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Contents lists available at ScienceDirect
Structures
journal homepage: www.elsevier.com/locate/structures
Effect of compressive strength of concrete on transmission length of pre-
tensioned concrete systems
Prabha Mohandoss, Radhakrishna G. Pillai
⁎
, Amlan K. Sengupta
Department of Civil Engineering, Indian Institute of Technology Madras, Chennai, India
ARTICLE INFO
Keywords:
Bond
Prestressed
Pretensioned concrete
Strand
Transmission length
ABSTRACT
This paper presents an experimental study on L
t
in pre-tensioned concrete (PTC) members made of four values of
compressive strength of concrete. Also, it presents a comparison between the formulations of transmission length
(L
t
) given in various codes and other literature. A test specimen consisted of a concrete prism
(100 ×100 × 2000) mm with a prestressed seven–wire strand (12.7mm diameter) at the center. The values of
average compressive strength for concrete at transfer (f
ci
) were 23, 28, 36, and 43 MPa. The challenges involved
with the measurements of DEMEC readings, difference between the readings from surface–mounted discs and the
inserts are discussed. The results indicate that L
t
could decrease by 33% when the f
ci
increases from about 23 to
43 MPa. Based on this data, a new bi-linear formulation to determine the L
t
as a function of f
ci
is proposed.
Further, it is shown that higher estimates of L
t
as per the available formulations, will lead to lower estimate of
bursting tensile stress in concrete generated during transfer. Hence, a precise estimate of L
t
as a function of the
strength of concrete at transfer is expected to provide more rational design of transmission zone reinforcement.
1. Introduction
Pre-tensioned concrete (PTC) elements with seven-wire strands are
widely used in many structures. The length required to transfer the
prestress (f
pe
) from strand to concrete is defined as the transmission
length (L
t
). Fig. 1 shows the schematic variations of stress and strain
along the length of a strand at the stage of transfer of prestress. As
shown by Line 1 (in Fig. 1(a)), the stress at the end of the member is
zero (at x= 0, f
ps
= 0), the stress gradually increases to the prestress at
transfer f
pe
, and then remains constant in absence of external load (as
depicted by Line 2, at x=L
t
,f
ps
=f
pe
). Similarly, the strain in the strand
and axial strain in concrete varies along L
t
(Fig. 1(b)). Typically, the
shear demand in the transmission zone (say, near support regions) of a
simply supported PTC element systems is high. Therefore, if prestress is
not transferred adequately within the desired distance from the end of
the member, the shear critical sections near the support may experience
shear failure. Fig. 2 shows an example of the shear cracks observed in
the end regions near the support on highway bridge girders. Such shear
cracks could occur due to the poor construction practices leading to
inadequate compressive strength or bond strength of concrete, in-
adequate amount of stirrups and inadequate design expression for L
t
.
Some codal provisions adopt empirical formulations based on the dia-
meter of strand and do not consider the various properties of concrete
to estimate L
t
. This may result in less conservative or less rational shear
design near the supports [1,2]. Hence, more rational formulations, by
incorporating the properties of both strand and concrete, and the
prestress level must be developed to determine L
t
. A previous paper
addressed the importance of determining L
t
precisely and the details of
the experimental programme conducted [3].
This paper presents the analysis of the experimental data, to study
the effect of compressive strength of concrete at transfer (f
ci
) on L
t
. A
bi–linear formulation of L
t
with respect to f
ci
was developed. The values
of L
t
from the proposed model were compared with those estimated
using IS 1343 (2012), ACI 318 (2014), AASHTO LFRD (2012),
AS 3600 (2009), and CSA A23 (2004) codes. Also, an example pre-
tensioned girder was studied to investigate the effect of over-estimation
of L
t
on the estimation of bursting tensile stress in the transmission
zone.
1.1. Mechanism of strand–concrete bond
The S–C bond is mainly governed by the adhesion, friction, and
mechanical interlock mechanisms [4,5]. Adhesion plays a minimal role
in transferring the prestress as the slipping of the strand with respect to
the hardened concrete during the stress transfer can destroy the ad-
hesive bond [1]. The friction is developed by the Hoyer effect of the
https://doi.org/10.1016/j.istruc.2019.09.016
Received 22 August 2019; Received in revised form 16 September 2019; Accepted 17 September 2019
⁎
Corresponding author.
E-mail address: pillai@iitm.ac.in (R.G. Pillai).
Structures 23 (2020) 304–313
2352-0124/ © 2019 Institution of Structural Engineers. Published by Elsevier Ltd. All rights reserved.
T
strand and confinement of concrete. During prestressing, the strand is
elongated and when the prestress is transferred to the surrounding
hardened concrete, the strand tries to shorten longitudinally and ex-
pand laterally due to Poisson’s effect. However, the surrounding con-
fined concrete provides wedge action and induces compressive stresses
perpendicular to the S-C interface. This increases the frictional force,
which in turn enhances the bond. This phenomenon is known as the
Hoyer effect [6]. In addition, the mechanical interlock occurs due to the
bearing stress provided by the grooves formed by the outer helically
twisted wires.
