14. Federico A. Tavarez is a graduate student in the Department of Engineering Physics
at the University of Wisconsin-Madison. He received his BS in civil engineering from
the University of Puerto Rico-Mayagüez and his MSCE from the University of
Wisconsin. His research interests include finite element analysis, the use of composite
materials for structural applications, and the use of discrete element methods for
modeling concrete damage and fragmentation under impact.
ACI member Lawrence C. Bank is a professor in the Department of Civil and
Environmental Engineering at the University of Wisconsin-Madison. He received his
PhD in civil engineering and engineering mechanics from Columbia University in
1985. He is a member of ACI Committee 440, Fiber Reinforced Polymer Reinforcement.
His research interests include FRP reinforcement systems for structures, progressive
failure of materials and structural systems, and durability of FRP materials.
Michael E. Plesha is a professor in the Engineering Mechanics and Astronautics
Program in the Department of Engineering Physics at the University of WisconsinMadison. He received his PhD from Northwestern University in 1983. His research
interests include finite element analysis, discrete element analysis, dynamics of
geologic media, constitutive modeling of geologic discontinuity behavior, soil structure
interaction modeling, and continuum modeling of jointed saturated rock masses.
developed in the shear span near the load points. The study
concluded that further research was needed to obtain a better
understanding of the stress state in the longitudinal reinforcement at failure to predict the correct capacity and failure
modes of the beams.
Further experimental tests on concrete beams reinforced
with three-dimensional FRP composite grids were conducted to
investigate the behavior and performance of the grids when
used to reinforce beams that develop significant flexural-shear
cracking.7 Different composite grid configurations were
designed to study the influence of the FRP grid components
(longitudinal bars, vertical bars, and transverse bars) on the
load-deflection behavior and failure modes. Even though failure
modes of the beams were different depending upon the
characteristics of the composite grid, all beams failed in their
shear spans. Failure modes included splitting and rupture of
the main longitudinal bars and shear-out failure of the
vertical bars. Research results concluded that the design
of concrete beams with composite grid reinforcements must
account for failure of the main bars in the shear span.
A second phase of this experimental research was performed
by Ozel and Bank5 to investigate the capacity and failure modes
of composite grid reinforced concrete beams with different shear
span-to-effective depth ratios. Three different shear spandepth ratios (a/d) were investigated, with values of 3, 4.5, and 6,
respectively.11 The data obtained from this recently completed
experimental study was compared with the finite element results
obtained in the present study.
Experimental studies have shown that due to the development of large cracks in the FRP-reinforced concrete beams,
most of the deformation takes place at a relatively small
number of cracks between rigid bodies.12 A schematic of this
behavior is shown in Fig. 2. As a result, beams with relatively
small shear span-depth ratios typically fail due to rupture of the
main FRP longitudinal reinforcement at large flexural-shear
cracks, even though they are over-reinforced according to
conventional flexural design procedures.5,7,13,14 Due to the
aforementioned behavior for beams reinforced with composite
grids, especially those that exhibit significant flexural-shear
cracking, it is postulated that the longitudinal bars in the
member are subjected to a uniform tensile stress distribution,
plus a nonuniform stress distribution due to localized rotations at large cracks, which can be of great importance in
determining the ultimate flexural strength of the beam. The
present study investigates the stress-state at the flexuralshear cracks in the main longitudinal bars, using explicit
finite element tools to simulate this behavior and determine
the conditions that will cause failure in the beam.
ACI Structural Journal/March-April 2003
Fig. 1—Structural members in composite grid reinforced
concrete beam.
Fig. 2—Deformation due to rotation of rigid bodies.
Numerical analysis of FRP composite grid
reinforced beams
Implicit finite element methods are usually desirable for
the analysis of quasistatic problems. Their efficiency and
accuracy, however, depend on mesh topology and severity
of nonlinearities. In the problem at hand, it would be very
difficult to model the nonlinearities and progressive damage/
failure using an implicit method, and thus an explicit method
was chosen to perform the analysis.15
Using an explicit finite element method, especially to
model a quasistatic experiment as the one presented herein,
can result in long run times due to the large number of time
steps that are required. Because the time step depends on the
smallest element size, efficiency is compromised by mesh
refinement. The three-dimensional finite element mesh for
this study was developed in HyperMesh16 and consisted of
brick elements to represent the concrete, shell elements to
represent the bottom longitudinal reinforcement, and beam
elements to represent the top reinforcement, stirrups, and
cross rods. Figure 3 shows a schematic of the mesh used for
the models developed. Beams with span lengths of 2300,
3050, and 3800 mm were modeled corresponding to shear
span-depth ratios of 3, 4.5, and 6, respectively. These models
are referred to herein as short beam, medium beam, and long
beam, respectively. The cross-sectional properties were
identical for the three models. As will be seen later, the longitudinal bars play an important role in the overall behavior of the
system, and therefore they were modeled with greater detail
than the rest of the reinforcement. The concrete representation
consisted of 8-node solid elements with dimensions 25 x 25 x
12.5 mm (shortest dimension parallel to the width of the beam),
with one-point integration. The mesh discretization was established so that the reinforcement nodes coincided with the
concrete nodes. The reinforcement mesh was connected to the
concrete mesh by shared nodes between the concrete and the
251
15. Fig. 3—Finite element model for composite grid reinforced concrete beam.
Fig. 4—Short beam model at several stages in simulation.
reinforcement. As such, a perfect bond is assumed between the
concrete and the composite grid.
The two-node Hughes-Liu beam element formulation with
2 x 2 Gauss integration was used for modeling the top longitudinal bars, stirrups, and cross rods in the finite element
models. In this study, each model contains two top longitudinal
bars with heights of 25 mm and thicknesses of 4 mm. The
models also have four cross rods and three vertical members
at each stirrup location, as shown in Fig. 3. The vertical
members have a width of 38 mm and a thickness of 6.4 mm.
The cross rod elements have a circular cross-sectional
area with a diameter of 12.7 mm. To model the bottom
longitudinal reinforcement, the four-node BelytschkoLin-Tsay shell element formulation was used, as shown
in Fig. 3, with two through-the-thickness integration points.
252
Boundary conditions and event simulation time
To simulate simply supported conditions, the beam was
supported on two rigid plates made of solid elements. The
finite element simulations were displacement controlled,
which is usually the control method for plastic and nonlinear
behavior. That is, a displacement was prescribed on the rigid
loading plates located on top of the beam. The prescribed
displacement was linear, going from zero displacement at t =
0.0 s to 60, 75, and 90 mm at t = 1.0 s for the short, medium,
and long beams, respectively. The corresponding applied
load due to the prescribed displacement was then determined
by monitoring the vertical reaction forces at the concrete
nodes in contact with the support elements.
The algorithm CONTACT_AUTOMATIC_SINGLE_
SURFACE in LS-DYNA was used to model the contact
ACI Structural Journal/March-April 2003
16. between the supports, load bars, and the concrete beam.
This algorithm automatically generates slave and master
surfaces and uses a penalty method where normal interface
springs are used to resist interpenetration between element
surfaces. The interface stiffness is computed as a function
of the bulk modulus, volume, and face area of the elements
on the contact surface.
The finite element analysis was performed to represent
quasistatic experimental testing. As the time over which the
load is applied approaches the period of the lowest natural
frequency of vibration of the structural system, inertial forces
become more important in the response. Therefore, the load
application time was chosen to be long enough so that inertial
effects would be negligible. The flexural frequency of vibration
was computed analytically for the three beams using conventional formulas for vibration theory. 17 Accordingly, it was
determined that having a load application time of 1.0 s
was sufficiently long so that inertial effects are negligible
and the analysis can be used to represent a quasistatic experiment. For the finite element simulations presented in this
study, the CPU run time varied approximately from 22 to
65 h (depending on the length of the beam) for 1.0 s of load
application time on a 600 MHz PC with 512 MB RAM.
