Engineering Structures 27 (2005) 891–908
www.elsevier.com/locate/engstruct
Localized nonlinearity and sizedependent mechanics of inplane
RC element in shear
Masoud Soltani
a,∗
,Xuehui An
b
,Koichi Maekawa
c
a
Department of Civil Engineering,Tarbiat Modares University,Jalaaleal Ahmad Ave.,Tehran,Iran
b
Department of Hydraulic Engineering,Tsinghua University,Beijing 100084,China
c
Department of Civil Engineering,The University of Tokyo,Hongo 731,Bunkyoku,Tokyo 113,Japan
Received 10 May 2004;received in revised form11 January 2005;accepted 14 January 2005
Available online 8 March 2005
Abstract
Structural nonlinearity and stresscarrying mechanics of reinforced concrete are greatly associated with local kinematics at crack planes
and nonuniform stress ﬁelds in un cracked concrete regions mobilized in the vicinity of cracks,especially when much less and/or highly
anisotropic amounts of reinforcement are placed in space.An exact local stress ﬁeld approach is implemented to inplane reinforced concrete
elements for deriving sizeindependent/dependent spatial averaged stress–strain relation,stress carrying mechanics and their interaction of
high complexity.Much attention is directed to size sensitivity of overall responses involving transient tension softening–stiffening,interlock
shear hardening–softening and average yield strength of reinforcing bars,which were previously formulated based on mere superimposition
of these constituent mechanics under uniaxial stresses.Single crack localization and transitory phase of lightly reinforced concrete inplane
elements are chieﬂy investigated since those are out of scope of nonlocalized stress ﬁeld approach with smeared crack concept.
©2005 Elsevier Ltd.All rights reserved.
Keywords:Size effect;Shear;Bond;Crack spacing;Crack width;Nonlinearity
1.Introduction
Spatial averaged constitutive modeling of inplane re
inforced concrete (RC) elements has brought about engi
neering success to nonlinear mechanics of reinforced con
crete [1] and its sizeindependency of smeared crack model
ing has been theoretically and experimentally veriﬁed within
some applicability conditions in terms of reinforcement ra
tios that guarantee distributed stable cracking in space.Here,
the modeling derives from simple superimposition of con
stituent mechanics of cracked concrete in tension,compres
sion and shear.Successful compression ﬁeld theory [ 2],its
subsequent modiﬁed version (MCFT) [ 3,4] and softened
truss method [5,6] can be classiﬁed in this category of non
localized stress ﬁeld approach.The compression,tension
and shear along cracking are formulated basically without
mutual interaction.
∗
Corresponding author.Fax:+98 21 4423660.
Email address:msoltani@modares.ac.ir (M.Soltani).
01410296/$  see front matter ©2005 Elsevier Ltd.All rights reserved.
doi:10.1016/j.engstruct.2005.01.011
As a further development,nonorthogonal multi
directional ﬁxed crack theory [ 1,7] was presented to in
stall mutual nonlinear shear interaction among cracking
with different orientations so that it can strictly deal with
the interacting kinematics of shear cracks that governs the
entire nonlinearity of elements.With this generalization,
nonorthogonal cracktocrack interaction problems can be
solved with reasonably larger sized ﬁnite elements [ 1].
However,stable crack dispersion is nevertheless assumed
for its multicrack formulation and simple superposition is
similarly applied around the axis of active cracking.Then,
when light reinforcement or highly anisotropic amount of
steel is placed in space,this nonlocalized stress ﬁeld
approach tends to deviate from the reality.For solving
this problem with engineering manner,discrete zoning
approach [8–10] was adopted for lightly reinforced concrete
structures so that sizedependent capacity of RC can be
simulated with comparatively smaller numbers of degree
of freedom.Here,size independent constitutive modeling
stated above is adopted in RC zones and sizedependent
892 M.Soltani et al./Engineering Structures 27 (2005) 891–908
Nomenclature
A
s
,I
s
area and moment of inertia of bar section
d diameter of reinforcing bar
e
c
,e
max
normalized compressive strain and its maxi
mumvalue
e
p
,k equivalent plastic strain and fracture parameter
E
s
elastic modulus of bar section
f
t
,f
c
tensile strength and cylindrical compressive
strength of concrete
f
y
yield strength of steel bar
g(ε
s
) strain reduction function in bond–slip model
G
f
fracture energy of plain concrete
G
f c
compressive fracture energy
K foundation stiffness of concrete
L
b
,L
c
bond deterioration and curvature inﬂuencing
length
L
c0
curvature inﬂuencing length in the elastic
range
L
cs
length of element in compressive principal
direction
L
e
length of reinforcing bars between two
adjacent cracks
L
θ
average crack spacing
M(x),V(x) bending moment and shear force along
the bar axis
s nondimensional slip
S local slip along reinforcing bar
S
cr
axial pullout of reinforcing bar at crack plane
x distance from the middle of rebars between
adjacent cracks
y local coordinate of steel ﬁber from center of
bar section
α inclination of reinforcement
β angle between reinforcement and crack
δ deﬂection of reinforcing bars
˜ε
xy
second order tensor of strain
ε
c
compressive strain correspond to peak stress
ε
f
,σ
f
ﬁber strain and stress of bar cross section
ε
s
,σ
s
local strain and stress of steel bar
¯ε
s
,¯σ
s
average strain and stress of steel bar
ε
x
,ε
y
,γ
xy
normal strains (ε) and shear strain (γ) of
element in global directions (x–y)
ε
1
,ε
2
,γ
12
normal and shear strains of concrete in local
coordinate system(1–2)
φ(x) curvature of reinforcing bar
λ parameter reﬂecting concrete ﬂaking
µ nondimensional damage parameter
θ directional angle normal to the crack plane
ρ reinforcement ratio
˜σ
xy
second order tensor of stress
˜σ
Cxy
second order tensor of stress of concrete
˜σ
Sxy
second order tensor of stress of reinforcing
bars
σ
br
,σ
d
bridging and dilatancy stresses transfer across
crack
σ
cr
s
local stress of steel bar at crack location
σ
sd
projection of dowel stress on 1 direction
σ
x
,σ
y
,τ
xy
normal stresses (σ) and shear stress (τ) of
element in global directions (x–y)
σ
1
,σ
2
,τ
12
normal and shear stresses of concrete in
local coordinate system(1–2)
τ(ε
s
,s),τ local bond stress
τ
agg
,τ
sd
shear stress transfer by aggregate interlock and
reinforcing bars
τ
st
contribution of steel to shear transfer across
crack
τ
0
(s) intrinsic bond stress for strain equal to zero
ω
av
,δ
av
average opening and shear slipping of crack
ζ reduction factor for fracture parameter
plain concrete modeling assuming an intrinsic single crack
is applied to the plain concrete (PL) one.
