Analytical Study of Force–Displacement Behavior and Ductility of Self-centering Segmental Concrete Columns
International Journal of Concrete Structures and Materials
Analytical Study of Force-Displacement Behavior and Ductility of Self-centering Segmental Concrete Columns
In this study the behavior of unbonded post-tensioned segmental columns (UPTSCs) was investigated and expressions were proposed to estimate their ductility and neutral axis (NA) depth at ultimate strength. An analytical method was first employed to predict the lateral force-displacement, and its accuracy was verified against experimental results of eight columns. Two stages of parametric study were then performed to investigate the effect of different parameters on the behavior of such columns, including concrete compressive strength, axial stress ratio, diameter and height of the column, axial stress level, duct size, stress ratio of the PT bars, and thickness and ultimate tensile strain of fiber reinforced polymer wraps. It was found that the column's aspect ratio and axial stress ratio were the most influential factors contributing to the ductility, and axial stress ratio and column diameter were the main factors contributing to the NA depth of self-centering columns. While at aspect ratios of less than ten, as the axial stress ratio increased, the ductility increased; at aspect ratios higher than ten, the ductility tended to decrease when the axial stress ratio increased. Using the results of parametric study, nonlinear multivariate regression analyses were performed and new expressions were developed to predict the ductility and NA depth of UPTSCs.
post-tensioned; self-centering; concrete columns; ductility; seismic response; force-displacement behavior
In the recent past considerable attention has been given to
the use of self-centering precast concrete members and
(Priestley et al. 1999; Kurama et al. 2002; Perez
et al. 2007; Kim and Choi 2015)
and bridge construction
(Dawood et al. 2011; Kwan and Billington 2003; Youssf
et al. 2015a, 2016; Kim et al. 2012; Kim 2013)
Self-centering behavior can be induced to the system by employing
unbonded post-tensioning (PT) steel. In these systems the
residual drifts and the damage to the structure in seismic
events are limited, resulting in reduced cost associated with
structural repair and business downtime
(Hassanli et al.
. Self-centering concrete systems can display a ductile
response and carry high levels of lateral loads (Henry 2011).
The conventional methods used to determine the lateral
strength of concrete columns cannot be applied to
self-centering columns due to the increase in the PT force that results
in variable axial load. Moreover, strain compatibility is not a
correct assumption for these columns
(ElGawady et al. 2010;
. As a result of the rotation of the columns
at their base, deformations of self-centering columns are
mainly due to the rocking mechanism (Wight et al. 2007).
This rocking mechanism leads to variation in the level of axial
stress at different drift ratios and hence the lateral load
behavior of the columns. The lateral displacement of the
selfcentering columns also consists of elastic flexural
deformation, shear deformation and relative movement of segments
due to sliding. Deformation due to shear and sliding are
usually ignored, however they might be significant in columns
with low aspect ratios
(Priestley et al. 1996a)
Recently, segmental precast concrete members have
become a subject of interest. In segmental systems with PT
bars/tendons, the PT force works as the clamping force
which keeps the segments together. Unbonded
post-tensioned segmental columns (UPTSCs) can be used in bridge
structures to reduce construction times and improve
structural quality. Monotonic, cyclic and dynamic behavior of
these columns have been studied experimentally by many
(Chang et al. 2002; Chou and Chen 2006;
Marriott et al. 2009; Yamashita and Sanders 2009; Wang
et al. 2008; Shim et al. 2008; Bu et al. 2015)
. While UPTSCs
are capable of withstanding large nonlinear displacement,
due to their low level of damage under seismic excitation,
they experience large displacements. To tackle this
drawback, different approaches have been developed to increase
the energy dissipation capacity of the rocking columns,
including the use of metallic yielding components
(ElGawady 2011; Ou et al. 2009, 2010)
, shape memory alloy
(Roh and Reinhorn 2010; Roh et al. 2012)
energy dissipating dampers
(Palermo et al. 2007)
columns with cast-in-place (emulative) base
(Kim et al. 2015;
Ou et al. 2013)
. To improve the behavior self-centering
precast members the effect of using confinement jackets
(Motaref et al. 2013; Hewes and Priestley 2002)
fiber-reinforced concrete (Shajil et al. 2016), and rubber or
elastomeric bearing pads
(ElGawady 2011; Motaref et al. 2010)
have also been investigated.
Research undertaken on the analytical approach to predict
the behavior of UPTSCs is limited.
Hewes and Priestley
proposed a simplified analytical model to predict the
lateral force- displacement relationship.
Ou et al. (2007)
extended the approach to hybrid UPTSCs (with longitudinal
mild steel across the segments’ joints), and using a modified
curvature to account for the effect of mild steel, they showed
the accuracy of the method for hybrid UPTSCs.
et al. (2007)
showed that a similar design procedure
considering lumped plasticity models can be used to design and
model hybrid bridge structures.
Chou et al. (2013)
considered two plastic hinges to account for the joint opening
between the column and its base as well as between the first
two column segments and demonstrated that their model is
capable of predicting pushover responses of UPTSCs.
Three dimensional finite element (FE) modeling has been
performed to predict the force–displacement behavior of
(Dawood et al. 2011; Ou et al. 2007; Chou et al.
2013; Sideris 2015; Youssf et al. 2015b)
. Ou et al. (2007)
demonstrated that three dimensional FE modeling can be
used to effectively predict the lateral response of UPTSCs.