Material models
Material Type 72 (MAT_CONCRETE_DAMAGE) in
LS-DYNA was chosen for the concrete representation in the
present study. This material model has been used successfully
for predicting the response of standard uniaxial, biaxial, and
triaxial concrete tests in both tension and compression. The
formulation has also been used successfully to model the
behavior of standard reinforced concrete dividing walls
subjected to blast loads.18 This concrete model is a plasticitybased formulation with three independent failure surfaces
(yield, maximum, and residual) that change shape depending
on the hydrostatic pressure of the element. Tensile and
compressive meridians are defined for each surface, describing
the deviatoric part of the stress state, which governs failure in
the element. Detailed information about this concrete material
model can be found in Malvar et al.18 The values used in
the input file corresponded to a 34.5 MPa concrete compressive
strength with a 0.19 Poisson’s ratio and a tensile strength of
3.4 MPa. The softening parameters in the model were chosen to
be 15, –50, and 0.01 for uniaxial tension, triaxial tension, and
compression, respectively.19
The longitudinal bars were modeled using an orthotropic
material model (MAT_ENHANCED_COMPOSITE_DAMAGE),
which is material Type 54 in LS-DYNA. Properties used for
this model are shown in Table 1. Because the longitudinal
bars were drilled with holes for cross rod connections, the
tensile strength in the longitudinal direction of the FRP bars
was taken from experimental tensile tests conducted on
notched bar specimens with a 12.7 mm hole to account
for stress concentration effects at the cross rod locations.
The tensile properties in the transverse direction were
taken from tests on unnotched specimens. 11 Values for
shear and compressive properties were chosen based on
data in the literature. The composite material model uses
the Chang/Chang failure criteria. 20
The remaining reinforcement (top longitudinal bars, stirrups,
and cross rods) was modeled using two-noded beam elements
using a linear elastic material model (MAT_ELASTIC) with
the same properties used for the longitudinal direction in the
bottom FRP longitudinal bars. A rigid material model
ACI Structural Journal/March-April 2003
Fig. 5—Experimental and finite element load-deflection
results for short, medium, and long beams.
Fig. 6—Typical failure of composite grid reinforced concrete
beam (Ozel and Bank5).
Table 1—Material properties of FRP bottom bars
Ex
26.7 GPa
Xt
266.8 MPa
151.0 MPa
Ey
14.6 GPa
Yt
Gxy
3.6 GPa
Sc
6.9 MPa
νxy
0.26
Xc
177.9 MPa
β
0.5
Yc
302.0 MPa
(MAT_RIGID) was used to model the supports and the
loading plates.
FINITE ELEMENT RESULTS AND DISCUSSION
Graphical representations of the finite element model for
the short beam at several stages in the simulation are shown
in Fig. 4. The lighter areas in the model represent damage
(high effective plastic strain) in the concrete material model.
As expected, there is considerable damage in the shear span
of the concrete beam. Figure 4 also shows the behavior of the
composite grid inside the concrete beam. All displacements
in the simulation graphics were amplified using a factor of 5
to enable viewing. Actual deflection values are given in
Fig. 5, which shows the applied load versus midspan deflection behavior for the short, medium, and long beams for the
experimental and LS-DYNA results, respectively. The
jumps in the LS-DYNA curves in the figure represent the
progressive tensile and shear failure in the concrete elements. As
shown in this figure, the ultimate load value from the finite
element model agrees well with the experimental result. The
model slightly over-predicts the stiffness of the beam, however,
and under-predicts the ultimate deflection.
The significant drop in load seen in the load-deflection
curves produced in LS-DYNA is caused by failure in the
253
17. Fig. 7—Medium beam model at several stages in simulation.
Fig. 8—Long beam model at several stages in simulation.
longitudinal bars, as seen in Fig. 4. The deformed shape
seen in this figure indicates a peculiar behavior throughout the length of the beam. It appears to indicate that after
a certain level of damage in the shear span of the model,
localized rotations occur in the beam near the load points.
These rotations create a stress concentration that causes
the longitudinal bars to fail at those locations. This deflection
behavior was also observed in the experimental tests.
Figure 6 shows a typical failure in the longitudinal bars
from the experiments conducted on these beams. 11 As
shown in this figure, there is considerable damage in the
shear span of the member. Large shear cracks develop in
the beam, causing the member to deform in the same
fashion as the one seen in the finite element model.
Figure 7 shows the medium beam model at several stages
in the simulation. The figure also shows the behavior of the
main longitudinal bars. Comparing this simulation with the
one obtained for the short beam, it can be seen that the shear
damage is not as significant as in the previous simulation.
The deflected shape seen in the longitudinal bars shows that
this model does not have the abrupt changes in rotation that
254
were observed in the short beam, which would imply that this
model does not exhibit significant flexural-shear damage. For
this model, the finite element analysis slightly over-predicted
both the stiffness and the ultimate load value obtained from
the experiment. On the other hand, the ultimate deflection
was under-predicted. Failure in this model was also caused
by rupture of the longitudinal bars at a location near the load
points. In the experimental test, failure was caused by a
combination of rupture in the longitudinal bars as well as
concrete crushing in the compression zone. This compressive
failure was located near the load points, however, and
could have been initiated by cracks formed due to stress
concentrations produced by the rigid loading plates. 11
Figure 8 shows the results for the long beam model.
Comparing this simulation with the two previous ones, it
can be seen that this model exhibits the least shear damage,
as expected. As a result, the longitudinal bars exhibit a
parabolic shape, which would be the behavior predicted
using conventional moment-curvature methods based on the
curvature of the member. Once again, the stiffness of the
beam was slightly over-predicted. However, the ultimate load
ACI Structural Journal/March-April 2003
18. Table 2—Summary of experimental and finite
element results
Total load capacity, kN
Tensile force in each
main bar, kN
Finite
element
analysis
Flexural
analysis
Finite
element
analysis
Beam
Short
value compares well with the experimental result. Failure in
the model was caused by rupture of the longitudinal bars.
Failure in the experimental test was caused by a compression
failure at a location near one of the load application bars,
followed by rupture of the main longitudinal bars. Figure 5
also shows the time at total failure for each beam, which can
be related to the simulation stages given in Fig. 4, 7, and 8
for the short, medium, and long beam, respectively.
To investigate the stress state of a single longitudinal bar
at ultimate conditions, the tensile force and the internal
moment of the longitudinal bars at the failed location for the
three finite element models was determined, as shown in
Fig. 9(a) and (b). It is interesting to note that for the short
beam model, the tensile force at failure was approximately
51.6 kN, while for the medium beam model and the long
beam model the tensile force at failure was approximately
76.5 kN. On the other hand, the internal moment in the short
beam model was approximately 734 N-m, while the internal
moment was approximately 339 N-m for both the short beam
model and the long beam model. It is clear that the shear
damage in the short beam model causes a considerable
localized effect in the stress state of the longitudinal bars,
which is important to consider for design purposes.
According to Fig. 9(a), the total axial load in the longitudinal
bars for the short beam model produces a uniform stress of
130 MPa, which is not enough to fail the element in tension
at this location. However, the ultimate internal moment
produces a tensile stress at the bottom of the longitudinal
bars of 141 MPa. The sum of these two components produces
a tensile stress of 271 MPa. When this value is entered in the
Chang/Chang failure criterion for the tensile longitudinal
direction, the strength is exceeded and the elements fail.
Using conventional over-reinforced beam analysis formulas,
the tensile force in the longitudinal bars at midspan would
be obtained by dividing the ultimate moment obtained from
the experimental test by the internal moment arm. This
would imply that there is a uniform tensile force in each
longitudinal bar of 88.1 kN. This tensile force is never
achieved in the finite element simulation due to considerable
shear damage in the concrete elements. As a result of this
shear damage in the concrete, the curvature at the center of
the beam is not large enough to produce a tensile force in the
bars of this magnitude (88.1 kN). The internal moment in the
longitudinal bars shown in Fig. 9(b), however, continues
to develop, resulting in a total failure load comparable to
the experimental result. As mentioned before, the force in
the bars according to the simulation was approximately
51.6 kN, which is approximately half the load predicted
using conventional methods. Therefore, the use of conventional
beam analysis formulas to analyze this composite grid reinforced
beam would not only erroneously predict the force in the
longitudinal bars, but it would also predict a concrete
ACI Structural Journal/March-April 2003
215.7
196.2
215.3
90.7
51.6
Medium
Fig. 9—(a) Tensile force in longitudinal bars; and (b) internal
moment in longitudinal bars.
Experimental
Flexural
analysis
143.2
130.8
161.9
90.7
76.5
Long
108.1
97.9
113.0
90.7
76.5
compression failure mode, which was not the failure
mode observed from the experimental tests.
The curves for the medium beam model and the long beam
model, shown in Fig. 9, show that for both cases, the beam
shear span-depth ratio was sufficiently large so that the stress
state in the longitudinal bars would not be greatly affected by the
shear damage produced in the beam. As such, the ultimate axial
force obtained in the longitudinal bars for both models was
close to the ultimate axial load that would be predicted by using
conventional methods.
