2.
Federico A. Tavarez is a graduate student in the Department of Engineering Physics
at the University of WisconsinMadison. He received his BS in civil engineering from
the University of Puerto RicoMayagüez 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 WisconsinMadison. 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 threedimensional FRP composite grids were conducted to
investigate the behavior and performance of the grids when
used to reinforce beams that develop significant flexuralshear
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
loaddeflection 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 shearout 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
spantoeffective 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 FRPreinforced 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 spandepth ratios typically fail due to rupture of the
main FRP longitudinal reinforcement at large flexuralshear
cracks, even though they are overreinforced 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 flexuralshear
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 stressstate 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/MarchApril 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 threedimensional 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
spandepth ratios of 3, 4.5, and 6, respectively. These models
are referred to herein as short beam, medium beam, and long
beam, respectively. The crosssectional 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 8node solid elements with dimensions 25 x 25 x
12.5 mm (shortest dimension parallel to the width of the beam),
with onepoint 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
3.
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 twonode HughesLiu 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 crosssectional
area with a diameter of 12.7 mm. To model the bottom
longitudinal reinforcement, the fournode BelytschkoLinTsay shell element formulation was used, as shown
in Fig. 3, with two throughthethickness 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 LSDYNA was used to model the contact
ACI Structural Journal/MarchApril 2003
4.
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
LSDYNA 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 LSDYNA. 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 twonoded 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/MarchApril 2003
Fig. 5—Experimental and finite element loaddeflection
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 LSDYNA results, respectively. The
jumps in the LSDYNA 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 overpredicts the stiffness of the beam, however,
and underpredicts the ultimate deflection.
The significant drop in load seen in the loaddeflection
curves produced in LSDYNA is caused by failure in the
253
5.
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 flexuralshear damage. For
this model, the finite element analysis slightly overpredicted
both the stiffness and the ultimate load value obtained from
the experiment. On the other hand, the ultimate deflection
was underpredicted. 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 momentcurvature methods based on the
curvature of the member. Once again, the stiffness of the
beam was slightly overpredicted. However, the ultimate load
ACI Structural Journal/MarchApril 2003
6.
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 Nm, while the internal
moment was approximately 339 Nm 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 overreinforced 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/MarchApril 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 spandepth 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 underpredicts
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 overpredicted 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 spandepth
ratio. The results for tensile load in the bars reported in this table
suggest that composite grid reinforced concrete beams
with values of shear spandepth 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 postfailure 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 postfailure 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
7.
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 crosssectional area, the thickness of
the flange was increased. A maximum pure bending moment
of 2.92 kNm 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 overreinforced. As such, conventional formulas
are used to ensure that the selected crosssectional 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 flexuralshear cracks begin to occur. The
ultimate moment in the beam is assumed to be related to
this shearcritical 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 shearcritical value, it is assumed (conservatively)
that the tensile force t, which is the force in each bar at the
shearcritical 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/MarchApril 2003
8.
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 fourpoint 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. (91) of ACI
440.1R01) 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 shearcritical 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/MarchApril 2003
Fig. 12—Tensionmoment 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 underpredict
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 flexuralshear 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 flexuralshear
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
9.
beams with a low shear spandepth 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 shearcritical 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 underpredict 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
flexuralshear 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 LSDYNA. 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,” FiberReinforcedPlastic (FRP) Reinforcement for Concrete Structures: Properties and
Applications, Developments in Civil Engineering, A. Nanni, ed., Elsevier,
Amsterdam, V. 42, 1993, pp. 355385.
2. Schmeckpeper, E. R., and Goodspeed, C. H., “FiberReinforced
Plastic Grid for Reinforced Concrete Construction,” Journal of Composite
Materials, V. 28, No. 14, 1994, pp. 12881304.
3. Bank, L. C.; Frostig, Y.; and Shapira, A., “ThreeDimensional FiberReinforced Plastic Grating Cages for Concrete Beams: A Pilot Study,” ACI
Structural Journal, V. 94, No. 6, Nov.Dec. 1997, pp. 643652.
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. 715.
5. Ozel, M., and Bank, L. C., “Behavior of Concrete Beams Reinforced
with 3D Composite Grids,” CDROM Paper No. 069. Proceedings of the
16th Annual Technical Conference, American Society for Composites,
Virginia Tech, Va., Sept. 912, 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. 8289.
7. Bank, L. C., and Ozel, M., “Shear Failure of Concrete Beams Reinforced
with 3D Fiber Reinforced Plastic Grids,” Fourth International Symposium on
Fiber Reinforced Polymer Reinforcement for Reinforced Concrete Structures,
SP188, C. Dolan, S. Rizkalla, and A. Nanni, eds., American Concrete
Institute, Farmington Hills, Mich., 1999, pp. 145156.
8. ACI Committee 440, “Guide for the Design and Construction of
Concrete Reinforced with FRP Bars (ACI 440.1R01),” American Concrete
Institute, Farmington Hills, Mich., 2001, 41 pp.
9. Nakagawa, H.; Kobayashi. M.; Suenaga, T.; Ouchi, T.; Watanabe, S.;
and Satoyama, K., “ThreeDimensional Fabric Reinforcement,” FiberReinforcedPlastic (FRP) Reinforcement for Concrete Structures: Properties
and Applications, Developments in Civil Engineering, V. 42, A. Nanni, ed.,
Elsevier, Amsterdam, 1993, pp. 387404.
10. Yonezawa, T.; Ohno, S.; Kakizawa, T.; Inoue, K.; Fukata, T.; and
Okamoto, R., “A New ThreeDimensional FRP Reinforcement,” FiberReinforcedPlastic (FRP) Reinforcement for Concrete Structures: Properties
and Applications, Developments in Civil Engineering, V. 42, A. Nanni, ed.,
Elsevier, Amsterdam, 1993, pp. 405419.
11. Ozel, M., “Behavior of Concrete Beams Reinforced with 3D
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. 252258.
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, SP188, C. Dolan, S. Rizkalla, and A. Nanni, eds.,
American Concrete Institute, Farmington Hills, Mich., 1999, pp. 157167.
14. Guadagnini, M.; Pilakoutas, K.; and Waldron, P., “Investigation on
Shear Carrying Mechanisms in FRP RC Beams,” FRPRCS5, FibreReinforced Plastics for Reinforced Concrete Structures, Proceedings of the
Fifth International Conference, C. J. Burgoyne, ed., V. 2, Cambridge, July
1618, 2001, pp. 949958.
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. 847873.
19. Tavarez, F. A., “Simulation of Behavior of Composite Grid Reinforced
Concrete Beams Using Explicit Finite Element Methods,” MS thesis, University
of WisconsinMadison, 2001.
20. Hallquist, J. O., LSDYNA Keyword User’s Manual, Livermore
Software Technology Corporation, Livermore, Calif., Apr. 2000.
ACI Structural Journal/MarchApril 2003
11.
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 largescale 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 ACIASCE
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/SeptemberOctober 2003
lowerbound 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 lowerbound 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
12.
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 loaddeformation response of
structural walls dominated by shearrelated 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
stressbased 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 RambergOsgood 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/SeptemberOctober 2003
13.
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/SeptemberOctober 2003
Fig. 4—Partial unloading/reloading for concrete in compression.
619
14.
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
tensionstiffened 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/SeptemberOctober 2003
15.
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 RambergOsgood
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/SeptemberOctober 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
16.
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 postcracking 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.
CRACKCLOSING 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 crackshear 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 crackfree concrete. Once the cracks completely close,
the stiffness assumes the initial tangent stiffness value. The
crackclosing 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 closingofcracks path is
then determined from the following expression
p
Fig. 10—Crackclosing model.
622
f c = E close ( ε c – ε c )
(31)
ACI Structural Journal/SeptemberOctober 2003
17.
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 closingofcracks 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 crackclosing 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 strainhardening portion. The hysteretic response of the
reinforcement has been modeled after Seckin,17 and the Bauschinger effect is represented by a RambergOsgood 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
twodimensional 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 totalload,
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 trussbar 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/SeptemberOctober 2003
(37)
Fig. 11—Hysteresis model for reinforcement, adapted from
Seckin (1981).
623
18.
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 shearrelated 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 stiffnessbased 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
RambergOsgood 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
crackclosing 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
strainhardening 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
crackclosing 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
REFERENCES
1. Nuclear Power Engineering Corporation of Japan (NUPEC),
“Comparison Report: Seismic Shear Wall ISP, NUPEC’s Seismic Ultimate
Dynamic Response Test,” Report No. NUSSWISPD014, Organization for
Economic CoOperation and Development, Paris, France, 1996, 407 pp.
2. Okamura, H., and Maekawa, K., Nonlinear Analysis and Constitutive
Models of Reinforced Concrete, Gihodo Press, University of Tokyo, Japan,
1991, 182 pp.
3. Sittipunt, C., and Wood, S. L., “Influence of Web Reinforcement on
the Cyclic Response of Structural Walls,” ACI Structural Journal, V. 92,
No. 6, Nov.Dec. 1995, pp. 745756.
4. Vecchio, F. J., “Finite Element Modeling of Concrete Expansion and
Confinement,” Journal of Structural Engineering, ASCE, V. 118, No. 9,
1992, pp. 23902406.
5. Vecchio, F. J., “Towards Cyclic Load Modeling of Reinforced Concrete,”
ACI Structural Journal, V. 96, No. 2, Mar.Apr. 1999, pp. 132202.
6. Mander, J. B.; Priestley, M. J. N.; and Park, R., “Theoretical StressStrain Model for Confined Concrete,” Journal of Structural Engineering,
ASCE, V. 114, No. 8, 1988, pp. 18041826.
7. Mansour, M.; Lee, J. Y.; and Hsu, T. T. C., “Cyclic StressStrain
Curves of Concrete and Steel Bars in Membrane Elements,” Journal of
Structural Engineering, ASCE, V. 127, No. 12, 2001, pp. 14021411.
8. Palermo, D., and Vecchio, F. J., “Behaviour and Analysis of Reinforced
Concrete Walls Subjected to Reversed Cyclic Loading,” Publication No.
200201, Department of Civil Engineering, University of Toronto, Canada,
2002, 351 pp.
ACI Structural Journal/SeptemberOctober 2003
19.
9. Palermo, D., and Vecchio, F. J., “Compression Field Modeling of
Reinforced Concrete Subjected to Reversed Loading: Verification,” ACI
Structural Journal. (accepted for publication)
10. Hognestad, E.; Hansen, N. W.; and McHenry, D., “Concrete Stress
Distribution in Ultimate Strength Design,” ACI JOURNAL, Proceedings V. 52,
No. 12, Dec. 1955, pp. 455479.
11. Popovics, S., “A Numerical Approach to the Complete StressStrain
Curve of Concrete,” Cement and Concrete Research, V. 3, No. 5, 1973, pp.
583599.
12. Vecchio, F. J., and Collins, M. P., “The Modified CompressionField
Theory for Reinforced Concrete Elements Subjected to Shear,” ACI
JOURNAL, Proceedings V. 83, No. 2, Mar.Apr. 1986, pp. 219231.
13. Karsan, I. K., and Jirsa, J. O., “Behaviour of Concrete Under
Compressive Loadings,” Journal of the Structural Division, ASCE, V. 95,
No. 12, 1969, pp. 25432563.
14. Bahn, B. Y., and Hsu, C. T., “StressStrain Behaviour of Concrete
Under Cyclic Loading,” ACI Materials Journal, V. 95, No. 2, Mar.Apr.
1998, pp. 178193.
15. Buyukozturk, O., and Tseng, T. M., “Concrete in Biaxial Cyclic
Compression,” Journal of Structural Engineering, ASCE, V. 110, No. 3,
Mar. 1984, pp. 461476.
16. Pilakoutas, K., and Elnashai, A., “Cyclic Behavior of RC Cantilever
Walls, Part I: Experimental Results,” ACI Structural Journal, V. 92, No. 3,
MayJune 1995, pp. 271281.
ACI Structural Journal/SeptemberOctober 2003
17. Seckin, M., “Hysteretic Behaviour of CastinPlace Exterior BeamColumn SubAssemblies,” PhD thesis, University of Toronto, Toronto,
Canada, 1981, 266 pp.
18. Yankelevsky, D. Z., and Reinhardt, H. W., “Model for Cyclic
Compressive Behaviour of Concrete,” Journal of Structural Engineering,
ASCE, V. 113, No. 2, Feb. 1987, pp. 228240.
19. Stevens, N. J.; Uzumeri, S. M.; and Collins, M. P., “Analytical Modelling
of Reinforced Concrete Subjected to Monotonic and Reversed Loadings,”
Publication No. 871, Department of Civil Engineering, University of
Toronto, Toronto, Canada, 1987, 201 pp.
20. Hordijk, D. A., “Local Approach to Fatigue of Concrete,” Delft
University of Technology, The Netherlands, 1991, pp. 210.
21. Yankelevsky, D. Z., and Reinhardt, H. W., “Uniaxial Behaviour of
Concrete in Cyclic Tension,” Journal of Structural Engineering, ASCE,
V. 115, No. 1, 1989, pp. 166182.
22. Gopalaratnam, V. S., and Shah, S. P., “Softening Response of Plain
Concrete in Direct Tension,” ACI JOURNAL, Proceedings V. 82, No. 3, MayJune 1985, pp. 310323.
23. Vecchio, F. J., “Nonlinear Finite Element Analysis of Reinforced
Concrete Membranes,” ACI Structural Journal, V. 86, No. 1, Jan.Feb.
1989, pp. 2635.
625
21.
ACI member Alexandros G. Tsonos is a Professor of reinforced concrete structures,
Department of Structural Engineering, the Aristotle University of Thessaloniki,
Thessaloniki, Greece. He received his PhD from the Aristotle University of Thessaloniki
in 1990. His research interests include the inelastic behavior of reinforced concrete
structures, structural design, fiberreinforced concrete, seismic repair and rehabilitation
of reinforced concrete structures, and the seismic repair and restoration of monuments.
Approximately 10 electricalresistance strain gauges were
bonded on the reinforcing bars of each subassemblage of
the program.
EXPERIMENTAL SETUP AND
LOADING SEQUENCE
The general arrangement of the experimental setup is
shown in Fig. 2(a). All subassemblages were subjected to
11 cycles applied by slowly displacing the beam’s free end
according to the load history shown in Fig. 2(b) without
reaching the actuator stroke limit. The amplitudes of the
peaks in the displacement history were 15, 20, 25, 30, 35, 40,
45, 50, 55, 60, and 65 mm. One loading cycle was performed
at each displacement amplitude. An axial load equal to 200 kN
was applied to the columns of the subassemblages and kept
constant throughout the test. The experimental loading
sequence used is a typical one, commonly used in previous
studies.1,11,13 It was not the objective of this study to investigate
the effect of other, nonstandard loading histories on the
response of the subassemblages.
As previously mentioned, all the subassemblages were loaded
slowly. The strain rate of the load applied corresponded to
static conditions. In the case of seismic loading, the strain
·
rate ε is higher than the rate corresponding to static conditions.
·
Soroushian and Sim14 showed that an increase in ε with
respect to static conditions leads to a moderate increase in
the strength of concrete
2
·
·
f c, dyn = [ 1.48 + 0.160 × log ε + 0.0127 ( log ε ) ] × f c, stat (1)
Scott et al.15 tested column subassemblages with various
amounts of hoop reinforcement under strain rates ranging
from 0.33 × 10–5 sec–1 (static loading), to 0.0167 sec–1
(seismic loading). Their test results conformed with the
results obtained from Eq. (1).
Using the aforementioned expression, it is estimated that for
·
a strain rate of ε = 0.0167 sec–1, concrete strengths increase
by approximately 20% (compared with the static one). An
expression similar to Eq. (1) can be found in the CEB code.16
Thus, the strengths exhibited by Subassemblages A1, E1,
E2, and G1 during the tests are somewhat lower than the
strengths they would exhibit if subjected to load histories
similar to actual seismic events.
EXPERIMENTAL RESULTS
Failure mode of Subassemblages A1, E1, E2, and G1
The failure mode of Subassemblages A1 and E2, as
expected, involved the formation of a plastic hinge in the
beam at the column face. The formation of plastic hinges
caused severe cracking of the concrete near the fixed beam end
of each subassemblage (Fig. 3). The behavior of Subassemblages A1 and E2 was as expected and as documented in the
seismic design philosophy of the modern codes as will be
explain in the following.47
Fig. 1—Dimensions and crosssectional details of: (a) Subassemblages A1 and E2; and (b) Subassemblages E1 and G1. (Note:
dimensions are in cm. 1 cm = 0.0394 in.)
ACI Structural Journal/JulyAugust 2007
469
22.
Fig. 2—(a) General arrangement of experimental setup and
photograph of test setup (dimensions are in m; 1 m = 3.28 ft);
and (b) lateral displacement history. (Note: 1 mm = 0.039 in.)
Fig. 4—Applied shear versus strain in beamcolumn joint
hoop reinforcement of: (a) Subassemblages A1 and E2;
and (b) Subassemblages E1 and G1. (Note: 1 kN = 0.225 kip.)
Fig. 3—Views of collapsed Subassemblages A1, E1, E2, and G1.
Significant inelastic deformations occurred in the beams’
longitudinal reinforcement in both Subassemblages A1 and E2
(strains of over 40.000με were obtained in the beams’
longitudinal bars), while joint shear reinforcement remained
elastic. Figure 4(a) shows strain gauge data of joint hoop
reinforcement for both Subassemblages A1 and E2. As is
clearly shown in Fig. 4(a), the maximum strain recorded in
470
the joint hoop reinforcement for both subassemblages was
below the yield strain of 2.500με, which was in agreement with
the observed failure modes of Subassemblages A1 and E2.17
One difference between the failure modes of Subassemblages
A1 and E2 was that hairline cracks appeared in the joint region
of E2, and partial loss of the concrete cover in the rear face of
the joint of E2 took place during the three last cycles of loading
(ninth, tenth, and eleventh) when drift Angle R ratios
exceeded 4.5 while the joint region of Subassemblage A1
remained intact at the conclusion of the test (refer to Fig. 3).
The connections of both Subassemblages E1 and G1,
contrary to expectations, exhibited shear failure during the
ACI Structural Journal/JulyAugust 2007
23.
Fig. 5—Maximum strain during each cycle of loading in
beam longitudinal reinforcement of Subassemblages A1, E1,
E2, and G1.
Fig. 6—Gradual cracking configuration of Subassemblage E1
during test.
early stages of cyclic loading. Damage occurred both in the
joint area and in the columns’ critical regions. Figure 4(b)
shows strain gauge data for the joint hoop reinforcement for
Subassemblages E1 and G1. As shown in Fig. 4(b), the
maximum strain recorded in the joint hoop reinforcement of
both Subassemblages E1 and G1 was significantly higher
than the yield strain 2.500 με. Joint shear damage has been
shown to occur after yielding of the joint hoop reinforcement,
which is in agreement with the damage observed in the joints
of these subassemblages.18 The maximum strain recorded in
the longitudinal bars of the beams of both Subassemblages
E1 and G1 was below 2.500με (refer to Fig. 5). In Fig. 6, the
progression of cracking of Subassemblage E1 during the test
is demonstrated.
Loaddrift angle curves
Plots of applied shear force versus drift angles for all the
Subassemblages (A1, E1, E2, and G1) are shown in Fig. 7.
The beam calculated flexural capacities of the subassemblages
are shown as dashed lines in Fig. 7.
ACI Structural Journal/JulyAugust 2007
Fig. 7—Hysteresis loops of Subassemblages A1, E1, E2, and
G1. (Note: 1 kN = 0.225 kip.)
A major concern in the seismic design of RC structures is
the ability of members to develop their flexural strength before
failing in shear. This is especially true for members framing
at a beam column joint (beams and columns), where it is
important to develop their flexural strengths before joint
shear failure. Moreover, by designing the flexural strengths
of columns in RC frame structures to meet the strongcolumn
weakbeam rule, all members against premature shear failure,
and by detailing plastic hinge (critical) regions for ductility,
RC frame structures have been shown to exhibit a controlled
and very ductile inelastic response.2,4,9
471
24.
As can be seen in Fig. 7, the beam of Subassemblage A1
developed maximum shear forces higher than those corresponding to its ultimate flexural strength until the sixth cycle
of loading. This is an indication of the flexural response of
this beam because it developed its flexural strength until a
drift Angle R ratio of 4.0 was reached and exceeded. Also, a
flexural failure was observed for this beam, caused by
crushing of the concrete cover of the longitudinal reinforcement,
and subsequent inelastic buckling of the longitudinal bars.
The beam of Subassemblage E2 also developed maximum
shear forces higher than those corresponding to its ultimate
flexural strength until the eleventh upper half cycle of loading
and until the seventh lower half cycle of loading. In particular,
during the final cycles of loading beyond drift Angle R ratios
of 4.5 when large displacements were imposed, crushing of
the concrete cover of the reinforcement took place and the
beam’s hoops could not provide adequate support to the longitudinal reinforcement. As a result, buckling of the beam longitudinal reinforcement in Subassemblages A1 and E2 occurred
after the sixth and seventh cycles of loading, respectively.
The beam of Subassemblage E1 developed maximum
shear forces very close to those corresponding to its ultimate
flexural strength only during the second and third cycle of
loading. For the remaining cycles (four through 11), the
premature joint shear failure did not allow the beam in this
subassemblage to develop its flexural capacity (Fig. 6 and 7).
The premature joint shear failure of Subassemblage G1
also did not allow the beam in this subassemblage to develop
its flexural capacity. As can be seen in Fig. 7, the beam of
Subassemblage G1 developed maximum shear forces
significantly lower than those corresponding to its ultimate
flexural strength.
One of the basic provisions of all modern structural codes
is to provide the structures with sufficient strength and sufficient
ductility to undergo postelastic deformations without losing
a large percentage of their strength.2,4,7,9 As can be seen in
Fig. 7, this criterion is fulfilled for Subassemblies A1 and E2.
By contrast, it is not fulfilled for Subassemblies E1 and G1
because they exhibited significant loss of strength during
cyclic loading.
The beamcolumn Subassemblages A1, E1, E2, and G1 are
similar to real modern frame structures. If the sequence in the
breakdown of the chain of resistance of these real frame
structures follows the desirable hierarchy during a catastrophic
earthquake, the formation of plastic hinges in the beams of
these structures would be expected, because the use of a
weak girderstrong column design philosophy is adopted by
the modern codes.2,4,5,7,9 The aforementioned desirable
failure mode (with formation of a plastic hinge in the beam)
was developed by Subassemblages A1 and E2. Thus, the
magnitude of loads resisted by Subassemblages A1 and E2
are consistent with the expected values from actual events.
Story drifts allowed by modern codes are on the order of 2%
of the story height.4,7,8 While it was reassuring that story
drifts of as much as 4% of the story height were achieved in
most reported tests referring to the seismic response of beamcolumn specimens, it should be remembered that drifts in
excess of 2% are not likely to be readily accommodated in
high rise frames. This is due to significant and detrimental
influence of PΔ phenomena on both lateral load resistance
and dynamic response.19
Subassemblages A1 and E2, which developed plastic
hinges in their beams (Fig. 3 and 7), showed stable hysteretic
behavior up to drift Angle R ratios of 4.0. They showed a
472
Table 1—Comparison of joint of Subassemblage A1
design parameters with ACI 318055 and ACI 352R026
Subassemblage
A1
γ
ldh, cm
Ash, cm
2
hbeam/
column bar
sh, cm diameter
5.0
0.67 < 17 (15.65)* 0.95
(0.66)*† (5.0)*†
(1.0)*†
(17)†
MR
30
1.72
(23.80)† (1.20)*†
*
Numbers inside parentheses are required values of ACI 31805.5
†Numbers inside parentheses are required values of ACI 352R02.6
Note: γ is shear strength factor reflecting confinement of joint by lateral members, ldh
is development length of hooked beam bars, Ash is total crosssectional area of transverse
steel in joint, and sh is spacing of transverse reinforcement in joint. Numbers outside
parentheses are provided values. 1 cm = 0.394 in.
Table 2—Comparison of joints of Subassemblages
E1 and E2 design parameters with Eurocode 84 and
Eurocode 23
Ash,
Subassemblage Vjh, kN cm2
Asv ,
cm2
dbl , mm
MR
lb,net,
cm
sw ,
cm
E1
14
2.60
126 < 6.85 3.08
45 (43)† 5 (5)*
(168)* (2.85)* (1.06)* (9.15)* (1.20)*
E2
75.6 < 6.85 3.08
14
3.30
45 (32)† 5 (5)*
(222)* (2.85)* (1.06)* (11.20)* (1.20)*
*Numbers inside parentheses are required values of Eurocode
8.4
Numbers inside parentheses are required values of Eurocode 2.3
Note: Vjh is horizontal joint shear force, Ash is total crosssectional area of transverse
steel of joint, Asv is vertical joint shear reinforcement, dbl is diameter of hooked beam
bars (in both E1 and E2 setup recommended by EC8 and shown in Fig. 5 was applied), lb,net
is development length of hooked beam bars, and sw is spacing of transverse reinforcement
of joint. Numbers outside parentheses are provided values. 1 m = 0.394in.; 1 mm =
0.039 in.; 1 kN = 0.225 kip.
