Engineering Structures 31 (2009) 758767

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Engineering Structures
journal homepage: www.elsevier.com/locate/engstruct
Buckling modeling of reinforcing bars with imperfections
Leonardo M. Massone
a,
, Daniel Moroder
b
a
Department of Civil Engineering, University of Chile, Blanco Encalada 2002, Santiago, Chile
b
Civil Engineering, University of Bologna, Viale Risorgimento 2, 40136, Bologna, Italy
a r t i c l e i n f o
Article history:
Received 29 April 2008
Received in revised form
3 November 2008
Accepted 20 November 2008
Available online 3 January 2009
Keywords:
Reinforcing bar
Buckling
Reinforced concrete
Model
a b s t r a c t
Reinforced concrete columns in seismic zones are subjected to combined actions, resulting in axial loads
in longitudinal reinforcing bars. Thus, knowing the bars response, especially when it is subjected to
important axial compressive forces that might lead to buckling, is important. A bar buckling model based
on concentrated plasticity and with the capability of introducing an initial imperfection is described.
The initial imperfection is imposed by bending the bar with a transversely applied nonpermanent force.
Additionally, a comprehensive study of the monotonic tensile response beyond the peak stress point
and a simple cyclic rule, complete the physical approach of the model. Comparisons of the model with
experimental results reveal that peak capacity (average axial stress) is well captured, as well as the post-
peak response shape (average axial stress versus strain), with differences observed basically in the peak
capacity for specimens with high bar imperfection-to-diameter ratio, and in the shape of the post-peak
response for specimens with low bar length-to-diameter ratio.
2008 Elsevier Ltd. All rights reserved.
1. Introduction
Reinforced concrete columns in seismic zones are subjected to
combined actions that include mainly axial, moment and shear
forces. Longitudinal reinforcing bars act as members that resist
axial loads, which also contribute in maintaining the columns
moment. Thus, the axial response of longitudinal bar becomes
relevant. In absence of buckling effects the axial response can
be associated with the monotonic or cyclic response of bars.
That situation, although ideal, might not represent all cases.
Reinforced concrete columns under cyclic lateral displacement,
which represents a seismic action, would remain elastic under
small displacements. Under severe loading, lateral displacement
would increase, and in combination with compressive axial forces,
deterioration of cover concrete that ends with spalling would
reveal part of the longitudinal bars whichare laterally supportedby
stirrups. A large distance between stirrups would trigger buckling
at lower loads, which also affects the columns response. Therefore,
a buckling modeling is required to establish a good understanding
of column behavior, especially when the longitudinal bar response
may be affected by relatively large stirrup separation.
The study of buckling begins with Euler in the 18th century.
He developed a simple equation to calculate the critical load for
the elastic case. More recent developments have included material
inelasticity. The application to reinforced concrete modeling

Corresponding author. Tel.: +56 2 9784984; fax: +56 2 6892833.