In summary, Table 2 presents the ultimate load capacity
for the three models, including experimental results, conventional flexural analysis results, and finite element results. As
shown in this table, conventional flexural analysis under-predicts
the actual ultimate load carried by the beams and a better
ultimate load prediction was obtained using finite element
analysis. The tensile load in the bars was computed (analytically)
by dividing the experimental moment capacity by the internal
moment arm computed by using strain compatibility. Although
the finite element results over-predicted the ultimate load for the
medium and long beams, the simulations provided a better
understanding of the complex phenomena involved in the
behavior of the beams, depending on their shear span-depth
ratio. The results for tensile load in the bars reported in this table
suggest that composite grid reinforced concrete beams
with values of shear span-depth ratio greater than 4.5 can be
analyzed by using the current flexural theory.
It is important to mention that the concrete material model
parameters that govern the post-failure behavior of the material
played a key role in the finite element results for the three finite
element models. In the concrete material formulation, the
elements fail in an isotropic fashion and, therefore, once an
element fails in tension, it cannot transfer further shear.
Because the concrete elements are connected to the reinforcement mesh, this behavior causes the beam to fail prematurely
as a result of tensile failure in the concrete. Therefore, the
parameters that govern the post-failure behavior in the
concrete material model were chosen so that when an element
fails in tension, the element still has the capability to transfer
shear forces and the stresses will gradually decrease to zero.
Because the failed elements can still transfer tensile stresses,
however, the modifications caused an increase in the stiffness
of the beam. In real concrete behavior, when a crack opens,
there is no tension transfer between the concrete at that
location, causing the member to lose stiffness as cracking
progresses. Regarding shear transfer, factors such as aggregate interlock and dowel action would contribute to transfer
shear forces in a concrete beam, and tensile failure in the
concrete would not affect the response as directly as in the
finite element model.
255
19. Stress analysis of FRP bars
As discussed previously, failure modes observed in experimental tests performed on composite grid reinforced concrete
beams suggest that the longitudinal bars are subjected to a
uniform tensile stress plus a nonuniform bending stress due
to localized rotations at locations of large cracks. This section
presents a simple analysis procedure to determine the stress
conditions at which the longitudinal bars fail. As a result of this
analysis, a procedure is presented to analyze/design a composite
grid reinforced concrete beam, considering a nonuniform stress
state in the longitudinal bars.
A more detailed finite element model of a section of the
longitudinal bars was developed in HyperMesh16 using shell
elements, as shown in Fig. 10. A height of 50.8 mm was
specified for the bar model, with a thickness of 4.1 mm. The
length of the bar and the diameter of the hole were 152 and
12.7 mm, respectively. The material formulation and properties
were the same as the ones used for the longitudinal bars in the
concrete beam models, with the exception that now the
unnotched tensile strength of the material (Xt = 521 MPa) was
used as an input parameter because the hole was incorporated in
the model.
The finite element model was first loaded in tension to
establish the tensile strength of the notched bar. The load
was applied by prescribing a displacement at the end of the
bar. Figure 10 shows the simulation results for the model at
three stages, including elastic deformation and ultimate
failure. As expected, a stress concentration developed on the
boundary of the hole causing failure in the web of the model,
followed by ultimate failure of the cross section. A tensile
strength of 274 MPa was obtained for the model. A value
of 267 MPa was obtained from experimental tests conducted on notched bars (tensile strength used in Table 2),
demonstrating good agreement between experimental
and finite element results.
A similar procedure was performed to establish the
strength of the bar in pure bending. That is, displacements
were prescribed at the end nodes to induce bending in the
model. Figure 11 shows the simulation results for the model
at three stages, showing elastic bending and ultimate failure
caused by flexural failure at the tension flange. As shown in
this figure, the width of the top flange was modified to
prevent buckling in the flange (which was present in the
original model). Because buckling would not be present in a
longitudinal bar due to concrete confinement, it was decided
to modify the finite element model to avoid this behavior. To
maintain an equivalent cross-sectional area, the thickness of
the flange was increased. A maximum pure bending moment
of 2.92 kN-m was obtained for the model.
Knowing the maximum force that the bar can withstand in
pure tension and pure bending, the model was then loaded at
different values of tension and moment to cause failure. This
procedure was performed several times to develop a tensionmoment interaction diagram for the bar, as shown in Fig. 12.
The discrete points shown in the figure are combinations of
tensile force and moment values that caused failure in the
finite element model. This interaction diagram can be used
to predict what combination of tensile force and moment
would cause failure in the FRP longitudinal bar.
Considerations for design
The strength design philosophy states that the flexural
capacity of a reinforced concrete member must exceed the
flexural demand. The design capacity of a member refers
256
Fig. 10—Failure on FRP bar subjected to pure tension.
Fig. 11—Failure on FRP bar subjected to pure bending.
to the nominal strength of the member multiplied by a strengthreduction factor φ, as shown in the following equation
φ Mn ≥ Mu
(1)
For FRP reinforced concrete beams, a compression failure
is the preferred mode of failure, and, therefore, the beam
should be over-reinforced. As such, conventional formulas
are used to ensure that the selected cross-sectional area of the
longitudinal bars is sufficiently large to have concrete
compression failure before FRP rupture. Considering a concrete
compression failure, the capacity of the beam is computed using
the following8
a
M n = A f f f d – --
2
(2)
Af ff
a = -------------β1 fc b
′
(3)
β1 d – a
f f = E f ε cu ----------------a
(4)
Experimental tests have shown, however, that there is
a critical value of shear Vscrit in a beam where localized rotations
due to large flexural-shear cracks begin to occur. The
ultimate moment in the beam is assumed to be related to
this shear-critical value and it is determined according to
the following equation
Mn = n ⋅ ( t ⋅ i e + m )
(5)
where n is the number of longitudinal bars. Once the beam has
reached the shear-critical value, it is assumed (conservatively)
that the tensile force t, which is the force in each bar at the
shear-critical stage, remains constant and any additional load is
carried by localized internal moment m in the longitudinal
bars. Furthermore, it is assumed that at this stage the concrete
is still in its elastic range, and, therefore, the internal moment
arm ie can be determined by equilibrium and elastic strain
compatibility. The tensile force t in Eq. (5) is computed
ACI Structural Journal/March-April 2003
20. Table 3—Summary of results for three beams using proposed approach
Beam
Experimental
ultimate
Theoretical shear
shear, kN
critical, kN
Total load capacity, kN
Equation for
moment capacity
Experimental
Analytical Tension in each
Pn = Mn /as main bar, kN
Short
108.1
88.1
Mn = t · ie + m
216
199
70.7
Medium
71.6
88.1
Mn = Af f f (d – a/ 2)
143
131
90.7
88.1
Mn = Af f f (d – a/2)
109
99
90.7
Long
54.7
according to the following equation for a simply supported
beam in four-point bending
crit
V s ⋅ as
t = --------------------ni e
(6)
where as is the shear span of the member. The obtained value
for the tension t in each bar is then entered in Eq. (7), which
is the equation for the interaction diagram, to determine the
ultimate internal moment m in Eq. (5) that causes the bar to
fail. In this equation, tmax and mmax are known properties of
the notched composite bar.
t- 2
m = m max 1 – -------- for t > 0 ; m > 0
t max
(7)
The aforementioned procedure is a very simplified analysis to
determine the capacity of a composite grid reinforced concrete
beam, and, as can be seen, it depends considerably on the shearcritical value Vscrit established for the beam. This value is
somewhat difficult to determine. Based on experimental data, a
value given by Eq. (8) (analogous to Eq. (9-1) of ACI
440.1R-01) can be considered to be a lower bound for
FRP reinforced beams with shear reinforcement.
crit
Vs
7 ρf Ef 1
- ′
= ----------------- -- f c bd
90 β 1 f c 6
′
(8)
where fc′ is the specified compressive strength of the concrete
in MPa. In summary, the ultimate moment capacity in the beam
is determined according to one of the following equations
crit
M n = A f f f d – a for V ult < V s
-
2
crit
M n = n ⋅ ( t ⋅ i e + m ) for V ult > V s
(9)
(10)
According to Eq. (9), if the ultimate shear force computed
analytically based on conventional theory does not exceed
the shear-critical value Vscrit, the moment capacity can be
computed from flexural analysis. On the other hand, if the
computed ultimate shear force is greater than Vscrit, Eq. (10)
is used. Table 3 presents a summary showing the load capacity
for the three beams obtained experimentally and analytically
using the present approach. As shown in this table, the equation
used to determine the flexural capacity depends on the ultimate
shear obtained for each beam.