†
considerable loss of strength, stiffness, and unstable hysteretic
behavior, but beyond drift Angle R ratios of 4.5 (Fig. 7).
Subassemblages E1 and G1, which exhibited premature joint
shear failure (refer to Fig. 3 and 7) showed a considerable loss of
strength, stiffness, and unstable degrading hysteresis beyond
drift Angle R ratios of 2.5 and 2.0%, respectively (Fig. 7).
CODE REQUIREMENTS
Despite the fact that all the subassemblages were designed
according to their corresponding modern codes, two developed
failure modes dominated by joint shear failure (Fig. 3). For
this reason, it is discussed how requirements of these codes
used for the design of the joints of Subassemblages A1,5,6 E1,
E23,4 (for DC”M” structures), and G17,8 were satisfied.
Table 1 clearly indicates that the joint of A1 satisfied the
design requirements of ACI 318055 and ACI 352R026 for
exterior beamcolumn joints for seismic loading.
Table 2 indicates that the joints of both E1 and E2 satisfied the
design provisions for exterior beamcolumn joints of
Eurocode 23 and Eurocode 84 for DC”M” structures.
In both subassemblages, two 8 mm diameter short bars
were placed and were tightly connected on the top bends of
the beam reinforcing bars and two on the bottom, running in
the transverse direction of the joint, as shown in Fig. 5. This
is the setup recommended by Eurocode 8 when the requirement
of limitation of beam bar diameter (dbl) to ensure appropriate
anchorage through the joint is not satisfied (refer to Table 2).
It was considered worthwhile, however, to determine the
beam bar pullout. Strain gauge measurements were used to
determine beam bar pullout. If the maximum strains in a
beam’s longitudinal bar during each two consecutive cycles
of loading remained the same or decreased, as long as buckling
ACI Structural Journal/JulyAugust 2007
25.
Fig. 8—(a) Exterior beamcolumn joint; (b) internal forces around exterior beamcolumn
joint as result of seismic actions;10,12 (c) two mechanisms of shear transfer (diagonal concrete
strut and truss mechanism);10,12,19 and (d) forces acting in joint core concrete through
Section II from two mechanisms.27,28
Table 3—Comparison of joint of Subassemblage G1
design parameters with ERC19957 and CDCS19958
Subassemblage
G1
Ash, cm2
2.01 (2.01)
lb,net, cm
*
45 (43)
*
MR
2.60 (1.40)†
*Numbers inside parentheses are required values of CDCS1995.8
†
Numbers inside parentheses are required values of ERC1995.7
Note: Ash is total crosssectional area of transverse steel of joint and lb,net is
development length of hooked beam bars. Numbers outside parentheses are provided
values. 1 cm = 0.394 in.
of this bar had not taken place, it was concluded that a pullout of this bar had occurred.13,18 As shown in Fig. 5, the
beam’s longitudinal reinforcement in Subassemblages E1
and E2 maintained adequate anchorage throughout the tests
due to the short bars placed and tightly connected under the
bends of a group of reinforcing bars (refer to Fig. 5).
Table 3 also clearly indicates that the joint of G1 satisfied
the design provisions for exterior beamcolumn joints of
both the new Greek codes.7,8
The codes prescribe minimum MR values. So, as can be
seen from Tables 1 through 3, the minimum value for the MR
ratio according to ACI 31805 and ACI 352R02, as well as
according to Eurocode 8 (DC”M”), is 1.20.46 The minimum
value for the MR ratio according to the new Greek Earthquake
Resistant Code is 1.40.7 Thus, a good target MR for most
structures is between 1.20 and 1.40.
Neither the New Greek Code for the Design of RC
Structures8 nor the new Greek Earthquake Resistant Code7
require limitations for the joint shear stress. Of course both of
these codes need to add requirements to limit joint shear stress.
ACI Structural Journal/JulyAugust 2007
THEORETICAL CONSIDERATIONS
A new formulation published in recent studies2026 predicts
the beamcolumn joint ultimate shear strength and was used in
the present study to predict the failure modes of Subassemblages
A1, E1, E2, and G1. A summary of this formulation is presented.
Figure 8(a) shows an RC exterior beamcolumn joint for a
moment resisting frame and Fig. 8(b) shows the internal
forces around this joint.10,12 The shear forces acting in the
joint core are resisted partly by a diagonal compression strut
that acts between diagonally opposite corners of the joint
core (refer to Fig. 8(c)) and partly by a truss mechanism
formed by horizontal and vertical reinforcement and concrete
compression struts.10,12,19 The horizontal and vertical
reinforcement is normally provided by horizontal hoops in
the joint core around the longitudinal column bars and by
longitudinal column bars between the corner bars in the side
faces of the column.10,12,27 Both mechanisms depend on the
core concrete strength. Thus, the ultimate concrete strength
of the joint core under compression/tension controls the
ultimate strength of the connection. After failure of the
concrete, strength in the joint is limited by gradual crushing
along the crossdiagonal cracks and especially along the
potential failure planes (Fig. 8(a)).
For instance, consider Section II in the middle of the joint
height (Fig. 8(a)). In this section, the flexural moment is
almost zero. The forces acting in the concrete are shown in
Fig. 8(d).27,28 Each force acting in the joint core is analyzed
into two components along the x and y axes (Fig. 8(d)). The
values of Ti are the tension forces acting on the longitudinal
column bars between the corner bars in the side faces of the
column. Their resultant is ΣTi. An equal and opposing
473
26.
compression force (–ΣTi) must act in the joint core to balance
the vertical tensile forces generated in the reinforcement.
This compression force was generated by the resultant of the
vertical components of the truss mechanism’s diagonal
compression forces D1, D2 …Dv.27 Thus, D1y + D2y + … +
Dvy = ΣTi = T1 + T2 + T3 + T4.27 The column axial load is
resisted by the compression strut mechanism.12 The summation
of vertical forces equals the vertical joint shear force Vjv
D cy + ( T 1 + ... + T 4 ) = D cy + D sy = V jv
↓
↓
(2a)
compression strut truss model
fc = K × fc ′
Also, f ′c is the concrete compressive strength and K is a
parameter of the model15 expressed as
ρ s × f yh
K = 1 + fc ′
D cy + D sy
V jv
σ =  = hc ′ × bc ′
hc ′ × bc ′
(3)
V jh
τ = hc ′ × bc ′
(4)
where h ′c and b ′c are the length and the width of the joint
core, respectively.
It is now necessary to establish a relationship between the
average normal compressive stress σ and the average shear
stress τ. From Eq. (3) and (4)
V jv
σ =  × τ
V jh
(5)
It has been shown that
V jv
hb
 =  = α
V jh
hc
(6)
where α is the joint aspect ratio.4,10,12
The principle (σI = maximum, σII = minimum) stresses are
calculated
2
σ σ
4τ
 σ I, II =  ±  1 + 2
2 2
σ
αγ
4
 ⎛ 1 + 1 + ⎞
2⎠
⎝
2 fc
α
(2b)
The vertical normal compressive stress σ and the shear
stress τ uniformly distributed over Section II are given by
Eq. (3) and (4)
(7)
Equation (8)29 was adopted for the representation of the
concrete biaxial strength curve30 by a fifthdegree parabola
(9b)
where ρs is the volume ratio of transverse reinforcement and
fyh is its yield strength.
Substituting Eq. (5) through (7) into Eq. (8) and using τ =
γ f c gives the following expression
The summation of horizontal forces equals the horizontal
joint shear force Vjh
D cx + ( D 1x + D 2x … + D vx ) = D cx + D sx = V jh
(9a)
5
5αγ
4
+  ⎛ 1 +  – 1⎞ = 1
2
⎝
⎠
fc
α
(10)
Assume herein that
αγ
x = 2 fc
(11)
4
αγ ψ =  1 + 2
2 fc
α
(12)
and
Then Eq. (10) is transformed into
5
( x + ψ ) + 10ψ – 10x = 1
(13)
The solution of the system of Eq. (11) to (13) gives the
beamcolumn joint ultimate strength τult = γult f c (MPa).
This system is solved each time for a given value of the joint
aspect ratio using standard mathematical analysis. The joint
ultimate strength τult depends on the increased joint concrete
compressive strength due to confining fc and on the joint
aspect ratio α. Thus, typical values of τult for comparison with
the values of ACI 31805,5 ACI 352R02,6 and Eurocode 84
are not possible to derive. A particular value, however, for
each joint would be calculated as in the following example.
Example for Subassemblage A1
The value α = 1.5 and the solution of the system of Eq. (11)
to (13) gives x = 0.1485 and y = 0.248; f ′c(A1) = 35 MPa,
K(A1) = 1.558 according to the Scott et al.15 model and
fc(A1) = K(A1) × f ′c(A1) = 54.53 MPa.
Equation (11) gives
2 ( 0.1458 ) 54.53
γ ult ( A1 ) =  = 1.46
1.5
(8)
and finally τult(A1) = 1.46 54.53 MPa = 10.78 MPa (refer to
Table 4).
where fc is the increased joint concrete compressive strength
due to confinement by joint hoop reinforcement, which is
given by the model of Scott et al.15 according to the equation
COMPARISON OF PREDICTIONS AND
EXPERIMENTAL RESULTS
The proposed shear strength formulation can be used to
predict the failure mode of the subassemblages and thus the
σ
σ II
– 10 I + fc
fc
474
5
= 1
ACI Structural Journal/JulyAugust 2007
27.
Table 4—Joint ultimate strength and ratios τpred /τexp
and γcal /γult for Subassemblages A1, E1, E2, and G1
According to Park and
Paulay10
According to proposed shear
strength formulation
Subassemblage τult , MPa τpred /τexp γcal /γult τult, MPa τpred /τexp γcal /γult
A1
6.05
1.19
1.0
10.78
1.17
0.47
E1
8.94
1.31
1.0
6.92
1.19
1.08
E2
5.96
1.24
1.0
10.78
1.20
0.46
G1
8.34
1.28
1.0
6.60
1.19
1.04
Note: 1 MPa = 144.93 psi.
actual values of connection shear stress. Therefore, when the
calculated joint shear stress τcal is greater or equal to the joint
ultimate strength (τcal = γcal f c ≥ τult = γult f c ), then the
predicted actual value of connection shear stress will be near
τult(τult = γult f c ). This is because the connection fails
earlier than the adjacent beam(s). When the calculated joint
shear stress τcal is lower than the connection ultimate
strength (τcal = γcal f c < τult = γult f c ), then the predicted
actual value of connection shear stress will be near τcal
because the connection permits its adjacent beam(s) to yield.
τult = γult f c is calculated from the solution of the system of
Eq. (11) to (13). The value of τcal is calculated from the
horizontal joint shear force assuming that the top reinforcement
of the beam yields (Fig. 8(a)). In this case, the horizontal
joint shear force is expressed as
V jhcal = 1.25A s1 × f y – V col
(14)
where As1 is the top longitudinal beam reinforcement (Fig. 8(a)),
fy is the yield stress of this reinforcement, and Vcol is the
column shear force (Fig. 8(a)). For Type 2 joints, the design
forces in the beam according to ACI 352R026 should be
determined using a stress value of α × fy for beam longitudinal
reinforcement, where α = 1.25.
The improved retention of strength in the beamcolumn
subassemblages, as the values of the ratio τcal/τult = γcal/γult
decrease was also demonstrated. For τcal/τult = γcal/γult ≤ 0.50,
the beamcolumn joints of the subassemblages performed
excellently during the tests and remained intact at the
conclusion of the tests.2026
The validity of the formulation was checked using test
data from more than 120 exterior and interior beamcolumn
subassemblages that were tested in the Structural Engineering
Laboratory at the Aristotle University of Thessaloniki,2026 as
well as data from similar experiments carried out in the U.S.,
Japan, and New Zealand.1,12,13,3136 A part of this verification
is presented in Table 5 where the comparison is shown
between experimental and predicted results by the preceding
methodology for 39 exterior and interior beamcolumn joint
subassemblages from the literature. A very good correlation
is observed (Table 5). In Table 5, the limiting values of joint
shear stress according to ACI 318055 and ACI 352R026
(1.0 f c ′ MPa for exterior beamcolumn joints and 1.25 f c ′
MPa for interior beamcolumn joints) are included for each
reference subassemblage. In Table 5, the limiting values of
joint shear stress according to Eurocode 84 (15τR MPa for
exterior beamcolumn joints and 20τR MPa for interior
beamcolumn joints) are also included.
The shear capacities of the connections of Subassemblages
A1, E1, E2, and G1 were also computed using the aforementioned methodology. One of the motivations behind this
ACI Structural Journal/JulyAugust 2007
study was the verification of the shear strength formulation
presented herein for beamcolumn joints designed according
to modern codes.
The horizontal joint shear stresses are mainly produced
by the longitudinal beam reinforcement as clearly
described by Eq. (14). The longitudinal beam reinforcement of
Subassemblages A1 and E2 was purposely chosen to
produce low joint shear stresses during the tests, that is, a ratio
τcal/τult = γcal/γult less than 0.5.
Table 6 shows that γcal/γult is equal to 0.47 in Subassemblage
A1 (that is, lower than 0.5) and γcal/γult is equal to 0.46 in
Subassemblage E2 (that is, lower than 0.5). Thus, the formation
of a plastic hinge in the beams near the columns is expected
without any serious damage in the joint regions and, as a
result, there will be satisfactory performance for both
Subassemblages A1 and E2. As predicted, both subassemblages
failed in flexure, exhibiting remarkable seismic performance
(Fig. 3 and 7). Values τpred of A1 and E2, which are shown
in Table 6, are equal to their τcal values (because γcal < γult)
and are significantly different from their τult values, which
are shown in Table 4.
The percentage of longitudinal beam reinforcement of
Subassemblages E1 and G1 was purposely chosen to be
higher than that of Subassemblages A1 and E2 to produce
higher joint shear stresses than those corresponding to their
ultimate capacities. The joint region of E1, however, satisfied
all the design requirements of Eurocode 23 and Eurocode 84
and the joint regions of G1 satisfied all the design requirements
of the two Greek codes.7,8
Table 6 also shows that for both Subassemblages E1 and
G1, the calculated joint shear stress τcal = γcal f c when the
beams reach their ultimate strength is higher than the joint
ultimate capacity τult = γult f c . Therefore, the joints of both
these subassemblages will fail earlier than their beams
according to the aforementioned methodology, because the
joints of both E1 and G1 reach their ultimate shear strength
during the tests before the beams reach their ultimate strength.
Thus, according to the aforementioned methodology, a joint
shear failure is expected for both Subassemblages E1 and G1
without any serious damage in their beams and, as a result, the
performance of both subassemblages will not be satisfactory.
As expected, both Subassemblages E1 and G1 demonstrated
premature joint shear failure starting from the early stages of
seismic loading and damage concentrated mostly in this
region (Fig. 3). As also predicted, both Subassemblages E1
and G1 exhibited poor seismic performance, which was
characterized by significant loss of strength, stiffness, and
energy dissipation capacity during the tests. Furthermore, the
volume ratios of joint transverse reinforcement for Subassemblages E1 and G1 were 0.025 and 0.017, respectively. Thus,
the joint of Subassemblage E1 was more confined than the
joint of Subassemblage G1, which explains why the hysteretic
response of the former was better than that of the latter (Fig. 7).
The concrete compressive strength significantly increases
the joint ultimate strength τult. Thus, if the Subassemblages
E1 and G1 had higher values with concrete compressive
strengths, they would have behaved as well as Subassemblages
A1 and E2. This would have happened for values with
concrete compressive strength of approximately 50 MPa,
which would have resulted in values of ratio γcal/γult lower
than 0.5. The value of concrete 28day compressive
strengths of 22 MPa for both Subassemblages E1 and G1,
however, is acceptable for Eurocode 2,3 Eurocode 8,4 and for
both Greek codes.7,8
475
28.
Table 5—Experimental verifications
Joint
Concrete
Type of
Longitudinal Joint
aspect compressive
subassem ratio
strength f ′ , τACI, τEC8, beam bar fy , hoop fy ,
Subc
*
α = hb/hc
MPa
MPa MPa
MPa
MPa γcal γexp
Reference assemblage blage
γult
Observed
Predicted shear
strength τpred, shear strength μ = τpred /
τexp
τexp, MPa
MPa
No. 1
1.00
31.10
5.58 7.80
391
250
0.78 0.88 0.92
4.46
5.03
0.89
No. 2
E
1.00
41.70
6.46 9.45
391
250
0.68 0.74 1.06
4.50
4.90
0.92
No. 3
E
1.00
41.70
6.46 9.45
391
250
0.68 0.67 1.06
4.50
4.43
1.01
No. 4
E
1.00
44.70
6.69 9.90
391
281
0.66 0.67 1.08
4.43
4.50
0.99
No. 5
E
1.00
36.70
6.06 8.63
391
281
0.74 0.69 0.99
4.50
4.20
1.07
No. 6
E
1.00
40.40
6.35 9.30
391
281
0.70 0.69 1.03
4.47
4.41
1.01
No. 7
34
E
E
1.00
32.20
5.67 7.95
391
250
0.77 0.82 0.93
4.47
4.76
0.94
No. 8
E
1.00
41.20
6.42 9.40
391
250
0.68 0.72 1.06
4.47
4.74
0.94
No. 9
E
1.00
40.60
6.37 9.30
391
250
0.69 0.67 1.05
4.51
4.40
1.03
No. 10
E
1.00
44.40
6.65 9.83
391
281
0.67 0.69 1.08
4.49
4.62
0.97
No. 11
E
1.00
41.90
6.47 9.48
391
281
0.69 0.70 1.05
4.49
4.55
0.99
No. 12
E
1.00
35.10
5.92 8.34
391
281
0.75 0.74 0.96
4.47
4.40
1.01
No. 13
E
1.00
46.40
6.81 10.16
391
250
0.64 0.64 1.12
4.47
4.47
1.00
No. 14
E
1.00
41.00
6.40 9.36
391
281
0.70 0.69 1.03
4.50
4.44
1.01
No. 15
E
1.00
30.70
5.54 7.74
391
281
0.71 0.74 1.02
3.95
4.12
0.96
No. 16
E
1.00
37.40
6.11 8.76
391
250
0.72 0.76 1.01
4.51
4.76
0.95
A1
I
1.14
40,20
7.93 12.33
1070
291
4.62 1.34 1.11
7.21
8.70
0.83
A2
33
I
1.14
40.20
7.93 12.33
409
291
1.76 1.23 1.11
7.21
7.99
0.90
A3
I
1.14
40.20
7.93 12.33
1070
291
4.62 1.34 1.11
7.21
8.70
0.83
A4
I
1.14
40.20
7.93 12.33
1070
291
4.48 1.33 1.14
7.62
8.88
0.86
1.14
30.00
5.48 7.65
1070
291
2.68 0.93 0.96
5.39
5.22
1.03
E
1.14
30.00
5.48 7.65
409
291
1.02 0,83 0.96
5.39
4.66
1.16
B3
E
1.14
30.00
5.48 7.65
1070
291
2.68 1.03 0.96
5.39
5.78
0.93
B4
E
1.14
30.00
5.48 7.65
1070
291
2.60 1.05 0.99
5.71
6.06
0.94
UNIT1
I
1.126
41.30
8.03 12.54
315
320
1.20 1.13 1.26
8.96
8.44
1.06
UNIT2
I
1.126
46.90
8.56 13.65
307
320
1.31 1.08 1.33
10.23
8.43
1.20
UNIT3
E†
1.126
38.20
6.18 8.70
473
321
1.17 0.90 1.09
7.06
5.85
1.21
UNIT4
E
1.126
38.90
6.23 8.55
473
321
2.32 0.90 1.11
7.29
5.91
1.23
SHC1
36
E
B2
12
B1
I
1.14
56.50
9.39 15.9
413
551
1.00 0.91 1.31
7.81
7.11
1.10
SHC2
I
1.14
59.50
9.64 16.5
413
551
0.97 0.91 1.36
7.90
7.41
1.07
SOC3
I
1.14
47.10
8.58 13.71
413
551
1.06 1.00 1.22
7.70
7.26
1.06
SP1
30.70
5.54 7.74
347
0
0.90 0.78 1.03
4.99
4.32
1.15
E‡
1.33
31.10
5.58 7.80
349
0
0.90 0.77 1.04
5.02
4.30
1.17
SP3
E
§
1.33
27.00
5.20 7.11
350
427
0.94 0.83 1.00
5.17
4.56
1.13
SP4
E§
1.33
31.00
5.57 7.79
349
379
0.86 0.87 1.09
5.13
5.19
0.90
SP5
E‡§
1.33
32.00
5.66 7.92
347
0
0.88 0.75 1.05
4.97
4.24
1.17
SP6
E
1.33
36.20
6.02 8.55
352
357
0.78 0.78 1.20
5.16
5.16
1.00
SP7
E
1.33
30.70
5.54 7.74
352
365
0.87 0.83 1.08
5.16
4.93
1.05
SP8
Total
1.33
SP2
35
E
‡§
E
1.33
26.30
5.13 7.00
352
365
1.19 1.02 1.11
6.44
5.92
1.09
39
Average
1.02
COV
0.10
*
I equals interior beamcolumn subassemblage; E equals exterior beamcolumn subassemblage.
Beam bars of UNIT3 were anchored in beam stub at far face of column.
joints.
§Subassemblages with one transverse beam for γ
cal < γult, γpred = γcal, τpred = τcal and for γcal ≥ γult, γpred = γult, τpred = τult.
Notes: τACI is the limiting values of joint stress according to ACI 318055 and ACI 352R02;6 τEC8 is the limiting values of joint shear stress according to Eurocode 8.4 Neither relevant Greek
codes7,8 provide information regarding limiting values for joint shear stress. All subassemblages have flexural strength ratios MR higher than 1.0. Overstrength factor a = 1.25 for
†
‡Unreinforced
beam steel is included in computations of joint shear stress τcal = γcal f c MPa. 1 MPa = 144.93 psi; 1.0 f c MPa = 12.05 f c psi.
476
ACI Structural Journal/JulyAugust 2007
29.
A question arises regarding how concrete slabs, which are
typical in buildings, affect the performance of the joints of
subassemblages such as A1, E1, E2, and G1. Ehsani and
Wight31 found that “the flexural strength ratio MR at the
connections is reduced significantly due to the contribution
of the slab longitudinal reinforcement.” They recommended
that, to ensure flexural hinging in the beam, flexural strength
ratios should be no less than 1.20.31 The flexural strength
ratios of all the Subassemblages A1, E1, E2, and G1 tested in
this study were significantly higher than 1.20 (refer to Fig. 1(a)
and (b)); thus, the presence of a concrete slab would not have
had any influence on the response of these subassemblages.
It would be of interest to learn whether simpler procedures
for arriving to the beamcolumn joint ultimate strength such
as that proposed by Park and Paulay,10 would lead to similar
findings as those derived from the solution of the system of
Eq. (11) to (13). To this end, Table 4 presents the joint ultimate
strength and ratios, τpred/τexp and γcal/γult for Subassemblages
A1, E1, E2, and G1 according to the aforementioned procedures.
The ultimate joint shear strengths of Subassemblages A1, E1,
E2, and G1 derived from the solution of the system of Eq. (11)
to (13) depend on the increased joint concrete compressive
strength due to confining fc, as well as on the joint aspect
ratio α. These values differ significantly from those of Park
and Paulay,10 which mainly depend on the percentage of top
longitudinal beam reinforcement. Thus, Table 4 shows that
the values of ultimate joint shear strengths of Subassemblages
A1 and E2 derived from the solution of the system of Eq. (11)
to (13) are higher than those of Subassemblages E1 and G1
derived by the same methodology. This clearly explains why
the Park and Paulay10 values of ultimate joint shear strength
in Table 4 are larger than the values from Eq. (11) to (13) for
E1 and G1 and less than the values from Eq. (11) to (13) for
A1 and E2. Finally, as can be seen from Table 4, the proposed
shear strength formulation predicted the failure mode for
Subassemblages A1, E1, E2, and G1 with significant accuracy,
while the Park and Paulay10 procedure predicted only the
failure mode of Subassemblages A1 and E2.
CONCLUSIONS
Based on the test results described in this paper, the
following conclusions can be drawn.
1. The behavior of Subassemblages A1 and E2 was as
expected and as documented in the seismic design philosophy
of ACI 31805,5 ACI 352R02,6 and Eurocode 8.4 The beamcolumn joints of both Subassemblages A1 and E2 performed
satisfactorily during the cyclic loading sequence to failure,
allowing the formation of plastic hinges in their adjacent beams.