E-mail address: lmassone@ing.uchile.cl (L.M. Massone).
appeared with Bresler and Gilbert [1], providing the information
about tie spacing requirements and the buckling behavior of
longitudinal reinforcement steel in compressed concrete members
based on critical load estimation at yielding. Further efforts have
been done by researchers in order to capture not only the buckling
capacity of longitudinal bars, but also to describe their monotonic,
as well as their axial cyclic response. Numerical simulations using
fiber discretization of beam-column elements with distributed
plasticity have been introduced (e.g., [2,3]), characterizing in part
the monotonic response. Cyclic response has also been estimated
based on calibration of monotonic experimental response of
bars subjected to buckling (e.g., [3,4]), allowing the introduction
of cyclic constitutive material laws for steel including buckling
into beam and column analysis. Other authors have adopted
different modeling approaches to introduce the buckling behavior
to beam and columns responses, such as introducing concentrated
plasticity models based on steel material constitutive laws
(e.g., [5,6]).
2. Research significance
The model described in this paper considers a similar approach
adopted by Restrepo [6] but with the capability of introducing
an initial imperfection, imposed by bending the bar with a trans-
versely applied nonpermanent force. Additionally, a comprehen-
sive study of the monotonic response pointed out the need for
defining the tensile response beyond the peak stress point, and a
consistent point of fracture. Altogether, with a reliable compres-
sive constitutive lawfor the steel material and a simple cyclic rule,
0141-0296/$ see front matter 2008 Elsevier Ltd. All rights reserved.
doi:10.1016/j.engstruct.2008.11.019
L.M. Massone, D. Moroder / Engineering Structures 31 (2009) 758767 759
this study provides an overall simple and physical approach to the
problem. Then, this model, verified with experimental tests from
the literature, can be implemented into column analysis.
3. Modeling studies
3.1. Tensile material model
Common mild reinforcing steel bars are usually characterized
by their monotonic stress versus strain response under tension.
The tensile monotonic bar response behaves linear-elastically
until yielding is achieved (f
y
,
y
in Fig. 1), maintaining an almost
constant stress until initiation of strain hardening is observed
(f
y
,
sh
in Fig. 1). A different steel composition may result in
a softer tensile response, showing a curved transition from the
linear-elastic zone to the strain hardening zone. Strain hardening
is understood as an increase of strength with an increment
of strain. The peak strength, or maximum strength (f
m
,
m
in
Fig. 1), is followed by a degradation of the strength, which is also
associated with strain localization that results in a local cross-
section reduction at weaker zones of the bar. Ultimate strength
(f
u
,
u
in Fig. 1) is observed before the axial strength drops to
zero at the onset of bar fracture. Fig. 1 shows a representation
of the overall stress versus strain response of reinforcing bars
under tension characterized by the model proposed by Mander
et al. [7]. The model by Mander presents an elastoplastic
response until reaching the hardening initiation (f
y
,
sh
in Fig. 1),
followed by a curve that describes the hardening range until the
peak strength point (f
m
,
m
in Fig. 1). The degrading zone, not
characterized by Mander, and usually by no-one for engineering
purposes, is assumed linear until bar fracture (f
u
,
u
in Fig. 1).
The experimental determination of the constitutive monotonic
material response for steel under tension once strain localization
begins (at peak strength) is spurious due to the fact that the strain
concentration might fall inside or outside the gauge length of
the instrument measuring axial deformation, resulting in either
larger or smaller deformations, respectively. Not only that, even
if the concentration of deformation is guaranteed to be localized
inside the instrumented length, the experimental stressstrain
response would be associated with the gauge length used for
deformation measurement. That is, if the strain concentration is
distributedover a length, l
p
, whichis smaller thanthe gauge length,
l
g
, the engineering strain, measured over the gauge length would
result in a smaller strain (Fig. 2). In general, if a section of the
longitudinal bars under axial load presents strain concentration
due to softening of the material, i.e. larger strain at lower stress,
all other points in the bar in order to maintain equilibrium, as a
beam-column element, are required to unload instead of overcome
the peak stress. In Fig. 2, a scheme of a reinforcing bar undergoing
tensile forces is shownassuming that the strainconcentrationzone
falls inside the displacement transducer length used to determine
the bar strain. The zone with strain concentration over the length
l
p
reveals the actual material constitutive stress versus strain
response in engineering coordinates, where after reaching the
peak strength (point 1) continues straining until point 2. Outside
the strain concentration zone, after reaching the peak strength
(point 1) instead of continue straining, undergoes unloading (point
2) due to material heterogeneity that results in slightly stronger
sections. In terms of the overall response, using a displacement
transducer over a length l
g
(larger than the strain localization zone)
to determine the bar relative displacement, and then the average
strain, results in smaller strain values after the peak stress than at
the strain localization zone. Thus, the ultimate strain defined for
the strain concentration zone,
u,p
, is determined by

u,p
=
m
+
_
_

u,g

m
_
l
g
+
_
f
u
f
m
E
s
_

_
l
g
l
p
_
_ _
l
p
(1)
Fig. 1. Monotonic tensile constitutive material model for steel.
Fig. 2. Strain localization of reinforcing steel in tension.
where
u,g
is the ultimate strain based on the gauge length, and
under the assumption that the unloading response follows the
initial elastic stiffness, E
s
. Common mild reinforcing steel bars
present a large ultimate strain, which leads to the approximation
of Eq. (1) given by