As seen in this procedure, the only difficulty in applying
these formulas is the fact that an equation needs to be determined
ACI Structural Journal/March-April 2003
Fig. 12—Tension-moment interaction diagram for longitudinal bar.
to compute the maximum moment that the bar can carry as a
function of the tensile force acting in the bar. If a specific bar
is always used, however, this difficulty is eliminated, and if
the flexural demand is not exceeded, a higher capacity can be
obtained by increasing the number of longitudinal bars in the
section. According to the results obtained for the three beams
analyzed herein, the proposed procedure will under-predict
the capacity of the composite grid reinforced concrete beam,
but it will provide a good lower bound for a conservative
design. Furthermore, it will ensure that the longitudinal bars
will not fail prematurely as a result of the development of
large flexural-shear cracks in the member, and thus the
member will be able to meet and exceed the flexural demand
for which it was designed.
CONCLUSIONS
Based on the explicit finite element results and comparison
with experimental data, the following conclusions can be made:
1. Failure in the FRP longitudinal bars occurs due to a
combination of a uniform tensile stress plus a nonuniform
stress caused by localized rotations at large flexural-shear
cracks. Therefore, this failure mode has to be accounted for
in the analysis and design of composite grid reinforced concrete
beams, especially those that exhibit significant flexuralshear cracking;
2. The shear span for the medium beam and the long beam
studied was sufficiently large so that the stress state in the
longitudinal bars was not considerably affected by shear
damage in the beam. Therefore, the particular failure mode
observed by the short beam model is only characteristic of
257
21. beams with a low shear span-depth ratio. Moreover, according
to the proposed analysis for such systems, both the medium
beam and the long beam could be designed using conventional
flexural theory because the shear-critical value was never
reached for these beam lengths;
3. Numerical simulations can be used effectively to understand the complex behavior and phenomena observed in the
response of composite grid reinforced concrete beams and,
therefore, can be used as a complement to experimental
testing to account for multiple failure modes in the design
of composite grid reinforced concrete beams; and
4. The proposed method of analysis for composite grid
reinforced concrete beams considering multiple failure
modes will under-predict the capacity of the reinforced
concrete beam, but it will provide a good lower bound for
a conservative design. These design considerations will
ensure that the longitudinal bars will not fail prematurely
(or catastrophically) as a result of the development of large
flexural-shear cracks in the member, and thus the member
can develop a pseudoductile failure by concrete crushing,
which is more desirable than a sudden FRP rupture.
ACKNOWLEDGMENTS
This work was supported by the National Science Foundation under
Grant. No. CMS 9896074. Javier Malvar and Karagozian & Case are
thanked for providing information regarding the concrete material formulation
used in LS-DYNA. Jim Day, Todd Slavik, and Khanh Bui of Livermore
Software Technology Corporation (LSTC) are also acknowledged for their
assistance in using the finite element software, as well as Strongwell
Chatfield, MN, for producing the custom composite grids.
NOTATION
a
as
b
d
=
=
=
=
Ef
Ex
Ey
Gxy
f ′c
ff
ie
Mn
m
n
Sc
t
Vscrit
=
=
=
=
=
=
=
=
=
=
=
=
=
Vult
Xc
Xt
Yc
Yt
β
β1
=
=
=
=
=
=
=
εcu
ρf
νxy
=
=
=
258
depth of equivalent rectangular stress block
length of shear span in reinforced concrete beam
width of rectangular cross section
distance from extreme compression fiber to centroid of tension
reinforcement
modulus of elasticity for FRP bar
modulus of elasticity in longitudinal direction of FRP grid material
modulus of elasticity in transverse direction of FRP grid material
shear modulus of FRP grid members
specified compressive strength of concrete
stress in FRP reinforcement in tension
internal moment arm in the elastic range
nominal moment capacity
internal moment in longitudinal FRP grid bars
number of longitudinal FRP grid bars
shear strength of FRP grid material
tensile force in a longitudinal bar at the shear critical stage
critical shear resistance provided by concrete in FRP grid reinforced concrete
ultimate shear force in reinforced concrete beam
longitudinal compressive strength of FRP grid material
longitudinal tensile strength of FRP grid material
transverse compressive strength of FRP grid material
transverse tensile strength of FRP grid material
weighting factor for shear term in Chang/Chang failure criterion
ratio of the depth of Whitney’s stress block to depth to neutral axis
concrete ultimate strain
FRP reinforcement ratio
Poisson’s ratio of FRP grid material
REFERENCES
1. Sugita, M., “NEFMAC—Grid Type Reinforcement,” Fiber-ReinforcedPlastic (FRP) Reinforcement for Concrete Structures: Properties and
Applications, Developments in Civil Engineering, A. Nanni, ed., Elsevier,
Amsterdam, V. 42, 1993, pp. 355-385.
2. Schmeckpeper, E. R., and Goodspeed, C. H., “Fiber-Reinforced
Plastic Grid for Reinforced Concrete Construction,” Journal of Composite
Materials, V. 28, No. 14, 1994, pp. 1288-1304.
3. Bank, L. C.; Frostig, Y.; and Shapira, A., “Three-Dimensional FiberReinforced Plastic Grating Cages for Concrete Beams: A Pilot Study,” ACI
Structural Journal, V. 94, No. 6, Nov.-Dec. 1997, pp. 643-652.
4. Smart, C. W., and Jensen, D. W., “Flexure of Concrete Beams
Reinforced with Advanced Composite Orthogrids,” Journal of Aerospace
Engineering, V. 10, No. 1, Jan. 1997, pp. 7-15.
5. Ozel, M., and Bank, L. C., “Behavior of Concrete Beams Reinforced
with 3-D Composite Grids,” CD-ROM Paper No. 069. Proceedings of the
16th Annual Technical Conference, American Society for Composites,
Virginia Tech, Va., Sept. 9-12, 2001.
6. Shapira, A., and Bank, L. C., “Constructability and Economics of FRP
Reinforcement Cages for Concrete Beams,” Journal of Composites for
Construction, V. 1, No. 3, Aug. 1997, pp. 82-89.
7. Bank, L. C., and Ozel, M., “Shear Failure of Concrete Beams Reinforced
with 3-D Fiber Reinforced Plastic Grids,” Fourth International Symposium on
Fiber Reinforced Polymer Reinforcement for Reinforced Concrete Structures,
SP-188, C. Dolan, S. Rizkalla, and A. Nanni, eds., American Concrete
Institute, Farmington Hills, Mich., 1999, pp. 145-156.
8. ACI Committee 440, “Guide for the Design and Construction of
Concrete Reinforced with FRP Bars (ACI 440.1R-01),” American Concrete
Institute, Farmington Hills, Mich., 2001, 41 pp.
9. Nakagawa, H.; Kobayashi. M.; Suenaga, T.; Ouchi, T.; Watanabe, S.;
and Satoyama, K., “Three-Dimensional Fabric Reinforcement,” FiberReinforced-Plastic (FRP) Reinforcement for Concrete Structures: Properties
and Applications, Developments in Civil Engineering, V. 42, A. Nanni, ed.,
Elsevier, Amsterdam, 1993, pp. 387-404.
10. Yonezawa, T.; Ohno, S.; Kakizawa, T.; Inoue, K.; Fukata, T.; and
Okamoto, R., “A New Three-Dimensional FRP Reinforcement,” FiberReinforced-Plastic (FRP) Reinforcement for Concrete Structures: Properties
and Applications, Developments in Civil Engineering, V. 42, A. Nanni, ed.,
Elsevier, Amsterdam, 1993, pp. 405-419.
11. Ozel, M., “Behavior of Concrete Beams Reinforced with 3-D
Fiber Reinforced Plastic Grids,” PhD thesis, University of WisconsinMadison, 2002.
12. Lees, J. M., and Burgoyne, C. J., “Analysis of Concrete Beams with
Partially Bonded Composite Reinforcement,” ACI Structural Journal,
V. 97, No. 2, Mar.-Apr. 2000, pp. 252-258.
13. Shehata, E.; Murphy, R.; and Rizkalla, S., “Fiber Reinforced
Polymer Reinforcement for Concrete Structures,” Fourth International
Symposium on Fiber Reinforced Polymer Reinforcement for Reinforced
Concrete Structures, SP-188, C. Dolan, S. Rizkalla, and A. Nanni, eds.,
American Concrete Institute, Farmington Hills, Mich., 1999, pp. 157-167.