Both subassemblages showed high strength without any appreciable deterioration after reaching their maximum capacity;
2. Despite the fact that Subassemblages E1 and G1 represented
beamcolumn subassemblages of contemporary structures, they
performed poorly under reversed cyclic lateral deformations.
The joints of both Subassemblages E1 and G1, contrary to
expectations based on Eurocode 2,3 Eurocode 8,4 and the two
Greek codes7,8 exhibited shear failure during the early stages
of cyclic loading. This happened because, for both Subassemblages E1 and G1, the calculated joint shear stress τcal
was higher than the joint ultimate strength τult (Table 6).
Damage occurred both in the joint area and in the columns’
critical regions. This effect cannot be underestimated as it may
lead to premature lateral instability in ductile momentresisting frames of modern structures; and
ACI Structural Journal/JulyAugust 2007
Table 6—Experimental and predicted values of
strength of Subassemblages A1, E1, E2, and G1
Joint
aspect
ratio
Subassem α =
blage hb/hc
K
γcal
Predicted Observed
μ=
shear
shear
strength strength τpred/ γcal/
γexp γult τpred, MPa τexp, MPa τexp γult
A1
1.50 1.558 0.685 0.584 1.46
5.05
E1
1.50 1.593 1.26 0.98 1.17
6.92
5.80
1.19 1.08
E2
1.50 1.558 0.675 0.554 1.46
5.00
4.10
1.20 0.46
G1
1.50
6.60
5.56
1.19 1.04
1.50 1.20 0.96 1.15
4.31
1.17 0.47
Notes: For γcal < γult, γpred = γcal, τpred = τcal and for γcal ≥ γult, γpred = γult, τpred = τult.
1 MPa = 144.93 psi; 1.0 f c MPa = 12.05 f c psi. Overstrength factor a = 1.25 for beam
steel is included incomputations of joint shear stress τcal = γcal
f c MPa.
3. It was demonstrated that the design assumptions of Eurocode 2,3 Eurocode 8,4 and those in the Greek codes7,8 did not
avoid premature joint shear failures because the resulting
design can not ensure that the joint shear stress will be significantly lower than the joint ultimate strength τult and did not
ensure the development of the optimal failure mechanism with
plastic hinges occurring in the beams while columns remained
elastic, according to the requisite strong columnweak beam.
Thus, provisions in Eurocode 23 and Eurocode 84 and those in
the two Greek codes7,8 related to the design of beamcolumn
joints need improvement.
∅
a
b ′c
f ′c
hb
h ′c
hc
MR
N
Vjh
Vjv
α
γcal
γexp
γult
τ
NOTATION
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
bar diameter
overstrength factor
width of joint core
compressive strength of concrete
total depth of beam
length of joint core
total depth or width of square column
sum of flexural capacity of columns to that of beam
applied column axial load during test
horizontal joint shear force
vertical joint shear force
hb/hc
design values of parameter [γcal = (τcal / f c )]
actual values of parameter [γexp = (τexp/ f c )]
values of parameter γ at ultimate capacity of connection [γult =
(τult/ f c )]
= joint shear stress
REFERENCES
1. Leon, R. T., “Shear Strength and Hysteretic Behavior of Interior BeamColumn Joints,” ACI Structural Journal, V. 87, No. 1, Jan.Feb. 1990, pp. 311.
2. Penelis, G. G., and Kappos, A. J., EarthquakeResistant Concrete
Structures, E&FN Spon, London, 1997, 572 pp.
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CEN, Berlin, Germany, 1991, 61 pp.
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Design of Structures—Part 1: General Rules and Rules for Buildings (ENV
199811/2/3),” CEN, Berlin, Germany, 1995, 192 pp.
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Concrete (ACI 31805) and Commentary (318R05),” American Concrete
Institute, Farmington Hills, Mich., 2005, 430 pp.
6. Joint ACIASCE Committee 352, “Recommendations for Design of
BeamColumn Connections in Monolithic Reinforced Concrete Structures
(ACI 352R02),” American Concrete Institute, Farmington Hills, Mich.,
2002, 37 pp.
7. “New Greek Earthquake Resistant Code (ERC1995),” Athens,
Greece, 1995, 145 pp. (in Greek)
8. “New Greek Code for the Design of Reinforced Concrete Structures
(CDCS1995),” Athens, Greece, 1995, 167 pp. (in Greek)
9. Hakuto, S.; Park, R.; and Tanaka, H., “Seismic Load Tests on Interior
and Exterior BeamColumn Joints with Substandard Reinforcing Details,”
ACI Structural Journal, V. 97, No. 1, Jan.Feb. 2000, pp. 1125.
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10. Park, R., and Paulay, T., Reinforced Concrete Structures, John Wiley
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11. Park, R., “A Summary of Results of Simulated Seismic Load Tests
on Reinforced Concrete BeamColumn Joints, Beams and Columns with
Substandard Reinforcing Details,” Journal of Earthquake Engineering, V. 6,
No. 2, 2000, pp. 147174.
12. Paulay, T., and Park, R., “Joints of Reinforced Concrete Frames
Designed for Earthquake Resistance,” Research Report 849, Department
of Civil Engineering, University of Canterbury, Christchurch, New
Zealand, 1984, 71 pp.
13. Ehsani, M. R., and Wight, J. K., “Exterior Reinforced Concrete
BeamtoColumn Connections Subjected to EarthquakeType Loading,”
ACI JOURNAL, Proceedings V. 82, No. 4, JulyAug. 1985, pp. 492499.
14. Soroushian, P., and Sim., J., “Axial Behavior of Reinforced Concrete
Columns under Dynamic Loads,” ACI JOURNAL, Proceedings V. 83, No. 6,
Nov.Dec. 1986, pp. 10181025.
15. Scott, B. D.; Park, R.; and Priestley, M. J. N., “StressStrain Behavior
of Concrete Confined by Overlapping Hoops at Low and High Strain
Rates,” ACI JOURNAL, Proceedings V. 79, No. 1, Jan.Feb. 1982, pp. 1327.
16. CEBFIP, “Model Code 1990,” Bulletin d’ Information, CEB, Lausanne,
Switzerland, 1993, 490 pp.
17. Mitchel, D., “Controversial Issues in the Seismic Design of Connections
in Reinforced Concrete Frames,” Recent Developments in Lateral Force
Transfer in Buildings, SP157, N. Priestley, M. P. Collins, and F. Seible,
eds., American Concrete Institute, Farmington Hills, Mich., 1995, pp. 7596.
18. Ehsani, M. R.; Moussa, A. E.; and Vallenilla, C. R., “Comparison of
Inelastic Behavior of Reinforced Ordinary and HighStrength Concrete
Frames,” ACI Structural Journal, V. 84, No. 2, Mar.Apr. 1987, pp. 161169.
19. Paulay, T., “Seismic Behavior of BeamColumn Joints in Reinforced
Concrete Space Frames, Stateofthe Art Report,” Proceeding of the Ninth
World Conference on Earthquake Engineering, V. VIII, Tokyo, Japan,
1988, pp. 557568.
20. Tsonos, A. G., “Towards a New Approach in the Design of R/C
BeamColumn Joints,” Technika Chronika, Scientific Journal of the Technical
Chamber of Greece, V. 16, No. 12, 1996, pp. 6982.
21. Tsonos, A. G., “Shear Strength of Ductile Reinforced Concrete
BeamtoColumn Connections for Seismic Resistant Structures,” Journal
of European Association for Earthquake Engineering, No. 2, 1997, pp. 5464.
22. Tsonos, A. G., “Lateral Load Response of Strengthened Reinforced
Concrete BeamtoColumn Joints,” ACI Structural Journal, V. 96, No. 1,
Jan.Feb. 1999, pp. 4656.
23. Tsonos, A. G., “Seismic Retrofit of R/C BeamtoColumn Joints
using Local ThreeSided Jackets,” Journal of European Earthquake
478
Engineering, No. 1, 2001, pp. 4864.
24. Tsonos, A. G., “Seismic Rehabilitation of Reinforced Concrete
Joints by the Removal and Replacement Technique,” Journal of European
Earthquake Engineering, No. 3, 2001, pp. 2943.
25. Tsonos, A. G., “Seismic Repair of Exterior R/C BeamtoColumn
Joints using TwoSided and ThreeSided Jackets,” Structural Engineering
and Mechanics, V. 13, No. 1, 2002, pp. 1734.
26. Tsonos, A. G., “Effectiveness of CFRPJackets and RCJackets in PostEarthquake and PreEarthquake Retrofitting of BeamColumn Subassemblages,” Final Report, Grant No. 100/11102000, Earthquake Planning and
Protection Organization (E.P.P.O.), Sept. 2003, 167 pp. (in Greek).
27. Paulay, T., “Equilibrium Criteria for Reinforced Concrete BeamColumn
Joints,” ACI Structural Journal, V. 86, No. 6, Nov.Dec. 1989, pp. 635643.
28. Park, R., “The Paulay Years,” Recent Developments in Lateral Force
Transfer in Buildings, SP157, N. Priestley, M. P. Collins, and F. Seible,
eds., American Concrete Institute, Farmington Hills, Mich., 1995, pp. 130.
29. Tegos, I. A., “Contribution to the Study and Improvement of EarthquakeResistant Mechanical Properties of Low Slenderness Structural
Elements,” PhD thesis, Appendix 13, V. 8, Aristotle University of Thessaloniki,
1984, pp. 185. (in Greek)
30. Kupfer, H.; Hilsdorf, H. K.; and Rusch, H., “Behavior of Concrete
under Biaxial Stresses,” ACI JOURNAL, Proceedings V. 66, No. 8, Aug.
1969, pp. 656667.
31. Ehsani, M. R., and Wight, J. K., “Effect of Transverse Beams and
Slab on Behavior of Reinforced Concrete BeamtoColumn Connections,”
ACI JOURNAL, Proceedings V. 82, No. 2, Mar.Apr. 1985, pp. 188195.
32. Durrani, A. J., and Wight, J. K., “Behavior of Interior BeamtoColumn
Connections under EarthquakeType Loading,” ACI JOURNAL, Proceedings
V. 82, No. 3, MayJune 1985, pp. 343349.
33. Fujii, S., and Morita, S., “Comparison Between Interior and Exterior
RC BeamColumn Joint Behavior,” Design of BeamColumn Joints for
Seismic Resistance, SP123, J. O. Jirsa, ed., American Concrete Institute,
Farmington Hills, Mich., 1991, pp. 145166.
34. Kaku, T., and Asakusa, H., “Ductility Estimation of Exterior BeamColumn Subassemblages in Reinforced Concrete Frames,” Design of
BeamColumn Joints for Seismic Resistance, SP123, J. O. Jirsa, ed.,
American Concrete Institute, Farmington Hills, Mich., 1991, pp. 167185.
35. Uzumeri, S. M., “Strength and Ductility of CastinPlace BeamColumn
Joints,” Reinforced Concrete Structures in Seismic Zones, SP53, American
Concrete Institute, Farmington Hills, Mich., 1977, pp. 293350.
36. Attaalla, S. A., and Agbabian, M. S., “Performance of Interior BeamColumn Joints Cast from High Strength Concrete Under Seismic Loads,”
Journal of Advances in Structural Engineering, V. 7, No. 2, 2004, pp. 147157.
ACI Structural Journal/JulyAugust 2007
32.
ACI member HungJen Lee is an Assistant Professor in the Department of Construction
Engineering and a Research Engineer of the Service Center for Construction Technology
and Materials in the National Yunlin University of Science and Technology, Yunlin,
Taiwan. He received his PhD from the National Taiwan University of Science and
Technology, Taipei, Taiwan, in 2000. His research interests include seismic design of
reinforced concrete structures, behavior of beamcolumn connections, reinforcement
detailing, and strutandtie models.
JenWen Ko is a PhD Student in the Department of Construction Engineering at the
National Taiwan University of Science and Technology. He received his MS from the
National Yunlin University of Science and Technology in 2005.
Fig. 1—Illustration of test specimens.
edge connection in an exterior momentresisting frame. The
researchers15,16 reported that the damage in the joint region
of these eccentric beamcolumnslab connections was not
as severe as that of previous tests without floor slabs.9,11
Including floor slabs significantly improves the overall
performance of eccentric connections and delays the
deterioration of joint stiffness and strength. LaFave et al.20
pointed out that including floor slabs in cruciform eccentric
connections would not only raise the joint shear demand but
would also reduce the effect of joint eccentricity and
enhance the joint shearresisting mechanisms.
Test and analytical results of another nine cruciform
eccentric beamcolumn connections were presented in the
13th World Conference on Earthquake Engineering.1719
Based on experimental results and finite element analysis
of three cruciform eccentric connections, Goto and Joh17
concluded that the joint shear strength decreases as the
joint eccentricity increases due to the stress concentration
on the eccentric side. Similar observations were also
concluded by Kusuhara et al.18 who tested two cruciform
eccentric connections (one with additional Ushaped reinforcement in the eccentric side). Finally, Kamimura et al.19 tested
four cruciform eccentric connections (three deep beamwide
column connections) and proposed an equation combining
shear and torsion to evaluate the joint shear strength.
Beamcolumn joints in RC buildings are probably
subjected to lateral loading in two principal directions during
an earthquake. Nevertheless, current ACI design procedures7,8
require that the joint shear strength be evaluated in each
direction independently and implicitly assume an elliptical
interaction relationship for biaxial loading. Notably, only
one value of permissible shear stress is selected for a joint
according to the effective confinement on the vertical faces
of the joint, even though the column cross section is rectangular.
Current ACI design procedures consider the effects of the
column’s aspect ratio and eccentric beam on joint shear
460
strength by limiting and reducing the effective joint width.
More experimental results are needed to verify the effective
joint width in eccentric connections.57 Thus, this experimental
program focuses on the behavior of eccentric corner connections
with rectangular columns because they have not been
experimentally verified.
RESEARCH SIGNIFICANCE
Current ACI design provisions for estimating joint shear
strength of eccentric beamcolumn connections are established
based on few experimental investigations. The effects of a
column’s aspect ratio and eccentric beam on joint shear
strength are evaluated by the effective joint width. Additional
experimental verification of the design provisions for eccentric
connections is needed, especially for eccentric corner
connections with rectangular columns. This paper presents
experimental results for five corner connections with one
concentric or eccentric beam framing into a rectangular joint
in the strong or weak direction. Experimental verifications
on the ACI approach provided contribution to the understanding of beamcolumn connections.
EXPERIMENTAL PROGRAM
Five RC corner beamcolumn connections were
designed, constructed, and tested under reversed cyclic
loading. A Tshaped assembly was used to represent the
essential components of a corner beamcolumn connection
in a twoway building frame subjected to lateral loading in
each principal direction. The primary test variables were
the lateral loading directions and the eccentricity between
the beam and column centerlines. Neither transverse beams
nor floor slabs were constructed to ease testing. As a result,
each subassembly had only one beam framing into one
corner column in each principal direction. For a corner,
interstory connections, floor slabs, and transverse beams
could not only introduce additional demand on joint shear
force but also reduce the effect of joint eccentricity. The
enhancement on the joint shear capacity from confinement
of floor slabs and transverse beams is questionable because
a corner joint is only confined on two adjacent faces and it
is likely to sustain biaxial loading. Further study on the
behavior of corner beamcolumnslab connections
subjected to biaxial loading is needed.
Specimen geometry and reinforcement
The experimental program was designed using a concrete
compressive strength f ′c of 30 MPa (4.35 ksi) and a reinforcement yield stress fy of 420 MPa (60.9 ksi). Cross sections and
reinforcement details of the five specimens, designated as
S0, S50 (Series S), W0, W75, and W150 (Series W), are
shown in Fig. 1. The first character (S or W) of the designation
represents one south or west beam framing into the rectangular
column in the strong or weak direction. The subsequent
numerals denote the eccentricity between the beam and
column centerlines in mm. Thus, two concentric (S0 and
W0) and three eccentric (S50, W75, W150) connections
were tested in total.
The corner column had a cross section of 400 x 600 mm
(16 x 24 in.) and used 12 D22 (No. 7) longitudinal bars
(gross reinforcement ratio of 1.9%) and D10 (No. 3) hoops
with crossties at a spacing of 100 mm (4 in.) throughout the
column. The total crosssectional area of the lateral reinforcement for each direction of the column was approximately equal
to the minimum amount required by ACI 31805, Section
ACI Structural Journal/JulyAugust 2007
33.
Table 1—Connection design parameters
Specimen
S0
S50
W0
W75
W150
Column width bc, mm (in.)
400 (16)
600 (24)
Column depth hc, mm (in.)
600 (24)
400 (16)
5.10
3.46
24
15
Moment strength ratio Mr
*
Provided embedment length db†
Joint shear demand Vu, kN (kips) 699 (157.1)
Joint eccentricity e, mm (in.)
Effective joint width bj318,
706 (158.7)
0 (0) 50 (2) 0 (0) 75 (3)
150 (6)
400
mm (in.) (16)
300
(12)
600
(24)
450
(18)
300
(12)
0.53
0.71
0.54
0.72
1.07
Vu
318
γ f c ′b j h c
Effective joint width b352, mm (in.)
j
350 (14)
450 (18)
360 (14.4)
Vu
352
γ f c ′b j h c
0.61
0.72
0.90
*M
r
†
= ΣMn(columns)/ΣMn(beams).
Embedment lengths required by ACI 31805 and ACI 352R02 are 14.2db and
16.8db, respectively.
Note: All values are computed with fc′ = 30 MPa (4.35 ksi) and fy = 420 MPa (60.9 ksi).
21.4.4.1.8 To control the demand of shear force acting on the
joint, the loading beam had a cross section of 300 x 450 mm
(12 x 18 in.) and used four D22 (No. 7) longitudinal bars (steel
ratio of 1.29%) at both top and bottom. To avoid beam shear
failure and ensure adequate confinement in the beam plastic
hinge region, closed overlapping hoops were provided
through the length of the beam. Figure 2 illustrates the
overall geometry of the specimens. The lengths of the beam
and column that were chosen to simulate the nearest inflection
points in the beam and column framing into the joint. In
general, the five specimens were nominally identical except
for the joint shear direction, the embedment lengths of the
hooked beam bars, and the eccentricity between the beam
and column centerlines.
Connection design parameters
Table 1 shows the main design parameters for the specimens.
Due to column bending in the strong or weak direction, the
ratios of columntobeam flexural strength Mr at the connections
of Series S and W were equal to 5.10 and 3.46, respectively.
Because both Mr values were much greater than the specified
value of 1.2, flexural hinging in the beam was anticipated.
To ensure the anchorage of beam longitudinal bars and to
promote the development of a diagonal compression strut
within the joint, the beam longitudinal bars were anchored
using a 90degree standard hook bent into the joint and
embedded as close as possible to the back of the column
(Fig. 2). Leaving a 70 mm (2.8 in.) back cover behind the
hook, the provided embedment lengths within the joint were
24db for Series S and 15db for Series W. The required
development lengths of hooked beam bars, measured from
the critical section, are given in ACI 31805, Section
21.5.4.1, and ACI 352R02, Section 4.5.2.4, for Type 2
connections. Per ACI 31805,8 the critical section is taken at
the beamcolumn interface. Per ACI 352R02,7 for Type 2
connections, it is taken at the outside edge of the column core.
For fy of 420 MPa (60.9 ksi) and f ′c of 30 MPa (4.35 ksi), the
embedment lengths required by ACI 31805 and ACI 352R02
are 14.2db and 16.8db, respectively. As shown in Table 1, the
provided embedment length within the joints in Series W is
ACI Structural Journal/JulyAugust 2007
Fig. 2—Overall geometry of test specimens.
105% of that required by ACI 31805 but only 89% of that
required by ACI 352R02.
Based on the capacity design concept, the demand of the
joint shear force Vu is dominated by the flexural capacity of
the beam. When computing Vu values, a probable strength
of 1.25fy for the beam longitudinal reinforcement was
included. Due to small differences in beam lengths, the
value of Vu is equal to 699 kN (157.1 kips) for the specimens in
Series S and 706 kN (158.7 kips) for the specimens in
Series W, respectively.
The current ACI design procedures for joint shear strength
are based on Eq. (1)
φV n = φγ f c ′b j h c ≥ V u
(1)
where φ is the strength reduction factor of 0.85; Vn is the
nominal joint shear strength; γ f c′ is the nominal joint shear
stress of 1.0 f c′ MPa (12 f c′ psi) for corner, interstory
connections; hc is the column depth (mm or in.) in the direction
of joint shear to be considered; and bj is the effective joint
width (mm or in.) calculated using the following equations
8
ACI 31805 :
318
bj
⎧ b b + 2x
⎪
= the smaller of ⎨ b b + h c
⎪
⎩ bc
(2)
461
34.
sheets and wetcured for 1 week. For each batch of concrete,
12 150 x 300 mm (6 x 12 in.) concrete cylinders were cast
and cured together with the beamcolumn assemblies. Three
cylinders were tested at 28 days and the rest were tested at
the testing date of each beamcolumn assembly. Table 2
summarizes the concrete compressive strengths at 28 days
and the testing date. The average of concrete compressive
strengths at the testing date are used for analytical f ′c in this
paper, because the variation of concrete compressive strengths
within each batch of concrete is small.
Fig. 3—Test setup for Series W (similar setup for Series S).
Table 2—Concrete compressive strengths
Specimen
S0
Concrete batch
S50
W0
1
28day f ′c, MPa (psi)
W75
W150
2
28.5 (4133)
25.2 (3655)
Test days
49
67
53
57
60
Test day f ′c, MPa (psi)
32.6
(4728)
34.2
(4960)
28.9
(4191)
30.4
(4409)
29.1
(4220)
Analytical f ′c, MPa (psi)
33.2 (4815)
352
ACI 352R027: b j
29.5 (4278)
⎧ (b + b ) ⁄ 2
c
⎪ b
⎪
mh
= the smaller of ⎨ b + Σ c (3)
2
⎪ b
⎪
⎩ bc
where bb is the beam width (mm or in.); x is the smaller
distance between the beam and column edges (mm or in.); bc
is the column width (mm or in.); and m is 0.3 when e is
greater than bc/8, otherwise m is 0.5. The summation term is
applied on each side of the joint where the column edge
extends beyond the beam edge. The joint eccentricity e was
designed to be bc/8 for Specimen S50 and W75, and to be bc/4
for Specimen W150.
As shown in Table 1, only Specimen W150 had a target
joint shear stress exceeding the nominal value of 1.0 f c′ MPa
(12 f c′ psi) for the effective joint width per ACI 31805.8
The other four specimens satisfied the requirement on the
joint shear stress when following ACI design procedures
with a strength reduction factor of 0.85.
Construction and material properties
Two sizes of standard reinforcement meeting ASTM A 706
were used for longitudinal and transverse reinforcement in all
specimens. The D22 (No. 7) longitudinal reinforcement had
an average yield stress of 455 MPa (66 ksi) and an average
ultimate strength of 682 MPa (99 ksi). The average yield and
ultimate strengths were 471 and 715 MPa (68 and 104 ksi) for
D10 (No. 3) transverse reinforcement, respectively.
Each specimen was cast in a wood form with the beam and
column lying on the ground and the exterior column side
(east side for Series S and north side for Series W) facing up.
Concrete was supplied by a local ready mix plant using
normal concrete aggregate and delivered by pump using a
125 mm (5 in.) diameter hose. Series S was cast at one time
using a single batch of concrete, and then Series W was cast
using another batch of concrete with the same mixture
proportions. The fresh concrete was covered with plastic
462
Test setup and loading sequence
Figure 3 shows the elevation views of the test setup. To
restrain the column for twisting about the column axis, each
beamcolumn assembly was rotated 90 degrees and tied
down to a strong floor with reaction steel beams, cover
plates, and rods. In addition, four onedimensional rollers
were seated beside the column to allow inplane rotation at
both ends of the column. This arrangement was chosen to
provide stability against torsional action. The actuator load
was applied at the beam centerline while the column axial
load was applied along the column longitudinal axis. Thus, a
twist of the column about its longitudinal axis was applied
for the eccentric connections.
To simulate the displacement reversal of beamcolumn
connections during earthquake events, the specimens were
subjected to reversed cyclic lateral displacements. Axial load
was applied at the beginning of a test and held at a level of
0.10Ag fc′ during testing. A typical lateral displacement
history consisting of three cycles at monotonically
increasing drift levels (0.25, 0.50, 0.75, 1.0, 1.5, 2, 3, 4, 5, 6,
and 7%) was used for all specimens. The actuator applied
each target displacement in a quasistatic manner at a speed
ranging from 0.05 to 1.40 mm/s (0.002 to 0.056 in./s). Target
displacement amplitudes at the beam tip Δ were computed
using the following equation
Δ
Drift ratio θ = L b + 0.5h c
(4)
where drift ratio θ is the angular rotation of the beam chord
with respect to the column chord; Lb + 0.5hc is the vertical
distance between the actuator and column centerlines, and it
is equal to 2.15 m (86 in.) for Series W and 2.075 m (83 in.)
for Series S (Fig. 2).