1.2. Factors affecting L
t
The L
t
of PTC members is highly dependent on the concrete and
strand properties and amount and method of prestressing operations.
The diameter of the strand (d
s
), the surface conditions of the strand, the
compressive strength of concrete at transfer (f
ci
), the confinement due
to the presence of closed stirrups, the method of prestress transfer
(gradual or sudden), and other time–dependent effects (creep and
shrinkage) can affect the L
t
in PTC members. These factors affecting the
L
t
could be classified into two broad categories: (i) strand and asso-
ciated properties and (ii) concrete and associated properties.
1.2.1. Strand and associated properties
The cross-sectional geometry of the strand determines the available
surface area to be bonded with the concrete to transfer the prestress.
Nomenclature
α
Proportionality constant and coefficient used in the L
t
equation by Barnes et al. (2003) and Pellegrino et al.
(2016)
α1
Factor to account the releasing method
α2
Factor to account the type of strand in EN 2 (2004) and the
action effect in fib-MC (2010)
α3
Factor for pretensioned strand in fib–MC (2010)
β, γ, δ
Coefficient used in the L
t
equation Pellegrino et al. (2016)
λFactor to account for the concrete type
Δε
p
Strain difference in prestressed strands
ΔlChange in distance (mm)
K
t
Factor to account the type of tendon in BS 8110 (1997)
d
s
Diameter of the prestressing strand (mm)
ε
c(x)
Strain on the concrete surface at xdistance
ε′
c(x)
Smoothened strain on the concrete surface at xdistance
ε
ce
Effective strain on the concrete surface
ε
pe
Effective strain in prestressing strands
b
w
Width of the cross section at the centroid (mm)
f
bpt
Bond stress at the time of releasing (MPa)
f
ci
Compressive strength of concrete at time of prestress
transfer (MPa)
f
ctd
Design tensile strength of concrete (MPa)
f
pe
Effective stress in prestressing steel after allowance for all
losses (MPa)
f
pi
Initial prestress of the strand (MPa)
f
pk
Characteristic tensile stress in strand (MPa)
f
s
Permissible stress in the reinforcement (MPa)
f
v
Transverse tensile stress at the centroid of the end face
(MPa)
f
t
Tensile stress (MPa)
k
1
Factor to account for the distance of section considered
from L
t
l
x
Distance of section considered from the starting point of
the L
t
(mm)
pPerimeter of the strand (mm)
ACross-sectional area of the concrete (mm
2
)
A
ps
Circumferential area of the strand (mm
2
)
A
sv
Area of vertical steel required (mm
2
)
DOverall depth of the member (mm)
Fbat Transverse tensile force (kN)
L
t
Transmission length (mm)
L
t, code
Calculated L
t
using codal formulations (mm)
L
t, measured
Experimentally obtained L
t
(mm)
L
t, estimated
Estimated L
t
based on proposed model (mm)
MInitial moment due to prestress (N mm)
Fig. 1. Schematic diagrams showing idealized variation of stresses and strains.
Fig. 2. Shear cracks in a highway bridge girder.
P. Mohandoss, et al. Structures 23 (2020) 304–313
305
The AASHTO LRFD (2012), ACI 318 (2014), AS 3600 (2009),
CSA 23 (2014), EN 2 (2004), IRC 112 (2011), fib–MC (2010), and
IS 1343 (2012) codes consider this in an empirical way – by using the
diameter of the strand (d
s
) and a multiplier in the equation for L
t
[7–14]. It is to be noted that, the pitch of the helical outer wires in-
fluences the S–C bond mechanism – longer the pitch, the lesser will be
the angle between the outer and center wires – leading to lower bearing
and mechanical interlock provided. Surface condition of the strand is
also critical to be considered as it influences the adhesion and frictional
mechanisms of S–C bond. Rough or indented surface can enhance the
friction and reduce the L
t
[1,15,16].
The method of prestressing can also influence the L
t.
The gradual
and sudden methods of transfer of prestress can have different dy-
namics of energy at stress transfer [17,18]. In a sudden release of the
stretched strand, the end portion of the strand does not get enough time
to transmit the energy – resulting in longer L
t
than that achieved in a
gradual release method [1,19]. Russell and Burns (1997) also reported
that the effect due to the releasing method becomes less significant as
the length of the PTC element increases [20]. However, the gradual
release method is recommended and it is followed in the current study.