14. Guadagnini, M.; Pilakoutas, K.; and Waldron, P., “Investigation on
Shear Carrying Mechanisms in FRP RC Beams,” FRPRCS-5, FibreReinforced Plastics for Reinforced Concrete Structures, Proceedings of the
Fifth International Conference, C. J. Burgoyne, ed., V. 2, Cambridge, July
16-18, 2001, pp. 949-958.
15. Cook, R. D.; Malkus, D. S.; and Plesha, M. E., Concepts and
Applications of Finite Element Analysis, 3rd Edition, John Wiley &
Sons, N.Y., 1989, 832 pp.
16. Altair Computing, HyperMesh Version 2.0 User’s Manual, Altair
Computing Inc., Troy, Mich., 1995.
17. Thompson, W. T., and Dahleh, M. D., Theory of Vibration with
Applications, 5th Edition, Prentice Hall, N.J., 1998, 524 pp.
18. Malvar, L. J.; Crawford, J. E.; Wesevich, J. W.; and Simons, D., “A
Plasticity Concrete Material Model for DYNA3D,” International Journal
of Impact Engineering, V. 19, No. 9/10, 1997, pp. 847-873.
19. Tavarez, F. A., “Simulation of Behavior of Composite Grid Reinforced
Concrete Beams Using Explicit Finite Element Methods,” MS thesis, University
of Wisconsin-Madison, 2001.
20. Hallquist, J. O., LS-DYNA Keyword User’s Manual, Livermore
Software Technology Corporation, Livermore, Calif., Apr. 2000.
ACI Structural Journal/March-April 2003
23. Daniel Palermo is a visiting assistant professor in the Department of Civil Engineering,
University of Toronto, Toronto, Ontario, Canada. He received his PhD from the University
of Toronto in 2002. His research interests include nonlinear analysis and design of
concrete structures, constitutive modeling of reinforced concrete subjected to cyclic
loading, and large-scale testing and analysis of structural walls.
ACI member Frank J. Vecchio is Professor and Associate Chair in the Department of
Civil Engineering, University of Toronto. He is a member of Joint ACI-ASCE
Committee 441, Reinforced Concrete Columns, and 447, Finite Element Analysis
of Reinforced Concrete Structures. His interests include nonlinear analysis and
design of concrete structures.
approach, to illustrate the analysis capability for arbitrary
loading conditions, including reversed cyclic loading. The
models presented herein have also been formulated in the
context of smeared rotating cracks, and are intended to build
upon the preliminary constitutive formulations presented by
Vecchio.5 A companion paper 9 documenting the results
of nonlinear finite element analyses, incorporating the
proposed models, will demonstrate accurate simulations
of structural behavior.
Compression response
First consider the compression response, illustrated in
Fig. 1, occurring in either of the principal strain directions.
Figure 1(a) and (b) illustrate the compressive unloading and
compressive reloading responses, respectively. The backbone
curve typically follows the monotonic response, that is,
Hognestad parabola 10 or Popovics formulation,11 and
includes the compression softening effects according to
the Modified Compression Field Theory. 12
The shape and slope of the unloading and reloading responses
p
are dependent on the plastic offset strain εc , which is essentially
the amount of nonrecoverable damage resulting from
crushing of the concrete, internal cracking, and compressing of
internal voids. The plastic offset is used as a parameter in
defining the unloading path and in determining the degree of
damage in the concrete due to cycling. Further, the backbone
curve for the tension response is shifted such that its origin
coincides with the compressive plastic offset strain.
Various plastic offset models for concrete in compression
have been documented in the literature. Karsan and Jirsa13
were the first to report a plastic offset formulation for concrete
subjected to cyclic compressive loading. The model illustrated
the dependence of the plastic offset strain on the strain at the
onset of unloading from the backbone curve. A review of
various formulations in the literature reveals that, for the
most part, the models best suit the data from which they were
derived, and no one model seems to be most appropriate. A
unified model (refer to Fig. 2) has been derived herein considering data from unconfined tests from Bahn and Hsu14 and
Karsan and Jirsa,13 and confined tests from Buyukozturk and
Tseng.15 From the latter tests, the results indicated that the
plastic offset was not affected by confining stresses or strains.
The proposed plastic offset formulation is described as
ε 2c 2
ε 2c
p
ε c = ε p 0.166 ------ + 0.132 ------
εp
εp
Fig. 1—Hysteresis models for concrete in compression: (a)
unloading; and (b) reloading.
(1)
where εcp is the plastic offset strain; εp is the strain at peak
stress; and ε2c is the strain at the onset of unloading from the
backbone curve. Figure 2 also illustrates the response of other
plastic offset models available in the literature.
The plot indicates that models proposed by Buyukozturk
and Tseng15 and Karsan and Jirsa13 represent upper- and
ACI Structural Journal/September-October 2003
lower-bound solutions, respectively. The proposed model
(Palermo) predicts slightly larger residual strains than the
lower limit, and the Bahn and Hsu14 model calculates
progressively larger plastic offsets. Approximately 50% of
the datum points were obtained from the experimental results
of Karsan and Jirsa;13 therefore, it is not unexpected that the
Palermo model is skewed towards the lower-bound Karsan
and Jirsa13 model. The models reported in the literature were
derived from their own set of experimental data and, thus,
may be affected by the testing conditions. The proposed
formulation alleviates dependence on one set of experimental
data and test conditions. The Palermo model, by predicting
Fig. 2—Plastic offset models for concrete in compression.
617
24. relatively small plastic offsets, predicts more pinching in
the hysteresis behavior of the concrete. This pinching
phenomenon has been observed by Palermo and Vecchio8 and
Pilakoutas and Elnashai16 in the load-deformation response of
structural walls dominated by shear-related mechanisms.
In analysis, the plastic offset strain remains unchanged
unless the previous maximum strain in the history of loading
is exceeded.
The unloading response of concrete, in its simplest form,
can be represented by a linear expression extending from the
unloading strain to the plastic offset strain. This type of
representation, however, is deficient in capturing the energy
dissipated during an unloading/reloading cycle in compression.
Test data of concrete under cyclic loading confirm that the
unloading branch is nonlinear. To derive an expression to
describe the unloading branch of concrete, a RambergOsgood formulation similar to that used by Seckin17 was
adopted. The formulation is strongly influenced by the
unloading and plastic offset strains. The general form of
the unloading branch of the proposed model is expressed as
f c ( ∆ε ) = A + B ∆ε + C ∆ε
N
stress point on the reloading path that corresponded to the
maximum unloading strain. The new stress point was assumed
to be a function of the previous unloading stress and the
stress at reloading reversal. Their approach, however, was
stress-based and dependent on the backbone curve. The
approach used herein is to define the reloading stiffness
as a degrading function to account for the damage induced in the
concrete due to load cycling. The degradation was observed to
be a function of the strain recovery during unloading. The
reloading response is then determined from
f c = f ro + E c1 ( ε c – ε ro )
(6)
where fc and εc are the stress and strain on the reloading path;
f ro is the stress in the concrete at reloading reversal and
corresponds to a strain of εro ; and Ec1 is the reloading
stiffness, calculated as follows
( β d ⋅ f max ) – f ro
E c1 = ----------------------------------ε 2c – ε ro
(7)
(2)
where
where fc is the stress in the concrete on the unloading curve,
and ∆ε is the strain increment, measured from the instantaneous
strain on the unloading path to the unloading strain, A, B,
and C are parameters used to define the general shape of the
curve, and N is the Ramberg-Osgood power term. Applying
boundary conditions from Fig. 1(a) and simplifying yields
1
β d = ----------------------------------------------0.5
1 + 0.10 (ε rec ⁄ ε p )
for ε c < ε p
(8)
1
β d = -------------------------------------------------0.6
1 + 0.175 (ε rec ⁄ ε p )
for ε c > ε p
(9)
and
N
( E c3 – E c2 )∆ε
f c ( ∆ε ) = f 2c + E c2 ( ∆ε ) + -------------------------------------N–1
p
N ( ε c – ε 2c )
(3)
where
and
∆ε = ε – ε 2c
(4)
and
p
( E c2 – E c3 ) ( εc – ε 2c )
N = --------------------------------------------------p
f c2 + E c2 ( ε c – ε 2c )
(5)
ε is the instantaneous strain in the concrete. The initial
unloading stiffness Ec2 is assigned a value equal to the
initial tangent stiffness of the concrete Ec, and is routinely
calculated as 2fc′ / ε′c . The unloading stiffness Ec3, which defines
the stiffness at the end of the unloading phase, is defined as
0.071 E c, and was adopted from Seckin. 17 f2c is the stress
calculated from the backbone curve at the peak unloading
strain ε 2c.