u,p

m
+
_

u,g

m
_

l
g
l
p
. (2)
The previous equations determine the real strain at the strain
concentration zone by correcting the strain values obtained
experimentally after the peak stress was reached, under the
assumption that the strain concentration falls inside the gauge
length.
In general, constitutive material models for reinforcing bars
have been calibrated using data before degradation is observed
(e.g., [7]) to avoid spurious results. Also, many analyses would
not require reaching such large strains. The model verification
presented in this study requires in some cases reaching large
tensile strains, which forces the analysis to calibrate the tensile
response of reinforcing bars after reaching the peak stress.
3.2. Compressive material model
The tensile stress versus strain response is usually also adopted
as the response for the steel in compression. However, it has
been shown that the use of engineering coordinates to estimate
the stress and the strain as well, that is using the initial cross-
sectional area and bar length to estimate such magnitudes, does
760 L.M. Massone, D. Moroder / Engineering Structures 31 (2009) 758767
not represent the true or actual stress or strain the material
is undergoing due to sequential increase of element length and
decrement of cross-sectional area while in tension. The use of
true coordinates or natural coordinates as indicated by Dodd
and Restrepo-Posada [8] provides stress and strain measurements
accounting for the current cross-sectional area and bar length.
These stress and strain measurements have been shown to provide
a good estimate for the stress versus strain response for bars
in compression, assuming an identical behavior of the steel
material in tension and compression in true coordinates. Hence,
the compressive response can be estimated from tensile tests.
According to Dodd and Restrepo-Posada [8] findings such analysis
gives good correlation with tests results performed in compression
until buckling is observed. The compressive stress versus strain
response (f
s,c
,
s,c
) can be determined by
f
s,c
= f
s,t
_
1 +
s,t
_
2
(3)

s,c
=

s,t
1 +
s,t
(4)
where f
s,c
and
s,c
are the stress and strain coordinates (engineer-
ing coordinates) in compression (negative) for the corresponding
stress, f
s,t
, and strain,
s,t
, coordinates (engineering coordinates) in
tension (positive), respectively. Thus, once the constitutive steel
material response is characterized in tension through tests, and
the post-peak is corrected in order to represent the strain values
in the strain concentration zone, Eqs. (3) and (4) can be used to
determine the compressive constitutive steel response. Although,
tensile tests end with bar fracture, that ultimate point value may
not be consistent with a failure mechanism in compression when
using Eqs. (3) and (4). Post-peak points for the tensile response can
be extrapolated assuming no fracture failure in order to estimate
the compressive response.
3.3. Simple cyclic material model
The previous analysis defines the monotonic behavior of
steel reinforcement. A cyclic response requires a more detailed
description. Many steel constitutive material models have been
proposed to predict cyclic response (e.g., [9,8,7]), although most
of them have assumed an identical compressive and tensile
response of the steel. Such an assumption is reasonable for
relative small strain values. However, the analysis of bars that
exhibit buckling with an initial imperfection shows relative high
axial strain even for the first loading steps (e.g., induction of
the initial imperfection imposed by bending the bar with a
transversely applied nonpermanent force at bar midheight). For
the analysis of reinforcing bar buckling under monotonic axial
loading, cyclic response of steel is required in order to describe
initial imperfections or model deviation from a uniform strain
distribution in the cross-section once buckling is onset. In this
case, usually no full cycles are achieved. Thus, a calibrated material
model, capable of reproducing few or incomplete cycles, for not
only small strains, but also relatively large strains is required to
capture the response of buckling bars that consider imperfections.
The suggested simple cyclic material model for steel is depicted
in Fig. 3. The model maintains both envelope monotonic responses
for steel in tension and compression. Once reversal loading occurs
from the envelope (e.g., f

r,1
,

r,1
or f
+
r,1
,
+
r,1
in Fig. 3) outside the
linear range a curve (called curve A) joins the current unloading
point (origin) and a point with the same strain coordinate of the
previous unloading point from the opposite envelope (end). The
end stress of curve A is determined based on the assumption that
straining in one direction shifts the origin of the opposite envelope
curve (dashed lines). The shifted envelope curve, that is connected
with an elastic stiffness to the unloading point, starts froma virtual
Fig. 3. Simple cyclic constitutive material model for steel.
plastic strain point (e.g., point (0,
+
p,1
) in Fig. 3) and defines the
new stress value. In case of unloading from the envelope for the
first time, the zero strain point in the opposite envelope is selected
as the previous unloading point in that branch. After following
curve A, it is considered for simplicity that the material model
follows the remaining envelope curve (initiation or connection to
the envelope curve is marked with a dot in Fig. 3). In the linear
range, i.e., before yielding, the response is maintained within the
linear-elastic behavior. In case of unloading or reloading within
curve A, a similar curve can be defined that joins the reversal from
curve Atothe previous unloading point fromthe opposite envelope
or another unloading point from curve A. For the purposes of this
study, it is considered that unloading or reloading within a curve
A forces joining to the previous point from the opposite envelope,
maintaining the same model parameters for curve A.
The curve A represents the Bauschinger effect, that is, softer
unloading and reloading branches affected by the strain previously
attained. Chang and Mander [10] present a formulation to
characterize curve A, based on the MenegottoPinto equation,
which allows defining, among others, the initial and final
unloading/reloading stiffness values. Although this formulation
is general, it presents the disadvantage of requiring a numerical
iterative scheme in order to connect initial and end points of curve
A. Such a formulation is simplified in this study by adopting a final
unloading/reloading stiffness value that guarantees connecting the
initial and end points of curve A. The modified stress (f
s
) versus
strain (
s
) expression that characterizes curve A is given by
f
s
= f
o
+ E
o
(
s

