Fracture Mechanics of Concrete and Concrete Structures Assessment, Durability, Monitoring and Retrofitting of Concrete Structures- B. H. Oh, et al. (eds)
ⓒ 2010 Korea Concrete Institute, Seoul, ISBN 978-89-5708-181-5
Bearing angle model for bond of reinforcing bars to concrete
O.C. Choi & H.J. Choi
Soongsil University, Seoul, Korea
G.H. Hong
Hoseo University, Asan, Republic of Korea
ABSTRACT: Experimental studies have demonstrated that bond strength increases with an increase in the
relative rib area bars under high confinement, but under low confinement, bond strength is independent of deformation pattern. This study is intended to explain the nature of the wedging action of reinforced bars as they
interact with concrete during bond failure. Analytical expressions to predict bond resistances for splitting failure of cover by fracture and shearing failure are derived, in which the bearing angle is a key variable. As the
bearing angle is decreased, the splitting bond resistance decreases while the shearing bond resistance increases. In the case of bars at a moderate level of confinement, the bearing angle is decreased to decrease the
splitting resistance and to increase the shearing resistance. The bearing angle model is useful to better understand bond mechanisms between reinforcing bars and concrete.
During the late 1950s and the 1960s, researchers
observed two phenomena accompanied by the slip of
ribbed bars: (1) concrete is split by the wedging action of the ribs and (2) concrete between the ribs is
crushed (Rehm 1957, Lutz & Gergely 1967). Researchers observed that the ribs act as wedges and
the concrete in front of the ribs crushes gradually,
resulting in a pullout-type failure and found that the
concrete in front of the ribs undergoes gradual
crushing, followed by a pullout failure (Fig. 1).
A number of researchers have derived analytical
expressions for bond mechanisms in splitting failure
(Tepfers 1979, Cairns 1979). Bond between steel bars
and concrete has been idealized in finite element
analyses. For the case of splitting failure, analytical
studies of interfacial bond have been performed to predict the bond strength of ribbed reinforcing bars (Choi
& Lee 2002), and in this paper, the fracture of concrete cover on bond behavior is addressed.
The rib geometry of deformed bars governs bond
behavior and is instrumental in guaranteeing adequate bond resistance. The influence of deformation
pattern on bond performance has been studied and
bond resistances have been observed to vary with
the rib characteristics (Tefers 1979, Skorobogatov &
Edwards 1979). Studies by Tholen & Darwin (1996)
have demonstrated that bond strength increases with
an increase in the relative rib area bars under high
confinement, but under low confinement, bond
strength is independent of deformation pattern.
With this information as background, this study is
intended to explain the nature of the wedging action
of ribbed bars as they interact with concrete during
bond failure. Analytical expressions to determine
bond resistances for splitting and shearing failures
are derived and used to predict bond strength. The
roles of the bearing angle, which is the key variable
in the expressions, are explored. The bearing angle
model is proposed for analyzing the bond behavior
of ribbed reinforcing bars to concrete and improving
the understanding of bond mechanisms of reinforcing steel in concrete structures.
face angle of crushed concrete( α )
lodged crushed concrete
rib spacing
A
rib face angle
diameter of bar
1 INTRODUCTION
rib
rib height
A
SECTION A-A
Figure 1. Flattened rib face angle by concrete crouching
(Tepfers 1979).
2 BOND RESISTANCES IN SPLITTING AND
SHEARING RAILURE
2.1 Bond resistance in splitting failure
Wedging action by the rigid steel rib of deformed
bars makes it possible to resolve bond forces into
J = − D ( hstress
, T ) ∇h σ
normal
as
n and tangential shear stress τ, (1)
shown in Figure 2. The resultant of normal components
the bar is what
places D(h,T)
the surrounding
Thealong
proportionality
coefficient
is called
concrete
in
tension.
