Modeling of Bond Stress–Slip Relationships of a Strand in Concrete during Steam Curing
International Journal of Concrete Structures and Materials
Modeling of Bond Stress-Slip Relationships of a Strand in Concrete during Steam Curing
The restrained thermal expansion of a pretensioned strand causes thermal prestress loss during steam curing until sufficient bond strength develops. The amount of thermal prestress loss is directly related to the characteristics of the interfacial bond stress-slip relationship at different maturity phases of concrete. For a rational assessment, the bond stress-slip relationship needs to be investigated experimentally at different maturity phases. In this study, a total of 12 pull-out tests were performed using seven-wire strand of 12.7 mm diameter, at different concrete equivalent ages of 7.8, 23.5, 53.8 and 85.2 h. Based on the test results, an empirical model of the bond stress-slip relationship was developed. The model comprised four segments: a curvilinear ascending region, a constant maximum region, a linearly descending region, and a region of constant frictional bond stress. The characteristic values in the model were expressed as functions of equivalent age. The model was able to predict the test results with reasonable accuracy.
prestressed concrete; prediction; steam curing; pull-out test; bondslip
Extensive experimental and theoretical studies have been
performed for several decades for prestressed concrete to
find its short and long-term behavior as well as seismic
(Freyssinet 1954; Huang 1980; Naaman and
Alkahiri 1991; Ahlborn et al. 2000; Lu et al. 2016; Singh
et al. 2013; Kim et al. 2012a, b, 2016; Jeon et al. 2015)
Unlike reinforced concrete, prestressed concrete suffers from
prestress loss. Accordingly, prediction of prestress loss has
been a key concern
(Rizkalla et al. 2011)
In mass fabrication of precast pretensioned concrete, steam
curing is used to accelerate the hydration process and to
achieve the specified release strength. Relatively recent
research has reported that thermal prestress loss occurs
during steam curing
(Ahlborn et al. 2000; Bruce et al. 2001;
Roller et al. 2003; Tadros 2003; Barr et al. 2005; Erkmen
et al. 2008; Newhouse and Wood 2008; Rizkalla et al. 2011)
Thermal prestress loss occurs in a strand during steam
curing because its axial thermal extension is restrained owing
to the fixing of its ends in the prestressing bed (Fig. 1).
During steam curing, thermal extension of the strand at time ti
causes local slip and bond stress at location x between the
strand and the surrounding concrete [s(x) and s(x) in Fig. 1,
respectively]. The amount of s(x) corresponding to s(x) is
determined by the bond stress–slip relationships developed at
time ti. As a sufficient bonding develops between the
embedded strand and the surrounding concrete, the strand
and concrete form a composite. Thereafter, the strand
becomes compatible with the surrounding concrete, without
local slippage. As a result, thermal prestress loss is locked in
and becomes irrecoverable in concrete
(Barr et al. 2005)
The amount of thermal prestress loss depends directly
upon the bond stress–slip relationships during the steam
curing process. Although it seems to be necessary priori
knowledge to understand the bond stress–slip relationships
of a strand in concrete at elevated temperature, no research
has been reported in the literature. Absent such knowledge,
inconsistent bonding time has been assumed for the purpose
of estimating thermal prestress loss in the literature.
et al. (2005)
estimated the thermal prestress loss of 3–7% of
the initial prestress if bonding occurred within the range of
6–10 h after casting.
Roller et al. (2003)
thermal prestress loss prior to release was 6% if bonding
occurred within 6 h after casting.
Erkmen et al. (2008)
assumed that thermal prestress loss occurred until the
concrete reached its maximum temperature. To validate the
theoretical estimation of thermal prestress loss, the
occurrence of bonding in different steam curing regimes must be
It has been reported that bond stress–slip relationships are
greatly influenced by the surrounding concrete strength
(Choi 1988; Eurocode2 2004; Campione et al. 2005;
Missouri University of Science and Technology 2012)
. In steam
curing, the development of concrete strength is affected by
both the temperature and the curing duration. Accordingly,
due consideration must be paid to concrete maturity in the
modeling of bond stress–slip relationships of a strand
embedded in concrete during the steam curing process. A
comprehensive literature search indicated that no
information was available regarding the bond stress–slip behavior of
strands in concrete at elevated temperature during steam
curing. Accordingly, in the present study pull-out tests were
performed of specimens subjected to various steam curing
conditions, and a model of bond stress–slip relationships for
strands was developed to serve as a tool in determining the
bonding time. This model was based upon the concept of an
The main purpose of the pull-out tests performed in this
study was to investigate the local bond-slip behavior of a
strand embedded in concrete under various conditions of
concrete maturity, which is influenced by both the curing
temperature and time duration. Accordingly, the temperature
of the concrete in the vicinity of the embedded strand was
recorded throughout the experimental process.