1.2.2. Concrete and associated properties
Based on the studies using low strength concretes prevailing in the
1950s, Hanson and Kaar (1959) found no relationship between L
t
and f
ci
– similar to some of the findings in this paper [5]. However, several
studies after that indicated that the L
t
decreases with increase in f
ci
[1,16,21–27]. Details on these studies and comparing their results are
provided later in this paper. The concrete composition – the amount of
cement and water–cement ratio used could also influence the L
t
as they
primarily contribute to the strength development [28]. However, some
researchers reported no significant change in L
t
with different type of
concrete due to relatively small specimen size, especially between the
self-consolidated concrete and conventional concrete [29–30]. The
stiffness and confinement of concrete, and their effects on L
t
tend to
increase as the concrete strength at transfer is higher, which in turn
results in increased friction and shorter L
t
[1,28,31]. In addition to the
lateral confinement provided by the concrete matrix, the closed stirrups
also provide confinement. Russell and Burns (1996) and Kim et al.
(2016) reported that the effect of stirrups on L
t
is negligible [32,33]. On
the other hand, Vázquez-Herrero et al., 2013 reported that it could help
to minimize variations in L
t
as a function of time – in laboratory studies
[34]. It should be noted that considering size effect, correlating the L
t
of
laboratory specimens with single-strand and a few stirrups with that of
structural elements with multi-strands and many stirrups is challenging
and is not a focus of the current study.
As the maturity of concrete continues to increase with curing, the
S–C bond improves. This, in long term, could help in counteracting the
possible adverse effects such as loss of prestress due to creep, shrinkage,
and relaxation [35–37]. One reviewed literature did not reveal any
consistent information on the systematic variation of L
t
with time. For
example, reported findings say that L
t
can increase or decrease or may
not change with time – as it depends on many other external conditions
[1,23,38–41]. Thus, the time-dependent effects on L
t
for concretes with
different compressive strengths are not well-reported/modelled in lit-
erature.
1.3. L
t
equations provided in the codes and literature
Table 1 provides the design L
t
equations from the codes and lit-
erature studied and Table 2 summarizes the different parameters con-
sidered in the codes. In this paper, L
t
calculated using various codal
equations is denoted as L
t, code
. ACI 318 (2014), AASHTO LFRD (2012),
AS 3600 (2009), CSA A23 (2004), and IS 1343 (2012) provide empirical
equations with d
s
as the only variable. The formula given in
ACI 318 (2014) are based on the experimental work by Hanson and
Kaar (1959). In this, the denominator ‘20.7’ represents the f
ck
of typical
concrete used in the 1950s [21]. However, the properties of concrete
and steel used today are different; hence, for a more rational design,
such equations need to be modified to reflect the properties of today’s
concrete and steel. BS 8110 (1997) code considers d
s
and f
ci
and not f
pe
[42]. Later, this code became obsolete and the EN 2 specifications was
adopted in general practice.
EN 2 (2004) and fib-MC (2010) consider f
ci
to compute the bond
stress. Considering the bond stress or S-C interface condition could re-
sult in a better estimation of L
t
[43]. Both EN 2(2004) and fib–MC
(2010) considered many factors to account for the effect of prestress
releasing method, type of strand, structural action (flexural/shear), and
position/profile of the strand. Both these codes consider upper and
lower bound values for L
t
to verify the design. EN 2 (2004) considers
0.8 and 1.2 times L
t
as its lower and upper bound values to check local
stresses at prestress transfer and ultimate limit states, respectively. On
the other hand, fib-MC (2010) considers factor of 0.5 and 1 as lower and
upper bound values for L
t
to check the design. ACI 318 (2014), AASHTO
LFRD (2012), AS 3600 (2009), and IS 1343 (2012) do not suggest
upper/lower bound values to check the design at transfer and ultimate.
In this case, the calculated L
t
using these codes could significantly in-
fluence the structural performance of PTC systems. The IRC 112 (2011)
has adopted the approach of EN 2 (2004). However, the suggested as-
sumptions/values on these factors could be questionable. More work
needs to be done in this area and is out of the scope of the current work.
Shahawy et al. (1992), Deatherage et al. (1994), and Bucker (1995)
suggested L
t
equations with f
pi
as a variable and not f
ci
(similar to ACI
318) [44–46]. On the other hand, Zia and Moustafa (1977), Cousins
et al. (1990), Mitchell et al. (1993), Lane (1998), Barnes et al. (2003),
Martí-Vargas et al. (2006), Kose and Burkett (2007), Ramirez and
Russell (2008), Pellegrino et al. (2015), and Ramirez–Garcia et al.
(2016) proposed different L
t
equations with f
ci
as a variable
[1,16,21–24,26–28,47,48].
To understand the effect of f
ci
on L
t
, about 200 experimental data
Table 1
Equations for L
t
in the studied literature.