When
a
reinforcing
barfunction
in tenmoisture permeability and it is a nonlinear
sion
concrete
under the
bearing
side ofT a(Bažant
rib is
of theP,relative
humidity
h and
temperature
placed
in
a
state
of
tri-axial
compression,
with
the
& Najjar 1972). The moisture mass balance requires
major
principal
stress,
the
bearing
stress,
σ
,
on
the
q
that the variation in time of the water mass per unit
rib
acting
parallel to
the content
bar axis.
volume
of concrete
(water
w)Normal
be equaltoto the
the
bearing
stress,
themoisture
minor principal
divergence
of the
flux J stress σr acts radially around the bar. The method of analysis (presented
here is a slightly revised and condensed form)
− ∂wbeen
= ∇ •used
J
has
previously by Choi & Lee (2002)(2)
to
∂t
evaluate the bond strength in splitting. The bond
force
the sumwofcan
thebe
bearing
stressason
sinTheequal
waterto content
expressed
thea sum
gle
rib
area
T
,
is
given
by
of the evaporable water w (capillary water, water
e
vapor, and adsorbed water) and the non-evaporable
T = Ar σq
(1)
(chemically
bound) water wn (Mills 1966,
Pantazopoulo & Mills 1995). It is reasonable to
in
whichthat
Ar =the
projected
area ofwater
rib parallel
to the bar
assume
evaporable
is a function
of
axis,
approximated
by
A
=
π
d
h
where
h
is
the
r
b
r
r
relative humidity, h, degree of hydration, αc, and
σqreaction,
= bearingαsstress
the bar rib
average
height,
, i.e. won
degree ofribsilica
fume
e=we(h,αc,αs)
acting
parallel
to
the
bar
axis.
The
frictional
force
= age-dependent sorption/desorption isotherm
between
concrete
andUnder
the steel
the inclined
(Norling the
Mjonell
1997).
this on
assumption
and
surface
of
the
rib
may
be
represented
using
the
by substituting Equation 1 into Equation 2 one
Mohr-Coulomb
relation.
obtains
−
∂w ∂h
e + ∇ • ( D ∇h) =db/2∂we
h
∂α
∂h ∂t
hr
∂w
e α& + w&
α&
+
c
s
n
d
∂α
c
s
(3)
db
θ
σq
hr
θ
θ
r
θ
n
,
∞ − α )h ⎥
(g α
⎢
c
c ⎥⎦
e
⎣
dFx = σr cot αhrdb
∞ − α )h ⎤
⎡
(g α
c
c − ⎥
⎢
K (α c α s ) e
⎢
⎥
,
1
1
π
Fx = ∫ π
2
−
2
10
,
10
1
1
+
(4)
(4)
1
1
⎣ in to Equation (4),
⎦ resultEquation (6) is substituted
ing in the final equation to predict bond resistance,
which
expressed
as follows.
where isthe
first term
(gel isotherm) represents the
physically bound (adsorbed) water and the second
(1 +isotherm)
c
µ cot α )
term
Tsplit = (capillary
Fxπ tan α
+ Arrepresents the capillary
(1 − µ tan α )is valid
sin αonly
(cos αfor
− µlow
sin αcontent
) (5)
water.
This expression
of SF. The coefficient G1 represents the amount of
water
heldforce
in thebygelfracture
pores atof100%
where per
Fx isunit
thevolume
confining
conrelative
humidity,
and
it
can
be
expressed
(Norling
crete cover or transverse reinforcement.
Mjornell 1997) as
ins shearing failure
2.2 Bond resistance
c
G (α α
)=
k α c+k α s
(5)
c s bars
1
vg bear
c against
vg s the concrete in front of
Deformed
the ribs, thus increasing shearing stress on the conand ksmay
material
parameters.
the
wherekey.
kcvgShear
crete
failure,
and the From
potential
vg arecause
maximum
amount
water per unit
volume
that
can
failure
plane
can beofestablished
for such
cases
along
fill all shear
pores stresses
(both capillary
pores
and gelinpores),
which
are high,
as shown
Figureone
3.
as onefailure
obtains
can calculate
K1shear
The
location of
surface along the possible shear crack depends on the rib geometry and the
⎡
⎛
⎞ ⎤horilevels of vertical force (confining
force)
∞ and
⎜ g α − α ⎟h ⎥
⎢
c
c
⎝
⎠
zontal force
Failure occurs when
w −(bond
α sforce).