Although socalled zoning method [9] implicitly results
in combination of sizedependent tension in concrete
and steel/bond sizeindependency and brings success
for sizeeffect analysis,geometrical zoning has to be
performed with engineering judgment based on detailed
arrangement of steel.Furthermore,highly nonlinear mutual
interaction among local shear stress transfer with contact
damaging [11],bond deterioration close to crack planes
and local curvature induced in reinforcing bars [12] cannot
be explicitly taken into account.These can be ignored for
the case of dispersed cracking ﬁeld,but are crucial for the
cases where a very few localized cracks may rule the whole
nonlinearity of RC domains.
Thus,the authors aim at explicitly deriving highly
localized kinematics of RC elements from the exact local
stress ﬁeld approach without any empirical smoothing of
stress ﬁelds [ 13],and offering a strict mesolevel modeling
for reevaluation of nonlocalized stress ﬁeld approach
with the most straightforward manner.It is also expected
that applicability of the nonlocalized stress ﬁeld approach
having high cordiality with nonlinear structural analyses
will be strictly and rationally deﬁned,and that the semi
theoretical and partially empirical sizedependent modeling,
such as crack shear transfer softening along cracking [1]
and averaged yield strength reduction of steel under non
orthogonal intersection with cracking [7],will be veriﬁed.
2.Exact local stress ﬁelds
RC panel is modeled as a multicomponent structural
systemcomposed of reinforcing bars and concrete as well as
their interaction.Structural nonlinearity and local stress ﬁeld
comprise deferent size dependent stress transfer mechanisms
and phenomena (Fig.1) such as:
M.Soltani et al./Engineering Structures 27 (2005) 891–908 893
Fig.1.Inplane stress transfer mechanism in RC domain and its size dependency.
(a) Local bond stress transfer along reinforcing bars.
(b) Shear and dilatancy stresses transfer through aggregate
interlock on crack planes.
(c) Bridging stress (softened tension) transfer across rough
crack.
(d) Dowel action and kinking of reinforcing bars at a crack
location.
(e) Stress transfer of concrete parallel to cracks (compres
sive stress in struts).
(f) Softened compression of concrete due to coexisting
transverse tensile deformation.
(g) Compressive damage localization between cracks.
Inplane RC structural nonlinearity can be mathemat
ically characterized by both local and space averaged
stress/strain deﬁned on gl obal Cartesian coordinate x–y
(Fig.2(a)) in terms of the secondorder tensors of stress and
strain as:
˜σ
xy
=
σ
x
τ
xy
τ
xy
σ
y
(1)
˜ε
xy
=
ε
x
0.5γ
xy
0.5γ
xy
ε
y
(2)
where σ
x
,ε
x
are spaceaveraged normal stress and strain
acting on the whole domain of RC element in x direction,
σ
y
,ε
y
are the ones in y direction and τ
xy
and γ
xy
are the
shear stress and strain in x–y direction,respectively.
Upon cracking,both local and spaceaveraged steel
stresses are activated.The crack direction is initially
assumed normal to the principal direction of the maximum
tensile stress (normal to 1 direction).At a distinct cracked
section,the local force normal to crack is carried by
reinforcing bars (σ
cr
s
) and bridging stress acting on the
concrete crack plane (σ
br
).The local shear stress along
the crack plane is carried by aggregate interlock (τ
agg
) and
894 M.Soltani et al./Engineering Structures 27 (2005) 891–908
Fig.2.Stress state in RC membranes:(a) applied stress;(b) principle
directions of applied stress;(c) local stress state at crack location.
dowel action (τ
sd
) of reinforcing bars.Due to interlocking
of aggregates at crack surfaces,the compression stress,so
called “dilatancy stress (σ
d
)”,is applied normal to the crack
surface.The local force developing along reinforcing bars
is partly transferred to the concrete between adjacent cracks
through bond stress between reinforcing bars and concrete,
while the bridging stress and stresses due to aggregate
interlock (crack dilatancy and shear stress) and dowel action
at a crack location are distinctly deﬁned just on the local
cracking planes.The equilibrium among average stress of
concrete,that of reinforcing bars and the overall applied
stress on RC in the global coordinate is expressed as:
˜σ
xy
= ˜σ
Cxy
+ ˜σ
Sxy
(3)
where ˜σ
Cxy
and ˜σ
Sxy
are tensors of concrete and reinforcing
bars in global coordinates,respectively,which represent
individual contributions of these two components to overall
response.The tensor expression yields:
˜σ
Cxy
= [R][σ
12
][R]
T
(4)
[σ
12
] =
σ
1
τ
agg
τ
agg
σ
2
(5)
[R] =
cos θ −sinθ
sin θ cos θ
(6)
where [R] is the matrix of coordinate transformation,[σ
12
]
is tensor of cracked concrete in its local coordinate system
(1–2 coordinate system) and θ is directional angle normal
to the crack plane deﬁned in counter clockwise direction
from x coordinate axis.σ
1
is spaceaveraged tensile stress
of concrete commonly crowned “tension stiffening” which
expresses the averaged total of tensile stress transfer from
reinforcing bars to concrete by bond and the bridging normal
stress transfer across cracks (local stresses).