The bond-slip tests were designed by using RILEM
7-II128 (1994) as a guide. According to the RILEM 7-II-128,
the reinforcing bar will be embedded in the concrete a total
length of 15 times the bar diameter to be tested. For a
bonded breaker, a length of 7.5 times the bar diameter is to
be placed so that the bar is unbonded from the bottom
surface to halfway in the concrete, leaving a bonded length of
7.5 times the bar diameter. Some details of the test method,
however, were slightly modified based upon previous work
Campione et al. (2005)
estimation of bond-slip relationships. In this experiment, the
bars were embedded in concrete a total length of 16.5 times
the bar diameter with a bond length of 5dps = 63.5 mm
(Choi 1988; Campione et al. 2005)
. The embedded and
bonded lengths adopted in this experiment has been reported
to produce approximately uniform bond-stress and slip
distribution, and to reduce the scatter of test results observed
with very short bonded lengths
(Eligehausen et al. 1983a, b)
2.1 Preparation of Experiments
Figures 2a, b illustrate the pull-out specimen and its test
setup. Table 1 lists specimen names and details on their test
conditions. Each specimen is named following the format of
‘PNR’, where the letter P represents a pull-out test specimen,
N the elapsed time in hours of steam curing (N = 4, 6, 8 and
10 for 4, 6, 8 and 10 h of steam curing, respectively) and R
distinguishes between replicate specimens prepared in the
Pull-out tests were performed upon seven-wire strands
each having a total length of 910 mm. A 210 mm portion
of its length was embedded and passed through a prismatic
concrete block of 210 9 210 9 210 mm. A bond of length
5dps = 63.5 mm was made within the embedded portion,
where dps (12.7 mm) is the diameter of strand; the
remaining parts were left unbonded. This bonded length of
63.5 mm was short enough to produce approximately
uniform bond stress and slip distributions, but long enough
to reduce the scatter of test results usually observed for
bonds of very short length
(Eligehausen et al. 1983a, b)
Confinement reinforcements (both vertical and transverse
bars) were provided in the specimen to prevent splitting
failure of the concrete (Fig. 2a). The diameter and yield
strength of the deformed bars used for confinement
reinforcement were 6 mm and 400 MPa, respectively. The
nominal tensile strength of the seven-wire strand used in
the pull-out tests was 1820 MPa. The concrete used in the
standard specimen had a 28-day compressive strength of
37 MPa. The maximum size of the aggregate in the
concrete was 19 mm.
A thermocouple was positioned near the bonded part of
the strand to measure temperature changes in the specimen
during steam curing process. Insulation tube was used to
prevent bonding outside of the designed bonding length
To determine the changes in bond-slip relationships under
different conditions of concrete maturity, each pull-out
specimen was removed from the steam chamber at a specific
elapsed time after the steam supply was initiated. Soon after,
the strand was pulled out monotonically. A pull-out force
was applied at one end and bar slippage was measured at the
other end (Fig. 2b).
2.2 Experimental Procedure
Experiments were performed in the following order: the
steam curing chamber was assembled, the confinement steel
cage was placed in the mold and the strand was positioned,
concrete was casted into the pull-out molds, steam was
emitted into the chamber according to the given steam curing
regime, and pull-out tests were performed at various
The procedures are described in detail as follows.
(1) A steam curing chamber was assembled from sandwich
panels made of 30 mm thick insulating polyethylene
foam boards and thin steel sheets for panel covers.
Specimens inside the chamber were supported with
wooden pallets during the steam curing process to keep
cooled condensed water from adversely affecting the
specimen temperature (Fig. 3a).
(2) Two plastic tubes 8 mm in diameter were placed
parallel to the width of the chamber. These tubes were
connected to a portable steam generator. To prevent
possible movement of tubes from their original position
during the steam curing process, they were fixed to the
rod to hold them straight (Fig. 3a). Holes were made
along the lengths of these tubes to allow emission of
steam. During testing, steam temperature was manually
monitored by measuring the temperatures at four
different locations in the chamber, namely points at
mid-height and at the bottom of either side of the
curing chamber (Fig. 3b).
