Engineering, 2013, 5, 850-855
Published Online November 2013 (http://www.scirp.org/journal/eng)
http://dx.doi.org/10.4236/eng.2013.511103
Open Access ENG
Computation of the Strand Resistance Using the Core
Wire Strain Measurement
Keunhee Cho, Sung Tae Kim, Sung Yong Park, Young-Hwan Park*
Structural System Research Division, Korea Institute of Construction Technology, Goyang-Si, Republic of Korea
Received August 5, 2013; revised September 5, 2013; accepted September 12, 2013
Copyright © 2013 Keunhee Cho et al. This is an open access article distributed under the Creative Commons Attribution License,
which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
ABSTRACT
This paper proposes a method enabling to compute the prestressing strand resistance using the strain measured on only
one core wire. Numerical analysis is conducted considering the pitch length of the strand and the diameters of the core
wire and helical wires as parameters. The results verify that the relation between the stresses of the core wire and helical
wires can be expressed in terms of the helical angle. Based on this observation, a formula computing directly the
prestress force in the strand from the strain measured in the core wire is suggested. Owing to the recently developed
measurement method for the core wire strain, the proposed formula can be exploited to determine the prestress of the
strand.
Keywords: Finite Element Method; Measurement; Prestress; Strand; Core Wire; Helical Wire
1. Introduction
Prestress is introduced in concrete structures to reduce
the quantity of materials and achieve longer span. The
evaluation of the changes in the prestress according to
the level of prestress and use of the structure is of utmost
importance to assess the state of the structure. This pre-
stress is generally introduced by means of strands. Ac-
cordingly, the resistance of the strand should be meas-
ured continuously along the service life to evaluate the
level of prestress.
Among the methods that have been developed to
measure the prestress of the strand, methods using guided
stress wave [1,2], elastomagnetic stress sensor [3], and
fiber optic sensor [4] are currently applied. Recently, a
method using fiber optic sensor and measuring the strain
of one wire instead of the whole strand has been deve-
loped [5]. This method differs from previous methods,
which measure the resistance of the whole strand, by
measuring the strain of only one wire constituting the
strand. Therefore, need is for a method estimating the
resistance of the whole strand from the strain measured
on one wire.
A strand is composed of a straight core wire sur-
rounded by helical wires. The difference in the shapes of
the core wire and helical wires provokes the development
of different stresses. Analytic studies [6,7] and numerical
studies [8-10] have been carried out to identify the be-
havior of the strand and its load-resisting characteristics.
However, these studies focused on the behaviors of the
whole strand or individual core wire or individual helical
wire without providing any method estimating the resis-
tance of the whole strand based on the strain of an indi-
vidual wire.
Accordingly, this study intends to propose a method
enabling to compute the prestress force of the strand by
measuring the strain of one wire. First, numerical analy-
sis is conducted for 7-wire strands with diameter of 15.2
mm and presenting diverse shapes. The results are used
to determine rational measurement positions on the wires
and to derive the relationship between the stresses of the
wires. Finally, a relation between the measured strain and
the resistance of the strand is suggested.
2. Selection of Analysis Parameters and
Material Properties
Parametric study is conducted to examine the behavioral
pattern of the strand according to the change of its shape.
The shape of the strand is determined with respect to the
diameter Dc of the core wire, the diameter Dh of the heli-
cal wires, and the pitch length Lp (Figure 1). In addition,
the shape can also be expressed in terms of the helical
*Corresponding author.