Chou et al. (2013)
Kim et al. (2010)
showed that two
dimensional FE modeling can also predict the
force–displacement behavior response of UPTSCs.
Using experimental results of eight columns, this paper
evaluates the accuracy of an analytical approach to predict
the behavior of UPTSCs. The concept of the analytical
method was similar to previous studies
(Hewes and Priestley
2002; Ou et al. 2007; Pampanin et al. 2001)
; however, the
displacement at the column top was considered as the
independent variable rather than the curvature at the base of
column. Moreover, the plastic hinge length expression
Hassanli et al. (2017)
was considered. Stress–strain
relationships for un-confined and FRP-confined concrete,
proposed respectively by
Kent and Park (1971)
, were adopted. The analytical approach was
developed based on the mechanics of rocking of columns
about their base and geometric compatibility conditions. In
this approach the column rotation was assumed to occur only
at the column base.
Using the validated analytical approach, Stage I of a
parametric study consisting of 43 columns was performed to
investigate the effect of different parameters on the behavior
of self-centering columns, including concrete compressive
strength, axial stress ratio, diameter and height of the
column, axial stress level, duct size, stress ratio of the PT bars,
and the thickness and ultimate tensile strain of fiber
reinforced polymer (FRP) wraps. The results of Stage I of the
490 | International Journal of Concrete Structures and Materials (Vol.11, No.3, September 2017)
parametric study were then used to generate the
configuration of the columns for Stage II of the parametric study. The
ductility of the columns was then calculated and nonlinear
multivariate regression analysis was performed to develop
an expression to estimate the ductility of UPTSCs.
2. Analytical Approach
The conventional strain compatibility method to determine
the lateral strength of concrete columns is not appropriate for
unbonded PT specimens
(Hewes and Priestley 2002; Bu and
, as explained in the introduction section.
et al. (2017)
developed an expression to predict the plastic
hinge length of masonry walls and incorporated it with a
step-by-step analytical procedure
analytical method they developed could effectively predict
the lateral behavior of the masonry walls. Hassanli et al.
(2016a) applied the procedure to concrete walls and showed
that it could accurately predict their lateral force behavior. In
this study the same procedure was considered, however, it
was modified to be appropriate for column members. The
steps of the analytical approach were as follows:
2.1 Force–Displacement Response at
The stress distribution over the cross section at the
decompression point is illustrated in Fig. 1. In the rocking
walls before the decompression point, as the whole cross
section is in compression, no elongation of the PT tendons/
bar and hence no increase in the PT force occurs
et al. 2016a, b)
. By taking moments about the column toe,
the base shear corresponding to the decompression point can
be calculated as:
ðfseAps þ N ÞðrÞ
where r is the radius and half of the length of the cross
section in a circular and rectangular cross sections,
respectively, N is the gravity load including the self-weight of the
column, fse is the initial stress in the pre-stressing steel after
fse Aps+ N
where fcoðxÞ and bðxÞ are respectively the concrete stress and
width of the element at distance x from the column heel, see
Fig. 1. The maximum compressive stress in the concrete at
the decompression point at the column toe, fco, is usually a
small value and a linear stress–strain relationship can be
considered before the decompression point
(Hassanli et al.
. Therefore, the compression stress at distance x from
the column heel can be expressed as,
fcoðxÞ ¼ 2r e0Ec
where e0 is the maximum compressive strain in the concrete
at the decompression point and Ec is the concrete elastic
modulus. Considering a triangular distribution of
compressive stress on the cross section (Fig. 1), the e0 can be
u0 ¼ e0=2r
fseAps þ N
fseAps þ N
e0 ¼ 0:5pr2Ec
rectangular cross sections
circular cross sections:
Using Eqs. 2 and 3, the values of cc0 and X were calculated
for rectangular and circular cross sections and are presented
Assuming a linear curvature variation along the height of
the column and by integrating the curvature function along
the height, the lateral displacement at the column top at the
decompression point is
(Priestley et al. 2007)
where u0 is the maximum wall curvature at decompression
Using Eqs. 1 and 6 a point of the force displacement
response corresponding to the decompression point, D0,V0,
can be determined.
2.2 Force–Displacement Response Beyond
The stress increment in the PT steel beyond the
decompression point needs to be taken into account. Considering a
rocking mechanism (Fig. 2) and assuming that the entire
rotation of the column occurs at the base, to determine the
force–displacement response of UPTSCs, the following
iterative procedure needs to be undertaken: (note that
immediate losses, Aps is the pre-stressing steel area, hc is the
column height, cc0 is the total compression force in concrete
at the decompression point and X is the corresponding lever
arm. cc0 can be calculated as (Fig. 1),
where fc0 is the axial compressive strength of unconfined
concrete (MPa), and fc is the axial stress level which is
assuming the entire rotation of the column occurs at the
column footing interface is reasonable if the column
segments are similar and the same column to footing connection
is used between the column segments. In other cases such as
in hybrid columns where the energy dissipation devices do
not extend to the top segments, the possibility of gap
opening between the column segments should also be
1. Assume a top displacement of D (due to rocking) and
calculate the corresponding column rotation, h (Fig. 2)
(h = D/hc).