Equation for L
t
Sources
Codes
d
fpe
20.7 s
ACI-318 (1963–2014)
60ds
AASHTO LFRD (2012) & AS 3600 (2009)
50ds
CSA A23 (2004)
30ds
IS 1343 (2012)
Ktds
fci
BS 8110 (1997)
d
1 2
fpi
fbpt s
EN 2 (2004), IRC:112 (2011)
1 2 3
fpi
fbpt
Aps
ds
fib MC (2010)
Research studies
80ds
Martin and Scott (1976)
1.5 d 117
fpi
fci s
Zia and Moustafa (1977)
0.33f d d fci
pi b 3
fci
'
fpi
20.7 s20.7
Mitchell et al. (1993)
d - 127
fci
4fpi
s
Lane (1998)
d
fpi
fci s
Barnes et al. (2003)
2.5Apsfpi
Pfci
2/3
Martí-Vargas et al. (2006)
0.045 fci
fpi(25.4 - ds)2
Kose and Burkett (2007)
d 40d
fci
315
s s
Ramirez and Russell (2008)
+ + +
edsfpi fci
Pellegrino et al. (2015)
25.7 d
fci
fpi
s
0.55
Ramirez-Garcia et al. (2016)
P. Mohandoss, et al.
Structures 23 (2020) 304–313
306
from literature were collected and plotted as a function of f
c
(see Fig. 3)
[16,22,27–30,32,40,44,48–50]. This shows that L
t
decreases as f
ci
in-
creases; therefore, f
ci
must be included as a variable in the design
equations to determine L
t
. Also, Fig. 3 shows a large scatter (say, about
1000 mm as indicated by the gap between two boundary curves) among
the values reported by different studies for the PTC systems. The data
from literature contain values of L
t
for different types of strand (coated
strands and FRP strands) and releasing methods (gradual/sudden).
Among these, the work by Cousins et al. (1990), Deatherage et al.
(1994) and Russell and Burns (1997), Mahmoud et al. (1999), Oh and
Kim (2000), Hodges (2006), Bhoem et al. (2010), and Myers et al.
(2012) used sudden prestress release using flame cut, which resulted in
longer L
t
(varying from 400 to 1500 mm). This difference in L
t
could be
seen in the range of f
ci
about 30–40 MPa. Some of these studies used
different types of strands with coatings and different material compo-
sition, which could also significantly affect the L
t
. Therefore, the values
of L
t
obtained from the various studies for the same f
ci
are not com-
parable. Hence, the effect of f
ci
could not be deduced/modelled using
these data. Ramirez–Garcia et al. (2016) proposed an equation to cal-
culate the L
t
based on their experimental data and reported data from
the literature. It was observed that there is significant scatter between
the reported experimental and literature data. In such case, proposing
an equation based on all these data may not be rational. Therefore, in
this study, the values of L
t
only for uncoated steel strands based on
gradual release method were considered to compare with the experi-
mental results.
2. Research significance
It is found that significant discrepancies exist among the design L
t
equations in the various codes and literature. Some codes provide
empirical formulations to estimate L
t
as a function of only the properties
of strands and do not consider that of concrete. This paper highlights
the influence of f
ci
on L
t
. Also, the influence of seemingly simple, but
complex test procedures on the determined L
t
and its scatter are ex-
plained. Based on the experimental results with various strength grade
concretes, a bi–linear model for L
t
as a function of the properties of both
strand and concrete is developed. It is anticipated that the proposed
equation would be incorporated in the design codes to achieve more
rational estimates of L
t
, and hence, more refined structural designs for
the end zone reinforcements of PTC members.
3. Experimental programme
The important information of the experimental programme is
briefly presented for ready reference. Three prism specimens, each for
Table 2
Parameters considered in the different equations for L
t.
Codes Effective prestress Diameter of strand Compressive strength of concrete at
transfer
Bond stress/
condition
Type of
tendon
Prestress releasing
method
AASHTO (2012) ✗✓✗ ✗ ✗ ✗
AS 3600 (2009)
CSA (2004)
IS 1343 (2012)
ACI 318 (2014) ✓ ✓ ✗ ✗ ✗ ✗
BS 8110 (1997) ✗✓ ✓ ✗✓✗
EN 2 (2004)/IRC 112
(2011)
✓ ✓ ✓ ✓ ✓ ✓
fib-MC (2010)
✓ Considered in the codes
✗Not considered in the codes
Fig. 3. Reported L
t
as a function of f
ci.
Fig. 4. The diagram of the prism specimen.
P. Mohandoss, et al. Structures 23 (2020) 304–313
307
f
ci
equal to 23, 28, 36, or 43 MPa were cast and tested for measuring L
t
.
Fig. 4 shows the diagram of a prism specimen [(100 × 100 × 2000)
mm size] with a concentrically embedded prestressing strand in con-
crete. Also, companion cube specimens [(100 × 100 × 100) mm size]
were cast and cured simultaneously, to determine f
ci
under compression
test.