⎥ the
c + α s s − G ⎢⎢ − e
⎥From
shear strength of the concrete key
is
overcome.
(6)
K (αforce
α ) =boundary conditions, ⎣an angle α is ⎦made
the
c s
⎛
⎞
∞
⎜ g α − α ⎟h
along the shear failuree surface,
the tangential
c cwhere
⎝
⎠ −
stresses and the radial stresses are in equilibrium.
Based
a studyparameters
by Birkeland
andBirkeland
ksvg and (1966),
g1 can
Theonmaterial
kcvg &
for
cracks
in
monolithic
concrete,
shearrelevant
strength
be calibrated by fitting experimental data
to
′
A
as
should
not
be
assumed
greater
than
0.2f
c
c
free (evaporable) water content in concrete at
shown
Equation
(6). & Cusatis 2009b).
variousinages
(Di Luzio
,
10
0
0.188
0.22
1
1
1
,
1
10
1
1
'
n = 0.2 f c Ac
V
2.2
Temperature evolution
(6)
Note that,
since the
chemical reactions
Ac is at
theearly
area age,
of cracked
surface.
where
associated
with
cement
hydration
SF reaction
defined
by the
The area of cracked surface Ac and
are
exothermic,
the
temperature
field
is
not
uniform
area of a cone with the angle of α ,
for non-adiabatic systems even if the environmental
temperature is constant. Heat conduction can be
described in concrete, at least forConcrete
temperature not
exceeding 100°C (Bažant & Kaplan 1996), by
Fourier’s law, which reads
q
= − λ∇Th
r
k
c
a
r
c
r
a
e
h
s
e
l
b
i
s
s
o
P
Proceedings of FraMCoS-7, May 23-28, 2010
,
α
min
α
e
t
e
r
c
n
o
c
d
e
h
s
u
r
c
r
we (h α c α s ) = G (α c α s )⎢ −
b
i
r
d
where ∂we/∂h is the slope of the sorption/desorption
σ
isotherm
(also called moisture capacity). The
governing equation (Equationσ 3) must be completed
τ
by appropriate boundary
and initial conditions.
P
The relation between the amount
of evaporable
σ
water and relative humidity is called ‘‘adsorption
isotherm” if measured with increasing relativity
humidityσ and ‘‘desorption isotherm”
in the opposite
α
case. Neglecting their difference (Xi et al. 1994), in
Figure
2. Stresses ‘‘sorption
acting on ribisotherm”
of bar (Cairns
the following,
will1979).
be used with
reference to both sorption and desorption conditions.
that ifthethe
stresses
along an
with
BySuppose
the way,
hysteresis
of interface
the moisture
α
,
defined
as
bearing
angle,
are
in
equian
angle
of
isotherm would be taken into account, two different
librium
the sliding
by σq and
the normal
relation,with
evaporable
waterstress
vs relative
humidity,
must
stress
by according
σn. The stress
σq sign
, is given
be used
to the
of thebyvariation of the
relativity humidity. The shape of the sorption
isotherm
⎞
(1 +HPC
µ cotis
α )influenced bycmany parameters,
σ q = ⎛ for
⎜⎜ σr those that influence
⎟ of (2)
+
especially
extent
and
rate
the
(1 − µ tan α ) sin α (cos α − µ sin α ) ⎟⎠
⎝
chemical reactions and, in turn, determine pore
structure and pore size distribution (water-to-cement
Equation
(3) ischemical
substitutedcomposition,
into Equation (1)
obtain
ratio,
cement
SFtocontent,
curing time and method, temperature, mix additives,
⎛ the(1literature
+ µ cot α )various formulations
c
etc.).
can⎞⎟ (3)be
T = AIn
r
r
+
found to⎜⎜⎝σdescribe
the
sorption
isotherm
of
normal
(1 − µ tan α ) sin α (cos α − µ sin α ) ⎟⎠
concrete (Xi et al. 1994). However, in the present
paper σtheacts
semi-empirical
expression
proposed
by
where
radically
around
the bar axis
on theit
r Mjornell
Norling
(1997)
is
adopted
because
concrete cover. The radial stress σ acts over a dis-
explicitly
accounts
tance
of dF
the the
rib, evolution
and exertsof a hydration
bursting
x below for
reaction
andconcrete
SF content.
sorption
force
on the
around This
the bar.