o
)
_

_
Q +
1 Q
_
1 +
_
E
o
_

o
f
f
f
o
__
R
_
1/R
_

_
(5)
where R is a parameter that represents the Bauschinger effect, E
o
is the initial unloading/reloading modulus of the steel bar, f
o
and

o
are the stress and strain coordinates of the origin of curve A, f
f
and
f
are the stress and strain coordinates of the end of curve A,
and Q is a parameter defined as Q =
E
sec
/E
o
a
1a
(with E
sec
=
f
f
f
o

f

o
and a =
_
1 + [E
o
/E
sec
]
R
_
1/R
) that warranties that curve A ends
at (
f
, f
f
). Eq. (5) describes a function that connects the origin and
end points by means of a variable radius of curvature (R), such
that small values of R result in a soft transition between an initial
stiffness E
o
, and a final stiffness. In the other hand, large values of
R (e.g., 25) result in a curve that closely follows two asymptotes
formed by the initial and final stiffness, that is, instead of gradually
changing the slope, curve A presents a kink characterized by two
slopes (the initial and final stiffness values). The previous function
L.M. Massone, D. Moroder / Engineering Structures 31 (2009) 758767 761
is then fully known after defining the parameter R and the stiffness
E
o
. According to the cyclic formulation by Chang and Mander [10]
and after calibration with experimental data of tests performed
by Panthaki (1991) (reported by Chang and Mander [10]), these
parameters for the unloading branch are
E
o
= E
s
(1 3) (6)
R = 16
_
f
y
E
s
_
1/3
(1 10) (7)
and for the reloading branch
E
o
= E
s
(1 ) (8)
R = 20
_
f
y
E
s
_
1/3
(1 20) (9)
where =

f

o

/2, and f
y
and E
s
represent the yield stress
and elastic stiffness of the steel bar.
In order to verify the agreement between the proposed cyclic
model and experimental results, a series of data reported by Chang
and Mander [10] are used. Representative results are presented
in Fig. 4. As it can be seen, the general trend is captured with
this simple model. However, slope discontinuities are expected
in transition zones from curve A to the monotonic envelope as
observed in Fig. 4(a) and (c).
3.4. Lumped plasticity buckling model for a reinforcing bar
Reinforced concrete columns are commonly constructed as
a series of longitudinal bars supported by stirrups or cross-
ties immersed in concrete, which are designed to withstand
axial (usually compressive loads), moment and shear forces. The
following discussion focuses on axial and moment action on
columns. Such actions transfer axial forces to the longitudinal
reinforcement together with transversal forces from concrete
core (inside stirrups) expansion and stirrup straining. The axial
forces in compression on reinforcing bars may lead to buckling
between two consecutive stirrups (Fig. 5). This behavior has been
captured by many researchers (e.g., experimental investigation on
bar buckling by Bayrak and Sheikh [11]; Bae et al. [12]) considering
variables such as bar diameter, stirrup spacing, as well as an initial
imperfection that deviates the bar from being straight.
In order to capture not only the bar critical load, that is,
the load required to buckle the bar, but also the overall stress
versus strain response, a sufficiently refined model is required.
The model described in this study was adapted from Restrepo [6]
to incorporate an initial imperfection, and was compared with
experimental results available in the literature (reflecting the
conditions imposed in the experiments).
The model for a bar of diameter d and length L between two
consecutive stirrups assumes fixedconditionat bothends, withthe
exception of the upper end, which is allowed to move vertically
(longitudinally, see Fig. 5). The initial imperfection, e, is included
as a transversal deviation from the vertical axis. All deformations
are concentrated in four plastic hinges located at both ends and
at both sides of the mid-length of the bar. The selection of the
location of the plastic hinges obeys the nature of the loading
conditions. For the selectedspecimens the imperfectionis obtained
after clamping both ends and applying a transversal point load at
bar mid-length, resulting in maximum moments at bar ends and
mid-length. Assuming uniform material properties along the bar,
the zone of maximum moment would result in concentration of
deformation once linear behavior is overcome. The symmetry of
the load application leads to the conclusion that the concentration
of deformation at bar mid-length can be divided into two plastic
hinges. After the imperfection is included, the progression of the
Fig. 4. Cyclic model comparison: (a) Kent and Park, 1973, specimen 8, (b) Ma,
Bertero and Popov, 1976, specimen1, and (c) Panthaki, 1991, specimenR5 (reported
by Chang and Mander, 1994).
axial load would deform the bar even further, but in this case
the axial load would result in a constant vertical force along the
length of the bar and a moment distribution similar to what is
762 L.M. Massone, D. Moroder / Engineering Structures 31 (2009) 758767
Fig. 5. Buckling representation of reinforcing bar with initial imperfection.
expected while inducing the imperfection. Thus, if axial strain and
curvature are concentrated only inside the plastic hinge length,
inducing the imperfection as well as the later application of
the axial load results in the same four hinge configuration. The
aforementioned plastic hinge formulation does not satisfy the
beam solution within the linear elastic range since it assumes that
there are always four hinges with concentrated rotation, which is
not consistent when loading the bar to induce the imperfection
while the material model remains elastic. Although this is an
approximation, it is shown in a later section, by comparing the
response of the analytical model with experimental results, that
the overall average stress versus strain response is captured,
including the peak stress and post-peak curve shape.
At this point, once the axial strain and curvature values are
known within the plastic hinge, a sectional analysis would allow
determining the axial stresses at different location of the bar cross-
section under the Bernoullis hypothesis (plane sections remain
plane after rotation), and using the uniaxial material constitutive
law. Axial resultant force and resultant moment are determined
based on integration of the uniaxial stresses and tributary areas.
3.4.1. Initial imperfection
The initial imperfection, e, can be approximately imposed by
forcing a uniform curvature,
e
, over the plastic hinge length, l
p
,
equal to