EXPERIMENTAL RESULTS
Experimental results showed that two joints of Series S
were capable of supporting the complete formation of a
beam plastic hinge. In contrast, three joints of Series W
exhibited significant damage and strength degradation after
the beam flexural yielding. Measured responses are summarized
and discussed in the following subsections. Results
presented include: 1) beam flexural failure for Series S; 2)
joint failure after beam yielding for Series W; 3) discussion
of joint shear capacity; and 4) effect of joint eccentricity. The
results are used to evaluate the influence of joint eccentricity
and loading directions on the seismic performance of corner
beamcolumn connections.
Beam flexural failure for Series S
Figure 4 depicts the normalized loaddisplacement hysteretic
curves for the test specimens. The actuator load P was
ACI Structural Journal/JulyAugust 2007
35.
Fig. 5—Final damage states for test specimens.
Fig. 4—Normalized load versus displacement response.
normalized to the nominal yield load Pn that was calculated
at a given strain of 0.004 for extreme compression fiber of
the critical beam section. When analyzing the beam section,
the measured material properties were used to model the
concrete and reinforcing bars. In addition, the beamtip
displacement Δ was also normalized to the drift ratio and
displacement ductility ratio. As shown in Fig. 4, the nominal
yield displacement Δy was determined by extrapolation from
measured displacement at 0.75Pn in the 1% drift cycle.
Table 3 reports the nominal yield load and displacement
for each specimen.
The loaddisplacement responses for Specimens S0 and
S50, as shown in Fig. 4, are very similar in stiffness, strength,
and ductility. Beam bars initiated yielding in the 1.0% drift
cycle and maximum load was recorded at 5% drift level. The
hysteretic curves show relatively little pinching, which is
typical for a flexuredominated system. The failure mechanisms for specimens of Series S were core concrete
crushing and subsequent buckling of longitudinal bars in the
beam plastic hinge region. The buckling of the beam bars in
eccentric Specimen S50 appeared earlier than that of concentric
Specimen S0.
The failure mode for the specimens in Series S was classified
as beam flexure failure (Mode B) due to buckling of the beam
bars. Figure 5 shows the final damage states for test specimens.
For Specimens S0 and S50, only hairline shear (diagonal)
cracks were observed on the east and west face of the joint
during testing. Concrete crushing in the beam plastic region
was evident, but only minor cover concrete spalling appeared
on the east face of the joint adjacent to the beamcolumn
interfaces. Further, the readings of shear deformations
ACI Structural Journal/JulyAugust 2007
Table 3—Test results
Specimen
Nominal yield load Pn, kN (kips)
S0
S50
W0
W75 W150
158 158 147 147 147
(35.5) (35.5) (33.0) (33.0) (33.0)
Nominal yield displacement Δ y , mm (in.) 18.9 20.1 23.5 23.5 24.8
(0.74) (0.79) (0.93) (0.93) (0.98)
Over strength factor Pmax /Pn
1.22 1.20 1.11
1.11
1.05
Ductility ratio Δmax /Δy
5.41 5.12 4.58
4.60
3.41
Maximum joint shear Vj,max, kN (kips) 827 814 778 781 739
(186) (183) (175) (176) (166)
V j, max
318
γ f c ′b j h c
0.60 0.78 0.60
0.80
1.13
V j, max
352
γ f c ′b j h c
0.68 0.67 0.80
0.80
0.94
BJ
BJ
Failure mode*
B
B
BJ
*Failure
Mode B means beam flexural failure and BJ means joint shear failure after
beam yielding.
Note: All values are computed with analytical f c (refer to Table 2) of concrete and
′
measured strengths of reinforcement.
measured on the east face of the joints remained in elastic
range during testing. Accordingly, it was concluded that both
joints of Series S were capable of maintaining joint integrity
and remaining elastic during the formation of adjacent beam
plastic hinges.
Joint failure after beam yielding for Series W
As shown in Fig. 4, the loaddisplacement responses for
the specimens in Series W were similar up to 4% drift cycles
after yielding of the beam bars (1% drift cycle) and joint
transverse reinforcement (2 to 3% drift cycle). All three
463
36.
Fig. 8—Measurement of joint shear deformation on north
face of Specimen W150: (a) cracking patterns on north face
at 5% drift; and (b) load versus joint shear deformation.
Fig. 6—Strain profiles of hooked beam bars for Specimens W0
and W150.
Fig. 7—Strain histories of Gauge 9 on hooked beam bars for
Specimens W0 and W150.
joints were capable of supporting beam flexural yielding up
to 4% drift; however, a considerable strength degradation
was observed after the maximum loads recorded at the 4%
drift level (Specimen W150) or 5% drift levels (Specimens W0
and W75). Eventually, specimens in Series W exhibited
significant pinching curves in Fig. 4, which were typical
responses of the shear or bondslip mechanism.
The beam bar strains were measured using electrical
resistance strain gauges attached to reinforcing bars at
selected locations. Figure 6 shows the strain distributions
along the beam bars at peak drift values for Specimens W0
and W150. The hooked beam bars initiated yielding at the
critical section (Gauge 10) during the 1% drift cycle, and
then spread plasticity into the plastic hinge region (Gauges 11
and 12) during the 2 and 3% drift cycle. Meanwhile, the
strain readings of Gauge 9 within the joint remained elastic
up to the 3% drift level. This denoted that some bond still
existed along the straight part of the bar embedded within the
joint. Figure 7 depicts the available strain histories of Gauge 9
for Specimens W0 and W150 during testing. Both gauge
464
readings remained elastic in the 3% drift cycles and then
went into yielding plateau in the first or second cycle of the
4% drift level. It is evident that the beam bar was adequately
developed up to 4% drift. The bond along the straight portion
of the bar was lost at this stage, and therefore the bearing
inside the bent portion of the hook resisted most of the
tension force. The stress of the bar would begin to drop after
crushing of the diagonal strut within the joint. In this paper,
this type of failure is classified as diagonal shear compression
failure of the joint rather than premature anchorage failure of
the beam bars.
Figure 8 shows the cracking pattern and measurement of
joint shear deformation on the north (flush) face of the joint
for Specimen W150. The initial joint shear cracks appeared
diagonally during the 0.5% drift cycle, followed by propagation
of diagonal cracks up to a 4% drift level. After strength
degradation commenced at 4% drift, however, no new joint
shear cracks appeared while crushing and spalling of
concrete started on the north face of the joint. The measured
joint shear deformation rapidly increased after the maximum
load recorded at 4% drift, followed by significant degradation
on strength and stiffness. The joint shear failure after beam
yielding (Mode BJ) was evident due to the nonlinear shear
deformation, wideopened diagonal shear cracks, and visible
expansion from crushing of concrete in the joint region.
Specimens W0 and W75, which had similar behavior with
Specimen W150, also failed in Mode BJ. Visible cracking,
crushing, and spalling of concrete in Specimens W0 and
W75 were less than those in Specimen W150 (Fig. 5). Due
to the distance between beam and column edges (Fig. 1), the
appearance of initial joint shear cracks on the north face of
the joint was delayed to the 1.0 and 1.5% drift cycle for
Specimens W75 and W0, respectively. Strength degradation
after the 5% drift cycle was attributed to the crushing of
concrete within the joints, followed by extensive pushout
cracks distributed on the east face of the joint behind the
hooked beam bars (Fig. 5). Due to crushing of the concrete
within the joint, the hooked beam bars might gradually lose
its bond and anchorage within the joint. As a result, the
pushout movement of the beam compression bars induced
the pushout cracks on the east face of the joints in Series W.
The joint failure and subsequent pushout cracks were
observed at a drift level of 5% or more, which is large for a welldesigned building system. Therefore, the observed behavior
appears to be acceptable for the seismic design purpose.
Discussion of joint shear capacity
Paulay et al.12 first discussed that there are two shearresisting
mechanisms exiting in joints, the truss mechanism and the
diagonal strut mechanism. The truss mechanism transfers the
ACI Structural Journal/JulyAugust 2007
37.
forces uniformly from the beam and column bars through the
bond mechanism. Adequate bond must exist between the
reinforcement and concrete to necessitate a truss mechanism,
which also requires considerable amounts of horizontal and
vertical tie forces in the truss panel to be in equilibrium.
Figure 9 illustrates a conceptual model for the degradation
of joint shear capacity under increasing drift or ductility
ratio. Joints subjected to inelastic displacement reversals
often undergo significant bond deterioration along the
reinforcing bars from the adjacent beam plastic hinge. At
this stage, a part of the joint shear is transferred through the
horizontal hoops with fanshaped struts, while the remainder
is carried by the diagonal strut. As the drift or ductility ratio
increases, the horizontal hoops would yield progressively,
the joint concrete may crack excessively, and the bond of the
reinforcing bars within the joint might be lost. Eventually,
the joint shear force is directly transferred by the diagonal
strut mechanism.
Real sheartransferring mechanisms in joints may be a
combination of the diagonal strut and the truss mechanism,
with the bond deterioration being at a certain degree of longitudinal reinforcement during cyclic loading (Fig. 9). Hence,
the joint shear capacity decreases as the cyclic inelastic
loading increases, which is referred to as the degradation of
the joint shear capacity. When the joint shear capacity falls
below the shear demand from beam hinging, the joint will
fail in the shear after beam yielding (Mode BJ). If the joint
shear capacity is greater than the demand, the maximum
strength is limited by the beam flexure capacity (Mode B).
Three levels of strength and ductility ratios for the test
specimens are shown in Table 3. Because the maximum
strengths of Specimens S0 and S50 were dominated by the
beam flexure capacity rather than the joint shear capacity,
Specimens S0 and S50 had overstrength factors of
approximately 1.2 and ductility ratios greater than 5. In contrast,
Specimens W0 and W75 had overstrength factors of approximately 1.1 and ductility factors of approximately 4.6 due to
the joint shear failure at 5% drift level. Further, the largejointeccentricity Specimen W150 barely reached the nominal
yield load and deteriorated at a ductility ratio of only 3.4.
Corresponding to the maximum actuator load, the
maximum shear force acting on the horizontal cross section
within the joint can be estimated by
L b ( L b + 0.5h c )
V j, max = T max – V col = P max ⎛  – ⎞
⎝ jd
⎠
Lc
(5)
where Tmax is the maximum force in the tension reinforcement
of the beam (N or lb); Vcol is the column shear in equilibrium
with the applied loading (N or lb); and jd is the internal level
arm of the beam section (mm or in.). From standard momentcurvature analysis for each specimen, jd is approximately 7/8 of
the effective depth of the beam section. Thus, jd is simply
assumed to be 350 mm (13.8 in.) for the following evaluation
of maximum joint shear forces.
Table 3 compares the maximum joint shear force with
the nominal joint shear strength following the methods in
ACI 318058 or ACI 352R02.7 When following ACI 31805,8
Specimens S0 and W0 had equal effective joint area.
Thus, the maximum joint shear forces were only 60% of
the nominal strength for concentric Specimens S0 and W0
(Table 3), but different failure modes occurred during
testing (Fig. 4). For the flexuredominated Specimen S0,
ACI Structural Journal/JulyAugust 2007
Fig. 9—Conceptual model for degradation of joint shear
capacity.
the maximum shear force acting on the joint was less than
the joint shear capacity (Fig. 9). In contrast, Specimen W0
failed in Mode BJ when the joint shear force reached the
joint shear capacity at 5% drift. Clearly, the joint shear
capacity in the strong direction of the rectangular joint
(Specimen S0) was greater than that in the weak direction
(Specimen W0). Comparing eccentric Specimens S50 and
W75 can also find similar observation. This point cannot be
rationally reflected on the calculation of a crosssectional
approach within the joint, especially for the effective joint
width given by Eq. (2).
When following ACI 352R02,7 the maximum joint shear
forces were approximately 70% of the nominal strengths for
Series S, 80% of those for Specimens W0 and W75, and 94%
of that for Specimen W150. Three levels of demandtocapacity
ratios reasonably reflected three levels of performance on
strength and ductility ratios shown in Table 3. This shows that
the effective joint width bj352 is more rational than bj318 for test
specimens. Although following the ACI 352R027 procedures
could not avoid joint shear failure at a large drift level of 4 or
5%, it is considered acceptable in a real structural system.
In this experimental program, each specimen was able to
carry the applied column axial load of 0.10Ag f c over the
′
entire displacement history. Strain readings of gauges confirmed
that all column longitudinal bars remained elastic during
testing. For a building frame during earthquake events,
however, the axial load in a corner column may be higher
than 0.10Ag f c′ , or even in tension, due to overturning moment
from lateral loads. Therefore, more research on the behavior
of eccentric beamcolumn connections under high axial
loads is still needed.
Effect of joint eccentricity
The relative energy dissipation ratio β and the equivalent
viscous damping ratio ξeq, as shown in Fig. 10(a), were used
to evaluate the energy dissipation capacities of the test
specimens. The first index β represents a fatter or narrower
hysteretic curve (pinching) with respect to an elastic perfectly
plastic model. Another quantitative index ξeq describes the
hysteretic damping (or energy dissipation per cycle) with
respect to an equivalent linear elastic system.
Average β and ξeq of three cycles at each drift level for
the test specimens are compared in Fig. 10(b) and (c).
Three performance levels of energy dissipation capacities
465
38.
Fig. 12—Strain profiles at central layer of joint shear
reinforcement in Series S.
Fig. 10—Normalized energy dissipation at each drift level
for test specimens.
Fig. 11—Strain profiles at central layer of joint shear
reinforcement in Series W.
are evident. The flexuredominated Specimens S0 and S50
had a highest performance while Specimen W150 had
lowest performance. A small joint eccentricity of bc /8
(Specimens S0 and W75) had a slight influence on this
experimental program. Obviously, the large joint eccentricity of
bc/4 had significant detrimental effects on the seismic
performance of Specimen W150.
Strain histories for the joint hoops and crossties were used
to plot the strain distribution along the joint width at peak
466
drift values. There were three layers of transverse reinforcement
at a spacing of 100 mm (4 in.) in each joint. Only the strain
profiles of the hoop legs and crossties in the direction of
shear and at the central layer of the transverse reinforcement
were compared in Fig. 11 and 12.
For the corresponding drift ratios shown in Fig. 11, the
strain readings of Gauge 24 in Specimens W75 and W150
were larger than those in Specimen W0. These profiles
confirm the observations of more extensive shear or torsion
cracks on the north side of the eccentric joints. On the south
side, the strain readings of Gauge 20 in Specimens W75 and
W150 were less than those in Specimen W0 because the
shear and torsional stresses counteract each other.11
The effective joint width bj352 is also displayed in Fig. 11.
For the joints of Series W, strain gauges on hoop legs and
crossties within bj352 yielded during the 2 or 3% drift cycles
while the outside strain gauges remained elastic at the same
drift level. During testing of Series W, crushing of joint
concrete was observed within bj352 on the west side of the
joint. These observations agreed well with the strain profiles
shown in Fig. 11.
For the specimens in Series S, all joint hoops and
crossties remained elastic over the entire displacement
history. Figure 12 shows the strain distributions of hoop
legs and crossties along the joint width. Due to torsional
stresses from joint eccentricity, Specimen S50 had asymmetric
strain distribution with respect to concentric Specimen S0. It
should be noted that the total crosssectional area of joint
transverse reinforcement in two principal directions was
different. Although the maximum joint shear forces in
ACI Structural Journal/JulyAugust 2007
39.
Series S and W were similar (Table 3), the joint shear
forces transferring by the lateral joint reinforcement in
Series S were obviously less than those in Series W. These
profiles agree well with Hwang and Lee’s model,22 which
proposed that the fraction of shear carried by the joint
transverse reinforcement depends on the inclination of the
diagonal strut. Due to a deeper joint depth, the joints in
Series S had a flatter diagonal strut that can resist horizontal
joint shear more efficiently.23 As a result, the shear forces
transferring by the lateral joint reinforcement was reduced
and then the lateral joint reinforcement remained elastic
during testing.
CONCLUSIONS
Based on the evaluation of the cyclic loading responses of
five reinforced concrete beamcolumn corner connections in
this experimental program, the conclusions are as follows:
1. The joint shear capacity in the strong direction of a
rectangular joint is greater than that in the weak direction. In
this experimental program, two joints subjected to lateral
loading in the strong direction were capable of supporting
the complete formation of a beam plastic hinge. The other
three joints exhibited significant damage at the joints with
the joint shear acting along the weak direction of the column;
2. Joint eccentricity between the beam and column centerlines had detrimental effects on the seismic performance of
beamcolumn connections. Slight influence on the connection
performance was found when the joint eccentricity was equal to
halfquarter width of the column. As the joint eccentricity
increasing to onequarter of the column width, significant
reductions in the strength, ductility, and energy dissipation
capacity was observed; and
3. Compared with seismic performance levels, strain
distributions, joint damage of the test specimens, the effective
joint width recommend by ACI 352R02 is a better choice
than that given in the ACI 318 code. Experimental verifications
show that the current ACI design procedures are acceptable for
seismic design purposes but could not prevent the failure of
corner connections at a large drift level of 4 or 5%.
ACKNOWLEDGMENTS
The authors are grateful to the funding support (NSC 932211E224010) of
the National Science Council in Taiwan. The assistance of graduate students
for the construction and testing of the beamcolumn connections in the
structural laboratory of the National Yunlin University of Science and
Technology is also acknowledged.
REFERENCES
1. Moehle, J. P., and Mahin, S. A., “Observations on the Behavior of
Reinforced Concrete Buildings during Earthquakes,” EarthquakeResistant
Concrete Structures—Inelastic Response and Design, SP127, S. K.
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2. Sezen, H.; Whittaker, A. S.; Elwood K. J.; and Mosalam, K. M.,
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ACI Structural Journal/JulyAugust 2007
BeamColumn Joints in Monolithic Reinforced Concrete Structures,” ACI
JOURNAL, Proceedings V. 73, No. 7, July 1976, pp. 375393.
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JOURNAL, Proceedings V. 82, No. 3, MayJune 1985, pp. 266283.
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352R91),” American Concrete Institute, Farmington Hills, Mich., 1991,
18 pp.
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(ACI 352R02),” American Concrete Institute, Farmington Hills, Mich.,
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8. ACI Committee 318, “Building Code Requirements for Structural
Concrete (ACI 31805) and Commentary (318R05),” American Concrete
Institute, Farmington Hills, Mich., 2005, 430 pp.
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BeamColumn Joints with Eccentricity,” Design of BeamColumn Joints
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Performance of an Eccentric BeamColumn Joint,” Central Laboratories
Report 9125126, Central Laboratories, Lower Hutt, New Zealand, Aug.
1991, 81 pp.
11. Raffaelle, G. S., and Wight, J. K., “Reinforced Concrete Eccentric
BeamColumn Connections Subjected to EarthquakeType Loading,” ACI
Structural Journal, V. 92, No. 1, Jan.Feb. 1995, pp. 4555.
12. Chen, C. C., and Chen, G. K., “Cyclic Behavior of Reinforced Concrete
Eccentric BeamColumn Corner Joints Connecting SpreadEnded Beams,”
ACI Structural Journal, V. 96, No. 3, MayJune 1999, pp. 443449.
13. Vollum, R. L., and Newman, J. B., “Towards the Design of Reinforced
Concrete Eccentric BeamColumn Joints,” Magazine of Concrete
Research, V. 51, No. 6, Dec. 1999, pp. 397407.
14. Teng, S., and Zhou, H., “Eccentric Reinforced Concrete BeamColumn
Joints Subjected to Cyclic Loading,” ACI Structural Journal, V. 100, No. 2,
Mar.Apr. 2003, pp. 139148.
15. Burak, B., and Wight, J. K., “Seismic Behavior of Eccentric R/C
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Directions,” ACI Fifth International Conference on Innovations in Design
with Emphasis on Seismic, Wind and Environmental Loading, Quality Control,
and Innovation in Materials/Hot Weather Concreting, SP209, V. M. Malhotra,
ed., American Concrete Institute, Farmington Hills, Mich., 2002, pp. 863880.
16. Shin, M., and LaFave, J. M., “Seismic Performance of Reinforced
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BeamColumn Joints,” Proceedings of the 13th World Conference on
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18. Kusuhara, F.; Azukawa, K.; Shiohara, H.; and Otani, S., “Tests of
Reinforced Concrete Interior BeamColumn Joint Subassemblage with
Eccentric Beams,” Proceedings of 13th World Conference on Earthquake
Engineering, Paper No. 185, Vancouver, British Columbia, Canada, 2004,
14 pp.
19. Kamimura, T.; Takimoto, H.; and Tanaka, S., “Mechanical Behavior
of Reinforced Concrete BeamColumn Assemblages with Eccentricity,”
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Paper No. 4, Vancouver, British Columbia, Canada, 2004, 10 pp.
20. LaFave, J. M.; Bonacci, J. F.; Burak, B.; and Shin, M., “Eccentric
BeamColumn Connections,” Concrete International, V. 27, No. 9, Sept.
2005, pp. 5862.
21. Paulay, T.; Park, R.; and Priestley, M. J. N., “Reinforced Concrete
BeamColumn Joints under Seismic Actions,” ACI JOURNAL, Proceedings
V. 75, No. 11, Nov. 1978, pp. 585593.
22. Hwang, S. J., and Lee, H. J., “Strength Prediction for Discontinuity
Regions by Softened StrutandTie Model,” Journal of Structural Engineering,
ASCE, V. 128, No. 12, Dec. 2002, pp. 15191526.
23. Hwang, S. J.; Lee, H. J.; Liao, T. F.; Wang, K. C.; and Tsai, H. H.,
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467
41.
Joseph Robert Yost is an Associate Professor of Civil and Environmental Engineering at
Villanova University, Villanova, Pa. His research interests include the use of innovative
materials in transportation infrastructure, nondestructive methods for health monitoring
of structures, and seismic design and analysis of bridges.
ACI member Shawn P. Gross is an Associate Professor in the Department of Civil
and Environmental Engineering at Villanova University. He is Secretary of Joint
ACIASCE Committee 423, Prestressed Concrete, and a member of ACI Committees 213,
Lightweight Aggregate and Concrete; 363, HighStrength Concrete; 435, Deflection of
Concrete Building Structures; 440, Fiber Reinforced Polymer Reinforcement; and
E803, Faculty Network Coordinating Committee. His research interests include the
design and behavior of reinforced and prestressed concrete structures, including the
use of highstrength concrete and fiberreinforced polymer reinforcement.
David W. Dinehart is an Associate Professor of Civil and Environmental Engineering at
Villanova University. His research interests include seismic evaluation of wood structures,
passive damping systems, and the design and behavior of concrete and steel structures.
Jason J. Mildenberg is a Structural Engineer with Schoor De Palma of Brick, Manalapan,
N.J. He received an MS in civil engineering from Villanova University.
BACKGROUND AND LITERATURE REVIEW
Nanni (2000) compared the behavior of fullscale simply
supported highway bridge deck panels strengthened in flexure
with either externally bonded CFRP laminates or internally
placed NSM CFRP bars. Failure of the CFRP laminate
reinforced deck spans was through a combination of rupture
and peeling of the CFRP laminates. The NSM CFRPreinforced
span failed by tensile rupture of the CFRP bars. Relative to the
capacity of an unstrengthened control deck, moment strength
increases of 17 and 29% were reported for decks retrofitted
with externally bonded CFRP laminates and internally placed
NSM CFRP bars, respectively.
DeLorenzis et al. (2000) tested three steelreinforced
concrete Tbeams strengthened in flexure with NSM glass
FRP (GFRP) and NSM CFRP bars. The CFRP retrofitted
beams experienced increases in strength of 30% (two No. 3
CFRP bars) and 44% (two No. 4 CFRP bars) over an
unstrengthened control specimen. Both CFRP strengthened
beams failed due to debonding of the NSM rods. The specimen
strengthened with two No. 4 GFRP bars also failed due to
debonding of the NSM GFRP bars at a load 26% higher than
the control specimen. The authors reported that bond is critical
to using this technology effectively. Bond failure of the
NSM FRP bars was also identified by DeLorenzis and Nanni
(2001) as in need of further investigation. Debonding of the
NSM FRP bars due to splitting of the epoxy used for holding
the rod in place was reported. It was suggested that this
failure limitstate could possibly be avoided by increasing
bond lengths or anchoring the NSM rods in the flange.