Reloading can sufficiently be modeled by a linear response
and is done so by most researchers. An important characteristic,
however, which is commonly ignored, is the degradation in
the reloading stiffness resulting from load cycling. Essentially,
the reloading curve does not return to the backbone curve at
the previous maximum unloading strain (refer to Fig. 1 (b)).
Further straining is required for the reloading response to
intersect the backbone curve. Mander, Priestley, and Park6
attempted to incorporate this phenomenon by defining a new
618
ε rec = ε max – ε min
(10)
βd is a damage indicator, fmax is the maximum stress in the
concrete for the current unloading loop, and εrec is the
amount of strain recovered in the unloading process and is
the difference between the maximum strain εmax and the
minimum strain εmin for the current hysteresis loop. The
minimum strain is limited by the compressive plastic offset
strain. The damage indicator was derived from test data on
plain concrete from four series of tests: Buyukozturk and
Tseng,15 Bahn and Hsu,14 Karsan and Jirsa,13 and
Yankelevsky and Reinhardt.18 A total of 31 datum points
were collected for the prepeak range (Fig. 3(a)) and 33 datum
points for the postpeak regime (Fig. 3(b)). Because there was a
negligible amount of scatter among the test series, the datum
points were combined to formulate the model. Figure 3(a) and
(b) illustrate good correlation with experimental data, indicating the link between the strain recovery and the damage due
to load cycling. βd is calculated for the first unloading/reloading
cycle and retained until the previous maximum unloading strain
is attained or exceeded. Therefore, no additional damage is
induced in the concrete for hysteresis loops occurring at strains
less than the maximum unloading strain. This phenomenon is
further illustrated through the partial unloading and partial
reloading formulations.
ACI Structural Journal/September-October 2003
25. It is common for cyclic models in the literature to ignore
the behavior of concrete for the case of partial unloading/
reloading. Some models establish rules for partial loadings
from the full unloading/reloading curves. Other models
explicitly consider the case of partial unloading followed
by reloading to either the backbone curve or strains in excess
of the previous maximum unloading strain. There exists,
however, a lack of information considering the case where
partial unloading is followed by partial reloading to strains
less than the previous maximum unloading strain. This more
general case was modeled using the experimental results of
Bahn and Hsu.14 The proposed rule for the partial unloading
response is identical to that assumed for full unloading;
however, the previous maximum unloading strain and
corresponding stress are replaced by a variable unloading
strain and stress, respectively. The unloading path is defined
by the unloading stress and strain and the plastic offset strain,
which remains unchanged unless the previous maximum
strain is exceeded. For the case of partial unloading followed
by reloading to a strain in excess of the previous maximum
unloading strain, the reloading path is defined by the expressions
governing full reloading. The case where concrete is partially
unloaded and partially reloaded to a strain less than the
previous maximum unloading strain is illustrated in Fig 4.
Five loading branches are required to construct the response
of Fig. 4. Unloading Curve 1 represents full unloading from
the maximum unloading strain to the plastic offset and is
calculated from Eq. (3) to (5) for full unloading. Curve 2
defines reloading from the plastic offset strain and is
defined by Eq. (6) to (10). Curve 3 represents the case of
partial unloading from a reloading path at a strain less than the
previous maximum unloading strain. The expressions used
for full unloading are applied, with the exception of substituting the unloading stress and strain for the current hysteresis
loop for the unloading stress and strain at the previous
maximum unloading point. Curve 4 describes partial
reloading from a partial unloading branch. The response
follows a linear path from the load reversal point to the
previous unloading point and assumes that damage is not
accumulated in loops forming at strains less than the
previous maximum unloading strain. This implies that the
reloading stiffness of Curve 4 is greater than the reloading
stiffness of Curve 2 and is consistent with test data reported
by Bahn and Hsu.14 The reloading stiffness for Curve 4 is
represented by the following expression
f max – f ro
E c1 = ---------------------ε max – ε ro
f c = f max + E c1 ( ε c – ε max )
(13)
The proposed constitutive relations for concrete subjected
to compressive cyclic loading are tested in Fig. 5 against the
experimental results of Karsan and Jirsa.13 The Palermo
model generally captures the behavior of concrete under cyclic
compressive loading. The nonlinear unloading and linear
loading formulations agree well with the data, and the plastic
offset strains are well predicted. It is apparent, though, that
the reloading curves become nonlinear beyond the point of
intersection with the unloading curves, often referred to as the
Fig. 3—Damage indicator for concrete in compression:
(a) prepeak regime; and (b) postpeak regime.
(11)
The reloading stress is then calculated using Eq. (6) for
full reloading.
In further straining beyond the intersection with Curve 2,
the response of Curve 4 follows the reloading path of Curve 5.
The latter retains the damage induced in the concrete from
the first unloading phase, and the stiffness is calculated as
β d ⋅ f 2c – f max
E c1 = ------------------------------ε 2c – ε max
(12)
The reloading stresses are then determined from the
following
ACI Structural Journal/September-October 2003
Fig. 4—Partial unloading/reloading for concrete in compression.
619
26. common point. The Palermo model can be easily modified to
account for this phenomenon; however, unusually small load
steps would be required in a finite element analysis to capture
this behavior, and it was thus ignored in the model. Furthermore, the results tend to underestimate the intersection of the
reloading path with the backbone curve. This is a direct result
of the postpeak response of the concrete and demonstrates the
importance of proper modeling of the postpeak behavior.
Tension response
Much less attention has been directed towards the modeling
of concrete under cyclic tensile loading. Some researchers
consider little or no excursions into the tension stress regime
and those who have proposed models assume, for the most
Fig. 5—Predicted response for cycles in compression.
part, linear unloading/reloading responses with no plastic
offsets. The latter was the approach used by Vecchio5 in
formulating a preliminary tension model. Stevens, Uzumeri,
and Collins19 reported a nonlinear response based on defining
the stiffness along the unloading path; however, the models
were verified with limited success. Okumura and Maekawa2
proposed a hysteretic model for cyclic tension, in which a
nonlinear unloading curve considered stresses through bond
action and through closing of cracks. A linear reloading path
was also assumed. Hordijk 20 used a fracture mechanics
approach to formulate nonlinear unloading/reloading rules
in terms of applied stress and crack opening displacements.
The proposed tension model follows the philosophy used to
model concrete under cyclic compression loadings. Figure 6 (a)
and (b) illustrate the unloading and reloading responses,
respectively. The backbone curve, which assumes the
monotonic behavior, consists of two parts adopted from the
Modified Compression Field Theory12: that describing the
precracked response and that representing postcracking
tension-stiffened response.
A shortcoming of the current body of data is the lack of
theoretical models defining a plastic offset for concrete in
tension. The offsets occur when cracked surfaces come into
contact during unloading and do not realign due to shear slip
along the cracked surfaces. Test results from Yankelevsky
and Reinhardt21 and Gopalaratnam and Shah22 provide data
that can be used to formulate a plastic offset model (refer to
Fig. 7). The researchers were able to capture the softening
behavior of concrete beyond cracking in displacementcontrolled testing machines. The plastic offset strain, in the
proposed tension model, is used to define the shape of the
unloading curve, the slope and damage of the reloading path,
and the point at which cracked surfaces come into contact.
Similar to concrete in compression, the offsets in tension
seem to be dependent on the unloading strain from the backbone curve. The proposed offset model is expressed as
p
2
ε c = 146ε1c + 0.523 ε 1c
(14)
where εcp is the tensile plastic offset, and ε1c is the unloading
strain from the backbone curve. Figure 7 illustrates very
good correlation to experimental data.
Observations of test data suggest that the unloading response
of concrete subjected to tensile loading is nonlinear. The
accepted approach has been to model the unloading branch
as linear and to ignore the hysteretic behavior in the concrete
Fig. 6—Hysteresis models for concrete in tension: (a)
unloading; and (b) reloading.
620
Fig. 7—Plastic offset model for concrete in tension.
ACI Structural Journal/September-October 2003
27. due to cycles in tension. The approach used herein was to
formulate a nonlinear expression for the concrete that would
generate realistic hysteresis loops. To derive a model consistent
with the compression field approach, a Ramberg-Osgood
formulation, similar to that used for concrete in compression,
was adopted and is expressed as
fc = D + F∆ε + G∆εN
(15)
where fc is the tensile stress in the concrete; ∆ε is the strain
increment measured from the instantaneous strain on the
unloading path to the unloading strain; D, F, and G are
parameters that define the shape of the unloading curve; and
N is a power term that describes the degree of nonlinearity.