2. Assume a value of neutral axis (NA) depth, c (Fig. 2).
3. Calculate the strain in the PT steel, eps, using Eq. 8.
eps ¼ hðr
cÞ=lps þ ese
where lps is the unbonded length of the PT steel and ese
is the effective strain in the PT steel after immediate
stress losses. Equation 8 implies that the strain in the PT
bar is the summation of the effective strain due to initial
post-tensioning, ese, and the strain developed due to the
4. Calculate the concrete strain at the extreme compressive
fiber, ec; using Eq. 9,
ec ¼ hc=Lpl þ e0
where Lpl is the plastic hinge length. No comprehensive
study was found in the literature evaluating the plastic
hinge length of UPTSCs, hence, the following
expression, which was originally developed for wall
(Hassanli et al. 2015, 2016a)
was adopted in
where Ac is the cross sectional area of concrete column.
5. Calculate the stress developed in the PT steel using an
appropriate constitutive model. In this study, the
following elasto-plastic material model with linear
kinematic hardening was considered,
6. Calculate the corresponding concrete stress. An
appropriate stress–strain relationship must be utilized. In this
study the following models were adopted for unconfined
and FRP-confined concrete,
Unconfined concrete In this study Kent–Park
stress–strain relationships was used (Eq. 13)
and Park 1971)
for unconfined concrete.
fcj ecj ¼ >fc0 1
where fcj ecj is the concrete stress (MPa), and ecj is
the concrete strain at distance xj from the neutral
axis (Fig. 3),
ecj ¼ ðxj=cÞec
(b) FRP-confined concrete Numerous models have
been proposed to predict the stress–strain response
of FRP-confined concrete. A comprehensive
review of 88 models developed to predict the
axial stress–strain behavior of FRP-confined
concrete in circular sections was performed by
Ozbakkaloglu et al. (2013)
. They considered both
design-oriented and analysis-oriented models and
to provide a comprehensive assessment of the
models, a large test database of 730 FRP-confined
concrete cylinders tested under monotonic axial
compression was collected. Comparing all models,
they concluded that the one developed by
was the most accurate model to
predict the ultimate strength. Therefore, this model
(provided below) was used in this study for
fc0 þ Ec2ecj
ðEc Ec2Þ2 ec2j ecj\ec1
fcj ecj ¼
ec1 ¼ Ec
Ec2 ¼ fc0c
f0 ¼ 0:872fc0 þ 0:371fl þ 6:258
( fc0 ð1 þ 3:3 flfuc0;aÞ fl
ecu ¼ 1:75 þ 5:53ðflfu;0aÞðeeFcRoPÞ0:45
where fc0c = axial compressive strength of
FRPconfined concrete (MPa); fl = confining pressure
provided by the FRP jacket (MPa); tFRP = thickness
of the FRP jacket; fFRP = ultimate tensile strength of
FRP material (MPa); flu;a = actual lateral confining
pressure at ultimate (MPa) (= 0.586 fFRP for CFRP
(Lam and Teng 2003)
); eFRP = ultimate tensile strain
of FRP material; and eco = axial strain of unconfined
concrete at fc0 and ecu = ultimate axial strain of
7. Calculate the total compression force, cc, and the total
tension force, T, using Eqs. 23 and 24.
cc ¼ 0c fcjdA ¼ 0c fcjbwdx
T ¼ rpsAps
where rps can be determined using Eq. 12, fcj can be
determined using Eq. 13 or 16 for unconfined and FRP
confined concrete, respectively, and bw can be calculated
form Eq. 32 or 33 for rectangular or circular cross
M ¼ rpsApsðd
cÞ þ N ðr
11. The point with the coordinate of (Df, V = M =hc)
corresponds to a point in the force–displacement curve.
To obtain another point, return to step 1. The
flowchart shown in Fig. 4 presents the proposed
procedure of the design approach.
It is worth noting that the presented force–displacement
procedure considers only elastic flexural deformation (before
decompression) and rocking response (beyond
decompression). The deformation due to shear and sliding is ignored. In
rocking columns with large aspect ratios (usually of higher
(Priestley et al. 1996a)
, shear and sliding
deformations typically have small magnitudes (Wight et al. 2007)
and hence can be neglected, otherwise the corresponding
displacement, Ds, needs to be considered to determine the
total displacement (Df ¼ D0 þ D þ Ds).
3. Verification of Analytical Approach
The accuracy of the presented analytical approach was
verified against the experimental results of eight columns
Hassanli et al. (2017)
. The properties of the test
specimens are shown in Table 1. All the columns had a
diameter of 150 mm and shear span of 1425 mm and were
tested under incrementally displacement increasing reverse
cyclic load. The top three segments in all columns were
constructed out of crumb rubber concrete (CRC). The
bottom-most segment in four column specimens (C50, C100,
CF50 and CF100) was constructed out of conventional
concrete (CC) and in the other four columns (CR50, CR100,
CRF50 and CRF100) it was constructed out of CRC. The
other variables were the level of initial PT force (50 kN in
columns C50, CF50, CR50 and CRF50, and 100 kN in
columns C100, CF100, CR100 and CR1050), and the effect
of confining the bottom-most segments (unconfined and
confined with 1 layer of FRP).
In the analytical approach, for both CC and CRC, the
concrete material model presented in Eq. 12 was adopted.