Significantly, the authors reported that, where debonding of
the NSM FRP bars is prevented, splitting of the concrete
cover surrounding the longitudinal steel bars might become
the controlling ultimate limitstate. Loss of anchorage was
observed in several of their test specimens. In a related
experimental bond study, DeLorenzis et al. (2004) state that
epoxy is superior to cement paste as the groove filler material,
a groove sizetobar diameter of 2.0 is optimal, and a smooth
grove surface yields slightly lower local bond strengths, but is
preferable because it yields a more ductile bondslip behavior.
Taljsten and Carolin (2001) evaluated four rectangular
concrete beams subjected to fourpoint bending and
monotonically loaded in deformation control. Three of the
test beams were strengthened with NSM CFRP strips and the
fourth served as a control specimen. Two of the three
strengthened beams used an epoxy for bonding the FRP and
the third used a cement grout. Test results showed that two of
the three retrofitted beams failed due to anchorage loss
ACI Structural Journal/JulyAugust 2007
Fig. 2—Test setup.
between the NSM FRP strips and concrete. The third
strengthened beam failed due to tensile rupture of the FRP
strip. Predicted failure loads overestimated measured strengths.
ElHacha and Rizkalla (2004) compared the behavior of
beams strengthened on an equal axial stiffness basis using
NSM FRP bars and strips and externally bonded FRP laminates.
Their research showed that higher ultimate strengths and
increased ductility were achieved by the NSM strengthened
specimens. They also noted that bond integrity of NSM FRP
bars was less effective than for NSM FRP strips.
Together, these research findings demonstrate that bond
integrity can not be taken for granted and that bond related
limit states must also be considered for NSM FRP. DeLorenzis
and Nanni (2002) suggest that bond performance will be
influenced by multiple factors including bond length, NSM
FRP bar diameter and surface characteristic, material characteristics of the FRP, groove geometry, and properties of the
epoxy grout. Their experimental bond tests showed three
bond related failure modes, namely, splitting of the epoxy
cover, cracking of the concrete surrounding the grove, and
pullout of the NSM FRP rod.
EXPERIMENTAL PROGRAM
This experimental investigation consisted of testing 15 simply
supported fullscale concrete beams in flexure and material
characterization of the CFRP, steel reinforcement, and concrete.
All test beams had a shearspantosteelreinforcementdepth
ratio av /ds of 8.4. This ratio was intentionally selected so that
ultimate strength would be controlled by flexural failure and
not shear failure. The test setup and associated specimen
details are shown in Fig. 2.
The 15 test beams were separated into three groups of five
beams, with all beams in a given group having the same
cross section and steel reinforcement ratio ρs. Within each
group of five beams, two beams had one CFRP strip
(designated 61Fa&b, 91Fa&b, and 121Fa&b), two beams
had two CFRP strips (designated 62Fa&b, 92Fa&b, and
122Fa&b), and one beam acted as a control with no CFRP
(designated 6C, 9C, and 12C). Note that beams identified as
a and b are replicate specimens. Thus, the two parameters
investigated in the study are the amount of steel and CFRP
reinforcements. Table 1 presents the unstrengthened steel
reinforcement ratio ρs relative to a balanced design ρs /ρsb.
431
42.
Table 1—Specimen design and predicted strength parameters
Specimen
ρs/ρsb*
Afb, mm2 (in.2)
0.684
–38.86 (–0.060)
Failure type†‡
NA
6C
61Fa&b
ffult, MPa (ksi) Mn, kNmm (kipin.) Pn , kN (kip)§
Af /Afb
SY/CC
NA
23,068 (204.2)
18.92 (4.25)
1
–0.85
CC
810 (117.4)
26,606 (235.5)
21.82 (4.91)
1.15
1.26
Pn/PnC
62Fa&b
–1.69
CC
709 (102.8)
29,168 (258.2)
23.92 (5.38)
9C
NA
SY/CC
NA
25,104 (222.2)
20.59 (4.63)
1
–31.66
CC
1276 (185)
31,221 (276.3)
25.61 (5.76)
1.24
1.41
91Fa&b
0.470
–1.04 (–0.0016)
92Fa&b
–63.31
CC
1091 (158)
35,415 (313.5)
29.05 (6.53)
12C
NA
SY/CC
NA
25,790 (228.3)
21.15 (4.76)
1
0.84
TR
1648 (239)
34,071 (301.6)
27.95 (6.28)
1.32
1.69
CC
1436 (208.2)
40,023 (354.2)
32.83 (7.38)
1.55
121Fa&b
0.353
38.94 (0.060)
122Fa&b
*ρ
s
†
= As /bds and ρsb = 0.85(fc′/fy)β1(εcu)/(εcu + εsy) is unstrengthened balanced reinforcement ratio.
SY = steel yield, CC = concrete compression failure, TR = tensile rupture of FRP.
‡For all samples with CC failure, steel has yielded at ultimate as per analysis of Eq. (4).
§P = M /1219 mm (M /48 in.).
n
n
n
Fig. 3—CFRP and tensile test results.
Fig. 4—Specimen preparation.
The ratios of 0.353, 0.470, and 0.684 were selected as typical
for existing structures.
All specimens were instrumented with a concrete strain
gauge located on the top compression fiber at the center
span. Strengthened Specimens 61Fb, 62Fb, 91Fb, 92Fb,
121Fb, and 122Fb had an additional strain gauge bonded to
the CFRP at the center span. Linear variable displacement
transducers (LVDTs) were used to measure displacement at
the center span.
Concrete for the test specimens was delivered to the
laboratory by a concrete supplier. The concrete was in
accordance with Pennsylvania Department of Transportation
(PennDOT) Class AAA, Concrete for Bridge Decks, with
design specifications and properties given in BD601M
432
(PennDOT 2001). The mixture design was selected as typical
for bridge decks and is given as follows: water 1530 N/m3
(263 lb/yd3), cement 3967 N/m3 (682 lb/yd3), coarse aggregate
1784 lb/yd3, fine aggregate 7242 N/m3 (1245 lb/yd3), air entrainment 30 N/m3 (3 oz/yd3), and retarder 196 N/m3 (20 oz/yd3). The
slump at specimen casting was 101.6 mm (4 in.), and the
33day compressive strength as determined by ASTM C 68499
(ASTM 1999) using 100 mm (4 in.) diameter by 200 mm (8 in.)
high cylinders was 37.2 MPa (5.4 ksi) for all beams. Yield
strength of the steel reinforcement was determined from
uniaxial coupon testing to be 510 MPa (74 ksi) for No. 4 bars
and 490 MPa (71 ksi) for the No. 5 bars. Elastic modulus Es
is taken as 200 GPa (29,000 ksi).
The CFRP strips have a thin rectangular cross section that
measures approximately 15 x 2.5 mm (0.60 x 0.10 in.), and the
surface of the wide face is roughened to enhance force
transfer with the concrete epoxy grout. A photo of the CFRP
reinforcement with associated instrumentation detail can be
seen in Fig. 3(a). The material composition is 60% 4137 MPa
(600 ksi) carbon fiber by volume in a bisphenol epoxy
vinylester resin matrix. The CFRP elastic modulus Ef and
ultimate tensile strength ffu were determined from testing
uniaxial coupon specimens according to ACI Committee
440 (2004). Test results are shown in Fig. 3(b) from which
Ef and ffu were determined to be 136 GPa and 1648 MPa
(19,765 and 239 ksi), respectively.
Installation of the NSM CFRP strips is shown in Fig. 4 and
described as follows. First, the beams were rotated 180 degrees
about the long axis so that the steel reinforcement was at the
top of the beam. Next a rectangular groove approximately
6.4 mm (1/4 in.) wide by 19 mm (3/4 in.) deep was cut
longitudinally in the concrete where the CFRP was to be
installed. The groove was cut using a handheld circular with
an 18 cm (7 in.) diameter diamondtooth, abrasive cutting
blade. The saw was fitted with a rip guide, so that the distance
from the edge of the beam to the blade could be set and
maintained during cutting. The depth of the blade was set to
19 mm (3/4 in.) by adjusting the saw. The saw blade was just
over 3.2 mm (1/8 in.) wide so that two passes were made to
achieve the required width. For test specimens having one
CFRP strip, the longitudinal groove was located at the center
of the cross section; and for specimens having two CFRP
strips, the grooves were located at the 1/3 points in the cross
section. Next, the groove was thoroughly cleaned of debris
with compressed air and then partially filled with a structural
epoxy material that bonds with the concrete and FRP to
ACI Structural Journal/JulyAugust 2007
43.
provide a mechanism for force transfer. The epoxy grout
used was a twopart epoxy. Finally, the FRP was depressed
into the groove, where care was taken to ensure that no air
voids were trapped within the epoxy gel. Excess epoxy gel
was then cleaned from the concrete surface and curing was
done for a minimum of 2 weeks.
All beams were tested monotonically from an uncracked
condition. Two 90 kN (10 ton) hydraulic cylinders, located
152 mm (6 in.) on either side of center span and controlled
by a manuallyoperated pump, were used to apply load at an
approximate rate of 4.5 kN/minute (1 kip/minute). A load
cell was located under each hydraulic cylinder to measure
applied load. Electronic signals from the strain gauges
(concrete and CFRP), LVDTs, and load cell were recorded
by a 16bit data acquisition system at a frequency of 1 Hz.
ANALYTICAL STRENGTH
Figure 5 illustrates the assumed basic analytical conditions
of internal strain, stress, and resultant force for a cracked
section at ultimate that is underreinforced with steel (ρs < ρsb)
and strengthened with FRP. From Fig. 5, the following
assumptions are implicit: strain varies linearly through the
cross section, the section is initially uncracked, perfect bond
exists between the steel and FRP reinforcements and
concrete, the concrete strain at compression failure is 0.003,
the Whitney rectangular stress block in the compression
zone is a valid substitution for a nonlinear stress distribution
at ultimate, and the steel stressstrain behavior is assumed to
be elasticplastic. Also noted in Fig. 5, because the section is
initially uncracked and df > ds, the FRP strain εf will slightly
exceed the steel strain εs.
The theoretical nominal flexural strength Mn of an initially
uncracked beam that is underreinforced with steel (ρs < ρsb)
and strengthened with FRP is dependent on the amount of
FRP provided (Af) relative to the FRP area corresponding to
a balancedstrengthened strain condition (Afb). In this context,
balancedstrengthened represents simultaneous tensile rupture
of the FRP and compression failure of the concrete. Again,
for an initially uncracked section with df > ds and εf = εfu in
Fig. 5, by default the steel for a balancedstrengthened
design will have yielded (εs > εsy). Using these assumptions
and strain limits, and considering compatibility and equilibrium,
the theoretical balancedstrengthened area of FRP is
A fb
⎧ ε cu ⎫
0.85f′ c bβ 1 d f ⎨  ⎬ – A s f y
⎩ ε cu + ε fu ⎭
= f fu
ACI Structural Journal/JulyAugust 2007
M n = Af f fu ⎛ d f – a⎞ + As f y ⎛ d s – a⎞ for A f < A fb
⎝
⎝
2⎠
2⎠
(2b)
For sections controlled by concrete crushing, the stress level
in the steel is initially unknown, as is shown in Fig. 5(b). It
can be determined by fixing the steel and concrete strains at
yield εsy and crushing εcu, respectively, calculating the steel
area corresponding to yield Asy, and comparing this with the
area of steel present As. From Fig. 5(b), this is as follows
A sy
ε cu
df
0.85f c ′bβ 1 d s ⎛  ⎞ – A f E f ε sy ⎛ ⎞
⎝ d s⎠
⎝ ε cu + ε sy⎠
= fy
(3)
Accordingly, for As ≤ Asy, the steel stress is equal to fy. Likewise,
for As > Asy, the steel stress is less than fy and must be determined
from compatibility and equilibrium. Using this procedure,
the steel stress at ultimate for all specimens controlled by
concrete failure in this study was equal to yield. With the
steel stress at yield, the compression block a, stress in the
FRP reinforcement ff, and nominal moment capacity Mn for
sections controlled by concrete failure are found from
compatibility and equilibrium as follows
2
( A f E f ε cu – A s f y ) + 4 ( 0.85 )f c ′bβ 1 A f E f ε cu d f – ( A f E f ε cu – A s f y )
a = ( 2 )0.85f c ′b
(4a)
( df – α ⁄ β1 )
f f = E f ε cu  ≤ f fu
α ⁄ β1
(4b)
M u = A f f f ⎛ d f – a⎞ + A s fy ⎛ d s – a⎞
⎝
⎝
2⎠
2⎠
(4c)
(1)
Using Eq. (1) as a theoretical FRP reinforcement limit, failure
will be tensile rupture of the FRP when Af > Afb , or compression
failure of the concrete, when Af < Afb. It is noted that Afb can be
either positive or negative, depending on the existing amount of
steel reinforcement present (As). For a negative result from
Eq. (1), Af provided will always be greater than Afb, indicating a
compression failure of the concrete. Strain distributions for FRP
failure, balancedstrengthened, and compression failure are
shown in Fig. 5(b). For sections controlled by FRP failure, the
compression block depth a and nominal moment strength at
ultimate Mn are calculated from equilibrium as follows
Af f fu + As f y
a =  for A f < A fb
0.85f c ′b
Fig. 5—Analytical model at ultimate.
(2a)
The preceding analysis is offered as an alternative to the trial
and error procedure set forth by ACI Committee 440 (2002)
and yields identical results as would be obtained using the
ACI 440.2R procedure. Table 1 summarizes relevant design
and strength parameters. Moment strength Mn was calculated
using the measured material strengths for the steel, CFRP,
and concrete. It is evident from Table 1 that, for a given area
of FRP Af , the relative increase in strength Pn/PnC is
inversely proportional to the amount of steel reinforcement.
TEST RESULTS
Loaddeflection and loadstrain results are shown in Fig. 6 and
summarized in Table 2. Typical photos at failure are shown in
Fig. 7. The applied cylinder loads plotted in Fig. 6 and recorded
in Table 2 have been corrected to include the selfweight
bending effects of the beam. Moment equivalence at center span
433
44.
Fig. 6—Loaddeflection and loadstrain results.
Fig. 7—Test specimens at failure.
Table 2—Summary of test results
Measured
Theory
Yield
Ultimate
Comparison
Pn, kN (kip) Py, kN (kip) Mechanism type* Pmax, kN (k) Py /PyC
6C (control) 18.9 (4.25) 19 (4.28)
SY/CC
21.12 (4.75)
1
Sample ID
61Fa
61Fb
62Fa
62Fb
21.8 (4.91)
23.9 (5.38)
20.9 (4.69)
CC
24.83 (5.58)
1.10
21.3 (4.78)
CC
23.24 (5.23)
1.12
24.4 (5.48)
CC
24.99 (5.62)
1.28
24.7 (5.56)
CC
26.94 (6.06)
1.30
SY/CC
25.29 (5.69)
1
9C (control) 20.6 (4.63) 22.4 (5.03)
91Fa
91Fb
92Fa
92Fb
25.6 (5.76)
29.0 (6.53)
25.3 (5.70)
CC
28.22 (6.34)
1.13
24.5 (5.50)
CC
27.93 (6.28)
1.09
27.7 (6.22)
CC
37.05 (8.33)
1.24
25.0 (5.63)
CC
35.82 (8.05)
1.12
SY/CC
23.52 (5.29)
1
12C (control) 21.2 (4.76) 21.5 (4.84)
121Fa
121Fb
122Fa
122Fb
27.9 (6.28)
32.8 (7.38)
24.7 (5.56)
TR
29.59 (6.65)
1.15
25.9 (5.81)
TR
31.01 (6.97)
1.20
26.5 (5.97)
CC
33.80 (7.60)
1.23
28.0 (6.30)
CC
41.77 (9.39)
1.30
Average
Pmax/PmaxC
Average
Pmax/Py
Average
1
1
—
1.11
—
1.11
1.29
1
1.11
1.18
1
1.18
1.27
1.18
1.10
1.18
1.28
1
1.12
1.10
1.47
1.42
1
1.26
1.32
1.44
1.78
1.14
1.23
—
1.11
1.44
—
1.29
1.61
1.19
1.09
1.02
1.09
1.13
1.11
1.14
1.34
1.43
1.09
1.20
1.20
1.27
1.49
1.14
1.06
—
1.13
1.38
—
1.20
1.38
Pmax/Pn
1.12
1.14
1.06
1.04
1.13
1.23
1.10
1.09
1.28
1.23
1.11
1.06
1.11
1.03
1.27
*
SY = steel yield, CC = concrete crushing, TR = CFRP tensile rupture.
was used to calculate an equivalent concentrated force Peq that
was added to all laboratory measured load data. Moment
equivalence at center span is expressed as {1/8wbeamL2} =
{Peqav}. From Fig. 3, Peq for the 152, 230, and 305 mm (6,
9, and 12 in.) wide specimens is calculated to be 0.50, 0.77,
and 1.0 kN (0.115, 0.172, and 0.230 kips), respectively.
From Fig. 6, the physical effects of supplemental strengthening
with CFRP are clearly evident when strengthened specimens
are compared with companion control (unstrengthened)
specimens. All specimens strengthened with CFRP showed
434
a significant increase in ultimate strength when compared
with the companion control specimens. To a lesser degree,
strengthening with CFRP increased stiffness and yield load.
Detailed discussions of the test results for control and
strengthened specimens are presented in the following sections.
Control specimens: 6C, 9C, and 12C
Referring to the loaddeflection behavior of control
Specimens 6C, 9C, and 12C, the ductile behavior characteristic of underreinforced steel flexural (ρs < ρsb) members
ACI Structural Journal/JulyAugust 2007
45.
is apparent. Initially, all sections are uncracked and gross
section properties apply (Ig). At the cracking load Pcr,
behavior changes from uncracked to crackedelastic. As load
is increased further, the section responds elastically until the
yield strength of the steel reinforcement fy is reached. At the
yield load Py, behavior changes from crackedelastic to
inelastic. For Specimens 6C, 9C, and 12C, steel yield
occurred at 19, 22.4, and 21.5 kN (4.28, 5.03, and 4.84 kips),
respectively. The yield load corresponds to a flattening of the
loaddeflection trace and simultaneous inflection in the
concrete loadstrain response. Yield is followed by a load plateau
where the moment capacity of the section remains roughly
constant. The load plateau is clearly visible for Specimens 9C
and 12C, and to a lesser degree for Specimen 6C.
At the ultimate load Pmax, failure occurred by concrete
crushing. Ultimate load for Specimens 6C, 9C, and 12C
was 21.1, 25.3, and 23.5 kN (4.75, 5.69, and 5.29 kips),
respectively. For all control specimens, the ultimate load
Pmax was approximately 12% greater than the yield load Py.
The measured failure loads for Specimens 6C, 9C, and 12C
were 12, 23, and 11%, respectively, greater than the theoretical
nominal capacity Pn.
Specimens strengthened with one CFRP strip:
61Fa&b, 91Fa&b, and 121Fa&b
For specimens strengthened with one CFRP strip, the change
from crackedelastic to inelastic behavior (yield point) is less
abrupt and the associated reduction in the slope of the loaddeflection curve is less than for the control specimens. This
is especially true for specimens with a large relative amount of
steel reinforcement ρs/ρsb. Referring to Fig. 6, for Specimens
61Fa&b, the change in stiffness at ensuing nonlinear loaddeflection response associated with steel yielding is negligible.
These specimens have the largest relative area of steel
reinforcement equal to 0.68ρsb. For Specimens 91Fa&b and
121Fa&b, however, the change in stiffness after steel yield
is more apparent. These specimens were reinforced with
0.47ρsb and 0.34ρsb, respectively.
The mechanism of failure at ultimate for all specimens in
this group is consistent with that predicted using the theory
outlined previously and summarized in Table 1. As can be
seen in Table 2, all 152 and 230 mm (6 and 9 in.) wide specimens
strengthened with one CFRP strip failed by crushing of the
concrete. For these specimens, the CFRP did not rupture
prior to concrete crushing, indicating that the strain level was
less than the ultimate material strength. For the 305 mm (12 in.)
wide specimens with one CFRP strip, however, the CFRP
reinforcement did rupture at ultimate. This was followed by
compression failure in the concrete. Thus, the bond between
the CFRP and concrete for Specimens 121Fa&b was able to
develop the tensile strength of the CFRP strip. Also, for all
samples in this group, no debonding or slip between the
CFRP strip and concrete was observed (refer to Fig. 7(b)).
When compared with control specimens, the average yield
and ultimate loads for Specimens 61Fa&b, 91Fa&b, and
121Fa&b increased by 11%, 11, and 18%, and 14%, 11, and
29%, respectively. Thus, the relative increase in yield Py and
ultimate Pmax loads for the 152 and 230 mm (6 and 9 in.)
wide specimens strengthened with one CFRP strip relative to
the respective control specimens (PyC and PmaxC) was roughly
the same and taken approximately as 11%. For the 305 mm
(12 in.) wide specimens, the yield load increased by 18% and
the ultimate load increased by 29%. Therefore, a greater
increase in both yield and ultimate load capacities was
ACI Structural Journal/JulyAugust 2007
achieved for the 305 mm (12 in.) wide specimens than for the
152 and 230 mm (6 and 9 in.) wide specimens. This is
verification that the increase in strength is inversely
proportional to the relative area of steel reinforcement (ρs/ρbs).
For the strengthened specimens in this group, the average
ultimate loads Pmax were between 13 and 20% greater than
the average yield loads Py. Thus, the strength increase
between yield and ultimate limit states is slightly greater for
these specimens than for the control specimens (which was
approximately 12%). This is expected and represents the
additional tensile capacity provided by the CFRP after steel
yield, which is not available for the control specimens.
All specimens failed at loads slightly in excess of their
respective predicted nominal flexural strength Pn. Referring
to Table 2, the measured failure loads Pmax were between 6%
(61Fb) and 14% (61Fa) greater than the theoretical strength
Pn. The magnitude and range of this comparison suggest that
the analytical model and associated assumptions used in Eq. (2)
and (4) are acceptable for predicting the flexural capacity of
these four test specimens.
Specimens strengthened with two CFRP strips:
62Fa&b, 92Fa&b, and 122Fa&b
Referring to Fig. 6, the change from crackedelastic to
inelastic behavior for the 230 and 305 mm (9 and 12 in.)
wide specimens reinforced with two CFRP strips can still be
seen. For the 152 mm (6 in.) wide specimens strengthened
with two CFRP strips, however, this change from elastic to
inelastic behavior is much less obvious from the loaddeflection
graphs. The loadstrain curve for Specimen 62Fb, however,
shows a clear redistribution of tensile force to the CFRP as a
result of steel yield. It is therefore concluded that the steel
did yield for these specimens (62Fa&b).
Failure of all 152, 230, and 305 mm (6, 9, and 12 in.) wide
specimens reinforced with two CFRP strips occurred by
concrete crushing. This is consistent with the failure mode
predicted in Table 1. After concrete crushing, the 305 mm
(12 in.) wide specimens were further deformed until rupture
of the CFRP occurred. This rupture is significant in that it
again confirmed that force transfer is sufficient to develop
the full tensile capacity of the CFRP strip.
For all specimens, there was a significant increase in yield
load Py relative to the respective companion control specimens
PyC. Referring to Table 2, the yield loads for 152, 230, and
305 mm (6, 9, and 12 in.) wide specimens reinforced with
two CFRP strips increased by 29, 18, and 27% over the
control, respectively. Comparing results, the yield load
increase for specimens with two CFRP strips was significantly
higher than for specimens with one CFRP strip. Relative to the
control specimens, the increase in ultimate load Pmax for the
152, 230, and 305 mm (6, 9, and 12 in.) wide specimens was
23, 44, and 61%, respectively. The trend in these values is
consistent with those listed in Table 1, where the gain in
ultimate strength increases with decreasing steel reinforcement
ratio. Thus, in design, the expected additional strength from
the CFRP must consider the existing relative amount of steel
in the unstrengthened condition.
For the 152 mm (6 in.) wide specimens with two CFRP
strips, the average ultimate load was only 6% greater than the
yield load. This indicates that at steel yield, the concrete
strain was near ultimate so that any increase in strength is
limited by the threshold level corresponding to concrete
compression failure. For the 230 and 305 mm (9 and 12 in.)
wide specimens, the average ultimate loads increased by
435
46.
Table 3—Ductility results
Yield
Sample ID
6C
61Fa
61Fb
62Fa
62Fb
9C
91Fa
91Fb
92Fa
92Fb
12C
121Fa
121Fb
122Fa
122Fb
*
E =
†
Δy , mm (in.)
22.17 (0.87)
19.51 (0.77)
23.06 (0.91)
24.66 (0.97)
25.26 (0.99)
21.05 (0.83)
21.14 (0.83)
24.16 (0.95)
20.76 (0.82)
22.15 (0.87)
17.55 (0.69)
19.50 (0.77)
20.56 (0.81)
20.23 (0.80)
19.90 (0.78)
Ey*,
Ultimate
*,
Deflection ductility
kNmm (kipin.)