Applying the boundary conditions from Fig. 6(a) and
simplifying yields
concrete due to load cycling. Limited test data confirm that
linear reloading sufficiently captures the general response of
the concrete; however, it is evident that the reloading stiffness
accumulates damage as the unloading strain increases. The
approach suggested herein is to model the reloading behavior
as linear and to account for a degrading reloading stiffness.
The latter is assumed to be a function of the strain recovered
during the unloading phase and is illustrated in Fig. 8 against
data reported by Yankelevsky and Reinhardt.21 The reloading
stress is calculated from the following expression
f c = β t ⋅ tf max – E c4 ( ε1c – ε c )
( β t ⋅ tf max ) – tf ro
E c4 = -------------------------------------ε 1c – t ro
(16)
where
∆ε = ε 1c – ε
(17)
(22)
where
N
( E c5 – E c6 )∆ε
f c ( ∆ε ) = f 1c – E c5 ( ∆ε ) + -------------------------------------p N–1
N ( ε 1c – ε c )
(21)
fc is the tensile stress on the reloading curve and corresponds
to a strain of εc. Ec4 is the reloading stiffness, βt is a tensile
damage indicator, tf max is the unloading stress for the current
hysteresis loop, and tfro is the stress in the concrete at reloading
reversal corresponding to a strain of tro. The damage parameter
βt is calculated from the following relation
1
β t = ---------------------------------------0.25
1 + 1.15 ( ε rec )
(23)
ε rec = ε max – ε min
and
(24)
p
( E c5 – E c6 ) ( ε 1c – ε c )
N = --------------------------------------------------p
E c5 ( ε 1c – ε c ) – f 1c
(18)
f1c is the unloading stress from the backbone curve, and Ec5
is the initial unloading stiffness, assigned a value equal to the
initial tangent stiffness Ec. The unloading stiffness Ec6, which
defines the stiffness at the end of the unloading phase, was
determined from unloading data reported by Yankelevsky and
Reinhardt.21 By varying the unloading stiffness Ec6, the
following models were found to agree well with test data
E c6 = 0.071 ⋅ E c ( 0.001 ⁄ ε 1c )
ε 1c ≤ 0.001
(19)
E c6 = 0.053 ⋅ E c ( 0.001 ⁄ ε 1c )
ε 1c > 0.001
(20)
The Okamura and Maekawa2 model tends to overestimate
the unloading stresses for plain concrete, owing partly to the
fact that the formulation is independent of a tensile plastic
offset strain. The formulations are a function of the unloading
point and a residual stress at the end of the unloading phase.
The residual stress is dependent on the initial tangent stiffness
and the strain at the onset of unloading. The linear unloading
response suggested by Vecchio5 is a simple representation of
the behavior but does not capture the nonlinear nature of the
concrete and underestimates the energy dissipation. The
proposed model captures the nonlinear behavior and energy
dissipation of the concrete.
The state of the art in modeling reloading of concrete in
tension is based on a linear representation, as described by,
among others, Vecchio5 and Okamura and Maekawa.2 The
response is assumed to return to the backbone curve at the
previous unloading strain and ignores damage induced to the
ACI Structural Journal/September-October 2003
where
εrec is the strain recovered during an unloading phase. It is
the difference between the unloading strain εmax and the
minimum strain at the onset of reloading εmin, which is
limited by the plastic offset strain. Figure 8 depicts good
correlation between the proposed formulation and the
limited experimental data.
Following the philosophy for concrete in compression, βt
is calculated for the first unloading/reloading phase and retained
until the previous maximum strain is at least attained.
The literature is further deficient in the matter of partial
unloading followed by partial reloading in the tension stress
regime. Proposed herein is a partial unloading/reloading
Fig. 8—Damage model for concrete in tension.
621
28. model that directly follows the rules established for concrete
in compression. No data exist, however, to corroborate the
model. Figure 9 depicts the proposed rules for a concrete
element, lightly reinforced to allow for a post-cracking response.
Curve 1 corresponds to a full unloading response and is
identical to that assumed by Eq. (16) to (18). Reloading from
a full unloading curve is represented by Curve 2 and is computed
from Eq. (21) to (24). Curve 3 represents the case of partial
unloading from a reloading path at a strain less than the
previous maximum unloading strain. The expressions for
full unloading are used; however, the strain and stress at
unloading, now variables, replace the strain and stress at
the previous peak unloading point on the backbone curve.
Reloading from a partial unloading segment is described
by Curve 4. The response follows a linear path from the
reloading strain to the previous unloading strain. The model
explicitly assumes that damage does not accumulate for
loops that form at strains less than the previous maximum
unloading strain in the history of loading. Therefore, the
reloading stiffness of Curve 4 is larger than the reloading
stiffness for the first unloading/reloading response of
Curve 2. The partial reloading stiffness, defining Curve 4,
is calculated by the following expression
tf max – tf ro
E c4 = -----------------------ε max – t ro
(25)
and the reloading stress is then determined from
f c = tf ro + E c4 ( ε c – t ro )
(26)
As loading continues along the reloading path of Curve 4,
a change in the reloading path occurs at the intersection with
Curve 2. Beyond the intersection, the reloading response
follows the response of Curve 5 and retains the damage induced
to the concrete from the first unloading/reloading phase. The
stiffness is then calculated as
β t ⋅ f 1c – tf max
E c4 = -------------------------------ε 1c – ε max
(27)
The reloading stresses can then be calculated according to
f c = tf max + E c4 ( ε c – ε max )
(28)
The previous formulations for concrete in tension are
preliminary and require experimental data to corroborate. The
models are, however, based on realistic assumptions derived
from the models suggested for concrete in compression.
CRACK-CLOSING MODEL
In an excursion returning from the tensile domain,
compressive stresses do not remain at zero until the
cracks completely close. Compressive stresses will arise
once cracked surfaces come into contact. The recontact
strain is a function of factors such as crack-shear slip.
There exists limited data to form an accurate model for
crack closing, and the preliminary model suggested
herein is based on the formulations and assumptions
suggested by Okamura and Maekawa. 2 Figure 10 is a
schematic of the proposed model.
The recontact strain is assumed equal to the plastic offset
strain for concrete in tension. The stiffness of the concrete during
closing of cracks, after the two cracked surfaces have come into
contact and before the cracks completely close, is smaller than
that of crack-free concrete. Once the cracks completely close,
the stiffness assumes the initial tangent stiffness value. The
crack-closing stiffness Eclose is calculated from
f close
E close = ----------p
εc
(29)
fclose = –Ec(0.0016 ⋅ ε1c + 50 × 10–6)
Fig. 9—Partial unloading/reloading for concrete in tension.
(30)
where
fclose , the stress imposed on the concrete as cracked surfaces
come into contact, consists of two terms taken from the
Okamura and Maekawa2 model for concrete in tension. The
first term represents a residual stress at the completion of
unloading due to stress transferred due to bond action.
The second term represents the stress directly related to
closing of cracks. The stress on the closing-of-cracks path is
then determined from the following expression
p
Fig. 10—Crack-closing model.
622
f c = E close ( ε c – ε c )
(31)
ACI Structural Journal/September-October 2003
29. After the cracks have completely closed and loading
continues into the compression strain region, the reloading
rules for concrete in compression are applicable, with the
stress in the concrete at the reloading reversal point assuming
a value of fclose.
For reloading from the closing-of-cracks curve into the
tensile strain region, the stress in the concrete is assumed to
be linear, following the reloading path previously established
for tensile reloading of concrete.
In lieu of implementing a crack-closing model, plastic offsets in tension can be omitted, and the unloading stiffness at
the completion of unloading Ec6 can be modified to ensure
that the energy dissipation during unloading is properly
captured. Using data reported by Yankelevsky and Reinhardt,21
a formulation was derived for the unloading stiffness at zero
loads and is proposed as a function of the unloading strain on
the backbone curve as follows
E c6 = – 1.1364 ( ε 1c
– 0.991
)
(32)
Implicit in the latter model is the assumption that, in an
unloading excursion in the tensile strain region, the compressive
stresses remain zero until the cracks completely close.