e
= tan
1
_
e
L/2 l
p
__
l
p
. (10)
Imposing an initial curvature over the plastic hinge length
would result in permanent moment at the plastic hinges. In order
to satisfy equilibrium at the beginning of the axial tests, that is, all
resultant moments need to become zero since there is no longer a
transversal force inducing the imperfection. This suggests that the
initial imposed curvature needs to be reduced in order to observe
unloading at different points in the cross-section that results in
zero moment. After unloading, the residual curvature becomes
e
,
which yields to a permanent transversal displacement e. Giving the
relatively high unloading stiffness, a small variation of curvature is
anticipatedandthe imperfectionvalue wouldvary slightly inmany
cases after unloading. It can be numerically solved for the initial
curvature required to obtain the desirable imperfection e.
The described procedure assumes that the impact of the axial
strain is small. If axial strain of the plastic hinge is measured at the
bar centroid, differences in the tensile and compressive envelope
stress versus strain responses would result in relatively small axial
strains. A similar phenomenon is anticipated when the curvature
is reduced due to unloading. Preliminary analyses suggest that
incorporating the axial flexibility results in small differences.
Fig. 6. Bar cross-section at plastic hinge zone: fiber discretization.
3.4.2. Application of axial force
The application of the axial force, p, changes the previous
equilibrium, modifying the axial strain, , and curvature, , in the
plastic hinge zone. At the cross-section level (Fig. 6), the uniaxial
strain at fiber i at a distance x
i
from the centroid (reference),
i
,
would vary assuming Bernoullis hypothesis by

i
= + x
i
. (11)
The uniaxial cyclic constitutive material model is used to
determined the stresses,
i
, at each fiber i. The resultant force (p)
and moment (m) are determined by
p =

i
A
i
(12)
m =

i
A
i
x
i
(13)
where A
i
is the tributary area of each fiber i.
Therefore, the application of the axial force induces a moment
resultant at the plastic hinge. The symmetry of the problem under
L.M. Massone, D. Moroder / Engineering Structures 31 (2009) 758767 763
Fig. 7. Buckling plastic hinge model of reinforcing bar with initial imperfection
(quarter bar).
study allows analyzing only one quarter of the bar that stands in
between two consecutive stirrups. Fig. 7 shows the element under
analysis. The upper end of the selected segment of the bar (quarter)
falls in the inflection point, resulting in no moment, but just axial
force. The other end, however, has a resultant moment m. From
equilibrium
m = p
(e + w)
2
(14)
where w is the additional transverse displacement at bar mid-
length. The transverse displacement is determined based on
the geometry of the deformation mechanism assuming that all
transversal deformations appear after rotation of the plastic
hinges. The total transverse displacement at mid-length can be
determined by
e + w =
sin
_