Δu, mm (in.)
Eu kNmm (kipin.)
μd = Δu/Δy
Ratio†
233 (2.07)
235 (2.08)
2823 (2.50)
353 (3.12)
354 (3.13)
280 (2.48)
323 (2.86)
331 (2.93)
344 (3.05)
323 (2.86)
228 (2.02)
296 (2.62)
317 (2.80)
334 (2.96)
334 (2.95)
30.23 (1.19)
28.98 (1.14)
29.30 (1.15)
26.19 (1.03)
31.04 (1.22)
47.03 (1.85)
36.80 (1.45)
44.45 (1.75)
40.81 (1.61)
47.87 (1.88)
44.68 (1.76)
44.09 (1.74)
47.36 (1.86)
46.10 (1.81)
58.55 (2.31)
395 (3.50)
455 (4.02)
423 (3.74)
389 (3.45)
503 (4.45)
909 (8.05)
729 (6.46)
863 (7.64)
989 (8.75)
1125 (9.96)
845 (6.80)
976 (8.64)
1081 (9.50)
1147 (10.15)
1732 (15.33)
1.36
1.49
1.27
1.06
1.23
2.23
1.74
1.84
1.97
2.16
2.55
2.26
2.30
2.28
2.94
1.00
1.09
0.93
0.78
0.90
1.00
0.78
0.82
0.88
0.97
1.00
0.89
0.90
0.89
1.16
Energy ductility
μE = Eu/Ey
1.69
1.93
1.50
1.10
1.42
3.24
2.26
2.61
2.87
3.49
3.70
3.29
3.42
3.43
5.19
Ratio†
1.00
1.14
0.88
0.65
0.84
1.00
0.70
0.80
0.88
1.08
1.00
0.89
0.92
0.93
1.40
∫ P dΔ .
Ratio = {strengthened sample}/{control sample}.
38% over the yield loads. This is expected and represents the
increased available capacity in the concrete at steel yield. This
behavior is reflective of the relative amounts of both steel and
CFRP reinforcement and how these reinforcement areas
compare with that required for a balancedstrengthened design.
Predicted flexural strength of all specimens with two
CFRP strips was less than measured values, indicating the
analytical model is conservative. Referring to Table 2, the
measured loads were between 3 and 28% greater than
predicted strengths. Thus, the model is an acceptable analytical
tool for strength prediction in design.
Ductility and energy
The reported effect of flexural strengthening with external
FRP reinforcement is a reduction in flexural ductility relative
to the unstrengthened condition (ACI Committee 440 2002,
Bencardino et al. 2002). Typically, ductility is calculated in terms
of dimensionless deflection or energy ratios. Using these parameters ductility μ relative to the yield condition is defined as
Deflection ductility: μd = Δu /Δy
(5a)
Energy ductility: μE = Eu /Ey
(5b)
In Eq. (5) Δu and Δy are the ultimate and yield centerspan
deflections, respectively, and Eu and Ey are the areas under
the loaddeflection diagrams at ultimate and yield, respectively.
Numerical integration of the measured loaddeflection
diagrams was used to determine Eu and Ey. Ductility results
are summarized in Table 3 where it is observed that most
specimens experience a decrease in both deflection ductility and
energy ductility relative to the control beams. The exceptions
are Specimens 61Fa and 122Fb, which experienced an
increase in both deflection and energy ductilities, and
Specimen 92Fb, which experienced a slight increase in
energy ductility. Under closer scrutiny, Specimen 122Fb,
experienced a major crack at approximately 35 kN (7.84 kips).
It could be argued that in a load controlled test this would have
been the ultimate limit state for which Δu, Eu, μd , and μE are
32.3 mm (1.27 in.), 724.2 kNmm (6.41 kin.), 1.62, and
2.17, respectively. This reduces the deflection ductility and
436
energy ductility ratios to 0.64 and 0.60, respectively,
resulting in a decrease in both ductility indexes.
The experimental ductility analysis presented previously
is subjective for two reasons. First, for some specimens, the
yield limit state is not an instantaneous condition that occurs
at a clearly defined load, deflection, or strain. Secondly, the
ultimate limit state is also subject to interpretation. Thus,
depending on the selection for the yield and ultimate limit
states, a range of ductility results can be expected that may
be slightly different from those reported in Table 3. The
general conclusion, however, must be that ductility is
decreased relative to the unstrengthened condition. Further
parametric investigation of ductility using theoretical
modeling to calculate deflection and strain is recommended.
CONCLUSIONS
The research presented in this study evaluated strength and
ductility of steel reinforced concrete beams strengthened with
near surface mounted CFRP strips. Experimental variables
were the amount of steel and CFRP reinforcements. Steel
reinforcement ratios ρs and concrete strength were selected
as typical for existing concrete flexural members that would
be found in nonprestressed bridge and building flexural
members. The conclusions reported are restricted to the
material properties (for concrete and CFRP), reinforcement
ratios (ρs and ρf), type of CFRP (thin rectangular strips), and
testing procedures that were used in this study. From the data
presented, the following conclusions are made.
1. The strengthened beams failed in flexure as predicted
according to the amounts of steel and CFRP reinforcement.
All 152 and 230 mm (6 and 9 in.) wide specimens, and 305 mm
(12 in.) wide specimens with two CFRP strips failed by steel
yield followed by concrete crushing. The CFRP remained
intact at concrete failure and no debonding was detected.
These beams were predicted to fail in compression. The
305 mm (12 in.) wide specimens strengthened with one
CFRP strip failed by steel yield followed by CFRP rupture.
These beams were predicted to fail by CFRP rupture. In all
cases, no debonding of the CFRP was detected;
2. All beams strengthened with CFRP failed at loads
greater than their respective control beams. Relative to
control specimen capacity, CFRP strengthened specimens
ACI Structural Journal/JulyAugust 2007
47.
had measured increases in yield strength ranging from 9 to
30%, and measured increases in ultimate strength ranging
from 10 to 78%. In general, the increase in strength was
inversely proportional to the relative amount of steel
reinforcement normalized to a balanced design ρs /ρsb;
3. The measured ultimate capacity of CFRP strengthened
beams was between 6 and 28% greater than the respective
predicted nominal strength. Nominal strength was calculated
using a simplified closedform analysis that yields identical
results to the trial and error procedure given in ACI 440.2R02.
For unstrengthened beams, the measured ultimate strength was
between 11 and 23% greater than the section’s predicted nominal
strength. These ratios suggest that the CFRP strengthened
section nominal flexural capacity is appropriately predicted using
the simplified closedform or ACI 440.2R02 methodologies;
4. Force transfer between the CFRP, epoxy grout, and
surrounding concrete was able to develop the full tensile strength
of the CFRP strips. Tensile rupture of the single CFRP strip was
achieved in the 305 mm (12 in.) wide specimens with no
apparent slip or damage to the concrete cover or epoxy grout.
For all other specimens where the CFRP did not fail, there
was no apparent loss in force transfer between the CFRP,
epoxy grout, and surrounding concrete. Thus, the CFRP strip’s
thin rectangular cross section and roughened surface provide
an effective mechanism of force transfer with this epoxy; and
5. For the specimens tested, there was no discernable trend
between the change in ductility (energy and deflection) and
the relative amount of steel reinforcement ρs/ρsb or CFRP
strengthening reinforcement Afrp. With the exception of two
strengthened beam, energy and deflection ductilities were
reduced for CFRP strengthened beams.
The authors suggest that additional research is required to
study the strength and ductility behavior of a beam strengthened
with wider range of combinations of steel and FRP reinforcement ratios. Furthermore, NSM FRP splice and bond behavior,
appropriate code mandated design limitations for strength,
deflection, and ductility need to be investigated.
ACKNOWLEDGMENTS
The authors wish to thank Hughes Brothers, Inc., for donating the CFRP
reinforcement and the Office of Research and Sponsored Projects at Villanova
University for providing financial support for this research.
NOTATION
Af , As
Afb
Asy
= area of CFRP and steel reinforcement, respectively
= balancedstrengthened area of CFRP
= steel area corresponding to simultaneous concrete crushing
and steel yielding
a, av
= depth of compression block at ultimate and shear span,
respectively
b, c
= beam width and depth on neutral axis, respectively
df , ds
= depth to CFRP and steel reinforcement, respectively
Ef , fc′
= FRP elastic modulus and concrete strength, respectively
ff , fs
= stress in CFRP and steel, respectively
= ultimate strength of FRP (1648 MPa [239 ksi]) and steel
ffu, fy
yield strength, respectively
ffult
= calculated CFRP stress at sections theoretical moment strength
= theoretical nominal moment strength
Mn
Pn
= theoretical applied load corresponding to Mn
PnC
= theoretical applied load for control specimens corresponding to Mn
Py , Pmax = measured load at steel yield and ultimate, respectively
PyC, PmaxC = measured load for control specimen at steel yield and ultimate,
respectively
Tf , T s
= tensile force in CFRP and steel, respectively
wbeam
= selfweight of beam
= ratio of a/c
β1
εf, εs
= strain in CFRP and steel, respectively
= ultimate strain of concrete (0.003) and FRP (0.012), respectively
εcu, εfu
ACI Structural Journal/JulyAugust 2007
ρs, ρf
ρsb
= steel As/bds and CFRP Af /bdf reinforcement ratio, respectively
= balanced steel reinforcement ratio for unstrengthened section
REFERENCES
ACI Committee 440, 2002, “Design and Construction of Externally
Bonded FRP Systems for Strengthening Concrete Structures (ACI 440.2R02),”
American Concrete Institute, Farmington Hills, Mich., 45 pp.
ACI Committee 440, 2004, “Guide Test Methods of FiberReinforced
Polymers (FRPs) for Reinforcing or Strengthening Concrete Structures (ACI
440.3R04),” American Concrete Institute, Farmington Hills, Mich., 40 pp.
ASTM C 68499, 1999, “Standard Test Method for Making, Accelerated
Curing, and Testing Concrete Compression Test Specimens,” ASTM
International, West Conshohocken, Pa., 10 pp.
Arduini, M., and Nanni, A., 1997, “Behavior of Precracked RC Beams
Strengthened with Carbon FRP Sheets,” Journal of Composites for
Construction, ASCE, V. 1, No. 2, pp. 6370.
Bencardino, F.; Spadea, G.; and Swamy, R., 2002, “Strength and Ductility of
Reinforced Concrete Beams Externally Reinforced with Carbon Fiber Fabric,”
ACI Structural Journal, V. 99, No. 2, Mar.Apr., pp. 163171.
Brena, S. F.; Bramblett, R. M.; Wood, S. L.; and Kreger, M. E., 2003,
“Increasing Flexural Capacity of Reinforced Concrete Beams Using Carbon
FiberReinforced Polymer Composites,” ACI Structural Journal, V. 100,
No. 1, Jan.Feb., pp. 3646.
DeLorenzis, L. A.; Nanni, A.; and Tegila, A. L., 2000, “Flexural and
Shear Strengthening of Reinforced Concrete Structures with Near Surface
Mounted FRP Bars,” Proceedings of the 3rd International Conference on
Advanced Composite Materials in Bridges and Structures, Ottawa, Canada,
Aug. 1518, pp. 521528.
DeLorenzis, L., and Nanni, A., 2001, “Shear Strengthening of Reinforced
Concrete Beams with NearSurface Mounted FiberReinforced Polymer
Rods,” ACI Structural Journal, V. 98, No. 1, Jan.Feb., pp. 6068.
DeLorenzis, L., and Nanni, A., 2002, “Bond between NearSurface
Mounted FiberReinforced Polymer Rods and Concrete in Structural
Strengthening,” ACI Structural Journal, V. 99, No. 2, Mar.Apr., pp. 123132.
DeLorenzis, L.; Lundgren, K.; and Rizzo, A., 2004, “Anchorage Length
of NearSurface Mounted FiberReinforced Polymer Bars for Concrete
Strengthening—Experimental Investigation and Numerical Modeling,”
ACI Structural Journal, V. 101, No. 2, Mar.Apr., pp. 269278.
ElHacha, R., and Rizkalla, S., 2004, “NearSurfaceMounted FiberReinforced Polymer Reinforcements for Flexural Strengthening of Concrete
Structures,” ACI Structural Journal, V. 101, V. 5, Sept.Oct., pp. 717726.
Grace, N.; AbdelSayed, G.; and Ragheb, W., 2002, “Strengthening of
Concrete Beams Using Innovative Ductile FiberReinforced Polymer Fabric,”
ACI Structural Journal, V. 99, No. 5, Sept.Oct., pp. 692700.
Mukhopadhyaya, P., and Swamy, R. N., 1999, “Critical Review of Plate
Anchorage Stresses in Premature Debonding Failures of Plate Bonded
Reinforced Concrete Beams,” Fourth International Symposium on Fiber
Reinforced Polymer Reinforcement for Reinforced Concrete Structures, SP188,
C. W. Dolan, S. H. Rizkalla, and A. Nanni, eds., American Concrete Institute,
Farmington Hills, Mich., pp. 359368.
Nanni, A., 2000, “FRP Reinforcement for Bridge Structures,” Proceedings,
Structural Engineering Conference, University of Kansas, Lawrence,
Kans., Mar. 16, pp. 15.
Nguyen, D.; Chan, T.; and Cheong, H., 2001, “Brittle Failure and Bond
Development Length of CFRPConcrete Beams,” Journal of Composites
for Construction, ASCE, V. 5, No. 1, pp. 1217.
Pennsylvania Department of Transportation (PennDOT), 2001, “The
Bridge Design Specification Sheet, BD601M,” Specifications for the
Concrete, Class AAA.
Rahimi, H., and Hutchinson, A., 2001, “Concrete Beams Strengthened
with Externally Bonded FRP Plates,” Journal of Composites for Construction,
ASCE, V. 5, No. 1, Jan., pp. 4455.
Saadatmanesh, H., 1994, “Fiber Composites for New and Existing
Structures,” ACI Structural Journal, V. 91, No. 3, MayJune, pp. 346354.
Sharif, A.; AlSulaimani, G. J.; Basunbul, I. A.; Baluch, M. H.; and
Ghaleb, B. N., 1994, “Strengthening of Initially Loaded Reinforced
Concrete Beams Using FRP Plates,” ACI Structural Journal, V. 91, No. 2,
Mar.Apr., pp. 160168.
Shin, Y. S.; and Lee, C., 2003, “Flexural Behavior of Reinforced Concrete
Beams Strengthened with Carbon FiberReinforced Polymer Laminates at
Different Levels of Sustaining Load,” ACI Structural Journal, V. 100, No. 2,
Mar.Apr., pp. 231239.
Taljsten, B., and Carolin, A., 2001, “Concrete Beams Strengthened with
Near Surface Mounted CFRP Laminates,” Proceedings of the NonMetallic
Reinforcement for Concrete Structures, FRP RCS5 Conference, July 1618,
Cambridge, UK, pp. 107116.
Teng, J. G.; Chen, J. F.; Smith, S. T.; and Lam, L., 2002, FRPStrengthened
RC Structures, John Wiley & Sons, West Sussex, UK, 266 pp.
437
49.
Table 1—Details of test specimens
KeunHyeok Yang is a Visiting Research Fellow at the University of Bradford, UK,
and an Assistant Professor at Mokpo National University, Korea. He received his MSc
and PhD from Chungang University, Korea. His research interests include ductility,
strengthening, and shear of reinforced, highstrength concrete structures.
HeonSoo Chung is a Professor at Chungang University, Korea. He received his MSc
and PhD from Tokyo Institute of Technology, Japan. His research interests include
flexure, shear, and bond behavior of reinforced, highstrength concrete members.
Ashraf F. Ashour is a Senior Lecturer at the University of Bradford, UK. He received
his BSc and MSc from Mansoura University, Egypt, and his PhD from Cambridge
University, UK. His research interests include shear, plasticity, and optimization of
reinforced concrete and masonry structures.
Details of shear reinforcement
fc′ ,
Specimen MPa
Horizontal
a/h
a/jd L, mm sh, mm
ρh
Vertical
sv , mm
ρv
L5NN
—
—
—
—
L5NS
—
—
120
0.003
0.006
L5NT
—
—
60
120
0.003
—
—
L5SS
120
0.003
120
0.003
—
L5SN
32.4
0.5
0.58
600
L5TN
60
0.006
—
L10NN
—
—
—
—
L10NS
—
—
120
0.003
0.006
L10NT
—
—
60
120
0.003
—
—
L10SS
120
0.003
120
0.003
L10TN
60
0.006
—
—
H6NN
—
—
—
—
H6NS
—
—
120
0.003
0.006
L10SN
H6NT
32.1
1.0
1.17
1200
—
—
60
120
0.003
—
—
H6SS
120
0.003
120
0.003
—
H6SN
65.1
0.6
0.7
720
H6TN
60
0.006
—
H10NN
—
—
—
—
H10NS
—
—
120
0.003
0.006
H10NT
—
—
60
120
0.003
—
—
H10SS
120
0.003
120
0.003
H10TN
60
0.006
—
—
H10SN
68.2
1.0
1.17
1200
Note: 1 MPa = 145 psi; 1 mm = 0.039 in.
Fig. 1—Geometrical dimensions and reinforcement of test
specimens. (Note: all dimensions are in mm and • indicates
locations of strain gauges. 1 mm = 0.039 in.)
configuration of shear reinforcement included four different
arrangements as shown in Fig. 1: none, only vertical, only
horizontal, and orthogonal reinforcement. The spacing of shear
reinforcement was chosen to be 60 and 120 mm (2.36 and
4.72 in.) and the corresponding shear reinforcement ratios, ρ
(= Aw /bw s, where Aw equals the area of shear reinforcement
at spacing s, and bw equals the beam width), were 0.003 and
0.006, respectively, to satisfy the maximum spacing specified in ACI 31805, Section 11.8, and the minimum amount
recommended in ACI 31805, Section A.3.3.2. The beam
notation given in Table 1 includes four parts. The first part
refers to the concrete design strength: L for low compressive
strength and H for high compressive strength. The second
part is used to identify the a/h. The third and fourth parts give
the amount of horizontal and vertical shear reinforcement,
ACI Structural Journal/JulyAugust 2007
respectively: N for no shear reinforcement, and S and T for
shear reinforcement ratios of 0.003 and 0.006, respectively.
For example, L5SS is a continuous deep beam having design
concrete strength of 30 MPa (4350 psi), an a/h of 0.5, and both
horizontal and vertical shear reinforcement ratios of 0.003.
All beams tested had the same section width bw of 160 mm
(6.3 in.) and overall section depth h of 600 mm (23.6 in.). Both
longitudinal top, ρs = (As′ /bwd), and bottom, ρs = (As /bwd),
′
reinforcement ratios were kept constant in all beams as 1%,
which were calculated from nonlinear FE analysis,15 to ensure
no flexural yielding of longitudinal reinforcement prior to
failure of concrete struts. The length of each span L varied
according to a/h, as given in Table 1. The clear covers to
longitudinal top and bottom reinforcement, and shear reinforcement were 35 and 29 mm (1.38 and 1.14 in.), respectively. The
longitudinal bottom reinforcement was continuous over the
full length of the beam and welded to 160 x 100 x 10 mm (6.3
x 3.9 x 0.39 in.) end plates, whereas longitudinal top reinforcement was anchored in the outside of the exterior
supports by 90degree hooks according to ACI 31805. The
vertical shear reinforcement was closed stirrups and the
horizontal shear reinforcement with 90 degree hooks was
arranged along the longitudinal axis in both sides of the beams.
Material properties
The mechanical properties of reinforcement are given in
Table 2. All longitudinal and shear reinforcing bars were
deformed bars of a 19 mm (0.75 in.) diameter, having a
nominal area of 287 mm2 (0.44 in.2) and yield strength of
562 MPa (81.6 ksi) and a 6 mm (0.23 in.) diameter, having a
nominal area of 28.2 mm2 (0.04 in.2) and yield strength of
421
50.
Fig. 2—Test setup. (Note: all dimensions are in mm. 1 mm =
0.039 in.)
Table 2—Mechanical properties of reinforcement
fy, MPa
εy
fsu, MPa
Es, GPa
*
6
483
0.0044
549
199
19
562
0.00284
741
198
Diameter, mm
*
Yield stress of 6 mm diameter reinforcement was obtained by 0.2% offset method.
Note: 1 mm = 0.039 in.; 1 MPa = 145 psi.
483 MPa (70 ksi), respectively. The yield strength of 6 mm
(0.23 in.) diameter reinforcement was obtained by 0.2%
offset method.
The ingredients of ready mixed concrete were ordinary
portland cement, fly ash, irregular gravel of a maximum size
of 25 mm (0.98 in.), and sand. The waterbinder ratios of the
Lseries added with fly ash of 12% and of the Hseries added
with fly ash of 20% were 0.41 and 0.27, respectively. All
specimens were cast in a vertical position in the same
wooden mold. Control specimens, which were 100 mm
(3.94 in.) diameter by 200 mm (7.87 in.) high cylinders, were
cast and cured simultaneously with beams to determine the
compressive strength. They were tested soon after the beam
test. The results of the cylinder compressive strength given
in Table 1 are the average value from testing nine cylinders.
Test setup
Loading and instrumentation arrangements are shown in
Fig. 2. All beams having two spans were tested to failure
under a symmetrical twopoint top loading system with a
loading rate of 30 kN/minute (6.7 kip/minute) using a 3000 kN
(675 kip) capacity universal testing machine (UTM). Each
span was identified as Espan or Wspan, as shown in Fig. 1.
The two exterior end supports were designed to allow
horizontal and rotational movements, whereas the intermediate
support prevented horizontal movement but allowed rotation.
To evaluate the shear force and loading distribution, 1000 kN
(225 kip) capacity load cells were installed in both exterior
end supports. At the location of loading or support point, a
steel plate of 100, 150, or 200 mm (3.94, 5.9, or 7.88 in.)
wide was provided to prevent premature crushing or bearing
failure, as shown in Fig. 2. All steel plates were 50 mm (1.97 in.)
thick and 300 mm (11.8 in.) long to cover the full width of
test specimen. All beams were preloaded up to a total load of
150 kN (33.7 kip) before testing, which wouldn’t produce
any cracks, to assure a similar loading distribution to supports
according to the result of the linear twodimensional finite
element (2D FE) analysis.
Vertical deflections at a distance of 0.45L to 0.47L from
the exterior support, which is the location of the maximum
422
Fig. 3—Crack patterns and failure of concrete strut. Numbers
indicate total load in kN at which crack occurred. (Note:
1 kN = 0.2248 kips.)
deflection predicted by the linear 2D FE analysis, and at the
midspan of each span were measured using linear variable
differential transformers (LVDTs). Both surfaces of the
beams tested were whitewashed to aid in the observation of
crack development during testing. The inclined crack width
of concrete struts joining the edges of load and support plates
was monitored by the πshape displacement transducers (PI
gauges) as shown in Fig. 2. The strains of shear reinforcement
were measured by 5 mm (0.2 in.) electrical resistance strain
gauges (ERS) at the region crossing the line joining the edges
of load and intermediate support plates as shown in Fig. 1. At
each load increment, the test data were captured by a data
logger and automatically stored.
Support settlements
Continuous deep beams are sensitive to differential
support settlements causing additional moment and shear.
To assess the effect of differential settlements on the beams
tested, a linear 2D FE analysis considering shear deformation
effect was performed on the beams shown in Fig. 1. For the
beams tested, sources of relative support settlements were
the elastic shortening of the load cell and plates and elastic
deformation of the bed of the testing machine. The second
moment of area of the testing machine bed cross section
about the bending axis was 3.2 × 1010 mm4 (7.69 × 104 in.4),
then the elastic deformation under a point load R (in kN) at a
distance 1500 mm (59 in.) from the center of the testing
machine is 0.000176R mm. The amount of elastic shortening
due to a load at the exterior and intermediate supports
involving the load cell and plates was considered in
designing the support size as follows. When a/h is 0.5, the
reactions of the exterior and intermediate supports due to the
total applied load P, from the linear 2D FE analysis, are 0.2P
and 0.6P, respectively. As the height of the intermediate
support was equal to that of the exterior load cell, the contact
area of the intermediate support with the bed of the testing
machine was designed to be three times wider than that of the
load cell at the exterior support to produce the same elastic
shortening. The pilot test results showed that the maximum
settlement of the exterior support relative to the intermediate
support was in order of L/25,000. For a differential settlement
between the exterior and intermediate supports of L/25,000,
the maximum additional shear forces obtained from linear 2D
FE analysis are 25 and 7 kN (5.62 and 1.57 kip) for beams
ACI Structural Journal/JulyAugust 2007
51.
