test results, shear
strengthwater.
Thissurface
expression
is valid
for lowstrength.
contenton
crack
is about
half onlyof tensile
represents
the amountof SF.means
The coefficientThisthat normalG1stressfor crack
surface of
is
water
per
unit
volume
held
in
the
gel
pores
100%
possible to keep crack strength under shear. at
Predictrelative
humidity,
and capacity
it can beofexpressed(Norling
ing
methodfor shear
R/ECC beams
isMjornell
1997)
asproposed based on the arch-truss method with thesummation of the effect of ECC. The effect is exc α c + k s thepressedreduction factor (0.5)(5)to) =multiplying
G1 (α c , α byk vg
sc vg αfroms s the evaluation of tentensile strength obtained
sile properties
using
bending test. The experimental
c
s
and
k
are
materialparameters.
wherekvg goodvgagreement
values show
withcalculatedFrom
ones.themaximum amount of water per unit volume that canfill all pores (both capillary pores and gel pores), one
can calculate K1 as one obtains
ACKNOWLEDGEMENT⎤⎡⎛∞ of ⎞Koken
The authors acknowledge the kind
support⎜ g α − α ⎟h ⎥
⎢
c
c
⎝
⎠
−
α sproducing
Sangyo Co.w Ltd.
for
⎢ − especimens. ⎥c + α s s − G the
10

0

0.188

K (α c α s ) =REFERENCES
,

1

0.22

⎛
10⎜
⎝

e

1

1

⎥
⎦

⎢
⎣

g αc − αc h
∞

1

1

⎞
⎟
⎠

(6)

−1

s
Architectural
Instituteof Japan (AIJ).
Guidelines
and Concrete
kDesign
The material
parameters
kcvg 1990.
vg and g1 canfor Earthquake Resistant Reinforced
Buildings
be Basedcalibrated
by fitting
experimental
data relevanttoon Ultimate
Strength
Concept, pp.106-116.
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content2007.
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2009b).
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moment-curvature
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Kanakubo,
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the chemical
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AdvancedNo.1, pp.3-17.
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notNo.2, pp.55-75.
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RILEM
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