Due to rocking of columns about their bases and damage in
only the bottom segments during the experiments, the
Calculate (Δ0 ,V0) of the decompression point
(Eqs. 1 and 6)
Calculate the total compression force (Eq. 23)
and total tension force (Eq. 24)
Select a value of top column displacement, Δ
Assume a value for NA depth, c
Calculate the strain in the PT steel,
Calculate the concrete strain at the extreme compressive
fiber, (Eq. 9)
Calculate the stress developed in PT bar using an appropriate
constitutive model (e.g. Eq. 12)
Calculate the concrete stress of element j,
(e.g. Eq. 13 for unconfined and Eq. 16 for FRP-confined concrete)
, using an appropriate
Assume a new
value of c
Calculate the total moment capacity (Eq. 25)
Plot pair of (∆= ,V= ) on the
a The unsaturated ultimate strength, elastic modulus, and thickness of the FRP sheets were 4950 MPa, 227 GPa, and 0.156 mm, respectively.
b The PT bars had 20 mm diameter, nominal tensile yield stress of fpy = 900 MPa and ultimate tensile stress of fpu = 1100 MPa, and total
unbonded length of 1950 mm.
CC conventional concrete, CRC crumb rubber concrete.
material properties of CC and CRC were used for columns,
C50, C100, CF50 and CF100 and columns CR50, CR100,
CRF50 and CRF100, respectively.
Figure 5 plotted the lateral force–displacement curves
obtained from the experimental study against those of the
analytical approach. As shown in the figures, the model was
able to correctly predict the column strength, initial stiffness,
and post-peak behavior. The model could also capture the
displacement recorded at different strengths.
The maximum lateral strengths of the columns obtained
from the force–displacement curves of the analytical
approach, VAnalysis, are compared with corresponding
experimental results, VEXP, in Table 2. As shown in the table,
the analytical approach could accurately predict the strength
of the walls. The predicted strength of the specimens using
the analytical approach falls within ±12% of the average of
the test results.
Figure 6 compares the force developed in the PT bars
obtained from the experimental work and the analytical
approach. As shown in the figure, in general, the analytical
model showed a good correlation with the experimental
results. Comparing the force developed in each PT bar at the
peak strength of the columns using the analytical approach,
TAnalysis, and experimental results, TEXP, revealed that using
the analytical approach, the PT force in the bars at the peak
strength was within ±12% of the average of the test results.
Note that the analytical model slightly overestimates the
force in the bars, especially at large drifts. For example for
column CR100 (see Fig. 6d), at large drift ratios, the PT
force predicted by the analytical approach deviated from the
experimental results. This overestimation can be attributed to
the losses that occurred in the PT bars during testing, which
are not considered in the analytical model. These losses
occur due to movement of the anchorage, deflection of
anchorage plates, elastic shortening of the column, the
friction between the PT tendon and the wall of the ducts at
higher drift ratios, which are very complicated and hence
difficult to quantify. However, this over-prediction of PT
stress is less sensitive when the force–displacement behavior
is considered. As shown in Fig. 6d, the analytical procedure
provided an acceptable prediction of the force–displacement
response of column CR100. Also note that in both Figs. 6d
and 7d the prediction from the analytical approach deviated
from the experimental results after the column had
experienced drift of higher than 4%. In practice, column drifts are
limited to smaller values. For example in a building
structure, due to the limited deformation capacity of
non-structural elements and to control damage at life safety design
level, the inter-story drift under the design level earthquake
should not exceed 2%
(IBC 2006; Priestley 2002)
. Due to
these reasons the over-prediction of PT stresses at large drifts
can be ignored.
The presented analytical procedure was developed for
selfcentering columns without energy dissipators; however, if
internal mild steel or external dissipators or other similar
devices are used, the approach can potentially be used if
modified to account for the contribution of the dissipators on
the moment capacity and stiffness of the column, as well as
the possibility of joint opening between segments. Previous
studies have shown that considering geometric compatibly
concepts, a similar analytical approach can be used to predict
the response of precast concrete hybrid frames and piers with
(Pampanin et al. 2001; Cao et al. 2015)
4. Parametric Study
As described, the presented analytical method was able to
accurately predict the force–displacement behavior of
UPTSCs and the force developed in the PT bars. The
validated analytical approach was then used to conduct a
parametric study to better understand the behavior of UPTSCs
and also to develop expressions to predict the ductility of
UPTSCs. The parametric study was performed in two stages
by considering two different sets of column models. The first
set was developed to examine the effect of different
UPTSCs. Subsequently, matrices of column models were
generated for the second stage (Stage II) of the parametric
study. Stage II of parametric study was performed on a set of
columns considering two parameters, axial stress ratio and
aspect ratio as main variables to investigate the ductility of
4.1 Parametric Study—Stage I
The first stage of the parametric study considered 43
columns in eight groups, shown in Table 3, to assess the
influence of different parameters on column strength,
deformation and ductility. As presented in Table 3, the
columns C1-1 to C1-6, C2-1 to C2-7, C3-1 to C3-6, C4-1
to C4-5, C5-1 to C5-4, C6-1 to C6-6, C7-1 to C7-5, and
C8-1 to C8-4 were used to study the effects of concrete
compressive strength, fc0 , axial stress ratio, fc=fc0 , axial
stress level, fc, duct size, column diameter, D, column
height, hc, FRP wrap thickness, t, and ultimate tensile
strain of FRP, on the strength and deformation of
Column C50 (Table 1) was considered as the ‘‘control’’
specimen. It had a height of 1425 mm and a 150 mm
diameter circular cross section. The concrete compressive
strength of 55 MPa, PT load of 50 kN, stub beam weight of
5 kN; and unbonded PT bar with the length of hc ? 800
(mm), cross sectional area of 314.4 mm2, yield strength of
901 MPa, ultimate strength of 1102 MPa and elastic
modulus of 200 GPa were considered for the control specimen.