K. CHO ET AL. 851
Figure 1. Shape and element meshing of the 7-wire strand
model.
angle (θ) between the axis of the core wire and the axis
of the helical wire. In general, the diameter of the core
wire is larger than that of the helical wire in order to re-
duce the frictional effect between the helical wires. The
KS standards (2011) [11] specify that, for a 7-wire strand,
the diameter of the core wire shall be larger by minimum
0.08 mm than that of the helical wire and that the pitch
length shall be 12 to 18 times larger than the diameter of
the 7-wire strand (D).
Accordingly, the difference in the diameters of the
core wire and helical wire and the pitch length can be
selected as variables for the shape of the 7-wire strand.
Therefore, the diameter of the 7-wire strand is fixed to
15.2 mm and the difference between the diameters of the
core wire and helical wire is set to range from 0.08 mm
to 0.20 mm by step of 0.04 mm (Table 1). The pitch
length is varied from 12 times to 18 times the diameter of
the 7-wire strand by unit step (Table 2). The analysis is
performed for 28 cases involving 4 cases with different
diameters between the core wire and helical wire and 7
different pitch lengths.
The core wire and helical wires of the 7-wire strand
are modeled using three-dimensional 20-node continuum
elements (Figure 1). The interaction between the core
wire and helical wires is represented by means of contact
and a coefficient of friction of 0.115 is applied [12].
Since the strand is actually tensioned to a level lower
than the yield strength, the material is modeled to be
linearly elastic. The elastic modulus is 200 GPa and the
Poisson’s ratio is 0.3.
The end sections of the strand are modeled as rigid
bodies. In order to prevent any effect of such end con-
straints on the analysis results, the length of the strand is
extended sufficiently to be twice the pitch length. All the
degrees-of-freedom on one end are restrained and the
degrees-of-freedom at the other end are restrained at the
exception of the degree-of-freedom in the longitudinal
direction. A load of 200 kN is applied in the longitudinal
direction.
3. Analysis Results
Contact between the helical wires occurred under the
application of the load for some cases where the differ-
ence between the diameters of the core wire and helical
wire was small. However, since the strand is fabricated to
Table 1. Parameter related to the difference between the
diameters of the core wire and helical wire (for 15.2 mm-
diameter strand).
Dc − Dh (mm) 0.08 0.12 0.16 0.20
Dc (mm) 5.12 5.15 5.17 5.20
Dh (mm) 5.04 5.03 5.01 5.00
Table 2. Pitch length parameter as multiple of the diameter
of the 7-wire strand (for 15.2 mm-diameter strand).
Pitch length
multiple 12 13 14 15 16 17 18
Lp(mm) 182.4197.6212.8 228.0 243.2 258.4273.6
prevent practically any contact between the helical wires,
the cases in which contact between the helical wires oc-
curred were discarded from the analysis (Table 3).
In order to compute the resistance of the strand based
upon the strain measured on the core wire, this measured
strain should be representative of the strain of the core
wire. If the resistance of the strand is computed using the
strain measured locally at a spot where strain concentra-
tion occurs due to contact or any other reason, erroneous
overestimation of the prestress would be achieved. There-
fore, the longitudinal distribution of the stress in the sec-
tion at mid-length of the strand is drawn to verify the
appropriate position on the core wire at which strain
should be measured (Figure 2). The stress in the core
wire exhibits nearly uniform distribution at the exception
of its circumferential contour in contact with the helical
wires. The stress in the helical wires appears to be prac-
tically constant with regard to the whole section in case
of small difference between the diameters but tends to
reduce as much as the distance from the center of the
strand increases in case of large difference between the
diameters.
To verify the consistency of this sectional stress dis-
tribution over the whole length of the strand, the longitu-
dinal stress of the core wire is plotted for the whole
length of the strand (Figure 3). Here, the normalized
distance corresponds to the longitudinal position on the
strand divided by the length of the strand. At the excep-
tion of some parts at the extremities, the stress at the cen-
tral axes of the core wire and helical wire are seen to
remain constant. Besides, the stress at the edge of the
core wire varies continuously due to the contact with the
helical wires. This stress varies in a range from 96.5% to
102.1% of the stress at the center axis of the core wire.
Accordingly, the measurement of the stress of the core
wire at the edge would result in an error of −3.5% to
2.1%.
4. Discussion
T
he objective of this study is to compute the resistance of
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K. CHO ET AL.
Open Access ENG
852
(a) (b)
(c) (d)
(e) (f)
(g) (h)
Figure 2. Longitudinal stress distribution at mid-length section of the strand. (a) Pitch length = 15, Dc − Dh = 0.12 mm; (b)
Pitch length = 15, Dc − Dh = 0.16 mm; (c) Pitch length = 16, Dc − Dh = 0.12 mm; (d) Pitch length = 16, Dc − Dh = 0.16 mm; (e)
Pitch length = 17, Dc − Dh = 0.08 mm; (f) Pitch length = 17, Dc − Dh = 0.12 mm; (g) Pitch length = 18, Dc − Dh = 0.08 mm; (h)
itch length = 18, Dc − Dh = 0.12 mm. P

K. CHO ET AL. 853
(a) (b)
Figure 3. Longitudinal stress distribution in the core wire and helical wire for the whole length of the strand (pitch length =
15 D, Dc – Dh = 0.16 mm). (a) Stresses at centers of core wire and helical wire; (b) Stresses at center and edge of core wire.
Table 3. Analysis cases excluding the cases in which contact
between the helical wires occurred.
Dc − Dh (mm) Pitch length multiple
0.08 12, 13, 14, 15
0.12 12
the strand by measuring the strain of the core wire. To
that goal, the mean strain of the core wire should be
measured or derived from the measurements and, the
corresponding stress of the helical wire should be esti-
mated from the measured strain of the core wire. Since,
apart from its edge, the core wire exhibits nearly constant
stress state, the mean strain can be obtained by measure-
ing the strain at the center of the core wire. Besides, the
mean stress of the helical wire cannot be obtained di-
rectly by analysis because the stress distribution shows
different patterns according to the difference in the di-
ameters.
This study intends to obtain the mean stress of the
helical wire as follows. First, the mean stress (c

) and
resistance (c
F
) of the core wire are calculated using the
strain (c

) measured at the center of the core wire.
cs
Ec

  (1)
cc
FA
c

 (2)
where
s
E is the Young’s modulus and, c
A
is the
cross-sectional area of the core wire. Thereafter, the
resistance (h
F
) of the helical wire is obtained by
subtracting the force (F) applied on the strand from the
so-obtained resistance of the core wire.
hc
F
FF (3)
Finally, the mean stress (h

) of the helical wire is ob-
tained by dividing the resistance of the helical wire by its
cross-sectional area (h
A
).
hhh
F
A