Table 3—Details of test results and predictions obtained from ACI 31805
Failure load Pn and ultimate shear
force (Vn)I at interior shear spans, kN
Load Pcr and shear force Vcr at first diagonal crack, kN
Wspan
Interior
Specimen (Pcr)I
(Vcr)I
Espan
Exterior
(Pcr)E (Vcr)E
Interior
(Pcr)I
(Vcr)I
ACI 31805
(Vn)I
Exterior
(Pcr)E (Vcr)E
Pn
Wspan
Espan
(Pn)Exp./ (Vn)IExp./
Pn, kN (Vn)I, kN (Pn)ACI (Vn)IACI
L5NN
852
255
902
180
816
244
937
187
1635
473
456*
1298
342
1.260
1.334
L5NS
849
247
1028
210
857
262
1330
281
1710
486
475*
1298
342
1.317
1.389
L5NT
1017
278
1380
284
850
230
1260
262
1789
512*
494
1298
342
1.378
1.498
L5SN
864
255
1268
252
867
257
927
179
1887
537*
546
1298
342
1.454
1.571
*
L5SS
814
247
990
192
980
293
1020
202
2117
607
583
1623
427
1.305
1.420
L5TN
912
266
1130
230
910
278
966
185
2317
655
*
640
1298
342
1.785
1.872
L10NN
537
173
—
—
537
171
—
—
880
264*
262
1000
265
0.880
0.997
L10NS
477
156
—
—
596
195
—
—
1153
349
*
348
1000
265
1.153
1.314
L10NT
635
206
1023
230
647
208
—
—
1541
446*
439
1000
265
1.541
1.684
L10SN
498
153
—
—
490
151
782
146
884
266
265*
1000
265
0.884
1.000
L10SS
521
166
—
—
452
148
713
129
1177
357
352*
1250
331
0.942
1.063
287
288*
1000
265
0.935
1.087
L10TN
538
175
—
—
621
193
775
143
935
*
H6NN
1046
305
1562
321
1236
303
1960
407
2248
633
634
2520
668
0.892
0.950
H6NS
1261
379
1646
316
978
300
2280
457
2289
684
683*
2520
668
0.908
1.023
H6NT
1116
324
2550
550
915
264
2480
531
2625
757
757*
2520
668
1.042
1.134
H6SN
1322
393
2420
517
1022
297
2420
513
2427
703*
708
2520
668
0.963
1.053
H6SS
1207
367
2630
548
825
256
2630
542
2763
792
799*
3150
834
0.877
0.958
H6TN
1442
439
—
—
980
297
2648
540
2966
854
852*
2520
668
1.177
1.276
H10NN
690
228
868
149
690
228
840
143
1276
373
372*
2124
563
0.601
0.661
H10NS
759
237
—
—
751
234
—
—
1443
413*
414
2124
563
0.679
0.734
H10NT
788
251
—
—
717
224
—
—
2116
638
637*
2124
563
0.996
1.132
H10SN
757
255
—
—
757
252
—
—
1309
387*
378
2124
563
0.616
0.688
H10SS
718
232
—
—
768
244
—
—
1575
492*
484
2655
703
0.593
0.699
393
*
2124
563
0.606
0.689
H10TN
754
234
—
—
704
220
—
—
1287
388
*
Failure occurred in this shear span.
Note: 1 kN = 0.2248 kips.
having an a/h of 0.5 and 1.0, respectively. This indicates that
the differential settlement had no significant effect on the
test arrangement.
EXPERIMENTAL RESULTS AND DISCUSSION
Crack propagation and failure mode
The crack propagation was significantly influenced by the
a/h as shown in Fig. 3 and Table 3. The crack pattern in the
Lseries was similar to that in the Hseries; therefore, it is not
shown in Fig. 3. For beams with a/h = 0.5, the first crack
suddenly developed in the diagonal direction at approximately
40% of the ultimate strength at the middepth of the concrete
strut within the interior shear span, and then a flexural crack
in the sagging region immediately followed. The first flexural
crack over the intermediate support generally occurred at
approximately 80% of the ultimate strength, and was less
than 0.2h deep at failure. As the load increased, more flexural
and diagonal cracks were formed and a major diagonal crack
extended to join the edges of the load and intermediate
support plates. A diagonal crack within the exterior shear
span occurred suddenly near the failure load. Cracks in
beams with a/h = 1.0 developed in a different order from that
described previously for beams with a/h = 0.5. In those
beams, the first crack occurred vertically in the hogging
ACI Structural Journal/JulyAugust 2007
zone, followed by a diagonal crack in the interior shear span,
and then a vertical crack took place in the sagging zone, but
diagonal cracks within exterior shear spans were seldom
developed. The influence of shear reinforcement on the first
flexural and diagonal crack loads was not significant (refer
to Table 3) as also observed in simple deep beams given in
Appendix A.
Just before failure, the two spans showed nearly the same
crack patterns. All beams developed the same mode of
failure as observed in other experiments.3 The failure planes
evolved along the diagonal crack formed at the concrete strut
along the edges of the load and intermediate support plates.
Two rigid blocks separated from original beams at failure
due to the significant diagonal cracking. An end block
rotated about the exterior support leaving the other block
fixed over the other two supports as shown in Fig. 3.
Load versus midspan deflection
The beam deflection at midspan was less than that
measured at 0.45L to 0.47L from the exterior support until
the occurrence of the first diagonal crack as predicted by the
2D FE analysis. After the first diagonal crack, however, the
midspan deflection was higher. Therefore, the midspan
423
52.
Fig. 5—Total applied load versus support reactions for Lseries
beams tested having a/ h of 0.5. (Note: 1 kN = 0.2248 kips.)
This stiffness reduction was prominent in case of lower
concrete strength and higher a/h.
Support reaction
Figure 5 shows the amount of the load transferred to the
end and intermediate supports against the total applied load
in the Lseries beams having a/h = 0.5. On the same figure,
the support reactions obtained from the linear 2D FE analysis
are also presented. The end and intermediate support reactions
of the Lseries beams having a/h = 1.0 and the Hseries
beams were similar to those of the Lseries beams having an
a/h = 0.5; therefore, not presented herein. Before the first
diagonal crack, the relationship of the end and intermediate
support reactions against the total applied load in all beams
tested shows good agreement with the prediction of the
linear 2D FE analysis. The amount of loads transferred to
the end support, however, was slightly higher than that
predicted by the linear 2D FE analysis after the occurrence
of the first diagonal crack within the interior shear span. At
failure, the difference between the measured end support
reaction and prediction of the linear 2D FE analysis was in
order of 7 and 12%, for beams with a/h = 0.5 and a/h = 1.0,
respectively. The distribution of applied load to supports was
independent of the amount and configuration of shear
reinforcement. This means that, although after the occurrence
of diagonal cracks the beam stiffness has reduced, as shown
in Fig. 4, the internal redistribution of forces is limited.
Fig. 4—Total load versus midspan deflection. (Note: 1 kN =
0.2248 kips.)
deflection of the failed span for different beams tested are
only presented in Fig. 4 against the total applied load: Fig. 4(a)
for beams in the Lseries and Fig. 4(b) for beams in the Hseries.
The initial stiffness of beams tested increased in accordance
with the increase of concrete strength and the decrease of the
a/h, but it seems to be independent of the amount and
configuration of shear reinforcement. The development of
flexural cracks in sagging and hogging zones has little influence
on the stiffness of beams tested. But the occurrence of diagonal
cracks in the interior shear span caused a sharp decrease in
the beam stiffness and an increase of the beam deflection.
424
Width of diagonal crack
Figure 6 shows the variation of the diagonal crack width in
the interior shear span according to the configuration of
shear reinforcement: Fig. 6(a) at the first diagonal cracking
load and Fig. 6(b) at the same load as the ultimate failure load
of the corresponding deep beam without shear reinforcement.
For the same concrete compressive strength, the larger the a/h,
the wider the diagonal crack width. Shear reinforcement had
an important role in restraining the development of the diagonal
crack width, which significantly depended on the a/h. A
more prominent reduction of diagonal crack width appeared
in beams with horizontal shear reinforcement only or
orthogonal shear reinforcement than in beams with vertical
shear reinforcement only when a/h was 0.5. On the other
hand, for beams with a/h = 1.0, a smaller diagonal crack
width was observed in beams with vertical shear reinforcement
only than in beams with orthogonal shear reinforcement, even
though the total shear reinforcement ratio in these beams was
the same (ρv + ρh = 0.006). It seems possible to reduce the
diagonal crack width by more than twice if shear reinforcement
is suitably arranged according to the variation of a/h.
ACI Structural Journal/JulyAugust 2007
53.
Fig. 7—Total load versus strains in shear reinforcement for
beams in Hseries. (Note: 1 kN = 0.2248 kips.)
Fig. 6—Configuration of shear reinforcement versus diagonal
crack width. (Note: 1 mm = 0.039 in.)
Figure 7 shows the strain in shear reinforcement against
the total applied load in the Hseries beams: Fig. 7(a) for
vertical shear reinforcement in beams having either vertical
or orthogonal shear reinforcement, and Fig. 7(b) for horizontal
shear reinforcement in beams having either horizontal or
orthogonal shear reinforcement. The relation between strains
in shear reinforcement and the total applied load in the Lseries
beams was similar to that in the Hseries beams; therefore,
not presented herein. The strains of shear reinforcement
were recorded by ERS gauges at different locations, as
shown in Fig. 1. Shear reinforcement was not generally
strained at initial stages of loading. However, strains suddenly
increased with the occurrence of the first diagonal crack. In
beams with a/h = 0.6, only horizontal reinforcing bars yielded,
whereas in beams with a/h = 1.0, only vertical reinforcing
bars yielded. This indicates that the reinforcement ability to
transfer tension across cracks strongly depends on the angle
between the reinforcement and the axis of the strut.
Ultimate shear stress
The normalized ultimate shear strength, λ = Vn/bwd f c′ ,
plotted against a/h, is given in Fig. 8: Fig. 8(a) for simply
supported deep beams given in Appendix A, and Fig. 8(b)
for continuous deep beams including the test results of
Rogowsky et al.1 and Ashour.2 It can be seen that the ultimate
shear strength of all beams without or with shear reinforcement
dropped due to the increase of a/h. The reduction of the ultimate
shear strength was also dependent on the configuration of shear
reinforcement. For deep beams without shear reinforcement,
the normalized ultimate shear strength λ in continuous deep
beams was less than that in simply supported ones by an
average of 26% due to higher transverse tensile strains
produced by the tie action of longitudinal top and bottom
ACI Structural Journal/JulyAugust 2007
Fig. 8—Normalized ultimate shear strength versus shear
spantooverall depth ratio.
reinforcement. When shear reinforcement is provided, the
normalized ultimate shear strength λ in continuous deep
beams matched that of the corresponding simply supported
ones. The influence of the horizontal and vertical shear
reinforcement on the ultimate shear strength is influenced by
the a/h. The lower the a/h, the more effective the horizontal
shear reinforcement and the less effective the vertical shear
reinforcement. When a/h was below 0.6, the shear strength
425
54.
Vn – ( Vn )W ⁄ O
Vs
 = Vn
Vn
(2)
The variations of Vs /Vn at the failed shear span against the
increase of a/h are given in Fig. 9: Fig. 9(a), (b), and (c) for
beams with vertical shear reinforcement only, with horizontal
shear reinforcement only, and with orthogonal shear reinforcement, respectively. On the same figure, the test results of
simply supported deep beams given in Appendix A, which
had the same material and geometrical properties as continuous
deep beams tested in the current study, are also presented.
The load transfer capacity of shear reinforcement is more
pronounced in continuous deep beams than that in simple ones.
The load transfer capacity of shear reinforcement is dependent
on a/h. The load transfer capacity of vertical shear reinforcement
was higher in beams having a/h = 1.0 than those having
a/h = 0.5 as shown in Fig. 9(a). On the other hand, the load
transfer capacity of horizontal shear reinforcement was
higher in beams having a/h = 0.5 than those having a/h = 1.0,
as shown in Fig. 9(b). Existing test results of continuous
deep beams carried out by Rogowsky et al.1 and Ashour,2
and the comments of ACI 31805, Section 11.8, have
suggested that horizontal shear reinforcement has little
influence on the shear strength improvement and crack
control. In the current tests, horizontal shear reinforcement is
more effective than vertical shear reinforcement for beams
with a/h of 0.5, as shown in Fig. 8 and 9.
Comparison with current codes
It has been shown by several researchers,1,2,4 that the
shear capacity prediction of reinforced concrete deep beams
obtained from ACI 3189916 (unchanged since 1983) was
unconservative. For the design of deep beams, ACI 31805
requires the use of either nonlinear analysis or strutandtie
model. Figure 10 shows a schematic strutandtie model of
continuous deep beams in accordance with ACI 31805,
Appendix A. The strutandtie model shown in Fig. 10 identifies
two main load transfer systems: one of which is the strutandtie
action formed with the longitudinal bottom reinforcement
acting as a tie and the other is the strutandtie action due to
the longitudinal top reinforcement. As the applied loads in
the twospan continuous deep beams are carried to supports
through concrete struts of exterior and interior shear spans
(refer to Fig. 10), the total load capacity of twospan continuous
deep beams Pn due to failure of concrete struts is
Fig. 9—Shear reinforcement ratios versus Vs / Vn.
Pn = 2(FE – FI)sinθ
of deep beams with minimum horizontal shear reinforcement
had an average value of 150% higher than the upper bound
value, 0.83 f c′ bwd, specified in ACI 31805, Section 11.8.3.
Load transfer capacity of shear reinforcement
The shear strength of deep beams Vn can be described
as follows
Vn = Vc + Vs
where FE and FI equal the load capacities of exterior and
interior concrete struts, respectively, and θ equals the angle
between the concrete strut and the longitudinal axis of the
deep beam, which can be expressed as tan–1(jd/a). The
distance between the center of top and bottom nodes jd could
be approximately assumed as the distance between the center
of longitudinal top and bottom reinforcing bars as
(1)
jd = h – c – c′
where Vc and Vs equal the load capacity of concrete and load
transfer capacity of shear reinforcement, respectively.
As the load capacity of concrete is usually regarded as the
strength of beams without shear reinforcement, (Vn)W/O, the
ratio of the load transfer capacity of shear reinforcement to
the shear strength of beams Vs/Vn is
426
(3)
(4)
where h equals the overall section depth and c and c′ equal
the cover of longitudinal bottom and top reinforcement,
respectively, as shown in Fig. 10.
The nodes at the applied load point could be classified as
a CCC type, which is a hydrostatic node connecting both
ACI Structural Journal/JulyAugust 2007
55.
exterior and interior compressive struts in sagging zone and
a CCT type for longitudinal top reinforcement in the hogging
zone. It was proved by Marti10 that the width of the strut at a
CCC node is in proportion to the principal stress normal to
the node face to make the state of stress in the whole node
region constant. To accommodate both CCC type and CCT
type, the loading plate width can be assumed to be subdivided
into two parts in accordance with the ratio of the exterior
reaction to the applied load β, each to form the node
connecting the exterior and the interior struts, respectively.
The β values of tested beams are 0.4 and 0.346 when a/h
ratios are 0.5 and 1.0, respectively, as estimated from the
linear 2D FE analysis. If enough anchorage of longitudinal
reinforcement is provided, average widths of concrete struts
in interior (ws)I and exterior shear spans (ws)E are
( w t ′ + 2c′ ) cos θ + [ 0.5 ( l p ) I + ( 1.0 – β ) ( l p ) P ] sin θ
( w s ) I = 2
(5a)
( w t ′ + 2c′ ) cos θ + [ ( l p ) E + β ( l p ) P ] sin θ
( w s ) E = 2
Fig. 10—Qualitative strutandtie model of continuous deep
beams according to ACI 31805.
(5b)
where (lp)P, (lp)E, and (lp)I equal the widths of loading, exterior
support, and interior support plates, respectively, and wt′
equals the smaller of the height of the plate anchored to
longitudinal bottom reinforcement wt and twice of the cover
of longitudinal bottom reinforcement 2c as shown in Fig. 10.
The load transfer capacity of the concrete strut depends on
the area of the strut and the effective concrete compressive
strength. Hence, the load capacities of the exterior and interior
concrete struts are
FE = ve f ′cbw(ws)E
(6a)
FI = ve f ′cbw(ws)I
(6b)
where ve equals the effectiveness factor of concrete. The
shear capacity at the interior shear span (Vn)I, where the
failure is expected to occur in continuous deep beams, can be
calculated from FI sinθ.
The minimum amount of shear reinforcement required in
bottleshaped struts, which is recommended to be placed in
two orthogonal directions in each face, is suggested by
ACI 31805 as follows
A si
∑ i sin αi ≥ 0.003
bw s
(7)
Fig. 11—Comparison of test results and predictions by
ACI 31805.
where Asi and si equal the total area and spacing in the ith
layer of reinforcement crossing a strut, respectively, and αi
equals the angle between ith layer of reinforcement and
the strut.
The effectiveness factors for concrete strength not
exceeding 40 MPa (5.8 ksi) in ACI 31805 are suggested as
0.75 and 0.6 when shear reinforcement satisfying Eq. (7) is
arranged and is not provided, respectively. The truss model
representing the load transfer mechanism of horizontal and
vertical shear reinforcement is not included in ACI 31805.
This implies that shear reinforcement satisfying Eq. (7)
enables the strength of beams to be increased by 25%.
Comparisons between test results and predictions obtained
from the strutandtie model recommended by ACI 31805
as developed previously are shown in Table 3 and Fig. 11:
Fig. 11(a) for simple deep beams given in Appendix A and
Fig. 11(b) for continuous deep beams including Rogowsky
et al.’s and Ashour’s test results. In simple deep beams, the
width of the strut can be calculated from wt′cosθ + (lp)Esinθ,
and the total load capacity is 2FEsinθ. Although Eq. (7)
proposed by ACI 31805 is recommended for deep beams
having concrete strength of less than 40 MPa, the load
capacity of the Hseries beams were also predicted using this
equation to evaluate its conservatism in case of highstrength
concrete deep beams. The mean and standard deviation of
the ratio, (Pn)Exp./(Pn)ACI, between the experimental and
predicted load capacities are 1.229 and 0.326, respectively,
for simply supported deep beams, and 0.969 and 0.306,
ACI Structural Journal/JulyAugust 2007
427
56.
respectively, for twospan continuous deep beams as shown
in Fig. 11. The ratio of the test result to prediction generally
dropped with the increase of a/h. This decrease rate was
more remarkable in continuous deep beams than that in
simple ones. In particular, the predictions for several continuous
deep beams having a/h exceeding 1.0 were unconservative,
even though the effectiveness factor used in the beams with
either horizontal or vertical shear reinforcement was 0.6
regardless of the amount of shear reinforcement. In addition,
for highstrength concrete continuous deep beams having a/h =
1.0, the ratio, (Vn)IExp./(Vn)IACI, between the experimental and
predicted shear capacities in the interior shear span was
generally below 1.0 as given in Table 3; namely, the strutandtie model recommended by ACI 31805 overestimated
the shear capacity of highstrength concrete continuous deep
beams having a/h = 1.0.
CONCLUSIONS
Tests were performed to study the influence of the amount
and configuration of shear reinforcement on the structural
behavior of continuous deep beams according to the variation
of concrete strength and a/h. The following conclusions
are drawn:
1. In beams having a/h of 0.6, only horizontal shear
reinforcement reached its yield strength with a sharp
increase of stress after the first diagonal crack. On the other
hand, only vertical shear reinforcement yielded in beams
with a/h of 1.0;
2. For deep beams without shear reinforcement, the
normalized ultimate shear strength was 26% lower in
continuous beams than that in simple ones. When shear
reinforcement was provided, however, the normalized ultimate
shear strength in continuous deep beams matched that in
simply supported deep beams;
3. The load transfer capacity of all shear reinforcement
was much more prominent in continuous deep beams than
that in simple ones. Horizontal shear reinforcement was
always more effective than vertical shear reinforcement
when the a/h was 0.5. However, vertical shear reinforcement
was more effective for a/h higher than 1.0;
4. In deep beams with a/h not exceeding 0.6, the critical
upper bound on shear strength suggested in ACI 31805,
0.83 f c ′ bwd, highly underestimated the actual measured
shear strength, as if it was a lower limit; and
5. The ratios of measured load capacity to that obtained
from the strutandtie model recommended by ACI 31805
dropped with the increase of the a/h. This decrease rate was
more remarkable in continuous deep beams than that in
simple ones. The strutandtie model recommended by ACI
31805 overestimated the shear capacity of highstrength
concrete continuous deep beams having a/h more than 1.0.
ACKNOWLEDGMENTS
This work was supported by the Korea Research Foundation Grant
(KRF2003041D00586) and the Regional Research Centers Program
(Biohousing Research Institute), granted by the Korean Ministry of Education
and Human Resources Development. The authors wish to express their
gratitude for financial support.
NOTATION
Ah
As
As′
Aw
a
bw
428
=
=
=
=
=
=
area of horizontal shear reinforcement
area of longitudinal bottom reinforcement
area of longitudinal top reinforcement
area of shear reinforcement
shear span
width of beam section
c
c′
d
h
Es
FE
FI
fc′
fsu
fy
jd
L
lp
Pcr
Pn
sh
sv
T
Vc
Vcr
Vn
Vs
ve
ws
wt
α
β
εy
λ
θ
ρh
ρst
ρst
′
ρv
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
cover of longitudinal bottom reinforcement
cover of longitudinal top reinforcement
effective depth of beam section
overall depth of beam section
elastic modulus of steel
load capacity of concrete strut in exterior shear span
load capacity of concrete strut in interior shear span
concrete compressive strength
tensile strength of reinforcement
yield strength of reinforcement
distance between center of top and bottom nodes
span length
width of loading plate
diagonal crack load
ultimate load at failure
spacing of horizontal shear reinforcement
spacing of vertical shear reinforcement
tensile force in longitudinal reinforcement
load capacity of concrete
diagonal crack shear force
ultimate shear force at failure
load transfer capacity of shear reinforcement
effectiveness factor of concrete
width of concrete strut
height of plate anchored to longitudinal reinforcement
angle between shear reinforcement and axis of concrete strut
ratio of exterior reaction to applied load
yield strain of reinforcement
normalized ultimate shear strength
angle between concrete strut and longitudinal axis of beam
horizontal shear reinforcement ratio (Ah/bwsh)
longitudinal bottom reinforcement ratio (As/bwd)
longitudinal top reinforcement ratio (As /bwd)
′
vertical shear reinforcement ratio (Av /bwsv)
REFERENCES
1. Rogowsky, D. M.; MacGregor, J. G.; and Ong, S. Y., “Tests of Reinforced
Concrete Deep Beams,” ACI JOURNAL , Proceedings V. 83, No. 4, JulyAug. 1986, pp. 614623.
2. Ashour, A. F., “Tests of Reinforced Concrete Continuous Deep
Beams,” ACI Structural Journal, V. 94, No. 1, Jan.Feb. 1997, pp. 312.
3. Subedi, N. K., “Reinforced Concrete TwoSpan Continuous Deep
Beams,” Proceedings of the Institution of Civil Engineers, Structures &
Buildings, V. 128, Feb. 1998, pp. 1225.
4. Tan, K. H.; Kong, F. K.; Teng, S.; and Weng, L. W., “Effect of Web
Reinforcement on HighStrength Concrete Deep Beams,” ACI Structural
Journal, V. 94, No. 5, Sept.Oct. 1997, pp. 572582.
5. Smith, K. N., and Vantsiotis, A. S., “Shear Strength of Deep Beams,”
ACI JOURNAL, Proceedings V. 79, No. 3, MayJune 1982, pp. 201213.
6. ACI Committee 318, “Building Code Requirements for Structural
Concrete (ACI 31805) and Commentary (318R05),” American Concrete
Institute, Farmington Hills, Mich., 2005, 430 pp.
7. Canadian Standards Association (CSA), “Design of Concrete Structures,”
A23.394, Canadian Standards Association, Rexdale, Ontario, Canada,
Dec. 1994, 199 pp.
8. FIP Recommendations: Practical Design of Structural Concrete. 1999.
9. MacGregor, J. G., Reinforced Concrete: Mechanics and Design, PrenticeHall International, Inc., 1997.
10. Marti, P., “Basic Tools of Reinforced Concrete Beam Design.” ACI
JOURNAL, Proceedings V. 82, No. 1, Jan.Feb. 1985, pp. 4656.
11. Schlaich, J.; Schafer, K.; and Jennewein, M., “Toward a Consistent
Design of Structural Concrete,” Journal of the Prestressed Concrete Institute,
V. 32, No. 3, MayJune 1987, pp. 74150.
12. Tjhin, T. N., and Kuchma, D. A., “Example 1b: Alternative Design
for the NonSlender Beam (Deep Beam),” StrutandTie Models, SP208,
K.H. Reineck, ed., American Concrete Institute, Farmington Hills, Mich.,
2002, pp. 8190.