e
+
p
_
cos
e
_
L
2
l
p
_
(15)
where
e
and
p
are the rotation due to the initial imperfection and
rotation after applying the axial load (p), respectively. All rotations
are assumed were formed by a uniform distribution of curvature
over the plastic hinge length (l
p
). The rotations are calculated by

e
=
e
l
p
= tan
1
_
e
L/2 l
p
_
(16)

p
=
p
l
p
(17)
where
e
and
p
are the curvature due to the initial imperfection
and after applying the axial load (p), respectively. It can be noticed
that the plastic hinge length value is maintained unchanged along
the entire loading procedure.
Giving the fixed rotational condition at both ends and the
force distribution, maximum moments (absolute values) are
obtained at both ends (symmetry) and bar mid-length. To maintain
symmetry for a homogeneous bar, four plastic hinges are placed at
maximummoment locations. It is assumed that such plastic hinges
concentrate all deformations.
For the vertical displacement (v), it is assumedthat deformation
due to axial strain () and curvature () can be decoupled in the
terms v

and v

, which are associated with the axial strain ()

and curvature (), respectively, which simplifies the numerical
procedure. Thus, the vertical displacement is determined by
v = v

+ v

=
_
L 2l
p
_
_
1
cos
_

e
+
p
_
cos
e
_
+ 4l
p

p
(18)
where
p
is the axial strain at the plastic hinge region.
Thus, the engineering average axial stress ( ) and average axial
strain ( ) are determined by
=
p