(4)

The ratio of the stress of the helical wire to the stress
of the core wire per shape of the strand computed
through this process ranges between 0.944 and 0.977
(Table 4). It can be seen that the stress of the helical wire
converges to the stress of the core wire as much as the
pitch length becomes longer, and that the difference be-
tween the diameters of the core and helical wires does
not have noticeable effect of the stress ratio.
The relation between the stresses of the core wire and
helical wire is formulated as function of the helical angle
(

), itself related to the pitch length and difference be-
tween the diameters. The helical angle is the angle de-
scribed by the longitudinal axis of the core wire and lon-
gitudinal axis of the helical wire, and can be obtained by
Equation (5).
2
ch
p
DD
L

tan
 (5)
Figure 4 plotting the stress ratio according to the heli-
cal angle is in good agreement with the analytic solution
(Costello, 1997). The stress ratio decreases gradually
with larger helical angle and is practically linearly pro-
portional to tan

. Accordingly, the relation between
the helical angle and stress ratio can be obtained through
linear regression analysis and is expressed in Equation
(6). This formula enables thus to estimate the stress of
the helical wire from the stress of the core wire by using
only the shape of the strand.
0.5214 tantan1.0388
h
c


 (6)
Finally, the prestress of the strand can be obtained as
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K. CHO ET AL. 854
Table 4. Mean stress and resistance of core wire and helical wire per shape of strand.
Pitch length multiple Diameter difference (mm) tan

c

(MPa) c
F
(kN) h
F
(kN) h

(MPa) h
c


12 0.16 0.1755 1505.2 31.6 168.4 1421.5 0.944
12 0.20 0.1757 1504.5 32.0 168.0 1426.5 0.948
13 0.12 0.1617 1490.0 31.0 169.0 1419.4 0.953
13 0.16 0.1620 1493.9 31.4 168.6 1423.5 0.953
13 0.20 0.1622 1494.1 31.7 168.3 1428.3 0.956
14 0.12 0.1502 1480.2 30.8 169.2 1421.1 0.960
14 0.16 0.1504 1481.5 31.1 168.9 1425.7 0.962
14 0.20 0.1506 1486.0 31.6 168.4 1429.8 0.962
15 0.12 0.1402 1473.2 30.6 169.4 1422.3 0.965
15 0.16 0.1404 1475.1 31.0 169.0 1426.8 0.967
15 0.20 0.1405 1479.6 31.4 168.6 1430.9 0.967
16 0.08 0.1312 1463.7 30.1 169.9 1419.1 0.969
16 0.12 0.1314 1467.8 30.5 169.5 1423.2 0.970
16 0.16 0.1316 1470.0 30.9 169.1 1427.7 0.971
16 0.20 0.1318 1474.4 31.3 168.7 1431.9 0.971
17 0.08 0.1235 1458.8 30.0 170.0 1419.9 0.973
17 0.12 0.1237 1461.3 30.4 169.6 1424.4 0.975
17 0.16 0.1238 1465.8 30.8 169.2
1428.5 0.975
17 0.20 0.1240 1470.2 31.2 168.8 1432.6 0.974
18 0.08 0.1167 1454.9 30.0 170.0 1420.6 0.976
18 0.12 0.1168 1457.9 30.3 169.7 1425.0 0.977
18 0.16 0.1170 1462.3 30.7 169.3 1429.1 0.977
18 0.20 0.1171 1466.7 31.1 168.9 1433.3 0.977
Figure 4. Relation between the helical angle and stress ratio.
follows by substituting Equations (1), (2), (3) and (4) into
Equation (3).
0.5214 1.0388
2
ch
sc ch
p
DD
FEA AL 




 









(7)
Equation (7) is involves only the strain measured at the
center of the core wire and the material and shape pa-
rameters of the strand. Following, this equation enables
to compute directly the prestress of the strand by simply
measuring the strain of the core wire.
5. Conclusion
This paper presented a method enabling to compute the
prestress of a 7-wire strand simply by measuring the
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K. CHO ET AL. 855
strain of the core wire. At the exception of the zones in
contact with the helical wires, the stress of the core wire
appeared to be nearly constant over the cross section and,
except the extremities, the stress of the core wire re-
mained also constant in the longitudinal direction. These
observations made it possible to presume that the mean
stress of the core wire could be directly obtained from
the strain measured at the mid-length center of the core
wire. On the other hand, since the stress varied inside the
section of the helical wire, the corresponding mean stress
was calculated indirectly. In addition, the relation be-
tween the mean stresses of the core and helical wires was
formulated using the helical angle. Based upon this rela-
tion, a formula computing the prestress of the strand di-
rectly from the strain measured on the core wire was
suggested. The proposed formula in combination with
the recently developed method measuring the strain of
the core wire can be exploited to determine the prestress
of the strand.
6. Acknowledgements
This research was supported by a grant from a Strategic
Research Project (Development of Smart Prestressing
and Monitoring Technologies for Prestressed Concrete
Bridges) funded by the Korea Institute of Construction
Technology.
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