(3) For pull-out test specimens, a prefabricated
confinement steel cage was placed in the pull-out steel mold. A
strand was placed through the holes in the mold; part of
the strand in the mold was covered with insulation tube
to prevent bonding; the middle portion of
5dps = 62.5 mm was left uncovered to allow bonding.
To measure the temperature in the vicinity of the
bonded portion of the strand, a thermocouple was
placed near this portion in each specimen.
(4) Concrete was cast into the steel mold and vibrated by
using portable rod vibrators. During the vibration,
caution was taken not to damage or displace the
unbonded part, the confinement reinforcements, and
(5) Specimens were carefully placed on the wooden pallets
in the curing chamber.
(6) For each specimen, two concrete cylinders 100 mm in
diameter and 200 mm long were fabricated for use in
measuring compressive strength at the time of the
pullout test. One thermocouple was embedded in each
cylinder to measure the temperature changes. These
cylinders were then placed inside the curing chamber.
(7) Steam was supplied by a steam generator connected to
the chamber by plastic tubes, and testing continued for
10 h following a typical 3–6–3 h steam curing regime
(Fig. 4a). The steam curing regime adopted in this
study consisted of four periods totaling 13 h: a 1 h
delay period at ambient temperature (approximately
20 C); a 3 h positive temperature ramp, 6 h of curing
at the constant maximum temperature of 60 C, and a
3 h negative temperature ramp for cooling. During the
steam curing process, thermocouples were used to
rs F se
a d on 5 6 1 6 6 2 5 9 0 4 9 3
on tre (
C s S
Fig. 3 Steam curing chamber. a Placement of pull-out
specimens in the curing chamber and b test setup for
curing and data collections.
measure the temperatures both inside the chamber and
inside the concrete in the pull-out specimens.
(8) Pull-out specimens and concrete cylinders were
removed from the curing chamber at specific curing
times (4, 6, 8 and 10 h) during the steam curing.
(9) After the completion of the steam curing process, the
pull-out specimens and concrete cylinders removed
from the curing chamber at the same age were
demolded. As soon as the specimens demolded,
pullout tests were performed on each pull-out specimen
and the concrete cylinders were simultaneously tested
for their compressive strength. To avoid undesirable
eccentricity during the compression test, the ends of the
cylinders were prepared by capping them with
compound rather than grinding them. This minimized
possible damage to the cylinders resulting from impact
during the grinding process, an important consideration
for testing of the immature concrete samples that
underwent insufficient curing.
2.3 Test Results
2.3.1 Temperatures and Concrete Strengths
Figure 4b shows typical histories of temperature changes
measured inside the chamber and inside the concrete in the
pull-out mold. At the beginning stage of steam curing, the
temperature increase of concrete within the specimen
followed that inside the chamber with a delay of approximately
1 h. As hydration of the concrete proceeded, the temperature
within the specimen exceeded the maximum temperature
inside the chamber (60 C), reaching the maximum
temperature of 76.2 C at the elapsed time of 7.7 h of steam
All tested specimens showed similar temperature changes,
because their mix proportions and the given steam curing
schedule were the same. After reaching the peak
temperature, the concrete in each specimen cooled gradually.
Table 1 lists the measured strengths of the concrete
specimens cured for different times. Average compressive
strength increased with curing time: 2.4, 8.5, 15.1, and
23.9 MPa for 4, 6, 8, and 10 h, respectively. The average
28-day compressive strength of two 100 mm 9 200 mm
concrete cylinders, cured at the ambient temperature of
22 C, was measured to be 37 MPa.
2.3.2 Bond Stress–Slip Relationships
It has been reported that the bond stress–slip relationship
of a deformed steel bar embedded in 28-day air-cured
concrete consists of an increasing curvilinear part, followed by a
constant part at the maximum bond stress, a linearly
decreasing part, and a constant tail part indicating the
frictional bond resistance
(Choi 1988; fib model code 2013)
Figure 5 shows the different experimentally observed bond
stress–slip relationships for seven-wire strands of diameter
12.7 mm that were embedded in the concrete specimens of
various maturities depending upon the steam curing
duration. Bond stress (s) in MPa was measured by dividing the
Fig. 5 Experimentally observed bond stress–slip
relationships measured at different maturity phases of
measured pull-out force by the circumferential surface area
of the bonded length of the embedded strand, which was
2532.3 mm2 for each strand used in the pull-out tests.