REINFORCEMENT MODEL
The suggested reinforcement model is that reported by
Vecchio,5 and is illustrated in Fig. 11. The monotonic response
of the reinforcement is assumed to be trilinear. The initial
response is linear elastic, followed by a yield plateau, and ending
with a strain-hardening portion. The hysteretic response of the
reinforcement has been modeled after Seckin,17 and the Bauschinger effect is represented by a Ramberg-Osgood formulation.
The monotonic response curve is assumed to represent the
backbone curve. The unloading portion of the response
follows a linear path and is given by
fs ( ε i ) = f s – 1 + Er ( ε i – εs – 1 )
(33)
where fs(εi) is the stress at the current strain of εi , fs – 1 and εs – 1
are the stress and strain from the previous load step, and Er
is the unloading modulus and is calculated as
Er = Es
if ( ε m – ε o ) < ε y
( Em – Er ) ( εm – εo )
N = -------------------------------------------fm – Er ( εm – εo )
(38)
fm is the stress corresponding to the maximum strain recorded
during previous loading; and Em is the tangent stiffness at εm.
The same formulations apply for reinforcement in tension
or compression. For the first reverse cycle, εm is taken as
zero and fm = fy, the yield stress.
IMPLEMENTATION AND VERIFICATION
The proposed formulations for concrete subjected to
reversed cyclic loading have been implemented into a
two-dimensional nonlinear finite element program, which
was developed at the University of Toronto.23
The program is applicable to concrete membrane structures
and is based on a secant stiffness formulation using a total-load,
iterative procedure, assuming smeared rotating cracks.
The package employs the compatibility, equilibrium, and
constitutive relations of the Modified Compression Field
Theory.12 The reinforcement is typically modeled as
smeared within the element but can also be discretely
represented by truss-bar elements.
The program was initially restricted to conditions of
monotonic loading, and later developed to account for
material prestrains, thermal loads, and expansion and
confinement effects. The ability to account for material
prestrains provided the framework for the analysis capability of
reversed cyclic loading conditions. 5
For cyclic loading, the secant stiffness procedure separates
the total concrete strain into two components: an elastic
strain and a plastic offset strain. The elastic strain is used to
compute an effective secant stiffness for the concrete, and,
therefore, the plastic offset strain must be treated as a strain
offset, similar to an elastic offset as reported by Vecchio.4
The plastic offsets in the principal directions are resolved
into components relative to the reference axes. From the
prestrains, free joint displacements are determined as functions
of the element geometry. Then, plastic prestrain nodal forces
can be evaluated using the effective element stiffness matrix
due to the concrete component. The plastic offsets developed in
(34)
ε m – εo
E r = E s 1.05 – 0.05 ---------------- if ε y < ( ε m – ε o ) < 4 ε y (35)
εy
Er = 0.85Es if (εm – εo) > 4εy
(36)
where Es is the initial tangent stiffness; εm is the maximum
strain attained during previous cycles; εo is the plastic offset
strain; and εy is the yield strain.
The stresses experienced during the reloading phase are
determined from
Em – Er
N
f s ( ε i ) = E r ( ε i – ε o ) + -------------------------------------- ⋅ ( ε i – ε o )
N–1
N ⋅ ( εm – εo )
where
ACI Structural Journal/September-October 2003
(37)
Fig. 11—Hysteresis model for reinforcement, adapted from
Seckin (1981).
623
30. each of the reinforcement components are also handled in a
similar manner.
The total nodal forces for the element, arising from plastic
offsets, are calculated as the sum of the concrete and reinforcement contributions. These are added to prestrain forces arising
from elastic prestrain effects and nonlinear expansion effects.
The finite element solution then proceeds.
The proposed hysteresis rules for concrete in this procedure
require knowledge of the previous strains attained in the history
of loading, including, amongst others: the plastic offset strain,
the previous unloading strain, and the strain at reloading reversal.
In the rotating crack assumption, the principal strain directions
may be rotating presenting a complication. A simple and
effective method of tracking and defining the strains is
the construction of Mohr’s circle. Further details of the
procedure used for reversed cyclic loading can be found
from Vecchio.5
A comprehensive study, aimed at verifying the proposed
cyclic models using nonlinear finite element analyses, will
be presented in a companion paper.9 Structures considered
will include shear panels and structural walls available in the
literature, demonstrating the applicability of the proposed
formulations and the effectiveness of a secant stiffnessbased algorithm employing the smeared crack approach. The
structural walls will consist of slender walls, with heightwidth ratios greater than 2.0, which are heavily influenced by
flexural mechanisms, and squat walls where the response is
dominated by shear-related mechanisms. The former is
generally not adequate to corroborate constitutive formulations
for concrete.
CONCLUSIONS
A unified approach to constitutive modeling of reversed
cyclic loading of reinforced concrete has been presented.
The constitutive relations for concrete have been formulated
in the context of a smeared rotating crack model, consistent
with a compression field approach. The models are intended
for a secant stiffness-based algorithm but are also easily
adaptable in programs assuming either fixed cracks or fixed
principal stress directions.
The concrete cyclic models consider concrete in compression
and concrete in tension. The unloading and reloading rules
are linked to backbone curves, which are represented by the
monotonic response curves. The backbone curves are adjusted
for compressive softening and confinement in the compression
regime, and for tension stiffening and tension softening in
the tensile region.
Unloading is assumed nonlinear and is modeled using a
Ramberg-Osgood formulation, which considers boundary
conditions at the onset of unloading and at zero stress.
Unloading, in the case of full loading, terminates at the plastic
offset strain. Models for the compressive and tensile plastic
offset strains have been formulated as a function of the
maximum unloading strain in the history of loading.
Reloading is modeled as linear with a degrading reloading
stiffness. The reloading response does not return to the backbone
curve at the previous unloading strain, and further straining is
required to intersect the backbone curve. The degrading
reloading stiffness is a function of the strain recovered
during unloading and is bounded by the maximum unloading
strain and the plastic offset strain.
The models also consider the general case of partial unloading
and partial reloading in the region below the previous maximum
unloading strain.
624
NOTATION
Ec =
Eclose =
Ec1 =
Ec2 =
Ec3 =
Ec4 =
Ec5 =
Ec6 =
Em =
=
Er
=
Es
Esh =
f1c =
f2c =
=
fc
=
f ′c
fclose =
=
fcr
=
fm
fmax =
=
fp
fro =
=
fs
fs – 1 =
=
fy
tfmax =
tfro =
tro =
βd
=
βt
=
∆ε =
ε
=
ε0
=
ε1c =
ε2c =
εc
=
ε′c =
p
εc
=
εcr =
ε i , εs =
εm =
εmax =
εmin =
εp
=
εrec =
εro =
εsh =
εs – 1 =
εy
=
initial modulus of concrete
crack-closing stiffness modulus of concrete in tension
compressive reloading stiffness of concrete
initial unloading stiffness of concrete in compression
compressive unloading stiffness at zero stress in concrete
reloading stiffness modulus of concrete in tension
initial unloading stiffness modulus of concrete in tension
unloading stiffness modulus at zero stress for concrete in tension
tangent stiffness of reinforcement at previous maximum strain
unloading stiffness of reinforcement
initial modulus of reinforcement
strain-hardening modulus of reinforcement
unloading stress from backbone curve for concrete in tension
unloading stress on backbone curve for concrete in compression
normal stress of concrete
peak compressive strength of concrete cylinder
crack-closing stress for concrete in tension
cracking stress of concrete in tension
reinforcement stress corresponding to maximum strain in history
maximum compressive stress of concrete for current unloading
cycle
peak principal compressive stress of concrete
compressive stress at onset of reloading in concrete
average stress for reinforcement
stress in reinforcement from previous load step
yield stress for reinforcement
maximum tensile stress of concrete for current unloading cycle
tensile stress of concrete at onset of reloading
tensile strain of concrete at onset of reloading
damage indicator for concrete in compression
damage indicator for concrete in tension
strain increment on unloading curve in concrete
instantaneous strain in concrete
plastic offset strain of reinforcement
unloading strain on backbone curve for concrete in tension
compressive unloading strain on backbone curve of concrete
compressive strain of concrete
strain at peak compressive stress in concrete cylinder
residual (plastic offset) strain of concrete
cracking strain for concrete in tension
current stress of reinforcement
maximum strain of reinforcement from previous cycles
maximum strain for current cycle
minimum strain for current cycle
strain corresponding to maximum concrete compressive stress
strain recovered during unloading in concrete
compressive strain at onset of reloading in concrete
strain of reinforcement at which strain hardening begins
strain of reinforcement from previous load step
yield strain of reinforcement
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