Figure isotherm
2 shows
reads
the
force, hr cot α exerted by σr on one rib over a
short length of the bar circumference. The component of force in the x-direction
and the summation
of
⎡
⎤
1 is given
the component force on the
⎢ perimeter
⎥ by
(7)
Sr
where q is the heat flux, T is the absolute
Bar
temperature, and λ is the heat conductivity;
in this
T
Figure 3. Shear cracks by the concrete key between bar ribs.
πdb hr
(7)
sin α
The concrete in contact with the bearing side of a
rib is in a state of triaxial compression and is subjected to very high compression from the confining
force Fx. This triaxility of stress increases the shear
strength of the concrete. The high compression is also beneficial to increase the shear strength, since the
high compressive stress modifies the magnitude and
direction of principal stress and increases the cracking load. Two parameters accounting for the increased shear strength from the tri-axial state and the
high compression, κ1 and κ2 are proposed.
Using Equation (6) and (7) and the two parameters,
the bond resistance in shearing failure is proposed by
Tshear = κ1κ 2
0.2 f 'cπdb hr
sin α
(8)
where κ1 = triaxial state parameter and κ2 = high
compression parameter. Information on these two
quantities shall be obtained from the results of future
analytical or experimental studies.
3 BEARING ANGLE MODEL
The friction coefficient µ is one of the key variables to
determine the bond resistance. Bond resistance increases as the friction coefficient increases. The contribution from cohesion to bond resistance is small and
diminishes as bars slip. The confinement force Fx , provided by fracture of concrete cover or transverse reinforcement, is proportional to the bond force. The capacity of the confinement force is made up of the
splitting resistance by concrete cover or by transverse
reinforcement, thus the confinement force has a limitation. When the confinement is determined by the structure itself, the bearing angle is the only variable in
Equation (5) corresponding to the change of bond resistance. The bearing angle of the failure surface of the
concrete in front of the ribs may be varied.
As in Equation (8), the shearing resistance is obtained by the concrete key which would be sheared
off, forming a cone with a length equal to several
times the rib height. The bearing angle is, again, the
key variable since the length of the cone is a function of the bearing angle. The bearing angle tends to
be decreased to a smaller value, to increase the
shearing bond resistance. There might be a lower
limit on the bearing angle and the minimum value of
the bearing angle can be obtained by the ratio of the
rib spacing to the rib height.
Bond strength is determined along the interface at
a state of resistance equilibrium under any failure
J = −the
D (h, Tweaker
) ∇h
condition. Normally,
mode of the two
failures, splitting and shearing failure, is considered
to govern bond strength,
but both failures
control D(h,T)
The proportionality
coefficient
bond strength because
two
failures
appears
to
moisture permeability and it isoccur
a nonlinea
simultaneously. In
cases, humidity
the bearing
angle
is
of these
the relative
h and
temperature
decreased to decrease
in
the
splitting
resistance
and
& Najjar 1972). The moisture mass balanc
increase in the shearing
resistance.inAs
that the variation
timetheofbearing
the water mas
angle reaches a certain
value
of
the
angle,
thew) be eq
volume of concrete (water then,
content
concrete key is sheared
off.ofThe
bearing is
divergence
the moisture
fluxdeterJ
mined so that the splitting resistance can be equal to
the shearing resistance,
finally the resistance it∂w = ∇and
• JTbond. Thus,
self becomes bond− strength
∂t
Tsplit = T
= T The
shear
bond
water content w can be expressed
a
(9)
wa
of the evaporable water we (capillary
vapor,
and adsorbed
the non-e
Equation (9) can
be solved
for the water)
bearingand
angle
(chemically
bound)
water
w
α . The solution for the bearing angle to determinen (Mil
& Mills
1995).