3.1. Materials
Seven-wire, prestressing steel strands with a nominal diameter of
12.7 mm, pitch of 184 mm, a modulus of elasticity of 196 GPa, and an
ultimate tensile strength of ≈1840 MPa were used in this study. Table 3
shows the chemical composition of the prestressing steel. The OPC 53S
grade cement (finely ground; IS 269, 2015) was used to achieve the
sufficient 3-day compressive strength (to transfer the prestress in
3 days) [51]. Polycarboxylate ether-based superplasticizer was used to
achieve the desired workability (say, slump of 100 ± 20 mm) for the
concrete. As per IS 1343 (2012), the grades of concrete recommended
for PTC construction are M35 and above. Hence, in this study, the
concrete mixes with an average f
ci
of 23, 28, 36, and 43 MPa (these
represent values close to those of grades M35, M45, M55, and M65,
respectively) were used and the mix details are provided in Table 4. The
moduli of elasticity of the concretes at 28 days for f
ci
= 23, 28, 36, and
43 MPa were 30, 33, 35, 39 GPa, respectively.
3.2. Preparation of prism specimens
Fig. 5 shows the setup with the prestressing bed and L
t
specimen.
The self-equilibrating prestressing bed consisted of a hollow steel sec-
tion with two end brackets. Fig. 5 also shows the close-up of the end
anchorages and the stress adjusting system (SAS). Further details on
this is provided in Mohandoss et al. (2018) [3] . The L
t
specimens were
prepared in three stages as follows.
3.2.1. Stage 1: Prestressing the strand
A 5.5 m long seven–wire strand was inserted in the through-holes of
the end brackets of the steel prestressing bed (Fig. 5). Then, an initial
prestress of about 0.75 f
pk
was applied. At the jacking end (JE), a hy-
draulic jack (300 kN capacity), was placed to apply the stress. Steel
chair, wedges and barrels, and load cell were also kept to, activate the
stress reaction, lock the stress, and monitor the load applied, respec-
tively. At the releasing end (RE), the stress adjusting system (SAS) and
wedges and barrels were placed to lock, adjust, and gradually release
the applied stress.
3.2.2. Stage 2: Placement of concrete and brass inserts
After stressing the strand, a custom-designed PVC mould was placed
around the strand (on the prestressing bed) to place the concrete.
Hexagonal brass inserts with bolt/head and nut/base were designed
and fabricated. These were affixed to acrylic strips with pre-defined
holes (50 mm c/c spacing) for accurate spacing and alignment. These
strips with brass inserts were then placed along the center line of the
two opposite, vertical surfaces on the inside faces of the PVC mould. It
should be noted that achieving uniform compaction of concrete along
the strand was crucial for the study. To achieve this, concrete in the
PVC mould was placed in a single layer of 100 mm height and com-
pacted uniformly (25 tampings for every 100 mm length). After 24 hrs,
the L
t
specimen and the accompanying cube specimens were de-
moulded. Then, the acrylic strips were removed from the concrete
surface (by unscrewing the bolt/head of the inserts). Then, the bolt/
head of the inserts was threaded again to the nut/base – to facilitate
DEMEC gauge measurements.
3.2.3. Stage 3: Curing of concrete and transferring of prestress
The specimens were cured (by covering with wet burlap and plastic
sheet) until 3 days or when the concrete attained about 60% of the
target compressive strength, whichever was earlier. After sufficient
curing the prestress was gradually transferred from the strand to the
concrete using the SAS system as shown in Fig. 5.
3.3. Strain measurements and estimation of L
t
The L
t
can be estimated using the strain on the strand surface or
concrete surface. Unlike in unstressed reinforcing bars, the accurate
measurement of stretched strand is difficult. This is because placing of a
strain gauge on the strand surface could locally affect the S–C bond [52]
and the gauge would snap during stretching. Also, Russell and Burns
(1997) suggested to use DEMEC measurements on concrete surface to
minimize manual errors. Hence, in this study, the strains measured on
the concrete surface using 150 mm gauge length was used to estimate
the L
t
. The measured data was smoothened by averaging the strain data
over three consecutive/overlapping 150 mm gauge readings. Then, the
average maximum strain (AMS), as defined by Russell and Burns
(1997), was computed by averaging all the strain measurements on the
plateau. The detailed description on this could be found in Mohandoss
et al. (2018). The L
t
is defined as the distance from the end of the
member to the point with average strain equal to the 95% of AMS. In
this way, L
t
at both ends of the specimen was calculated and the average
of the two values was defined as the L
t
of the specimen.