For RC element,multidirectionally reinforced in η =
i,j,...,directions,contribution of reinforcing bars to
overall response is expressed as:
˜σ
Sxy
=
η=i,j,...,
ρ
η
¯σ
sη
cos
2
α
η
η=i,j,...,
ρ
η
¯σ
sη
sinα
η
cos α
η
η=i,j,...,
ρ
η
¯σ
sη
sin α
η
cos α
η
η=i,j,...,
ρ
η
¯σ
sη
sin
2
α
η
+[R]
0 τ
st
τ
st
0
[R]
T
(7)
where ρ
η
is reinforcement ratio in η direction and α
η
is
directional angle of reinforcement placed at η direction
similarly deﬁned as θ.¯σ
sη
is the spatial average stress of
reinforcement along its axis and τ
st
is the contribution of
steel to shear transfer across the crack as:
τ
st
=
η=i,j,...,
ρ
η
τ
sdη
cos
2
β
η
(8)
where β
η
is angle of crack with respect to rebars direction
(β
η
= θ − α
η
) and τ
sdη
is shear stress of steel bar placed
at η direction.As the local and average stress states in RC
domain shall be in strict equilibrium(Fig.3),we have:
σ
1
=
η=i,j,...,
ρ
η
(σ
cr
sη
− ¯σ
sη
) cos
2
β
η
+σ
br
+σ
d
+σ
sd
(9)
where σ
cr
sη
is local steel stress at crack location.σ
sd
is the
projection of dowel stress on 1 direction as:
σ
sd
=
η=i,j,...,
ρ
η
τ
sdη
cos β
η
sin β
η
.(10)
The average stress of reinforcing bars ( ¯σ
sη
) and concrete
(σ
1
:tension stiffening) over the element depends on the
stress–strain proﬁle along th e reinforcing bars and the
amount of stress transferred.The tensile stress transferred
from reinforcing bars to concrete is dependent on the
bar cross sectional size,length of bar over which stress
is transferred to concrete and the volume of surrounding
concrete (size dependent) as well as bond characteristics
(dependent on material properties and casting conditions of
concrete).Eq.(9) also shows the dependency of tension
stiffening on the crack and reinforcement directions.
3.Modeling stress–strain proﬁle along reinforcement
Bond property controls crack development in RC domain
with respect to crack spacing and width,which in turn
inﬂuences local concrete stre ss transfer across cracks
(aggregate interlock and bridging stress).
The steel stress at crack plane is gradually transferred
to concrete over the span of reinforcing bars,which is
M.Soltani et al./Engineering Structures 27 (2005) 891–908 895
Fig.3.Equilibrium conditions:(a) average stress state;(b) local stress state
at crack plane.
limited by adjacent cracks (crack spacing) or anchorage
length of rebars (when a single crack is developed in RC
domain).Compatibility conditions can be written based
on either average strains of concrete as formulated by
Soltani et al.[13] or local deformations at crack location.In
considering the local deformation in the vicinity of cracks
as shown in Fig.4,we have the compatibility condition
among local deformation of crack planes,axial deformation
of concrete and steel deformation which are expressed as:
S
cr
η
= ω
av
cos β
η
−δ
av
sin β
η
+ε
2
L
θ
tanβ
η
sin β
η
(11)
δ
η
= ω
av
sin β
η
+δ
av
sin β
η
(12)
where S
cr
η
and δ
η
are axial pullout and transverse shear
displacement of reinforcing bars at crack plane,respectively.
L
θ
is average crack spacing or projection of length of
reinforcing bars on 1 direction for the case of single crack
localization.ω
av
and δ
av
are average opening and sliding of
crack planes,respectively.
Steel stress and strain proﬁle can be computed by the
exact local stress ﬁeld analysis based on the compatibility
and equilibrium conditions along the bar.Equilibrium
requirement on a small segment along the reinforcing bar
yields (Fig.5(a)):
dσ
sη
dx
η
=
πd
η
A
sη
¯τ (13)
where dσ
sη
/dx
η
is gradient of steel stress along the bar,d
η
and A
sη
are diameter and cross sectional area of reinforcing
bar,respectively and ¯τ is the average bond stress along the
segment which can be computed by the amount of local slip
along the bar.Herein the bond–slip–strain model proposed
by Shima et al.[14] (can be found also in [1,7]),which
is applicable for both elastic and inelastic ranges,is fully
utilized.The local bond stress τ(ε
s
,s
η
) is expressed not only
by the local bond slip but also the local steel strain as:
τ
ε
s
,s
η
= τ
0
(s
η
)g(ε
s
) (14)
τ
0
(s
η
) = 0.73 f
c
[ln(1 +5s
η
)]
3
(15)
g(ε
s
) =
1
1 +10
5
ε
s
(16)
where τ
0
(s) is intrinsic bond stress when the steel strain is
zero,f
c
is compressive strength of concrete,d
η
and ε
s
are
diameter and steel strain respectively,s
η
is nondimensional
bond slip equal to 1000S
η
/d
η
.S
η
deﬁned as pullout slip
is computed by integrating the local steel strain over the
span length of rebar starting fromthe ﬁxed point (zero slip
point) along the bar.The local bond performance near the
interface may easily deteriorate due to splitting and crushing
of concrete around the bar [12,15].In order to consider this
effect,bond deterioration zone (L
bη
) equal to 5 times bar
diameter is considered beside the crack surface (5d
η
).Bond
stress is assumed to linearly decrease to zero at a distance
L
bη
from the crack surface,and drops to zero at a distance
L
bη
/2 from the crack surface as experimentally shown by
Qureshi and Maekawa [1,7,12] (see Fig.5(a)).
The bond stress and the local bond slip at the midway
between two adjacent cracks or at the end of anchorage must
be zero and satisﬁed as boundary conditions for the ﬁrst
ﬁn ite segment.By assuming the local strain at the boundary,
local stress and strain proﬁles of steel are computed
by solving the equilibrium and bond slip compatibility
equations segment by segment along reinforcement.For any
assumed strain at boundary,the stress–strain proﬁle along
the bar is obtained.An iterative computation is performed
until the computed pullout slip at crack location satisﬁes
the compatibility condition of deformation represented by
Eq.(11).Comparison between experiment [1,7,14] and
the exact local stress ﬁeld anal ysis along reinforcing bars
(Fig.5(b)) elucidates reliability of the model for both elastic
and inelastic ranges.After yielding of reinforcement,some
areas close to cracks come into the plastichardening zone,
while the remaining parts are still in elasticity.Therefore,
the stress–strain relationship of reinforcing bars on average
basis shows lower averaged yield stress and post yield
stiffness compared to a single bare bar behavior.
The magnitude of bond stress transfer is thought highly
size dependent,as stress transferred per unit area of smaller
bars is higher than that of bars with larger sectional area.
