Engineering Structures 27 (2005) 891–908

www.elsevier.com/locate/engstruct

Localized nonlinearity and size-dependent mechanics of in-plane

RC element in shear

Masoud Soltani

a,∗

,Xuehui An

b

,Koichi Maekawa

c

a

Department of Civil Engineering,Tarbiat Modares University,Jalaale-al Ahmad Ave.,Tehran,Iran

b

Department of Hydraulic Engineering,Tsinghua University,Beijing 100084,China

c

Department of Civil Engineering,The University of Tokyo,Hongo 7-3-1,Bunkyo-ku,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 stress-carrying mechanics of reinforced concrete are greatly associated with local kinematics at crack planes

and non-uniform 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 in-plane reinforced concrete

elements for deriving size-independent/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 in-plane

elements are chieﬂy investigated since those are out of scope of non-localized 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 in-plane re-

inforced concrete (RC) elements has brought about engi-

neering success to nonlinear mechanics of reinforced con-

crete [1] and its size-independency 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.

E-mail address:msoltani@modares.ac.ir (M.Soltani).

0141-0296/$ - see front matter ©2005 Elsevier Ltd.All rights reserved.

doi:10.1016/j.engstruct.2005.01.011

As a further development,non-orthogonal 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,

non-orthogonal crack-to-crack interaction problems can be

solved with reasonably larger sized ﬁnite elements [ 1].

However,stable crack dispersion is nevertheless assumed

for its multi-crack 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 non-localized 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 size-dependent 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 size-dependent

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 non-dimensional slip

S local slip along reinforcing bar

S

cr

axial pullout of reinforcing bar at crack plane

x distance from the middle of re-bars 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 non-dimensional 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 so-called zoning method [9] implicitly results

in combination of size-dependent tension in concrete

and steel/bond size-independency and brings success

for size-effect 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 meso-level modeling

for re-evaluation of non-localized stress ﬁeld approach

with the most straightforward manner.It is also expected

that applicability of the non-localized stress ﬁeld approach

having high cordiality with nonlinear structural analyses

will be strictly and rationally deﬁned,and that the semi-

theoretical and partially empirical size-dependent 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 multi-component 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.In-plane 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.

In-plane 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 second-order 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 space-averaged 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 space-averaged 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 space-averaged 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,multi-directionally 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 re-bars 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 re-bars (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 non-dimensional

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 re-bar 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 plastic-hardening 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 re-bar;(b)

shear deformation of re-bar.

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].

non-dimensional 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 non-linearity

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 non-dimensional 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,in-plane 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 curvature-inﬂuence

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,so-called

“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

re-bars) 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 re-bar 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 size-dependent.

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 two-dimensional 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 size-independent.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 space-averaged

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 in-plane 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 one-dimensional 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 micro-cracks 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,elasto-plastic

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 energy-based 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 un-cracked 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 un-cracked 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 in-plane

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 crack-based 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 post-yield 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.Re-evaluation of non-localized 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

non-localized stress ﬁeld approach by assuming smeared

cracking but the criterion of distinct crack localization has

been rather soft in two-dimensional RC domains except for

uni-axial 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 in-plane 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 size-effect can be seen only in the unstable state where

subsequent multiple cracking is being created just after the

ﬁr st cracking.Scaled-up 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 scaled-up 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 bar-size 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 non-localized 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

multi-directions 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 post-peak 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 re-evaluation of non-localized 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 post-cracking phase of

larger strains and inﬂuence of structural geometry on the RC

nonlinearity.

The response of an RC in-plane 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 (one-dimensional 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,non-localized uniform stress ﬁeld approach

keeps rationality of size-independency.

The size-dependent post-peak 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 multi-directional 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 size-dependency.

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 in-plane 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 two-dimensional states.

(2) The exact stress ﬁeld approach mathematically proves

the size-independency 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 non-localized 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,

size-dependency 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 size-dependent degradation of

interlock stress transfer has been empirically taken into

account in the past non-localized 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 post-peak analysis.The exact local stress

ﬁeld approach offers much stricter explanation on this

aspect.

References

[1] Maekawa K,Pimanmas A,Okamura H.Nonlinear mechanics of

reinforced concrete.SPON Press;2003.

[2] Collins MP.Toward a rational theory for RC members in shear.

Journal of Structural Division,ASCE 1978;104(4):649–66.