4. Results and discussions
4.1. Challenges in strain measurements
Strain measurements on concrete surfaces are heavily dependent on
the type of gauges used and bond conditions, especially for long-term
measurements. In this study, initially DEMEC discs were glued onto the
concrete surface using acrylic based adhesive. It was observed that this
glue could not provide sufficient bond between these discs and concrete
– probably due to the moisture curing practices and/or smooth/ma-
chined surfaces of the discs. Also, the force exerted by the technician
while placing the gauge onto the DEMEC disc could also lead to de-
bonding. To avoid such issues, custom–made brass inserts were used to
obtain quality data, especially for long–term measurements. A brass
insert was profiled with a 6 mm wide groove to ensure adequate bond/
grip [3].
For a comparative study, the strain measurements were taken using
Table 3
Chemical composition of prestressing steel.
Element C Si Mn P Cr Ni Al Co Cu S Fe
Conc. (%) 0.72 0.21 0.94 0.02 0.27 0.02 0.01 0.02 0.02 0.002 97.8
Table 4
Concrete mix details.
Ingredients Values of f
ci
(MPa)
f
ci
23 f
ci
28 f
ci
36 f
ci
43
Cement (kg/m
3
) 380 380 420 420
Water – cement ratio 0.50 0.45 0.40 0.35
10 mm aggregate (kg/m
3
) 432 436 428 433
20 mm aggregate (kg/m
3
) 648 655 641 649
Fine aggregate (kg/m
3
) 750 758 743 752
Polycarboxylate ether-based Superplasticizer (% bwoc) 0.8 0.3 0.6 0.5
P. Mohandoss, et al. Structures 23 (2020) 304–313
308
discs and inserts placed on the same surface of a L
t
specimen. As shown
in the top-right corner of Fig. 6, inserts were placed along the mid–-
depth on opposite side faces of the specimen (indicated as I and I
/
) and
the discs were placed adjacent to the inserts (indicated as D and D
/
);
and companion readings were taken. The two curves with filled and
unfilled markers in Fig. 6 indicate that discs can significantly under-
estimate the strain measurements, when above 300 με. For example, at
about 600 mm from the end of the member, the strain values obtained
using inserts and discs were about 550 and 400 με, respectively – in-
dicating a reduction of about 25%. The strain values calculated using
Hooke’s law, applied stress, and the measured elastic modulus of con-
crete were closer to the strains obtained using the inserts than that
using the discs. Hence, the inserts were used in the remainder of this
study.
4.2. Influence of f
ci
on L
t
Table 5 presents the f
ci
,f
pe
at transfer, and measured L
t
(denoted as
L
t, measured
) at both jacking and releasing ends of all the specimens. Fig. 7
shows the smoothened variation of strain, (ε’
c
)
x
, along the length of the
specimens. The values of L
t, measured
at both the jacking and releasing
ends were found to be similar as gradual stress releasing method was
adapted in this study [53]. At transfer, the strain on the concrete sur-
face increased along the length of the member from the end and became
constant. The average maximum strain (AMS), as defined by Russell
and Burns (1997), was computed by averaging all the strain measure-
ments on the plateau. The L
t
is defined as the distance from the end of
the member to the point with average strain equal to the 95% of AMS.
The average strain for all the strengths of concrete at transfer was about
500–530 µε. As the stress is transferred from the strand to concrete at
3 days, the strain variations between the different specimens could be in
the range of 5 to 20 microstrains, which could be not exactly captured
by DEMEC gauge. The average L
t
for specimens with f
ci
= 23, 28, 36,
and 43 MPa strength concretes were 580, 565, 465, and 385 mm,
respectively, with coefficients of variations of about 2 to 10%, which is
reasonable. Also, it was observed that the scatter in the values of L
t
was
less with increasing strength of concrete.
Fig. 8(a) indicates the variation of L
t
with compressive strength of
concrete at transfer. Then the measured values of L
t
were normalized
w.r.t f
pe
and d
s
to model L
t
as a function of only f
ci
. This was based on
the assumptions that L
t
varies linearly with respect to f
pe
and d
s
alone.
As L
t
decreased with increasing L
t
, to capture the effect of f
ci
, a scatter
plot between the inverse of f
ci
and normalized L
t, measured
w.r.t f
pe
and d
s
was developed (see Fig. 8(b)). From the plot, it was observed that the
trend of L
t
as a function of 1/f
ci
was bi-linear. For a lower f
ci
(23 and
28 MPa), L
t
did not decrease significantly with increase in f
ci
. However,
for concrete with strength higher than 28 MPa, significant reduction
(say, upto 32%) in L
t
was observed. Therefore, a bilinear model was
Fig. 5. Sketches and photograph of prestressing bed with PTC specimen.
Fig. 6. Difference in strains from inserts and discs (Inset shows section view).
P. Mohandoss, et al. Structures 23 (2020) 304–313
309
developed as given in Eqs. (1) and (2).