i
A
i
=
p
A
(19)
=
v
L
. (20)
The present study validates a plastic hinge formulation capable
of reproducing the average axial stress versus average axial strain
response of reinforcing bar with an initial imperfection under
compression. The methodology, although described and compared
with experimental evidence on isolated reinforcing bars, can be
used to study column performance. Two different approaches
from the literature could be adopted to obtain moment versus
curvature responses: the Gomes and Appleton [5] formulation and
the Bayrak and Sheikh [11] formulation. Gomes and Appleton [5]
developed a stress versus strain constitutive lawfor reinforcement
under compression incorporating bar buckling as three plastic
hinges that formonce spalling of cover concrete occurs. The plastic
hinges are defined based on a fully plasticized cross-sectional area.
The compressive constitutive law for the reinforcement is then
applied into a sectional analysis by limiting the cyclic response that
would be obtained if no buckling were present. Thus, an identical
procedure can be followed, replacing the compressive envelop for
the longitudinal bar with the proposed approach.
Bayrak and Sheikh [11] followed a different direction. In
their formulation experimental stress versus strain responses for
reinforcing bar affected by buckling are used to predict sectional
response. In this case, initiation of bar buckling occurs after
spalling of the cover concrete. At that point, ties are strained
and confined concrete tends to push and bend the longitudinal
reinforcement between ties outwards. The acting transverse
force on the longitudinal reinforcement generates a midheight
deflection, calculus based on an assumed shape function for
the force distribution along the bar. The midheight deflection is
set as the initial imperfection assuming that further actions are
controlled by the axial force. Thus, an identical procedure can be
followed, replacing the compressive envelop for the longitudinal
bar with the proposed approach for a predefined imperfection.
3.5. Numerical implementation
The previously described model concentrates all deformations
in four plastic hinges without distinction between elastic or plastic
deformations, and maintaining the plastic hinge length constant.
This, although allows using the model even for the initial loading
stages while the material remains elastic, introduces differences
with a model that treats the elastic deformation as part of the
entire bar length, rather than just the plastic hinge region. It is
shown, in a later section, that the overall model response presents
good agreement with experimental results, revealing that the
assumption of concentrating all deformation at the plastic hinges
has little impact on the general behavior.
3.5.1. Initial imperfection
The numerical procedure that includes the initial imperfection
is described in Fig. 8. The model has an iterative scheme over
one variable: the initial curvature (
1
), that is, the curvature
required to induce deformation in the bar by the externally applied
load. Defined the initial curvature, and evaluated at the fiber and
section levels, the curvature obtained after unloading is set as the
curvature (
e
), which once the externally applied load is removed,
results in a permanent mid-length transversal deformation e.
An alternative scheme (Alternative 1, see Fig. 8) is also
presented to guarantee force equilibrium in the axial direction.
764 L.M. Massone, D. Moroder / Engineering Structures 31 (2009) 758767
Fig. 8. Numerical procedure to impose initial imperfection to plastic hinge model.
Using a zero axial strain would result in a small axial force, due to
the asymmetry of the material model in tension and compression,
which has little impact on the overall response. Incorporating the
axial strain can be done in the same numerical schemes by adding
this new variable. Most nonlinear numerical procedures can be
used to solve the problem, such as NewtonRaphson, bisection
method, etc.
The specimens that presented no initial imperfection can be
treated as bars with small imperfections in order to observe buck-
ling (with transversal displacement), which deviates fromthe triv-
ial solution that basically reproduces the compressive constitutive
material response with signs of only axial displacement.
3.5.2. Application of axial force
The numerical procedure that applies the incremental axial
force is described in Fig. 9. The scheme allows incrementally
determining different loading stages by increasing the average
axial strain ( ) in the bar. The new strain value results in a new
equilibrium, which is solved iteratively. The model has an iterative
scheme over one variable: the additional curvature (
p
), that is, the
additional curvature induced in the bar by the axial load, which
already has the imperfection included. As in the previous section,
most nonlinear numerical procedures can be used to solve the
problem.
4. Model correlation with test results
The following section includes a comparison between the
described model and experimental results from the literature.
4.1. Overview of tests
A series of tests carried out by Bayrak and Sheikh [11] are
considered for comparison with the described model. The test
program considered two important characteristics that made it
suitable for numerical comparison. The test program included
different imperfection magnitudes and the tensile coupon tests
were strained beyond the peak stress point. The experimental
program carried out by Bayrak and Sheikh [11] was performed
Fig. 9. Numerical procedure for buckling analysis of the plastic hinge model.
Fig. 10. Test setup: (a) transversal deformation (initial imperfection), and (b) axial
deformation.
using Grade 400 (f
y,nominal
= 400 MPa) 20 M (d = 19.5 mm)
steel reinforcing bars. Seven different tie spacing to longitudinal
bar diameter ratios L/d were used, starting from4 and ending with
10. For each L/d ratio, four different levels of initial imperfection
(e) were tested, with ratios e/d ranging from 0 to 0.3. Initial
imperfections were introduced into the bars, which had the
two ends restrained against rotation, by pushing at the middle
length of the bar with an external force, yielding to the desirable
initial imperfection-over-diameter ratio (Fig. 10(a)). The axial
displacement was measured by four linear variable differential
transducers (Fig. 10(b)). Companion specimens were tested to
validate the repeatability of the tests.
L.M. Massone, D. Moroder / Engineering Structures 31 (2009) 758767 765
Fig. 11. Stressstrain material calibration (tension and compression).
4.2. Model results
The material model was calibrated to a monotonic response
including the modification in the degrading zone. The strain
concentration zone, once degradation occurred, was set as the
diameter length, which is consistent with the assumption of plastic
hinge length of one diameter considered for the bar buckling
model (i.e., l
p
= d). The fracture strain was determined according
to Eq. (2) for a 50 mm gauge length. The experimental tensile,
analytical tensile (with and without the post-peak correction)