[3] Vecchio FJ,Collins MP.The response of reinforced concrete to in-

plane shear and normal stresses.Publication No.82-03,University of

Toronto;1982.

[4] Vecchio FJ,Collins MP.The modiﬁed compression ﬁeld theory for

reinforced concrete elements subjected to shear.Journal of American

Concrete Institute 1986;83(2):219–31.

[5] Hsu TTC.Nonlinear analysis of concrete membrane elements.ACI

Structural Journal 1991;88(5):552–61.

[6] Pang X,Hsu TTC.Behavior of reinforced concrete membrane

elements in shear.ACI Structural Journal 1995;92(6):665–79.

[7] Okamura H,Maekawa K.Nonlinear analysis and constitutive models

of reinforced concrete.Tokyo (Japan):Gihodo-Shuppan;1991.

[8] Maekawa K,An X.Shear failure and ductility of RC column after

yielding of main reinforcement.Engineering Fracture Mechanics

2000;65:335–68.

[9] An X,Maekawa K,Okamura H.Numerical simulation of size effect in

shear strength of RC beams.Journal of Materials,Concrete Structures

and Pavements 1997;35(564):297–316.

[10] Okamura H,Kim IH.Seismic performance check and size effect

FEManalysis of reinforced concrete.Engineering Fracture Mechanics

2000;65:369–89.

908 M.Soltani et al./Engineering Structures 27 (2005) 891–908

[11] Bujadham B,Maekawa K.The universal model for stress transfer

across cracks in concrete.Proceedings of JSCE1992;17(451):277–87.

[12] Qureshi J,Maekawa K.Computational model for steel embedded

in concrete under combined axial pullout and transverse shear

displacement.Proceedings of JCI 1993;15(2):1249–54.

[13] Soltani M,An X,Maekawa K.Computational model for post cracking

analysis of RC membrane elements based on local stress–strain

characteristics.Engineering Structures 2003;25(8):993–1007.

[14] Shima H,Chou L,Okamura H.Micro and macro models for bond in

reinforced concrete.Journal of Faculty of Engineering,The University

of Tokyo (B) 1987;39(2):133–94.

[15] Goto Y.Cracks formed in concrete around deformed tension bars.

Journal of American Concrete Institute 1971;68(4):244–51.

[16] Dei Poli S,Di Prisco M,Gambarova PG.Shear response,deformation

and subgrade stiffness of a dowel bar embedded in concrete.ACI

Structural Journal 1992;89(6):665–75.

[17] Dulacska H.Dowel action of reinforcement crossing cracks in

concrete.Journal of American Concrete Institute 1972;69(12):754–7.

[18] Ushida Y,Rokugo K,Koyanagi W.Determination of tension

softening diagrams of concrete by means of bending tests.

Proceedings of JSCE 1991;14(426):203–12.

[19] Maekawa K,Okamura H.The deformational behavior and constitutive

equation of concrete using the elasto-plastic and fracture model.

Journal of Faculty of Engineering,The University of Tokyo (B) 1983;

37(2):253–328.

[20] Markset G,Hillerborg A.Softening of concrete in

compression–localization and size effects.Cement and Concrete

Research 1995;25(4):702–8.

[21] Rokugo K,Koyanagi W.Role of compressive fracture energy of

concrete on the failure behavior of reinforced concrete beams.

In:Carpenteri A,editor.Application of fracture mechanics to

reinforced concrete.Elsevier Applied Science;1992.p.437–64.

[22] Rizkalla S,Hwang L.Crack prediction for members in uniaxial

tension.Journal of American Concrete Institute 1984;88(44):572–9.

[23] Vecchio FJ,Chan CCL.Reinforced concrete membrane elements

with perforations.Journal of Structural Engineering 1990;116(9):

2344–60.

[24] Zararis PD.Concrete shear failure in reinforced concrete elements.

Journal of Structural Engineering 1996;122(9):1006–15.

[25] Fantilli AP,Ferretti D,Iori I,Vallini P.Behavior of R/C elements

in bending and tension:The problem of minimum reinforcement

ratio.In:Carpiniteri A,editor.Minimum reinforcement in concrete

members,1998.

[26] Li B,Maekawa K,Okamura H.Contact density model for stress

transfer across cracks in concrete.Journal of the Faculty of

Engineering,University of Tokyo (B) 1989;40(1):9–52.

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