Moreover, the duct size factor, Df , of 0.03 was considered
for the control specimen. The duct size factor, Df , is defined
where dduct and db are the diameter of the duct and tendon/
PT bar, respectively.
The effect of duct size is particularly significant for hollow
sections. For example, when concrete masonry blocks or
precast concrete hollow sections are used, the movement of
the PT steel within the duct must be considered.
4.1.1 Effect of Compressive Strength, fc0
To investigate the effect of fc0 , columns C1-1 to C1-6 with
compressive strength of 25–75 MPa were generated. The
responses are plotted in Fig. 7. As shown in the figure, as the
compressive strength increased, the lateral strength and the
normalized PT force increased and the NA depth decreased.
As shown in Fig. 7b, columns with lower fc0 exhibited a
decreasing trend of force developed in the PT steel at large
drift ratios. This is attributed to the earlier damage of
concrete in compression zone in columns with lower fc0.
However, at higher fc0, the columns were intact for longer time at
higher drift ratios as observed in specimens C1-5 and C1-6.
The moment capacity of the columns increased from 3.8 to
4.7 kN m (24% increase) when the compressive strength
increased from 25 to 75 MPa, see Fig. 7d.
4.1.2 The Effect of Axial Stress Ratio, fc=fc0
The axial stress ratio is defined as the initial stress on the
column, fc, divided by fc0. The initial stress on the column
was calculated as the summation of the initial PT force and
the self-weight of the column and the loading beam, divided
by the column cross sectional area, Ag. In this parametric
study, the axial stress ratio, fc=fc0 , was varied from 0.032 to
0.263 in columns C2-1 to C2-7, by keeping the value of fc0
constant and changing the PT force. As shown in Fig. 8 the
level of fc=fc0 , has significant effect on the lateral capacity,
PT force and NA depth. Although in columns with higher
fc=fc0 , the lateral force capacity was higher, they exhibited a
higher rate of post-peak strength degradation. This is
attributed to the increased probability of concrete crushing at
high stress levels that results in faster degradation of the
lateral strength. The normalized PT force in the columns
with lower fc=fc0 was significantly higher than that in
columns with high levels of fc=fc0 . While the NA depth
comprised about 0.2D in the column C2-1 with fc=fc0 ¼
0:032 it increased to 0.45D in the column C2-7 with
fc=fc0 ¼ 0:263, see Fig. 8c. As shown in Fig. 8d, the moment
capacity of column C2-1 was 3.2 kN m, however; it
increased to 10.3 kN m in column C2-7. From all these
comparisons it can be concluded that the axial stress ratio
has significant effect on the capacity and behavior of
4.1.3 The Effect of Axial Stress Level, fc
Five different values of axial force were applied to
columns C3-1 to C3-6, providing an axial stress ranging from
1.4 kN in column C3-1 to 14.2 kN in column C3-6. To
maintain similar axial stress ratio (of 0.15) in all columns of
this category, the value of fc0 was varied accordingly. As
shown in Fig. 9, as the axial stress level increased the lateral
force capacity increased and the normalized PT force
decreased. According to Fig. 9c, the NA depths of the
columns of this category were similar and were about 0.35D at
peak load for all columns. Consequently, as long as the axial
stress ratio was constant, the NA depth was independent of
the axial stress level. The moment capacity of the columns
increased approximately linearly from 1.4 to 14.1 kN m as
the axial stress level increased from 1.4 to 14.2 kN, see
4.1.4 The Effect of Duct Size, Df
If an oversized duct is used to accommodate the PT bar,
then due to the rocking mechanism the PT bar moves
towards the compression zone until it touches the duct’s
inner wall. To investigate the effect of oversized ducts,
five duct size factors, Df (see Eq. 26) were used ranging
from 0 in column C4-1 to 0.13 in column C4-5. As shown
in Fig. 10a, as the duct size factor increased, the lateral
force capacity and the force developed in the PT steel
decreased. This is attributed to the reduced lever arm. At
peak strength, the NA depth was approximately similar in
all columns of this category. Increasing the duct size factor
from 0 to 0.13 caused the moment capacity to reduce from
5 to 3.2 kN m (about 36% reduction). Thus it is concluded
that if an oversized duct or concrete hollow section is
used, its negative effect on the effective PT depth, and
hence the lateral strength of the columns, must be taken
4.1.5 The Effect of Column Diameter, D
To investigate the effect of the diameter on the behavior of
UPTSCs, columns C5-1 to C5-4 having diameters ranging
from 150 to 450 mm were considered. As expected
increasing the column diameter had significant effect on its
behavior. The lateral strength, the PT force and the moment
capacity increased as the diameter increased, as shown in
Fig. 11. By increasing the column diameter, the lever arm of
the concrete cross-section increases. This increases the
moment capacity as well as the lateral strength of the column
section and delays the column deterioration under cyclic
loading, and hence results in a higher force in the PT bars
4.1.6 Effect of Column HEIGHT, hc
Columns C6-1 to C6-6 (Table 3) having heights ranging
from 0.5 to 3.0 m were generated to investigate the effect of
the column’s height on the behavior of UPTSCs. As the
height increased the lateral load capacity and the PT force
decreased due to the decrease in the column stiffness by
increasing its height. However, the moment capacity
decreased slightly from 5.0 in column C6-1 to 4.1 in column
C6-6. This can be attributed to the longer unbonded length
of PT steel in longer columns compared with that of shorter
columns which results in a lower PT strain, and hence lower
PT forces compared with shorter columns. The decreased PT
force results in reduced moment capacity in longer columns.