It is reas
bond strength byPantazopoulo
the bearing angle
model
is scheassume
that
the
evaporable
water
matically illustrated in Figure 4. As in cases of is a fu
humidity,
h, degree
of hydration
moderate or highrelative
confinement,
when
the splitting
, i.e. we=w
degree
of
silica
fume
reaction,
resistance is higher than the shearing resistance,αsthe
=
age-dependent
sorption/desorption
splitting resistance decreases with decreasing the
(Norling
Mjonell
1997).
Under this assum
bearing angle. As
in cases
of low
confinement,
by substituting
1 into
when the shearing
resistance isEquation
higher than
the Equati
obtains
splitting resistance, the shearing resistance tends to
be minimized and the splitting resistance tends to
∂as
∂w ∂the
we
be maximized keeping
angle∂was
h + bearing
e αhigh
&
α&s + w
+
∇ • ( D ∇h ) =
− e
c
possible.
h
∂α
∂α
∂h ∂t
c
s
where ∂we/∂h is the slope of the sorption/
isotherm (also called moisture capac
governing equation (Equation 3) must be
by appropriate boundary and initial conditi
The relation between the amount of e
T
water
and relative humidity is called ‘‘
T
isotherm”
if measured with increasing
T
humidity and ‘‘desorption isotherm” in th
case. Neglecting their difference (Xi et al.
the following,
‘‘sorption
isotherm” will be
α
α
α
reference
to both sorption
and desorption c
Bearing
Angle(α)
By the
way,
if the hysteresis of the
Figure 4. Schematicisotherm
for determination
bond strength
by
would beof taken
into account,
two
bearing angle model relation,
(different confinement).
evaporable water vs relative humi
be used according to the sign of the varia
relativity humidity. The shape of the
4 DISCUSSIONS
isotherm for HPC is influenced by many p
especially those that influence extent and
The bearing angle
model isreactions
proposed and,
for analyzing
chemical
in turn, determ
the bond behaviorstructure
of ribbed
reinforcing
to con- (waterand pore size bars
distribution
crete. Bearing angle
maycement
be reduced
so thatcomposition,
splitting
ratio,
chemical
SF
strength is maintained
to
be
less
than
pullout mix
curing time and method, temperature,
strength. The bearing
so that formulatio
the
etc.).angle
In theis determined
literature various
splitting resistance
can
be
equal
to
the
shearing
found to describe the sorption reisotherm
sistance and theconcrete
resistance
itself
becomes
bond
(Xi et al. 1994). However, in th
strength. Bearingpaper
angle the
is only
a single variable
to
semi-empirical
expression
pro
relate failure modes.
Norling Mjornell (1997) is adopted b
Splitting Bond Strength
Ac =
n
m
l
3
2
1
Proceedings of FraMCoS-7, May 23-28, 2010
J = −1.
D (Confinement
h, T )∇h effects for bars with the same rib height.
Table
(1)
Cases
Crushing Shape
The proportionality coefficient D(h,T) is called
moisture
Low Conf. permeability and it is a nonlinear function
of the relative humidity h and temperature T (Bažant
& Najjar 1972). The moisture mass balance requires
that the variation in time of the water mass per unit
Med
Conf.of concrete (water content w) be equal to the
volume
divergence of the moisture flux J
High
− ∂wConf.
= ∇•J
∂
t
Splitting
we (h α c α s ) =
,
vapor, and adsorbed water) and the non-evaporable
bound) water wn (Mills 1966,
Pantazopoulo & Mills 1995). It is reasonable to
assume that the evaporable water is a function of
Med
Conf.humidity, h, degree of hydration, αc, and
relative
degree of silica fume reaction, αs, i.e. we=we(h,αc,αs)
= age-dependent sorption/desorption isotherm
(Norling
High
Conf. Mjonell 1997). Under this assumption and
by substituting Equation 1 into Equation 2 one
obtains
(chemically
Low
Conf.