Similar shape of bond stress–slip relationships of strands in
concrete can be observed in Fig. 5 compared with that of a
deformed bonded bar tested 28 days after casting. As the
elapsed time of steam curing increased, the resulting
specimens showed increasing compressive strength, and
increased bond strength and initial stiffness were observed in
the bond stress–slip relationships.
All the measured bond-slip relationships could be
characterized by four different regions, regardless of their
maturity: a prepeak ascending portion, an approximately
constant plateau at the maximum stress, a gradually
descending region, and an approximately constant residual
region. Table 1 presents measured values of concrete
strength and characteristic points (bond stress and
corresponding slip) as well as calibrated ones for ab, g and m given
by Eqs. (5a), (7a) and (7b), respectively. In the prepeak
ascending region, adhesion bond strength was shown to
increase for samples subjected to longer curing. After the
failure of the adhesion bond, bond stress further increased
with the increased slip. Accordingly, the stiffness in the
ascending portion of the bond stress and slip relationships
was reduced. For all cases, a constant peak region appeared
at slip values between 5 and 10 mm. The maximum bond
stress was maintained throughout an additional slip of about
5 mm after reaching the peak value. The bond stress then
began to decrease, reaching residual value at high values of
3. Modeling of Bond Stress–Slip
3.1 Equivalent Age
To evaluate the effect of varying temperature and time
upon the development of concrete compressive strength, a
maturity method has been suggested by
regarding accelerated curing
methods. Based on their ideas, the Nurse–Saul maturity
function in Eq. (1) was suggested; this equation sums
temperature and time intervals during the curing period to
yield the maturity index, also called the temperature–time
where M is the maturity index in C h or C day, T is the
average concrete temperature in C during the time interval
Dt, To is the datum temperature, usually taken to be -10 C,
t is the elapsed time in h or days, and Dt is the time interval
in h or days.
Soon after the introduction of Eq. (1), it was recognized
that the linear approximation might not be reliable for wide
ranges of temperature change.
Freiesleben Hansen and
) suggested a new function to calculate a
temperature–time factor from a history of curing concrete
temperature; this function was derived from the Arrhenius
(Brown and LeMay 1988)
, which is used to
consider the effects of curing temperature upon the chemical
reaction rate. This function is given in Eq. (2) and gives a
quantity termed the equivalent age of concrete (te in h).
Xt e ER ðTþ1273 Trþ1273Þ Dt;
where E is the apparent activation energy (J/mol), R is the
universal gas constant (8.3144 J/mol/K), and Tr is the
absolute reference temperature in C.
The development of the equivalent age function overcame
one of the major limitations of the Nurse–Saul maturity
function by providing a nonlinear function to describe the
initial rate of development of concrete compressive strength
and curing temperature. Although both Eqs. (1) and (2) were
suggested in ASTM C1074-11 (2011), comparative research
has shown that the latter is superior to the Nurse–Saul
(Byfors 1980; Carino 1982)
Equation (2) converts the actual concrete age to its te at the
reference temperature, Tr. In the present study, the reference
temperature was taken as the measured ambient temperature
of 20 C. The value of E was taken as 44,072 J/mol, which
is in the range of the value suggested in ASTM C1074-11
(2011). In Table 1, the values of te calculated from Eq. (2)
are listed for all specimens; these values ranged between 7.8
and 85.2 h at Tr. Figure 6a illustrates the relationships
between the actual elapsed time (t in h) and te.
3.2 Development of Concrete Compressive
A mathematical model of the rate of relative increase in
concrete compressive strength with respect to its limiting
compressive strength (Su MPa) was suggested by
as given in Eq. (3). In Eq. (3), the positive-valued
reaction coefficient r reflects the retardation of continuous
hydration as curing elapses. Equation (3) assumes that the
rate of strength increase relative to Su decreases from the rate
constant (k) to zero as the strength (S) approaches Su during
3.3 Bond Stress–Slip Relationships of Strands
In the experiments, similar pull-out behavior was observed
for the strands embedded in concrete under the various
curing conditions and for deformed rebar in concrete at the
28-day curing condition (Fig. 4). Therefore, the model given
in Eq. (5) was adopted in the present study for modeling the
bond stress–slip relationships of the strands after curing
under various conditions. It is worth noting that Eq. (5) was
suggested for the modeling of bond stress–slip relationships
for a deformed steel bar embedded in concrete at 28 days of
(fib model code 2013)
; thus, the compressive strength
given in this equation is independent of te.