However,it should be noted that when the bar diameter and
anchorage length of the bar are proportionally scaled,local
steel stress and strain proﬁles along th e bars exhibit almost
the same (Fig.6),because relation of local bond stress and
896 M.Soltani et al./Engineering Structures 27 (2005) 891–908
Fig.4.Compatibility conditions between axial and shear deformation of steel and deformation of surrounding concrete:(a) axial deformation of rebar;(b)
shear deformation of rebar.
Fig.5.Compatibility and equilibrium along the bar:(a) local equilibrium along segments to compute local stress–strain proﬁle along bars;(b) comp arison
between local analysis and experiment [14].
nondimensional steel slip is almost independent on the bar
size as experimentally shown by Shima et al.This aspect
will be discussed later for the size dependency of overall
response.
The reinforcing bars under coupled axial force and
transverse displacement are bent in the vicinity of cracks
within the length of “curvature inﬂuence zone (L
cη
)” as
shown in Fig.7(a) and (b) [12].Due to localization of
curvature in the reinforcement close to the crack plane,
the axial stiffness and the mean yield strength of the
reinforcements are reduced,at the same time,some amount
of shear stress is transferred by reinforcing bars called dowel
action.In a small displacement when both the reinforcing
bar and concrete are in elasticity,the size of curvature
inﬂuence zone derives from modeling of the bar as a
classical beam resting on an elastic foundation [12,16] as:
L
c0η
=
3π
4
4
4E
s
I
sη
Kd
η
,K =
150 f
0.85
c
d
η
(17)
where K is the ﬁctitious foundation stiffness for concrete
and I
sη
and E
s
are the moment of inertia and elastic
modulus of bar section,respectively.In a large deﬂection,
M.Soltani et al./Engineering Structures 27 (2005) 891–908 897
Fig.6.Scale effect on stress distribution along the bar.
reinforcing bar and supporting concrete show nonlinearity
and consequently the curvature zone increases [12].On
the other hand,when the reinforcement is oblique to shear
plane and pushing against the less conﬁned free surface
of the supporting concrete due to ﬂaking of concrete,both
deterioration and curvature zone expand in geometry and
lead to smaller shear capacity for embedded bar [12,17].
An empirical nondimensional damage parameter is used to
consider the inﬂuence of damage on the size of curvature
inﬂu ence zone as proposed by Qureshi and Maekawa [12]
and Soltani et al.[13]:
L
cη
= µ
η
L
c0η
(18)
µ
η
= λ
η
1 +3
1 +150
S
cr
η
d
η
δ
η
2d
η
−0.02
0.85
(19)
where S
cr
η
is the axial steel pullout slip and δ
η
are local
transverse shear displacement of the reinforcing bar at the
interface as formulated in Eqs.(11) and (12).λ
η
reﬂects
concrete ﬂaking,which is considered equal to 1 for the
case of pushing bar at right angles to shear plane and equal
to (1 + 2β
η
/π) when the dowel bar pushes against a less
conﬁned surface ( Fig.7(c)).
The shape of curvature distribution φ(x
η
),within L
cη
is
mathematically modeled by a skew parabolic form [12] as
shown in Fig.7(d),which was developed so as to satisfy the
compatibility condition as:
δ
η
= 2
L
eη
2
φ(x
η
)dx
η
(20)
where L
eη
is the length of reinforcing bars between two
adjacent cracks (Fig.7(b)).
Due to the local curvature developed in the reinforcing
bar in the vicinity of crack,local strains within the bar cross
section are not uniform and the strain and stress at outer
ﬁbers of bar cross section are different from inner ones.
Then,inplane section hypothesis yields:
ε
f
= ε
s
(x
η
) +φ(x
η
)y (21)
where y is the local coordinate of steel ﬁber that is the
distance from the gravity center of the bar cross section,
ε
s
(x) is the averaged tensile strain of the steel cross section
and ε
f
is the ﬁber strain of the reinforcing bar.For any
ﬁber strain,the ﬁber stress (σ
s f
) and its average over the
cross section area are computed.The computed average
tensile strain of each cross section in the curvatureinﬂuence
zone shall satisfy the stress equilibrium along the steel bar
(Eq.(13)).The internal cross sectional forces,including the
bending moment M(x) and shear force V(x) along the bar
axis are also computed as follows:
M(x
η
) =
d/2
−d/2
σ
f
(φ(x
η
),y) · y · dA
sη
(y) (22)
V(x
η
) =
dM(x
η
)
dx
η
.(23)
Then,shear force carried by reinforcing bars,socalled
“dowel action”,is explicitly calculated by Eq.( 23) in terms
of local curvature and kinking of steel around intersecting
crack plane in shear.Also,the effect of shear stress due to
bending curvature on the yield stress of the bar (kinking of
rebars) can be taken into consideration by applying the Von
Mises yield criterion and isotropic hardening rule details of
which can be found in [12,13].Using the above iterative
method,strain and stress proﬁles along the bar as well as the
induced curvature and subsequent bending force and dowel
action for any amount of crack opening and shear sliding
(Eqs.(11) and (12)) can be computed in a strict manner.
The typical distribution of these parameters along the bar is
shown in Fig.8.The proposed local analysis procedure has
also this advantage that the inﬂuence of axial stress on the
shear transfer ability of reinforcing bars and its nonlinearity
is directly taken into account in the analysis.
4.Local stress transfer across crack surfaces
4.1.Shear stress transfer
When shear displacement is mobilized at the crack
plane,shear stress identiﬁed as dowel action at the crack
plane is transferred by sectional shear which is equilibrated
with distributed bending moment (associated with local
curvature) along reinforcing bars as stated in Section 3.At
the same time,large magnitude of shear stress is transferred
through aggregate interlocking between rough surfaces
of the crack.Shear interlocking of aggregate particles
is normally accompanied by widened cracks (dilatancy),
which subsequently increases in the axial stress when
reinforcement is placed across cracking.