For f
c i
≤ 28 MPa
= +L f d 0.028 0.24
f
t pe s
ci
(1)
For 28 ≤ f
c i
≤ 43 MPa
= +L f d 0.0036 0.94
f
t pe s
ci
(2)
Fig. 9 shows the correlation between the L
t
estimated using Eqs. (1)
and (2) and L
t, measured
. Many data points lie between the two dashed-
lines (one standard deviation away from the average line) indicating
reasonable prediction. Also, the mean absolute percent error (MAPE) of
the model is found to be 3%, – indicating reasonable prediction accu-
racy.
4.3. Comparison of L
t
obtained from the proposed model and various codes
Fig. 10 shows the comparison of L
t
obtained from the proposed
models and various codes (considering f
ci
). The straight lines indicate
the L
t
obtained using AASHTO, AS, ACI, CSA, and IS codes (i.e., not a
function of f
ci
). However, L
t
curves based on EN 2 (2004), fib–MC
(2010), and the proposed model show a downward trend as the con-
crete strength increases.In general, PTC members will use concrete
with f
ci
more than 20 MPa. Considering this, Fig. 10 indicates that all
the codes, except IS 1343 (2012), and the proposed model overestimate
the L
t
. Overestimation is not a conservative approach for design of
stirrups for bursting stress near the ends and this is explained in the
following section. Hence, a rational approach in estimating L
t
is re-
quired based on f
ci
. Also, concrete strength achieved could be lower
than the desired or target strength due to poor construction materials
and practices, which could result in longer L
t,
which could reduce the
available shear stress in the transmission zone.
The values of L
t
based on proposed model were compared with
values of L
t
from the literature (see Fig. 11). Certain individual data
points from Mitchell et al. (1993), Hegger (2007), NCHRP-603 (2008),
and Ramirez-Garcia et al. (2016) studies matched with the proposed
model. This could be due to gradual releasing method used in those
studies, as adapted in the present work. Russell and Burns et al. (1996)
adapted a combination of gradual and sudden release methods and
resulted slightly longer than the measured L
t
. All these studies de-
termined the L
t
based on the strain measurement on the concrete sur-
face except Martí-Vargas et al. (2012). Martí-Vargas et al. (2012) de-
termined the L
t
by measuring the effective force on the strand using
AMA systems, which is a virtual part of the specimen, whose stiffness
would be more than the concrete part that was replaced. In addition,
specimen size used in different studies are different which could also
affect the measurement of L
t
.
There are some scatters due to the difference in the geometry and
materials considered in the various studies. Therefore, the values of L
t
obtained from the various studies for the same f
ci
are not comparable.
Ramirez-Garcia et al. (2016) reported that L
t
is not varied significantly
when the f
ci
was greater than 34 MPa. Based on their work, this maybe
applicable for 15.2 mm strands but not for 12.7 mm strand as f
ci
greater
than 35 MPa was not considered for 12.7 mm strands. However, the
reported literature data and proposed model show a significant
Table 5
Values of transmission length for different values of compressive strength of
concrete at transfer.
Specimen ID f
ci
(MPa) f
pe
(MPa) L
t, measured
(mm) L
t, avg.
(mm)
Std. dev.
(mm)JE RE Avg.
f
ci
23 – 1 23.5 1270 630 565 598 580 32
f
ci
23 – 2 22.8 1189 520 570 545
f
ci
23 – 3 22.1 1230 625 580 600
f
ci
28 – 1 28.0 1179 550 570 560 565 17
f
ci
28 – 2 29.0 1209 540 560 550
f
ci
28 – 3 28.7 1220 565 600 585
f
ci
36 – 1 37.1 1148 480 500 490 465 28
f
ci
36 – 2 36.0 1179 460 470 465
f
ci
36 – 3 36.8 1120 425 445 435
f
ci
43 – 1 43.4 1169 385 370 375 385 9
f
ci
43 – 2 42.6 1189 385 405 395
f
ci
43 – 3 43.7 1108 400 375 390
Fig. 7. Variations of strains on the concrete surfaces for different f
ci.
P. Mohandoss, et al. Structures 23 (2020) 304–313
310
reduction in L
t
for f
ci
greater than 34 MPa. This indicates that f
ci
greater
than 35 MPa has an effect on L
t
, as it depends on the bond between the
strand and concrete.
4.4. Effect of overestimation of L
t
on the design of stirrups for bursting
stresses
In the transmission zone of the PTC beams, transverse reinforcement
is necessary to prevent the concrete cracking due to large radially
outward stresses. The crack occurs when the maximum stresses exceed
the tensile strength of the concrete. At the end of the member, the
transverse tensile stress is maximum and gradually gets reduced along
the length in the transmission zone [54]. However, for the design of end
reinforcement, this transverse tensile stress was approximated as a
linear variation over half the L
t
[55]. The transverse tensile force and
area of the vertical reinforcement are computed as follows using Eqs.
Fig. 8. Variation of L
t
with the compressive strength of concrete at transfer.
Fig. 9. Correlation between L
t, estimated
and L
t, measured.