Figure 12c shows that the NA depth of columns C6-1 to
C66 remained relatively similar. At peak load the NA depth was
about 0.20–0.25D for all columns in this category.
4.1.7 Effect of FRP Thickness, tFRP
As presented in Table 3, columns C7-1 to C7-5 were
generated to investigate the influence of the FRP thickness,
tFRP, on the behavior of FRP-confined specimens. As
mentioned before for the FRP-confined concrete, the model
Lam and Teng (2003)
was used. In this model the
effect of FRP thickness (Eq. 20) is reflected in the stress–
strain behavior of FRP-confined concrete (Eq. 16).
As shown in Fig. 13a, the stiffness of the columns remained
relatively unchanged; however, as the FRP-wrap thickness
increased the lateral force capacity and the level of PT force
increased. By increasing the FRP thickness, the stiffness of
Axial Stress Level, fc (MPa)
the FRP jacket increases and its hoop strain decreases, and
hence the micro cracks in the concrete core decrease. This
keeps the column cross-section intact for a longer time at
higher drifts which increases the column lateral strength and
PT force development. The NA depth at the peak load was
0.2D for all columns of this category (Fig. 13c). As shown in
Fig. 13d while the moment capacity of unconfined column,
C7-1 was 4.5 kN m, it increased to 8.2 kN m in the
FRPconfined column C7-5 with tFRP of 0.4 mm.
4.1.8 Effect of FRP Ultimate Tensile Strain, eFRP
A wide ultimate tensile strain range of 0.003–0.029 was
reported for different types of FRP material
(Lam and Teng
. Hence, FRP-confined columns C8-1, C8-2, C8-3, and
C8-4 with eFRP ¼ 0.003, 0.01, 0.02, and 0.03, respectively,
were considered to investigate the effect of eFRP on the
behaviour of UPTSCs. Note that the effect of FRP ultimate
tensile strain is reflected in ecu in Eq. 22 which influence the
stress–strain response of FRP-confined concrete (Eq. 16).
As shown in Fig. 14, increasing the FRP ultimate tensile
strain resulted in an increase in the lateral force and PT force
due to the increased deformation capacity of FRP with
higher tensile strain. However, the NA depth at peak strength
was about 0.2D in all columns. The moment capacity
increased from 5.7 kNm in C8-1 to 6.6 kN m in column
C84 (an increase of about 15%).
To determine the displacement ductility of the UPTSCs,
the capacity curves were first developed using the analytical
method presented previously. The idealized bilinear curves
of the capacity curves were then obtained using the
following procedure, which has been widely used and adopted
in previous studies
(Priestley and Park 1987; Park and
Paulay 1975; Ho and Pam 2002)
. The ultimate displacement,
Du, was taken as the displacement when the lateral strength
of the column dropped by 20%. The yield displacement, Dy,
yield strength,Vy, and the effective yield stiffness, ke, was
obtained using bilinear approximation of the
force–displacement response of the columns, and the post-yield
stiffness had a zero value (perfectly plastic). An iterative
procedure was used to determine the idealized bilinear
curves adopting the equal energy concept. The yield
displacement and yield strength were determined such that the
area under the idealized and capacity curves were equal and
the two lines intersected at a strength of 0.75Vy.
Unlike structural elements that have bonded
reinforcement, the pseudo yielding point in the idealized backbone
curve corresponds to the point where significant nonlinearity
(Hassanli et al. 2016b)
. In unbonded post-tensioned
segmental systems, the nonlinearity occurs when the
interface joint at the base of the column significantly opens
leading to stiffness softening. Hence, the pseudo yielding
point does not necessarily correspond to yielding of any
502 | International Journal of Concrete Structures and Materials (Vol.11, No.3, September 2017)
l ¼ DDuy :
bars, but rather it relates to stiffness softening of the column.
The parameters obtained from bilinearization of the capacity
curves are presented in Table 3. The displacement ductility
values, l, presented in the table were calculated using the
(Priestley et al. 2007)
The effect of different parameters on the displacement
ductility is presented in Fig. 15 and Table 3. As the fc0
increased the ductility increased, and the ductility was highly
sensitive to high levels of fc0 (Fig. 15a). By increasing the fc0 ,
the effective stiffness and the ultimate displacement
increased; however, yield displacement decreased (Table 3).