As observed in pull-out tests, the model consists of four
different branches: an increasing curvilinear part followed by
a constant part at maximum bond stress, a linearly
decreasing part, and a constant tail part indicating the
frictional bond resistance.
sb ¼ smax ðs=s1Þab for 0
sb ¼ smax for s1
sb ¼ smax
Depending upon the type of concrete, curing conditions,
or experimental parameters under consideration, various
modifications of Eq. (3) have been suggested
(Yi et al. 2005;
Carino and Tank 1992; Kim et al. 1998, 2001; Kwon et al.
. Bernhardt (1956) and
use of r = 2 after empirically investigating the development
of strength in ordinary concrete exposed to various curing
temperatures. As an extension of the basic equation (3) with
r = 2,
Tank and Carino (1991)
proposed the rate constant
model given in Eq. (4) to estimate the relative strength gain
of concrete based upon its te.
SðteÞ ¼ Su 1 þ kr ðte
where kr is the rate constant (h-1) at the reference
temperature, Tr, Su is the ultimate strength of concrete, and tor (h) is
the age at the start of strength development at the reference
Based upon the measured value of concrete compressive
strength (fc0t) and the corresponding calculated values of te,
regression analysis was performed to find the best-fitting
values of tor, kr and Su in Eq. (4); these were respectively
determined to be 4.99 h, 0.0149 h-1 and 41.3 MPa.
Equation (4) was able to predict the development of concrete
strength in terms of te with reasonable accuracy (Fig. 6b).
The average and standard deviation of the ratios of the
where sb, smax and sf are respectively the bond stress at
slippage s, the maximum bond stress, and the frictional bond
stress (MPa), and s, s1, s2 and s3 are respectively the
slippage, the slippage at characteristic point (1), the slippage at
characteristic point (2) and the slippage at characteristic
point (3) (mm). In Eq. (5), the value ab (0 B ab B 1)
determines the degree of convexity of the initial ascending
curve in bond stress–slip relationships.
Based upon the pull-out tests shown in Fig. 5, the
pullout model illustrated in Fig. 7 was used in the present
study. For this model, three characteristic points [(1)–(3) in
Fig. 7] are determined based upon the empirically observed
stress–slip relationships for each specimen: two
characteristic bond stresses (smax for the maximum bond stress and
sf for the frictional bond stress), three characteristic slip
values (s1 for slip at smax, s2 at the end of the plateau with
smax and s3 at the initiation of sf), and the curve parameter
It has been reported that, in general, characteristic slip
values are independent of the diameter of the reinforcement
and the compressive strength of 28-day air-cured concrete
(fib model code 2013)
. Similarly, Fig. 5 also shows that the
characteristic slip values vary only marginally between
different specimens with different concrete strengths or
different te. The values of smax and sf, however, seemed to be
influenced by the concrete strength and thus by te at the time
of the pull-out test.
Three characteristic slip values were determined from the
average values of test results obtained from the same te.
However, it is worth mentioning that these values were
repeatedly calibrated with other characteristic and parametric
values in order for the bond-slip model to better predict the
overall bond-slip relationships observed from the test results.
Regression analysis was performed to determine the
relationships between each of the characteristic values of slip
and te. Linear regression analysis of 12 measurements
collected at the characteristic points tested in this study yielded
the best-fit line given in Eq. (6):
s1 ¼ 0:0086 te þ 6:203;
0:014 te þ 12:983;
0:0104 te þ 17:417:
Figure 8a illustrates the comparisons between the
measured characteristic slip values and those predicted by
Eq. (6). The averages and standard deviations of the ratios of
the Eq. (5) predictions to the experimentally observed slip
values were 1.05 and 0.25, 1.01 and 0.12, and 1.02 and 0.13
for s1, s2 and s3, respectively (Table 2).
The test results showed that the magnitudes of smax and sf
were influenced by the developed concrete strength, as
indicated by the different values of te. Also, the value of sf
seemed to be dependent upon the smax value. Accordingly,
Eq. (7) was used to approximate the smax and sf as functions
smax ¼ gðteÞ
sf ¼ m smax;
where g is a coefficient representing the bond condition and
m is a coefficient representing the frictional bond resistance.
Using the measured values of smax and fc0t; 12 values of the
coefficient g were calculated for each specimen based on
Eq. (7a) and listed in Table 1. As illustrated in Fig. 8b, these
values were observed to be almost linearly proportional to te.