Fig. 10. Comparison of L
t
models as function of f
ci.
fci (MPa)
10 20 30 40 50 60 70 80 90 100
Lt (mm)
0
250
500
750
1000
1250
1500
1750
2000 Mitchell et al. (1993)
Russell and Burns (1996)
Rose and Russell (1996)
Hegger (2007)
NCHRP-603 (2008)
Marti-Vargas et al. (2012)
Ramirz-Garcia et al. (2016)
Dang et al. (2017)
Proposed model
Fig. 11. Comparison of proposed model with L
t
from literature.
P. Mohandoss, et al. Structures 23 (2020) 304–313
311
(3) and (4).
=F1
2fL
2b
bat v(max)
t
w
(3)
=A2.5M
f D
sv
s
(4)
These tensile stresses are dependent on the prestress level, rate of
prestress transfer along the length, etc., which influence the L
t
. As an
example study, the tensile stress (f
t
) distribution at the level of the
centroid of strands along the length of a PTC beam was investigated to
determine the effect of overestimation of L
t
on the transverse tensile
stress at the end of the member. Fig. 12 shows the cross section of the
beam considered for the study.
Fig. 13(a) presents the tensile stress distribution in the transmission
zone for different L
t
ranging from 300 to 1500 mm. At the end of the
member, the tensile stress was maximum and it gradually decreased to
zero at the end of transmission zone. In general, overestimation of
transmission length is considered as a safe design. Therefore, im-
portance was not given to this aspect. The authors’ previous paper
(Mohandoss et al. (2018)) addressed the significance of under-
estimation of transmission length and its effect on the shear capacity of
the members. This paper attempted to highlight only the case with
overestimation of L
t
.Fig. 13(b) shows the maximum tensile stress as a
function of the L
t
for the f
ci
= 42 MPa. It indicates that if the L
t
is
overestimated, then the maximum tensile stress is under estimated and
vice versa. As a consequence, this reduces the area and increases spa-
cing of the stirrups in the transmission zone, resulting in less shear
reinforcement. In this case, it could result in bursting and/or cracking at
prestress transfer. Hence, it is important not to overestimate the
transmission length beyond certain limit. According to the current
study, an overestimation up to 30% seems safe. This scenario could
become worse, if the strength of concrete is lower than 42 MPa. This is
critical, especially in narrow web pretensioned beams/girders, where
the strands would be closely placed and congested [56]. Therefore, the
overestimation of L
t
beyond 30% of the values from tests should be
avoided while designing PTC members.
5. Summary and conclusions
The transmission lengths (L
t
) for pre–tensioned concrete (PTC)
members for several values of the compressive strength of concrete at
transfer (f
ci
), were experimentally obtained from twelve specimens.
Based on this experimental data, a bi–linear model was developed for
determination of L
t
as a function of f
ci
(Eqs. (1) and (2)). The values
from the proposed model were compared with those obtained from
several codes. Also, an example study was done to understand the
significance of the overestimation of L
t
. The following conclusions were
drawn from the present study.
•The compressive strength of concrete at transfer has a significant
influence on L
t
. When f
ci
increased from 23 to 43 MPa, the possible
range in practice, the values of L
t
decreased from about 600 to
400 mm (33% reduction). Also, the scatter in the values of L
t
was
less with increasing strength of concrete.
•For the range of concrete compressive strength at transfer used in
PTC systems, the AASHTO, ACI, AS, CSA, EN and fib codes over-
estimate the values of L
t
, when compared with those estimated using
the proposed model. Only the Indian standard underestimates the
value of L
t
.
•From the case study, it was observed that the overestimation of L
t
is
not advisable for the design of the transmission zone reinforcement
required for the bursting stresses. Hence, the proposed model that
considers the compressive strength of concrete at transfer to
Fig. 12. Cross section of a simple girder.
Fig. 13. Bursting tensile stresses in the transmission zone.
P. Mohandoss, et al. Structures 23 (2020) 304–313
312
estimate L
t
precisely is expected to provide a more rational design of
the transmission zone reinforcement.
Declaration of Competing Interest
The authors declare that they have no known competing financial
interests or personal relationships that could have appeared to influ-
ence the work reported in this paper.
Acknowledgements
The authors acknowledge the financial support through the ‘Fund
for Improvement of Science & Technology Infrastructure (FIST)’ Grant
ETII-054/2012 from the Department of Science and Technology (DST),
and other financial support from the Ministry of Human Resource
Development (MHRD), Govt. of India, through the Department of Civil
Engineering, Indian Institute of Technology Madras (IITM), Chennai.
The authors express their gratitude to Prof. Ravindra Gettu, president,
RILEM for his valuable feedback and suggestions on this research work.
The authors also acknowledge the assistance from the laboratory staff
and students in the Construction Materials Research Laboratory at
IITM.
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