As shown in Fig. 15b, the ductility decreased as fc=fc0
increased and was very sensitive to lower levels of fc=fc0 . As
shown in Fig. 15c for the range of fc considered, the ductility
values remained relatively unchanged (category C3 of
columns were considered in Fig. 15c, in which both fc and fc0
were varied, to maintain a constant fc=fc0 ). It can be
concluded that as long as fc=fc0 , is constant, fc and, fc0 had nearly
no effect on the ductility. Hence, out of three parameters, fc,
fc0 and fc=fc0 , only fc=fc0 need to be considered to determine
the ductility of UPTSCs. As shown in Fig. 15d, the duct size
factor has only a slight effect on ductility. Generally, the duct
size factor is small (unless a hollow section is used) and
1 2 3
Column Height (m)
hence can be ignored. As shown in Fig. 15 and Table 3, as
the column diameter reduces and the column height
increased, the ductility increased. Instead of considering the
effect of diameter and height separately, aspect ratios (height
to the diameter ratio) of the columns of categories C4 and C5
are plotted against ductility. It can be seen that the ductility
of the columns with larger aspect ratio is higher, and
considering the smooth trend of data in Fig. 15g, it can be
concluded that among the three parameters, diameter, height
and aspect ratio, only the aspect ratio needs to be considered
for ductility evaluation.
As shown in Fig. 15h and i adding FRP wrap resulted in
decreased ductility. Also, the variation of the FRP thickness
and ultimate tensile strain had negligible effect on ductility,
and hence can be ignored. Consequently, for FRP-confined
specimens, regardless of the thickness and type of FRP
material, a minimum ductility of two can be considered. In
the experimental study conducted by Hadi
same adverse effect of FRP wrapping on the ductility of
columns, beams and beam-column members was observed.
This can be attributed to the method of bi-linearization of the
capacity curve as well as the sudden rupture of the FRP layer
at high drifts, which results in rapid strength degradation
after the peak strength. Note that although the FRP-confined
columns presented a low level of ductility they exhibited a
large displacement capacity (Table 3).
4.2 Parametric Study—Stage II
In the previous section it was concluded that the ductility
values of the specimens without FRP wrap depended mainly
on two factors, the level of axial stress ratio and the column’s
aspect ratio. The configuration of the columns in Stage II of
the parametric study, which included 36 columns, was
determined according to the conclusions obtained from
Stage I of the parametric study. As mentioned, stage II of
parametric study was performed on a set of columns
considering two parameters, axial stress ratio and aspect ratio as
main variables to investigate the ductility of UPTSCs. Axial
stress ratios of 0.032, 0.042, 0.057, 0.109, 0.212 and 0.263
were considered by applying different PT forces. The aspect
ratio was varied from 3.3 to 20, by considering six different
column heights ranging from 0.5 to 3.0 m. The other
parameters were fixed in all columns and were similar to the
control specimen described in Stage I. Similar to Stage I, the
force–displacement responses of the columns were first
determined using the analytical procedure previously
developed, and these responses where then used to calculate
the ductility of the columns according to the procedure
The ductility of columns is plotted against aspect ratio,
hc/D, for columns with different axial stress ratios in
Fig. 16a. As shown, the columns with low and high axial
stress ratio exhibited different ductility behavior. In the
columns with low axial stress ratios, as the aspect ratio
increased, the ductility increased; however, in the columns
with high axial stress ratio, the aspect ratio had an adverse
effect on ductility. This ductility behavior of self-centering
columns is different from what has been reported for
conventional columns. In conventional columns as the
aspect ratio increases the ductility decreases regardless of
the level of axial stress ratio
(Priestley et al. 1996b)
shown in Fig. 16a, the ductility of columns with low axial
stress ratios was strongly sensitive to the aspect ratio. For
the axial stress ratios of 0.032 and 0.042, the ductility was
9.3 and 6.0, respectively, for a column with hc/D = 3.3;
however, it increased to 44.7 and 33.7 for a column with
hc/D = 20. This significant effect of hc/D on the ductility
at low axial stress levels can be attributed to the unbonded
length of PT steel. At the same drift ratio, the rotation of
the columns about their footing was similar resulting in
equal elongation of the PT steel. Due to the greater heights
of the columns with greater aspect ratio, the unbonded
lengths of PT steel in these columns were larger, resulting
in a reduced strain developed in these columns compared
with columns with low hc/D. Lower strain caused less axial
stress, leading to less damage and more ductility exhibited
by the columns with high hc/D. As shown in Fig. 16a
while in the column with an aspect ratio of 20 by applying
an axial stress ratio of 0.042, the ductility reached a high
value of more than 30, by slightly increasing the axial
stress ratio to 0.057, the ductility dropped down
significantly to 15.6. The sensitivity of the ductility to the level of
axial stress in columns with high axial stress ratio was
considerably lower than that in the columns with low axial
stress ratios. However, in the columns with high axial
stress ratios the aspect ratio had adverse effect on the
ductility. While the ductility of the columns with axial
stress ratios of 0.212 and 0.264 was 12.8 and 13.7 for
columns with hc/D = 3.3, respectively, it reduced to 9.8
and 7.4 for columns with hc/D = 20.