and compressive responses are depicted in Fig. 11 in engineering
coordinates. Tensile responses are shown until fracture is set,
whereas the compressive response is not shown in full range in
order to select a reasonable scale.
The numerical procedure is performed using the calibrated
material model and the cyclic model described in previous
sections, discretizing the cross-section in twenty fibers, which
showed to be enough refinement. The average stress versus
average strain response is compared with the experimental data in
Fig. 12. All cases with different imperfection values are considered,
however the numerical procedure adopted did not consider
Alternative 1 (see Fig. 8), that is, no axial force equilibrium is
guaranteed (p = 0). In order to numerically obtain the buckling
response for bars with no imperfection a small perturbation was
imposed. In this case, an imperfection-to-diameter ratio of 0.01
was used (i.e. e/d = 0.01 > 0). Regarding the length-to-diameter
ratio (L/d) only four values were considered for comparison: 4, 6,
8 and 10.
Fig. 12 shows reasonably good correlation for most cases.
The peak capacity is better captured in the cases with lower
imperfections. Differences are usually in the range from 5%
to 15% when comparing the peak stress between the model
and experimental results. Only the specimen with the largest
imperfection and lowest length, i.e., e/d = 0.3 and L/d =
4, presents an analytical peak stress at a large strain, whereas
the experimental result has already degraded presenting large
differences. The post-peak response recovers the shape observed
in the experiments. Differences are observed for specimens with
low length-to-diameter ratio (e.g., L/d = 4, 6); where the model
overestimates the stress, since it presents a less pronounced
degradation than the tests. In the cases with relatively larger
length-to-diameter ratio (e.g., L/d = 8, 10), the post-peak
differences are usually less than 10% when comparing the stress
between the model and experimental results, all the way up
to the maximum experimental average axial strain commonly
in the range between 20% and 30%. Regarding the overall
response for different imperfection values introduced in the
specimens, the analytical prediction reveals similar trends as the
experimental response; that is, the larger the imperfection the
softer the response (lower stress). This is also consistent with other
authors observations (e.g., [13]), whichindicates that imperfection
influences the bars response.
Additional comparison analyses were performed to establish
the relevance of guaranteeing axial force equilibrium during the
imperfection induction process. Equilibrium of the axial force for
that stage was included as described in Fig. 8 as Alternative 1.
As expected, due to the flexural nature of the loading condition,
the impact of such consideration was minor. Fig. 13 includes a
similar comparison to Fig. 12, for two representative cases: with
an imperfection-to-diameter ratio of 0 (the nominal 0 value was
an actual value of 0.01) and 0.3 (extreme cases). In this case,
the additional dashed curves considered the correct equilibrium.
As can be seen, the responses are almost identical, especially
for the case with a low imperfection value (e/d = 0). The
largest difference is seen for the case with the largest imperfection
value (e/d = 0.3) and the smallest bar length value (L/d =
4), where the maximum axial average stress does not differ in
more than 2% to the case that does not satisfy initial axial force
equilibrium, yielding a response closer to the experimental data.
Another difference, which is expected due to the nature of the
numerical implementation, is the initial stress value. The model
without Alternative 1 (not satisfying initial axial force equilibrium)
presents initial axial stresses, which are small for the case with a
low imperfection value (e/d = 0), and increase for the case with
the largest imperfection value (e/d = 0.3) and the smallest bar
length value (L/d = 4), where the initial stress value does not
exceed the maximum axial average stress in more than 3%.
5. Summary and conclusions
Amodel capable of representing the buckling of longitudinal re-
inforcing bars with induced initial imperfections was developed
and compared with available experimental data. The model con-
siders concentrated plasticity at four plastic hinges. The locations
of plastic hinge correspond to zones of maximummoment that oc-
curs during the induction of the imperfection as well as the de-
formation due to the axial load. The imperfections were induced
by clamping the bar ends and applying a transversal force yield-
ing in a residual maximum transversal displacement. Once the
imperfection is set, the axial load acts on the specimen main-
taining both ends fixed, which results in a moment configura-
tion that is consistent with the stage of induction of imperfection.
Thus, the mechanical model is maintained, and a sectional analy-
sis, based on uniaxial constitutive material laws for steel, is used.
The monotonic tensile response is characterized by common pa-
rameters, but the post-peak behavior is additionally considered
in order to guarantee a good response of the model. Giving the
little information on this aspect, a linear response is proposed and
a correction of the ultimate or fracture strain is defined based on
strain concentration at the necking zone of the bar. The monotonic
compressive response is characterized based on the tensile curve,
assuming that tensile and compressive responses are identical
in the true or natural coordinates, except that the compressive
response does not present fracture. The cyclic behavior of the
steel material was based on a simple curve that incorporates the
Bauschinger effect, which agrees reasonably well with cyclic bar
tests with few cycles.
The model assumes that all deformations are concentrated at
the plastic hinges, resulting in an approximation of the response
in the elastic range. Although such an approximation exists,
comparisons of the model with experimental results reveal that
peak capacity is well captured, as well as the post-peak response
shape. Differences are observedbasically withthe peak capacity for
specimens withhighimperfectionvalues, andwiththe shape of the
post-peak response for specimens with low length-to-diameter
ratio.
766 L.M. Massone, D. Moroder / Engineering Structures 31 (2009) 758767
Fig. 12. Average stressaverage strain bar buckling response: model without alternative 1 (initial axial force equilibrium).
Fig. 13. Average stressaverage strain bar buckling response: model with and without alternative 1 (initial axial force equilibrium).
L.M. Massone, D. Moroder / Engineering Structures 31 (2009) 758767 767
Additionally, it was also observed from the comparison tests
that the numerical procedure that includes initial axial force
equilibrium at the imperfection induction stage (indicated as
Alternative 1) has a small impact in the overall response, especially
for the case of a low imperfection value.
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