In this study,aggregate interlock involving shear
and dilatancy is modeled with the concept of contact
density [11],and implemented as detailed by the authors
898 M.Soltani et al./Engineering Structures 27 (2005) 891–908
Fig.7.(a) Local deformation of reinforcing bars at crack;(b) modeling of rebar by BEF method;(c) ﬂaking of concrete for oblique bars;(d) model of in duced
curvature in the vicinity of crack [12].
in Ref.[13].The stress transfer is computed based on the
kinematical relative displacement of crack faces (opening
and sliding) with consideration of the path dependency
of local contact plasticity and irrecoverable damaging of
individual contact units which is expressed by the softened
nature of local contact stress–displacement relation.
Local bond pullout slip is highly related to the bar
size.Since the pullout is also the source of shear transfer
stiffness as a conﬁ nement agent to the crack dilatancy,
local shear stress versus shear relative displacement
relation along a crack is thought to be sizedependent.
To clarify this nature,numerical simulation is shown in
Fig.9 where a scale effect on the shear stress transfer,
involving both aggregate interlock and dowel action for
an RC interface (a discrete crack) subjected to shear and
associated axial pull out (induced by dilatancy stress),
is identical.The length of examined domains was set
forth to be 50 times the diameter of reinforcing bars
(anchorage length = 25d),which is large enough to have
full stress development.This is a twodimensional case
where both concrete cross section and axial length are
proportionally scaled.The result indicates that while the
nominal shear stress transfer across crack versus shear
displacement is highly dependent on the diameter of
bars (cross section size dependent),its value versus the
normalized shear displacement (shear displacement divided
by bar diameter) as well as versus average tensile strain
(Fig.10(b)) is almost sizeindependent.The amount of
conﬁnement at the interface is dependent on the bond.
Smaller bars offer higher conﬁnement at the crack interface,
which reduces the crack opening,and in turn increases the
stiffness of shear transfer against shear slip.
On the other hand,when the bar diameter and anchorage
length are simultaneously scaled up,the steel stress and
strain distributions along the bar show almost the same
proﬁle as experimentally shown by Shima et al.[ 14] (see
Fig.8.Typical distribution of stress–strain,curvature,bending moment and
induced shear force along the bar.
also Fig.6).Thus,if the same steel stress at crack location
is induced by dilatancy stress,the average strain of bars
becomes similar no matter how large the bar diameter is.
This numerical observation also coincides well with the size
independent experimental results on RC interface conducted
by Qureshi and Maekawa [12].
In Fig.10,the same cross section as Fig.9 is analyzed
again while the constant element length equal to 50
times the largest diameter is assumed for all cases (one
dimensional scaled).It can be seen that the shear stress–slip
relation is not affected by the element length.Increase
in the element length has no inﬂuence on the amount
of conﬁnement to the interface.This leads to the same
crack opening path at the interface as the previous one.
But,the different length results in dissimilar spaceaveraged
relation in tension.The size dependency of local stress
transfer at the crack surface is a source of this size
dependency and nonlinearity of RC members both for
joint interface (discontinuity between beam–columns or
column–foundations) and inplane members when a single
M.Soltani et al./Engineering Structures 27 (2005) 891–908 899
Fig.9.Scale effect on total shear stress transfer across crack (aggregate interlock and dowel action).
Fig.10.Two and onedimensional scale effect:(a) geometry for case 1 and 2;(b) size independency of shear stress–average tensile strain for case 1;(c) total
shear stress versus shear deformation (comparison between case 1 and 2);(d) size dependency of shear stress–average tensile strain for case 2.
crack is activated herein.Further discussion will be made in
subsequent sections.
4.2.Tensile stress transfer
When a discrete crack starts to localize,a narrow band
with microcracks as fracture process is reported prior to
development of a single continuous crack plane.Here,some
residual tensile stresses remain normal to cracks.As a result,
tensile stress at the crack plane does not completely drop
to zero and concrete shows softened tension after cracking.
Uchida’s model [18] expressing a relationship between crack
width and bridging stress is adopted in this work (Fig.11).
σ
br
= f
t
1 +0.5ω
av
f
t
G
f
−3
(24)
where f
t
is tensile strength of concrete and G
f
is fracture
energy of plain concrete.
5.Concrete in compression
For concrete continuum between cracks,elastoplastic
and fracture model (EPF) [7,19] is adopted.The softened
900 M.Soltani et al./Engineering Structures 27 (2005) 891–908
Fig.11.Bridging stress.
feature of cracked concrete in compression is expressed in
terms of coexisting transverse tensile strain as:
σ
2
= −2k(e
c
−e
p
) f
c
k = ζ exp
−0.73e
max
1 −exp(−1.25e
max
)
e
p
= e
max
−
20
7
1 −exp(−0.35e
max
)
(25)
where σ
2
is compressive stress,e
c
is normalized compressive
strain (e
c
= ε
2
/ε
c
).ε
c
is strain corresponding to peak
stress and e
max
is the maximum normalized strain in
the past loading history,e
p
and k are equivalent plastic
strain and fracture parameter,respectively.The reduction
of compressive stiffness and strength after propagation
of transverse cracks is mathematically considered by
reduction factor ζ to the fracture parameter as a
function of tensile strain perpendicular to the crack
plane (Fig.12(a)).The abovementioned relationship is
not an energybased formulation but just consistent with
a speciﬁc size (about 200 mm) of elements.With the
same manner as compression damage model (CDZ) by
Markest and Hillerborg [20],the EPF model is modiﬁed
based on the hypothesis of the characteristic compressive
damaging size.The modiﬁed equivalent fracture parameter
is used instead of original fracture parameter in Eq.(25)
as:
k
c
= ζ
c
k
ζ
c
= e
1−
(1.6ε
c
+0.001) f
c
L
cs
G
f c
1.5
c
,e
c
≥ 1 (26)
where G
f c
is compressive fracture energy which is assumed
to be a material property.L
cs
is length of element in
compressive principal direction (length of struts in crack
direction).For L
cs
less than damage zone (which is assumed
herein constant equal to 200 mm),ζ
c
is considered equal
to 1.0 as damage is localized in the whole length.The
comparison between the model and experimental works of
Rokugo and Koyanagi [21] shows the reliability of the model
as shown in Fig.12(b).
Fig.12.(a) The inﬂuence of coexis ting tensile strain on compressive
behavior of concrete;(b) size dependency of response (comparison with
experiment).