As confinement increases, bearing angles reduced
∂w
∂w
∂w ∂h
as
illustrated
in Table 1. When
is
e α& + pullout
e α& +resistance
w&n
+ ∇ • ( D ∇h ) =
− e
(3)
c
s
α
∂α
∂h ∂t bearinghangle ∂decreases
constant,
as confinement
inc
s
creases. When splitting resistance is constant, bearing
angle increases as pullout resistance increases as in
slope of the sorption/desorption
where ∂we/∂h is the
Table 2. Behavior
matches experimental observations
isotherm
(also
called
moisture capacity). The
that high rib face angle is flattened by crushed concrete
governing
equation
(Equation
3) must be completed
wedge. The bearing angle model is useful to simulate
by
appropriate
boundary
and
initial
conditions.
ribbed bars-concrete interface behavior and response.
The relation between the amount of evaporable
water and relative humidity is called ‘‘adsorption
isotherm” if measured with increasing relativity
5 CONCLUSIONS
humidity and ‘‘desorption isotherm” in the opposite
case. Neglecting their difference (Xi et al. 1994), in
Analytical expressions to determine the bond resistances
the following, ‘‘sorption isotherm” will be used with
for splitting and shearing failures are derived where the
reference to both sorption and desorption conditions.
bearing angle is a key variable. As the bearing angle is
By the way, if the hysteresis of the moisture
decreased, the splitting bond resistance decreases while
isotherm would be taken into account, two different
the shearing bond resistance increases. In the case of bars
relation, evaporable water vs relative humidity, must
at a moderate level of confinement, which represents the
be used according to the sign of the variation of the
practice, the bearing angle is decreased to decrease the
relativity humidity. The shape of the sorption
splitting resistance and to increase the shearing resisisotherm for HPC is influenced by many parameters,
tance, until reaching a certain value of angle. Bearing anespecially those that influence extent and rate of the
gle model is useful to simulate ribbed bars-concrete interchemical reactions and, in turn, determine pore
face behavior and response.
structure and pore size distribution (water-to-cement
ratio, cement chemical composition, SF content,
curing time and method, temperature, mix additives,
ACKNOWLEDGEMENT
etc.). In the literature various formulations can be
found to describe the sorption isotherm of normal
This research was supported by a grant (code 09F07)
concrete (Xi et al. 1994). However, in the present
from Technology innovation Program funded by
paper the semi-empirical expression proposed by
Ministry of Land, Transport and Maritime Affairs of
Norling Mjornell (1997) is adopted because it
Korean government.
,
Splitting
(2)
The
water
content
expressed
as the sum
Table
2. Rib
height
effects w
forcan
barsbe
with
high confinement.
(capillary
water,
water
of
the
evaporable
water
w
e
Cases
Crushing
Shape
Proceedings of FraMCoS-7, May 23-28, 2010
explicitly accounts for the evolution of hydration
reaction
content. This
isotherm
Modeand SF α
Fx sorption
Bond Strength
reads
Pullout
High
Low
Low
⎡
⎤
1
⎢
⎥
G1 (α c , α s )⎢1 −
∞
10(g α
− α c )h ⎥⎥
Med ⎢⎣ e Med1 c Med
⎦
⎡ 10(g α ∞
K1 (α c , α s )⎢e 1 c
Low ⎢⎣
High
− α c )h
+
(4)
⎤
− 1⎥
High ⎥⎦
where the first term (gel isotherm) represents the
physically bound (adsorbed) water and the second
α
Mode
Bondthe
Strength
term (capillary
isotherm)
represents
capillary
water. This expression is valid only for low content
Splitting
Lowthe amount of
of SF. The
coefficientHigh
G1 represents
water per unit volume held in the gel pores at 100%
relative Splitting
humidity, andMed
it can be expressed
(Norling
Med
Mjornell 1997) as
c
s
G (α c αPullout
s ) = k vg α c c + kLow
vg α s s
,
1
High
(5)
where kcvg and ksvg are material parameters. From the
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