Figure 16b indicates the effect of axial stress ratio and
aspect ratio on the moment capacity of the columns of Stage
II of the parametric study. As shown in the figure, increasing
the axial stress ratio resulted in an increase in the moment
capacity of the column. This behavior was nearly
independent of the column’s aspect ratio, especially for the columns
with higher axial stress ratios. At low axial stress ratios, the
moment capacity of columns with lower hc/D was slightly
higher than those with higher hc/D. This can be attributed to
the higher level of force in the PT steel in columns with
lower hc/D. Between two identical columns with different
heights, the unbonded length of the PT steel is usually
smaller in the shorter one, resulting in a higher level of force
developed in the PT steel. Of two identical columns with
different diameters, at the same drift ratio, the elongation of
the PT steel in the column with greater diameter is higher
(due to greater lever arm), leading to a higher level of force
in the PT steel.
The moment capacity of the columns with low axial
stress ratio is comparatively small, as shown in Fig. 16b.
In practice to reach the required strength and to provide an
economical design the axial stress ratio is usually higher
(Hassanli et al. 2016b)
. Moreover, a minimum
level of axial stress ratio is required to prevent sliding of
the segments and provide enough shear capacity
transferred between the segments and at the column-footing
By ignoring the columns with the axial stress ratio of less
than 5%, the range of the ductility of the columns reduces to
5.2 to 15.6, as shown in Fig. 17a. As shown in the figure, a
l = 5 can be considered as the minimum ductility of
As shown in Fig. 17a the ductility values of columns with
hc/D = 10, was approximately similar, regardless of the
level of axial stress ratio. Considering a linear relationship
between ductility and hc/D, Eq. 28 can be used to estimate
the ductility of UPTSCs.
l ¼ 11 þ a
where a is the slope of the l - hc/D lines. To develop an
equation to estimate the a value, a nonlinear multiple
regression analysis was performed for the columns of the
Stage II of the parametric study, excluding the ones with
axial stress ratio of less than 0.05, and the following
equation was obtained,
Axial Stress Level, fc (MPa)
100 200 300 400
Column Diameter (mm)
Stress Ra o, fc/f'c
Duct Size Factor
Column Height (m)
Axial stress ra o
method is presented in Fig. 18a. As shown, the model was
able to effectively predict the ductility of the columns.
However, Eq. 28 slightly overestimates the ductility of
some columns. Consequently, to provide a safer design,
Eq. 30 (Fig. 17c) is proposed in lieu of Eq. 28 to predict
the ductility of unconfined UPTSCs.
l ¼ 9 þ a hDc
The predicted ductility obtained using Eq. 30 is plotted
against the ductility values obtained using analytical method
in Fig. 18b. As shown, the equation could provide a
conservative prediction of the ductility of UPTSCs.
4.3 Neutral Axis (NA) Depth
If the depth of neutral axis is known, the elongation of
the PT steel and the concrete deformation at the
compression zone at different column rotations can be
determined accordingly. As shown in Figs. 7, 8, and 9 at peak
5 10 15
Column Duc lity
5 10 15
Column Duc lity
Fig. 19 Predicted NA depth.
strength, the concrete compressive strength and axial stress
ratio influence the NA depth however, the level of axial
stress seems to have no effect on the NA depth. The
column height, FRP thickness and FRP ultimate tensile
strains have negligible effect on the NA depth (see
Figs. 12, 13, 14), hence, can be ignored, however as
shown in Fig. 11, column diameter does have considerable
an effect on the NA depth. Using the column matrix of
Stage I (Table 3), multivariate regression analysis was
performed and the following equation was obtained to
determine the NA depth of UPTSCs,
f þ 0:12 D
The predicted NA depth obtained using Eq. 31 is plotted
against the corresponding values obtained using the
analytical method in Fig. 19. As shown, the equation provides an
acceptable prediction of the ductility of UPTSCs.
Equation 31 can be used for both confined and unconfined
circular cross sections, however, its accuracy for square cross
sections and sections with diameter of larger than 450 mm
needs to be examined.
In this study the behavior of unbonded post-tensioned
segmental columns (UPTSCs) was investigated using an
analytical method. The accuracy of the method was first
verified against experimental results, followed by a
parametric study to investigate the effect of different parameters
on the behavior and strength of UPTSCs. Multivariate
regression analysis was also performed to develop
expressions to determine the ductility and NA depth of UPTSCs.
The following conclusions were drawn:
1. The analytical approach considering single joint rotation,
could effectively predict the response of UPTSCs. It was
shown that the moment capacity of UPTSCs was highly
sensitive to the level of axial stress and axial stress ratio,
and less sensitive to the concrete compressive strength.
2. Oversized duct has negative effect on the effective depth
of the PT bars, and hence the lateral strength of the
columns. Hence, the strength reduction due to oversized
ducts must be taken into account.
3. The ductility of unbonded post-tensioned columns is
strongly affected by two factors, the level of axial stress
ratio and the aspect ratio. However, as long as the axial
stress ratio, fc0 =fc, is constant, the level of axial stress, fc,
and compressive strength of concrete, fc0 , has nearly no
effect on the ductility.
4. For unconfined UPTSCs, for high levels of aspect ratio
(hc/D [ 10), the ductility decreased as the axial stress
ratio increased; however, at low levels of aspect ratio
(hc/D \ 10) the ductility increased as the axial stress
ratio increased. Equation 30 was proposed to estimate
the displacement ductility of those columns.
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license, and indicate if changes were made.
bðxÞ ¼ bw
And the corresponding lever arm can be calculated as,
Substituting Eqs. 3, 32 and 33 in 2, gives,
International Journal of Concrete Structures and Materials (Vol.11, No.3, September 2017) | 509
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