Based on this observation, Eq. (8) was developed to express
g as a linear function of te. Equation (8) limits the maximum
value of g to 2.5 because this value expresses good bond
condition in well-confined concrete
(fib model code 2013)
gðteÞ ¼ 0:0159 te þ 0:2594
The average and standard deviation of the ratios of the
model predictions by Eq. (7a) to the experimentally
observed values of smax were 1.01 and 0.20, respectively. It
is worth noting that the measured value of fc0t from specimen
P4A was 1.8 MPa, which was about 62% less than the
2.9 MPa measured for its replicate specimen P4B. The tests
were performed at t = 4 h (or te = 7.8 h) and difficulties
were involved in measuring bond stress–slip relationships
accurately for these immature specimens.
Based on Eq. (7b) and the measured values of of smax and
sf, the coefficient of frictional bond stress (m) was obtained
and tabulated in Table 1. The average value of 0.75 was
taken as the value of m. The average and standard deviation
of the ratios of the Eq. (7b) model predictions to the
measured values of sf were 1.04 and 0.28, respectively.
The initial ascending curves in the bond stress–slip
relationship became increasingly convex upward for increasing
values of te (Fig. 5). However, the degree of this increase
was reduced with increasing te, and the curves were nearly
the same for te of 8 h or greater. This was reflected in the
function for ab, in which ab decreases exponentially from 1.0
and approaches 0.16 asymptotically at large te. Based on the
test results, the values of ab were estimated based on
Eq. (5a). From the regression analysis on these values,
Eq. (9) was then suggested to express ab as a function of te.
Equation (9) closely predicted the experimentally obtained
values of ab (Fig. 8c).
ab ¼ 0:16 þ 0:84 e 0:08 te :
The average and standard deviation of the ratios of
predicted values of ab to the observed ones were 1.03 and 0.20,
respectively (Table 2).
3.4 Comparisons Between Model Predictions and Test Results
The model predictions were compared to the measured
bond stress–slip relationships of the strand pull-out at
different curing conditions having different values of te
(Fig. 9). In general, the model was able to predict the overall
bond stress–slip relationships obtained from the pull-out
Because the thermal loss of the strand in pretensioned
prestressed concrete members during steam curing is directly
related to the bond conditions between the strand and the
surrounding concrete, the model developed herein can be a
useful tool in theoretically assessing the thermal prestress
loss of strand during the curing at elevated temperature.
In this study, pull-out tests at elevated temperatures were
performed for different tes with its maximum value of
85.2 h. According to
Barr et al. (2005)
, thermal prestress
loss occurs approximately within the range of 6–10 h in real
time after casting. This would correspond to te between 44
and 100 h at a typical curing regime of 3–6–3 h with
maximum temperature of 60 C. Equation (6) shows that the
values of s1 and s2 become identical at te equal to 300 h,
which seems to be far beyond the approximate maximum te
of 100 h at bonding. However, the developed bond-slip
model presented herein is recommended for its application to
the case when the formation of sufficient bonding is
expected to occur before the approximate upper bound of
100 h in equivalent age.
Relationships of bond stress versus slip were
experimentally observed by means of pull-out tests of strands
embedded in concrete under various curing conditions, and
were then modeled as a function of equivalent age. The
following conclusions were drawn from this study.
(1) Regardless of the elapsed time of steam curing, all
measured bond-slip relationships exhibited four
distinctive regions: a prepeak curvilinear ascending
region, a near constant region at peak stress, and a
gradually descending region followed by a region of
constant frictional bond resistance.
(2) Bond strength and initial stiffness tended to increase
with increasing equivalent age. In particular, the
prepeak ascending curve in the bond stress–slip
relationships tended to become more convex with
increasing equivalent age.
(3) The stresses smax and sf were influenced by te at the
time of the pull-out test. Both smax and sf were
observed to increase proportionally to the product of a
linearly increasing function of equivalent age and the
square root of concrete strength.
(4) The characteristic slip values varied only marginally
between different specimens with different concrete
strengths or different te. This observation conforms
with the observation made for a deformed bar pulled
out from 28-day air-cured concrete.
(5) In general, the empirical bond stress–slip model
developed based upon the equivalent age reasonably
predicted the local bond stress–slip relationships of
seven-wire strands embedded in concrete under various
(6) In future study, the developed model can be used for a
tool for predicting the time of bonding between the
strand and the surrounding concrete at various curing
This Research was supported by the Basic Science Research
Program through the National Research Foundation of
Korea (NRF) funded by the Ministry of Education, Science
and Technology (NRF-2013R1A2A2A01011563) and by the
Chung-Ang University Graduate Research Scholarship in
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