6.Exact local stress ﬁeld analysis with crack propaga
tion
By satisfying the equilibrium and compatibility condi
tions at a crack plane as well as uncracked concrete,the
local stress and crack deformation can be obtained with the
size dependent constitutive models of local stress transfer
discussed as before and the authors’ exact local stress ﬁeld
analysis [13] is hereafter summarized as follows.
Prior to cracking,concrete is modeled as an elasto
plastic and fracture continuum by using biaxial EPF model
[7,19].The biaxial stiffness and biaxial Poisson’s ratios are
formulated dependent on the strain–stress paths.The ﬁrst
crack is introduced in the direction of principle tensile stress
when its value reaches the cracking stress.After cracking,
the implicit iterative solution is conducted in terms of local
deformation of crack (ω
av
and δ
av
) and ε
2
as primary un
known parameters:
(a) The axial and shear deformations of steel bars are given
by Eqs.(11) and (12) and subsequently the local steel
stress–strain proﬁle along th e bars as well as the average
stress and strain are obtained.
(b) The local stress conditions at crack planes are computed
by the amount of local deformation of crack planes.
The average tension in concrete continuum between
adjacent cracks (tension stiffening) is obtained from
the local steel stress as formulated in Eq.(9).The
possibility of a newly formed crack is checked with the
maximum local concrete stress in uncracked concrete
M.Soltani et al./Engineering Structures 27 (2005) 891–908 901
region.When the maximum tensile stress in concrete
reaches cracking stress,a newcrack is introduced in RC
domain and computation is carried out with the updated
crack spacing.Since,in reality,it is not possible for
two cracks to initiate simultaneously,as experimentally
shown by Goto [15],Rizkalla and Hwang [22],it is
assumed in each cracking state that just one new crack
is numerically created,and the average crack spacing
(L
θ
) is computed by dividing the length of element in
the direction normal to cracks by the number of already
generated cracks.
(c) The average strain of RC element in global coordinate is
obtained by transformation of average concrete strains
on the local coordinate along a crack here noting that
ε
1
= ω
av
/L
θ
and γ
12
= δ
av
/L
θ
.The concrete and steel
stress computed in (a),(b) should be in equilibriumwith
applied stress deﬁned on the global coordinate (Eq.( 1)).
Then,for any given applied stresses,the unknown
average strains can be determined in an iterative way.
An example of comparison between the exact local
stress ﬁeld analysis and experimental results of Vecchio
and Chan [23] is shown in Fig.13.All selected panel
elements were orthogonally reinforced with deformed bars
having 5.75 mmdiameter and subjected to different inplane
stresses.Reinforcement ratio in x and y were 1.65% and
0.825% respectively.Good agreement of experiment and
analysis is shown.Other systematic veriﬁcation was reported
in Ref.[13].In the exact local stress ﬁeld a pproach,local
equilibrium at a crack location and global equilibrium with
average stresses and strains are strictly satisﬁed ( Figs.2
and 3) and contribution of different mechanisms to overall
response can be expressed on the local stress at crack
location and/or average stress conditions as it is computed
in the smeared crackbased FEM.For more clariﬁcation,
comparison between the exact local stress ﬁeld analysis
and experimental results for RC panel PC7 [23] is shown
in Fig.14,which identiﬁes the individual contribution of
cracked concrete and steel bars to overall response in terms
of both local stress at crack plane (Fig.14(c)) and average
stress (Fig.14(d)).The structural nonlinearity consists of
cracking process under propagation,a stabilized post
cracking in which stress transferred is not enough to create
further cracking,yielding of reinforcing bars in the weaker
direction and ﬁnally the postyield hardening part of RC
element in both directions.Local stress–strain conditions in
lightly reinforced domain rule the structural nonlinearity and
failure which can numerically be investigated through this
approach.
7.Reevaluation of nonlocalized stress ﬁeld approach
Fig.15 shows the response of an examined RC
element orthogonally reinforced in both directions under
proportionally applied biaxial and shear stresses.The
parametric investigation is carried out by changing the
reinforcements in two directions.The shear resistance and
mode of failure are summarized as shown in Fig.15(b).
For highly antisotropic arrangement of reinforcing bars,
there is the possibility for rotation of crack from its initial
orientation which results in failure of RC element [24].
This can be checked by local concrete stress in its principle
direction.As seen in this ﬁgure,depending on the amount
of reinforcement ratios in two directions as well as material
properties and loading conditions,different failure modes
can generally be recognized (Fig.15(c)) as:
(a) Yielding of bars in both directions.
(b) Shear failure of bars after yielding of reinforcing bars in
one or both directions.
(c) Compressive failure after yielding of reinforcing bars in
one or both directions.
(d) Shear failure before yielding of reinforcing bars.
(e) Compressive failure before yielding of reinforcing
bars.
These types of failure can be also speciﬁed by the
nonlocalized stress ﬁeld approach by assuming smeared
cracking but the criterion of distinct crack localization has
been rather soft in twodimensional RC domains except for
uniaxial stress state.The key issue is that the exact local
stress ﬁeld approach presents a theoretically strict criterion
of localization in RC space with interacting nonlinearity of
steel and crack as explained before.
If the length of reinforcing bars is sufﬁciently large,such
that the steel stress could be transferred over it (adequate
anchorage length),the criterion for single crack localization
can be expressed by extracting the stress equilibrium
(Fig.16(a)) and noting that the local concrete stress has a
small value compared to the yield strength of reinforcing
bars:
f
t
>
η=i,j,...
ρ
η
f
y−η
cos
2
(θ−α
η
)
.(27)
In such a situation the steel stress at crack location is not
sufﬁci ent for further crack development in RC domain,and
the structural response is characterized by local deformation
at a single crack plane no matter how long the element size
is formed.In Eq.(27) f
t
is the tensile strength of concrete
and f
y−η
is yield strength of reinforcing bars in η direction.
Fig.16(b) shows this condition for RC element reinforced
in two perpendicular directions.It should be noted that,the
borderline between localized crack and smeared cracking
(line 1–1 in Fig.16(b)) can be affected by changing the
bar diameter with the same steel area or by changing the
fracture energy with the same concrete tensile strength and
Eq.(27) is just a simple criterion with the abovementioned
assumptions.
7.1.Smeared crack development
For RCelements adequately reinforced by steel deformed
bars,cracks can be propagated in a smeared conﬁguration
902 M.Soltani et al./Engineering Structures 27 (2005) 891–908
Fig.13.Comparison between local stress ﬁeld analysis and experiment [ 23].
Fig.14.Contributions of different mechanisms to the overall response:(a) examined element and loading conditions;(b) stress states at crack location and on
average;(c) contribution of different mechanisms in terms of local stresses;(d) contribution of different mechanisms in terms of average stresses.
due to bond stress transfer.However,strictly speaking in
general,the number of generated cracks,the crack spacing
and local deformation of cracks are dependent on the length
of reinforcing bars (element size) over which the steel stress
is transferred and bond characteristics which are affected
by the steel bar size.Fig.17 shows the results of the exact
local stress ﬁeld analysis fo r examined RC panel elements
under inplane stress.The scale effect was investigated
M.Soltani et al./Engineering Structures 27 (2005) 891–908 903
Fig.15.Inﬂuence of reinforcement ratio on crack process and failure mode:( a) examined element;(b) failure mode and counter of capacity;(c) typical
stress–strain relations and the failure mode.
by changing the length of elements normal to the crack
inclination.Provided that the same bar diameter is given,
simulation results show almost the same crack spacing
independent of the length of element and illustrate that a very
few sizeeffect can be seen only in the unstable state where
subsequent multiple cracking is being created just after the
ﬁr st cracking.Scaledup element brings about the higher
local stresses transfer,which shows further crack generation
accompanying a more unstable transition part of cracking.
However,after several cracks are formed,the effect of
element size can rightly be disregarded in an engineering
view point.Thus,this size independency of overall stiffness
and material relations on which the uniform stress ﬁeld
approach has been based is proved to be quite rational and
acceptable.
The inﬂuence of bar diameter is examined for different
element sizes as shown in Fig.18.The length of examined
RC domain is equal to 60 times bar diameter.Here,the
scaling is considered for both the length of reinforcing
bars and bar size.Different bar diameters show differing
magnitude of bond stress transfer,which in turn may affect
crack spacing,opening and sliding.It can be also seen
that the overall stiffness,tension stiffness and yielding
point (yielding of transverse bars) are apparently size
independent when shear strain is less than 1% (maximum
Fig.16.Single crack localization:(a) stress equilibrium;(b) criterion for
RC element reinforced in orthogonal directions.
level experienced in normal structural concrete).Even
though smaller crack spacing associated with the smaller
904 M.Soltani et al./Engineering Structures 27 (2005) 891–908
Fig.17.Size effect on overall response and average material relations for the case of constant bar diameter.
bar size leads to the higher local stress,after propagation
of cracks and its stabilization,the total amounts of stress
transferred fromreinforcing bars to concrete which controls
the overall stiffness is the same as scaledup elements
irrespective of the size.
After yielding of reinforcing bars in the weaker direction,
the amount of shear stress at crack location highly increases
to balance the local steel stress at crack location,which
accompanies increase in crack sliding and opening.At
the same time,with increasing in the crack width,the
shear stress transfer ability of concrete (aggregate interlock)
decreases (Fig.18(b)).In such a case,the overall stiffness is
reduced as the local stress at crack location cannot sustain
further load.Therefore,the response exhibits shear stress
degradation with barsize dependency.
The reduction of overall stiffness after yielding of
reinforcing bars in the weaker direction is highly related
to the local deformation of concrete at crack location.
Generally when the mode of failure is shear,the smaller bar
diameters show the smaller crack opening and sliding,and
the structural response shows evidence of higher ductility
or may prevent the shear failure as shown for the examined
element with 6 mm bar diameter (Fig.18(a)).This means
that smaller sized bars for scaled down tests (like laboratory
experiments) may cause higher shear resistance even though
the dense cracks are developed.This is one source of size
dependency in structural ductility,which can be numerically
modeled only if the localized relative displacement (not the
average strain) is taken into account in the exact local stress
ﬁeld a pproach.
Fig.18.Effect of size of steel bars on shear capacity,degradation and
ductility of RCelement:(a) overall shear stress versus shear strain;(b) stress
transfer ability of concrete at crack location.
The shear transfer across cracks is also highly associated
with roughness of crack surfaces.This factor cannot be
explicitly taken into account in the nonlocalized stress
M.Soltani et al./Engineering Structures 27 (2005) 891–908 905
Fig.19.Effect of coarse aggregate size on shear degradation.
ﬁeld a pproach.It is obvious that the contact of aggregates
particles cannot be produced if the crack width is larger
than the roughness of crack surfaces.The size of coarse
aggregates used may be changed but not exactly proportional
to the scale of elements in practice.As a matter of fact,
correlation of shear degradation and the coarse aggregates
size has not been rationally recognized in previous
researches [1–10].To clarify this issue,the previously
discussed RC element reinforced by 30 mmreinforcing bars
is numerically examined again.All parameters and the size
of element are deﬁned the same and constant,while the size
of aggregates is changed in the range from8 to 20 mm.The
analytical results shown in Fig.19 illustrate the dependency
of shear degradation (ductility) on the course aggregate
size for the case where mode of failure is shear.Smaller
aggregates size compared to the crack width provides less
contact [11] which in turn reduces the shear transfer ability
of cracked concrete.
7.2.Single crack localization
RC elements lightly reinforced by steel in single or
multidirections shows distinct crack localization in the
principal direction of tensile stress at cracking.In the last
decade,growing attention has been evidenced for studying
deformability of tensile or ﬂexural reinforced concrete
members.The applicability of bond–slip–strain model of
Shima et al.[14] in the post yield range of steel and
reliability of the universal shear transfer model of rough
crack [11] in the greater magnitude of shear slide make
it possible to numerically investigate the transitory phase
of lightly reinforced concrete membranes after cracking,
failure mode and average constitutive relations.Although
this high nonlinearity provoked by much less reinforcement
under extraordinarily higher strain is rarely experienced in
structural reinforced concrete shells and linear members,
it would sometimes arise in the postpeak conditions of
members with localized failure [1].In these conditions,
uniform stress ﬁeld approach has been applied with some
modiﬁcation based on the ﬁrst order simpliﬁcation.The
exact local stress ﬁeld approach is here advantageous for the
strict reevaluation of nonlocalized stress ﬁeld approach.
It should be noted that the cracking strength of structural
concrete could be affected by element sizes and structural
details as reported by Okamura and Maekawa [7],Fantili
et al.[25] due to drying shrinkage,casting works at fresh
stage and environmental conditions.Then,the authors’
discussion is chieﬂy directed to the postcracking phase of
larger strains and inﬂuence of structural geometry on the RC
nonlinearity.
The response of an RC inplane element lightly
reinforced in orthogonal directions under pure shear is
shown in Fig.20(a).The average shear stress–average shear
strain relation and contribution of steel and concrete to the
stress carrying mechanics of RC elements are illustrated.
The tension stiffening is attributed to bridging stress (which
vanishes with increase in crack opening) and the bond
stress transfer.Bond stress transfer also manipulates the
average yield stress and post yield stiffness of steel bars.
The dependency of post cracking response and also average
constitutive models of cracked concrete and steel bars to the
size of the examined element is shown in Fig.20(b)–(d).
At the same loading states (same local steel stresses at the
crack location),the corresponding average deformations of
concrete and steel in terms of average strain are lower for
larger element sizes.In other words,for larger element size,
steel bars start to yield at a lower average strain.
This size dependency seen in lightly reinforced concrete
can be generally illustrated for any loading conditions and
reinforcement arrangements when a single crack is localized
in the RC domain.Let us consider the case of anisotropic
arrangement of reinforcing bars,where lightly reinforced
in one direction.In such a case,shear failure may happen
before or just after local yield of reinforcement due to large
crack opening and sliding.Fig.21 shows the scale effect
on structural nonlinearity and capacity of lightly reinforced
elements.Here,it was assumed that the bar diameter
is also scaled corresponding to the element size.When
the bar diameter and anchorage length are simultaneously
scaled,the steel stress and strain distributions along the bar
show almost the same proﬁle a long the bar as explained
before.Then,the average stress–strain relationships of
steel and concrete and subsequently the structural stiffness
are independent on the element size.But as the local
deformation of crack (opening and sliding) is dependent
on the bar diameter,larger sized RC elements show
comparatively wider crack opening which in turn reduces
the shear resistance of RC elements.This means the
deformability of element highly affected by the diameter of
reinforcing bars.
On the other hand,shear critical RC elements reinforced
with the same bar and aggregate size,but different element
size (onedimensional scaled) show the same crack opening
and sliding path if reinforcing bars have enough anchorage
length.Even though the stiffness in the transition as well
as softening part is dependent on the element size (see also
Fig.20),the structural resistance and failure mode are size
independent as shown in Fig.22.As far as the capacity
906 M.Soltani et al./Engineering Structures 27 (2005) 891–908
Fig.20.Size dependency of overall response and average material relationships for lightly reinforced concrete element:(a) contribution of different size
dependent mechanisms to the overall response;(b) effect of size on transition part of overall response;(c) effect of element size on tension stiffening;(d) effect
of element size on average stress–strain relationship of steel bar.
is concerned,nonlocalized uniform stress ﬁeld approach
keeps rationality of sizeindependency.
The sizedependent postpeak softening in shear is
mainly rooted in nonlinearity of aggregate interlock.One
is the effect of crack contact loss caused by relatively
larger crack opening.The other is the contact fracturing
at each contact point on the crack roughness.If this local
nonlinearity would not be considered as the original contact
density model by Li and Maekawa [26] is,the shear
softening is much suppressed as shown in Fig.23,the size
dependency hardly vanishes.The multidirectional ﬁxed
crack modeling adopts the original contact density model
by Li et al.[26] for its simplicity and accuracy under
the strain less than 0.4% in shear and further magniﬁes a
reduction factor [1] to equivalently take into account the
local contact inefﬁciency stated above.The exact local stress
ﬁeld approach of this study presents the strict solution in
terms of mean strain with sizedependency.
8.Conclusions
The exact stress ﬁeld approach,which strictly con
siders the local nonlinearity at a crack plane and ex
plicitly formulates the local proﬁle of steel stress and
strain over the element,was applied to RC inplane el
ements in shear.The parametric study was performed
in terms of the size of reinforcing bars and that of
ﬁnite element domains and t he following conclusions
were earned.
(1) The exact stress ﬁeld approach conﬁrms two modes of
deformational ﬁelds,i.e.,di stributed smeared cracking
and distinct crack localization.The criterion of their
boundary is successfully presented with a strict but
simple manner under twodimensional states.
(2) The exact stress ﬁeld approach mathematically proves
the sizeindependency of spatial averaged constitutive
laws of RC irrespective of bar size and the volume
of ﬁnite elements when dist ributed smeared cracking
is produced.Although the local stress of steel and
displacement of individual crack is highly size
dependent,the space averaging over the element domain
put the size dependency out of sight.Thus,the
basis of nonlocalized stress ﬁeld approach was again
substantiated with strong reliability.
(3) The exact stress ﬁeld approach reveals that size
dependency arises in both shear and tension softening
in averaged stress–strain relation when cracking starts
to be localized and/or strong anisotropic reinforcement
is assumed.When shear strain develops in middle scale,
sizedependency on bar diameter and element length is
mild but it becomes strong in the large magnitude of
deformation.
M.Soltani et al./Engineering Structures 27 (2005) 891–908 907
Fig.21.Scale effect on transitory phase of lightly reinforced concrete:(a)
RC element reinforced in orthogonal directions;(b) RC element reinforced
in only one direction.
Fig.22.Independency of post cracking resistance to element size for
constant bar diameter.
(4) The size dependency concluded above is investigated
being largely rooted in the aggregate interlock and the
pullout nature of deformed bars.When the local contact
fracturing at the interface contact point is ignored or the
crack contact efﬁciency loss at larger crack opening is
neglected in analysis,the size dependency was found
to almost vanish.This sizedependent degradation of
interlock stress transfer has been empirically taken into
account in the past nonlocalized stress ﬁeld approach
Fig.23.The inﬂuence of contact fract uring and loss of contact,in large
crack opening,on degradation of response of RC element failed in shear:
(a) RC element with smeared cracks;(b) RC element with localized crack.
especially for postpeak analysis.The exact local stress
ﬁeld approach offers much stricter explanation on this
aspect.
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