13. Yang, K. H., “Evaluation on the Shear Strength of HighStrength
Concrete Deep Beams,” PhD Thesis, Chungang University, Korea, Feb.
2002, 120 pp.
14. ACI Committee 445, “Shear and Torsion,” StrutandTie Bibliography,
ACI Bibliography No. 16, American Concrete Institute, Farmington Hills,
Mich., Sept. 1997, 50 pp.
15. Cervenka, V.; Jendele, L.; and Cervenka, J., “ATENA Computer
Program Documentation: Part 1,” Cervenka Consultant, 2003, 106 pp.
16. ACI Committee 318, “Building Code Requirements for Structural
Concrete (ACI 31899) and Commentary (318R99),” American Concrete
Institute, Farmington Hills, Mich., 1999, 369 pp.
ACI Structural Journal/JulyAugust 2007
59.
José R. MartíVargas is an Associate Professor of civil engineering in the Department of
Construction Engineering and Civil Engineering Projects, Polytechnic University of
Valencia (UPV), Valencia, Spain. He is member of the Institute of Science and Concrete
Technology (ICITECH) at UPV. He received his degree in civil engineering and his PhD
from UPV. His research interests include bond behavior of reinforced and prestressed
concrete structural elements, durability of concrete structures, and strutandties models.
César A. Arbeláez is a PhD Assistant Researcher in the Department of Construction
Engineering and Civil Engineering Projects at Polytechnic University of Valencia. He
is member of ICITECH at UPV. He received his civil engineering degree from Quindío
University, Armenia, Quindío, Colombia, and his PhD from UPV. His research interests
include bond properties of prestressed concrete structures and the use of advance
cementbased materials in structural applications.
Pedro SernaRos is a Professor of civil engineering in the Department of Construction
Engineering and Civil Engineering Projects at Polytechnic University of Valencia. He
is a member of ICITECH at UPV. He received his degree in civil engineering from
UPV and his PhD from l’Ecole National des Ponts et Chaussées, Paris, France. His
research interests include selfconsolidating concrete, fiberreinforced concrete, and
bond behavior of reinforced and prestressed concrete.
Carmen CastroBugallo is a PhD candidate in the Department of Construction
Engineering and Civil Engineering Projects at Polytechnic University of Valencia.
She is member of ICITECH at UPV. She received her degree in civil engineering from
UPV. Her research interests include bond properties of reinforced concrete and
prestressed concrete structures and strutandties models.
Table 2—Proposed equations for transfer length
from strand end slip
Equation
no.
Equation (U.S. units)
Reference
Equation (SI units)
δ
δ
L t = K
K
K = 0.00009 in.–1 for K = 0.0000035 mm–1
0.5 in. sevenwire
for 12.7 mm sevenstrand
wire strand
Lt =
Marshall and
Krishnamurthy25
(3)
Balázs26
(4)
δ Lt = 218db 4 f ci
′
11
Balázs
(5)
24.7δ
L t = 0.4
0.15 ⎛ f pi ⎞
f ci
′
⎝ E p⎠
111δ
L t = 0.4
0.15 ⎛ f pi ⎞
f ci
′
⎝ E p⎠
Rose and
Russell22
(6)
E
L t = 2δ p + 5.4
f pi
E
L t = 2δ p + 137.16
f pi
3⁄2
0.625
3⁄2
δ Lt = 105db 4 f ci
′
0.625
Notes: For U.S. units: fpi, f ′ci , and Ep in ksi; db, δ, and Lt in inches; for SI units: fpi,
f ′ci, and Ep in MPa; db, δ, and Lt in mm. 1 in. = 25 mm; 1 MPa = 0.145 ksi.
Some researchers conducted experimental studies to obtain
the transfer length from the strand end slip at the free end in
hollowcore slabs,7,1315 in beams,16,18,19,22,2729 in piles,20,30
in prisms,31 and in specimens to simulate bond behavior along
transfer length.32
Several authors7,14,20,30 have established an allowable
free end slip as the strand end slip which results in a transfer
length equal to that computed by the ACI provisions for
transfer length (Eq. (7)).1 By setting Eq. (2) to be equal to the
Eq. (7) and substituting α = 2 and α = 3 in Eq. (2), the
implied allowable value of end slip can be calculated by
Eq. (8) and (9), respectively.
1
1L t =  f se d b (U.S. units) L t =  f se d b (SI units)
3
20.7
1 f pi
δ all2 =  f se d b (U.S. units)
6 Ep
δ all2
488
1  f pi
=  f se d b (SI units)
41.4 E p
(7)
(8)
1 f pi
δ all3 =  f se d b (U.S. units)
9 Ep
δ all3
(9)
1 f pi
=  f se d b (SI units)
62.1 E p
where fse is the effective stress in the prestressing strand after
allowance for all prestress losses, db is the nominal diameter of
prestressing strand, δall2 is the implied allowable value of free
end slip when α = 2, and δall3 is the implied allowable value of
free end slip when α = 3 (Lt , fpi , and Ep, as previously described).
To apply Guyon’s end slip theory to determine transfer
length is easy, but the measurements of slips are affected by
the local bond loss at the ends. Equations (1) to (6) are not
applicable to elements of a poor bond quality.14 In this case,
greater slips are measured resulting in incorrect transfer
length estimation.
The other disadvantages of Guyon’s method are larger
scatter of experimental results,15 difficulty to measure
accurately smaller slips,13 breakage of gauges to measure
the strand end slip when a flame cutting process is applied,27
and excessive free end slip in prestressed members with poor
concrete consolidation around the strand.7
RESEARCH SIGNIFICANCE
This research study provides information on the transfer
length of a sevenwire prestressing strand in twelve
concretes of different compositions and properties. A test
method based on the measurement and analysis of the force
supported by the strand has been used. This paper analyzes
the reliability of transfer length determination from free end
slips according with proposed expressions in the literature.
Findings of the research are presented in procedures for the
experimental determination of transfer length measuring forces
or slips. The information is valuable for all parties involved
in the precast/prestressed concrete industry: manufacturers,
producers, designers, builders, and owners.
EXPERIMENTAL INVESTIGATION
An experimental program has been conducted to determine
the transfer length of prestressing strands: the ECADA*
test method3334 (*Ensayo para Caracterizar la Adherencia
mediante Destesado y Arrancamiento [Test to Characterize
the Bond by Release and Pullout]).
Materials
Twelve different concretes with a range of watercement
ratios (w/c) from 0.3 to 0.5, cement content from 590 to
843 lb/yd3 (350 to 500 kg/m3) and a compressive strength at the
time of testing fci from 3.5 to 8 ksi (24 to 55 MPa) were tested.
′
Concrete components were a) cement CEM I 52.5 R;35 b)
crushed limestone aggregate (0.275 to 0.472 in. [7 to 12 mm]);
c) washed rolled limestone sand (0 to 0.157 in. [0 to 4 mm]);
and d) policarboxilic ether highrange waterreducing additive.
The mixtures of the tested concretes are shown in Table 3.
The prestressing strand was a lowrelaxation sevenwire
strand specified as UNE 36094:97 Y 1860 S7 13.036 with a
guaranteed ultimate strength of 270 ksi (1860 MPa). The
main characteristics were adopted from the manufacturer:
diameter 0.5 in. (12.7 mm), crosssectional area 0.154 in.2
(99.69 mm2), ultimate strength 43.3 kips (192.60 kN), yield
stress at 0.2% 40 kips (177.50 kN), and modulus of elasticity
28,507 ksi (196,700 MPa). The prestressing strand was
ACI Structural Journal/JulyAugust 2007
60.
Table 3—Concrete mixtures from test program
f ci (at time of
′
Gravel/sand testing, 24 hours),
ratio
ksi (MPa)
Cement,
Designation lb/yd3 (kg/m3)
w/c
M3500.50
0.50
3.8 (26.1)
0.45
5.4 (37.3)
M3500.45
590 (350)
M3500.40
0.40
6.8 (46.7)
M4000.50
0.50
3.5 (24.2)
0.45
4.1 (28.3)
M4000.45
M4000.40
674 (400)
M4000.35
M4500.40
M4500.35
0.35
758 (450)
M5000.30
1.14
0.40
6.0 (41.4)
843 (500)
Fig. 1—Test equipment layout.
6.6 (45.3)
5.3 (36.3)
0.35
6.7 (46.6)
0.40
M5000.40
M5000.35
0.40
4.5 (30.8)
0.35
6.8 (46.6)
0.30
7.9 (54.8)
tested in the asreceived condition (free of rust and free of
lubricant). The strand was no treated in any special manner.
The strand was stored indoors, and care was taken not to drag
the strand on the floor.
Testing technique
The ECADA test method is based on the measurement and
analysis of the force supported by the strand in a series of
pretensioned concrete specimens with different embedment
lengths. Figure 1 shows the test equipment layout.
An anchoragemeasurementaccess (AMA) system is
placed at one end (stressed end) of a pretensioning frame to
simulate the sectional stiffness of the specimen. The AMA
system is made up of a sleeve in the final stretch of the specimen
to prevent the influence of the confinement caused by the
end frame plate, the stressed end frame plate, and an
anchorage plate supported on the frame by two separators.
The stepbystep test procedure was described in detail in
MartíVargas et al.,34 and may be summarized as follows:
Preparation stage—
1. The strand is placed in the frame;
2. Strand tensioning;
3. Strand anchorage by an adjustable strand anchorage;
4. The concrete is mixed, placed into the formwork in the
frame, and consolidated; and
5. After concrete placement, the specimen is cured to
achieve the desired concrete properties at the time of testing.
Testing stage—
1. The adjustable strand anchorage is relieved using the
hydraulic jack; and
2. Strand release is produced at a controlled speed, and the
prestressing force transfer to the concrete is performed. The
strand is completely released. The specimen is supported at
the stressed end frame plate.
Stabilization period—The level of force during this time is
zero at the free end. The force in the strand at the stressed end
depends on the strain compatibility with the concrete specimen.
This force requires a stabilization period to guarantee its
measurement. The strand force in the AMA system is
recorded continuously during the test.
Although it is not included in this study, the test can
continue with the pullout operation positioning the hydraulic
jack at the stressed end to increase the force in the strand,
separating the anchorage plate of the AMA system from
the frame.
ACI Structural Journal/JulyAugust 2007
Fig. 2—LVDT at free end of specimen.
Fig. 3—LVDT at stressed end of specimen.
Test parameters
The specimens had a 4 x 4 in.2 (100 x 100 mm2) cross section
with a concentrically located single strand at a prestress level
before release of 75% of guaranteed ultimate strand strength.
All specimens were subjected to the same consolidation and
curing conditions. Release was gradually performed 24 hours
after concreting at a controlled speed of 0.18 kips/s (0.80 kN/s).
A stabilization period of 2 hours from release was established.
With these test parameters, visible splitting cracks have not
happened in any of the tested specimens.
Instrumentation
The instrumentation used was a hydraulic jack pressure
sensor to control tensioning and release operations; a hollow
force transducer included in the AMA system to measure the
force supported by the strand; and two linear variable differential
transducers (LVDTs), one at the free end (Fig. 2) to measure
the drawin (δ, free end slip), and another at the stressed end
(Fig. 3) to measure the strand slip to the last embedment
concrete crosssection of the specimen (δl, stressed end slip).
No internal measuring devices were used in the test specimens
so as to not distort the bond phenomenon.
Criterion to determine transfer length
With the ECADA test method, the transfer length is
obtained with a series of specimens with different embedment
489
61.
Fig. 4—Determination of transfer length through ECADA
test method.
Fig. 5—Force loss versus embedment length for Concrete
M3500.50.
lengths. For each specimen, the strand force loss in the AMA
system directly after the stabilization period is measured.
The force loss values are arranged according to the specimen
embedment length (Fig. 4). The obtained curve shows a
bilinear tendency. The transfer length corresponds to the
smallest specimen embedment length that marks the beginning
of the horizontal branch.33,34
The resolution in the determination of the transfer length
will depend on the sequence of lengths of the specimens
tested. For an embedment length sequence of 2 in. (50 mm),
the transfer length obtained by the ECADA test method is
repeated when a same concrete mixture is tested.34
Transfer length overestimation
The ideal AMA system must have the same sectional
rigidity as the specimen. This rigidity depends on the
concrete properties, the age of the concrete at the time of
testing, and the specimen cross section. It would not really be
feasible to design a system for each specific test conditions.
For this reason, in this experimental work, the rigidity of
the AMA system designed is slightly greater than the
sectional rigidity of the specimens. A discontinuity section is
generated in the border between the specimen and the AMA
system. In these conditions, the strand force measured in the
AMA system after release will be slightly higher than the
effective prestressing force of the strand in the specimen.
This difference of forces gives rise to a small overestimation of
the real transfer length.34 Consequently, even if the specimen
embedment length is greater than the transfer length, a small
slip of the strand at the stressed end is registered.
490
EXPERIMENTAL RESULTS AND DISCUSSION
Determination of transfer length
Transfer length is determined for each concrete mixture in
accordance with the exposed criterion. As an example, Fig. 5
shows the results of force loss versus the embedment length
for the concrete M3500.50 (designation according with
Table 3). Two curves are shown, one with the force losses
registered just after release (ΔP), and another with the force
losses registered after the stabilization period (ΔP). Both
curves present a bilinear tendency with a descendent initial
branch with a strong slope and a practically horizontal
branch starting from 21.7 in. (550 mm) embedment length.
The transfer length determined by the ECADA test method
for this concrete mixture is 21.7 in. (550 mm).
The difference between the two curves corresponds to the
increment of force loss registered during the stabilization
period. When specimens have an embedment length below
21.7 in. (550 mm), the force loss after the stabilization period
is greater than the force loss registered just after release.
When specimens have an embedment length equal to or
greater than 21.7 in. (550 mm), however, the force loss is
similar at both points of time.
As it can be observed in Fig. 5, for this concrete, the
beginning of the horizontal branches coincides at both
points of time. In some cases, however, increases of force
loss have taken place during the stabilization period in the
first point of the resulting horizontal branch just after release.
For this reason, the transfer length must be always determined
on the curve measured after the stabilization period.
Comparison of test results with Guyon’s formula
Figure 6 shows the transfer length results obtained by the
ECADA test method for each concrete mixture, as well as
the transfer length obtained from the free end slips by
applying Guyon’s formula (Eq. (2)). This formula has been
applied to free end slips registered after the stabilization
period in specimens with an embedment length equal to or
greater than the transfer length. Between four and 18 specimens
for each concrete mixture, with a total of 121 specimens,
have been considered.
Two intervals are drawn for each concrete mixture. The
interior interval corresponds to the extreme transfer length
values obtained by applying Guyon’s formula with α = 2.8
(adopted by RILEM3) to the minimum and maximum free
end slips. The exterior interval corresponds to the extreme
transfer length values according to the hypotheses by Guyon
obtained as follows: the lower limit was calculated by
applying α = 2 to the minimum free end slip, and the upper
limit was calculated by applying α = 3 to the maximum free
end slip.
The amplitude of the transfer length intervals is very variable
for the different concrete mixtures, as shown in Fig. 6. The
results obtained by the ECADA test method are located
within both intervals in all cases except for the M5000.30
concrete mixture for the interior interval.
Figure 7 shows the transfer length results obtained by the
ECADA test method in the corresponding series versus the
free end slip registered after the stabilization period in each
specimen. Only the specimens with an embedment length
equal to or greater than the transfer length have been
included. The predicted transfer lengths by Guyon’s formula
are also plotted in Fig. 7. It is shown that 38.8% of the
experimental results fall outside the limits (33.0% show a
transfer length greater than the predicted maximum values,
ACI Structural Journal/JulyAugust 2007
62.
Fig. 6—Graphical comparison between experimental transfer
length and predicted transfer lengths from Guyon’s formula
and RILEM provisions.
Fig. 8—Comparison between results of present tests and
those of other researchers.
Fig. 9—Comparison of measured transfer lengths with
calculated values according to Eq. (6).
Fig. 7—Transfer length versus free end slip for specimens with
embedment length equal to or greater than transfer length.
Table 4—Comparison between measured
and calculated transfer lengths
Equation no.
Average
Lt (calculated)/L t (measured) Coefficient of correlation R2
(3)
and 5.8% show a transfer length smaller than the predicted
minimum values). A value of α = 2.44 from the regression
analysis of the test results has been obtained.
Figure 8 shows the experimental transfer lengths versus
the registered free end slips obtained in beams by several
authors. The predicted transfer length according to ACI 318051
(LtACI) and the allowable free end slips δall2 (Eq. (8)) and δall3
(Eq. (9)) are also plotted in Fig. 8. The LtACI, δall2, and δall3
values have been calculated by considering that fpi = 202 ksi,
fse = 0.8fpi = 162 ksi, Ep = 28,528 ksi and db = 0.5 in. (fpi =
1395 MPa, fse = 0.8fpi = 1116 MPa, Ep = 196,700 MPa and
db = 12.7 mm). The percentages of results included in each
sector delimited by LtACI, δall2, and δall3 are indicated in Fig. 8.
The range of free end slip registered is very ample for one
same transfer length, as observed in Fig. 8. Also the range of
transfer length values is very variable for one same free end
slip. Figure 8 also shows that when a transfer length is smaller
than LtACI, the δall2 limit is exceeded in 2.8% of the cases, and
the δall3 is exceeded in 32.3% of the cases (2.8% + 29.5%).
On the other hand, for registered free end slips smaller than
δall3 or δall2, transfer lengths greater than LtACI are measured
in some cases (2.3 and 4.6%, respectively). Consequently, the
use of an assurance procedure for bond quality based on a limit
value for the allowable free end slip is not completely reliable.
ACI Structural Journal/JulyAugust 2007
1.18
0.07
(4)
1.17
0.54
(5)
1.11
0.35
(6)
1.01
0.21
(2) with α = 2.44
0.95
0.20
Comparison of test results with other expressions
The experimental results obtained with both the ECADA
test method and the theoretical predictions from Eq. (3) to (6)
have been compared. As an example, Fig. 9 illustrates the
comparison with Eq. (6). Table 4 summarizes these
comparisons. Besides, the comparison with Eq. (2) by
substituting α = 2.44 (obtained value from the experimental
results of this study) is included. It can be observed that the
expressions based on Guyon’s formula (Eq. (6) and Eq. (2)
with α = 2.44) show a good prediction of the average
measured transfer length. The coefficient of correlation
improves when the expressions include, in addition to the
slips, other parameters like the concrete compressive strength.
Use of end slips sequences to determine
transfer length
The possibility of determining the transfer length from the
sequences of end slip values at both ends versus the embedment
length of specimens was considered.
491
63.
Fig. 10—Free end slip versus embedment length for
Concrete M3500.50.
Fig. 11—Stressed end slip versus embedment length for
Concrete M3500.50.
Table 5—Transfer length obtained from three
sequences of results (ΔP, δ, and δl)
Transfer length, in. (mm)
Designation ECADA test method ΔP Free end slip δ Stressed end slip δl
M3500.50
21.7 (550)
21.7 (550)
21.7 (550)
M3500.45
21.7 (550)
21.7 (550)
21.7 (550)
M3500.40
21.7 (550)
21.7 (550)
21.7 (550)
M4000.50
25.6 (650)
—
25.6 (650)
M4000.45
21.7 (550)
—
21.7 (550)
M4000.40
21.7 (550)
21.7 (550)
21.7 (550)
M4000.35
19.7 (500)
19.7 (500)
17.7 (450)
M4500.40
21.7 (550)
—
21.7 (550)
M4500.35
19.7 (500)
19.7 (500)
19.7 (500)
M5000.40
23.6 (600)
—
23.6 (600)
M5000.35
17.7 (450)
17.7 (450)
17.7 (450)
M5000.30
15.7 (400)
15.7 (400)
15.7 (400)
Figure 10 shows the free end slip results versus the
embedment length for the concrete mixture M3500.50.
Two curves are shown, one with the free end slip just after
release δ, and the other with the free end slip registered after the
stabilization period δ. Both curves present a bilinear tendency,
with a descendent initial branch and a practically horizontal
branch starting from 21.7 in. (550 mm) embedment length.
This embedment length coincides with the result obtained by
the ECADA test method (refer to Fig. 5). The free end slip
increases during the stabilization period in all the specimens.
492
Fig. 12—Ratios ΔP/ΔPAVE , δ/δAVE , and δl/δlAVE versus
embedment length for Concrete M3500.50.
Similarly, Fig. 11 shows the stressed end slip just after
release δl, and the stressed end slip after the stabilization
period δl versus the embedment length for the same concrete
mixture. Both curves present a bilinear tendency. The beginning
of the horizontal branch coincides with the result obtained by
the ECADA test method (21.7 in. [550 mm]). In regard to the
force losses, the stressed end slip only increases during the
stabilization period in specimens whose embedment length
is smaller than the transfer length.
Figure 12 summarizes the results of the three variables
(force loss and slip at both ends) for the concrete M3500.50
after the stabilization period. The shown ratios are the
quotient between each specimen test result (ΔP, δ, and δl),
and the average test results (ΔPAVE, δAVE, and δlAVE) of
specimens with an embedment length equal to or greater
than the transfer length. Again, a bilinear tendency is observed
with a descendent initial branch and a perceptibly horizontal
branch from 21.7 in. (550 mm) embedment length. Although
the slope of the descendent initial branch is very pronounced
in the cases of force loss and stressed end slip, it is very weak in
the case of free end slip. Consequently, the beginning of the
horizontal branch is more easily identifiable by analyzing the
force loss and stressed end slip than the free end slip.
This procedure of test results analysis for each concrete
mixture has been applied. The transfer lengths from the three
sequences of results obtained from the test instrumentation
(ΔP, δ, and δl) versus the embedment length have been
determined. Table 5 summarizes the obtained results. The
transfer lengths obtained from the stressed end slip and by
the ECADA test method coincide in 11 out of the 12 concrete
mixtures, and only a 2 in. (50 mm) difference is observed in
the concrete M4000.35. The transfer lengths obtained from
the free end slip coincide in eight out of the 12 concrete
mixtures. Given the wide dispersion of the measured free
end slip, no bilinear behavior was detected in the remaining
cases (see range of free end slip to one same transfer length
in Fig. 7). It was not possible to determine the transfer length
if the beginning of the horizontal branch was not clearly
defined. These cases correspond to concrete mixtures with
greater water content in their mixture.
CONCLUSIONS
Based on the results of this experimental investigation, the
following conclusions are drawn:
1. The feasibility of applying the ECADA test method to
determine the transfer length of prestressing strands has been
verified, even in concretes with a low compressive strength;
ACI Structural Journal/JulyAugust 2007
64.
2. An average value of α = 2.44 for Guyon’s formula has
been obtained from the experimental results of this study. An
ample range of free end slip values has been obtained for one
same transfer length. Furthermore, the range of transfer
length values for one same free end slip is very variable;
3. Consequently, a great variability of results for one same
concrete mixture has been observed in transfer length estimation
from the experimental free end slips when Guyon’s formula
was applied;
4. The prediction range of transfer lengths from expressions
proposed by several authors relating the transfer length to the
free end slip is very ample;
5. Determining transfer length from the free end slip is
relatively easy, although it can lead to a false perception that
the transfer length value is very variable;
6. Using a limit value for the allowable free end slip as an
assurance procedure for bond quality may give rise to
uncertain situations;
7. In relation to the results from the ECADA test method,
the sequence of stressed end slip values versus the embedment
length is a reliable assurance procedure for the experimental
determination of transfer length; and
8. The sequence of free end slip values versus the embedment
length is not a reliable assurance procedure for the experimental
determination of transfer length. The beginning of the horizontal
branch is not clearly defined when the dispersion of measured
free end slip is wide. This particularly occurs when concrete
has a low compressive strength.
ACKNOWLEDGMENTS
The contents of this paper are within the framework of a line of research
that is currently being carried out by the Concrete Technology and Science
Institute (ICITECH) of the Polytechnic University of Valencia, Valencia, Spain,
in collaboration with the companies PREVALESA and ISOCRON. Financial
support provided by the Ministry of Education and Science and FEDER funds
(Project MAT200307157 and Project BIA200605521) made this research
possible. The authors appreciate the collaboration of the aforementioned
companies and organizations, as well as the participation of the technical
staff of the Concrete Structures Laboratory at the Polytechnic University of
Valencia for their assistance in preparing and testing specimens.
NOTATION
db
Ep
fci
′
= nominal diameter of prestressing strand
= modulus of elasticity of prestressing strand
= compressive strength of concrete at time of prestress transfer
(cylinder)
fpi
= strand stress immediately before release
fse
= effective stress in the prestressing strand after allowance for all
prestress losses
Lt
= transfer length
LtACI = predicted transfer length according ACI 31805
α
= coefficient to take into account assumed shape of bond stress
distribution
δ
= strand end slip at free end
δall2 = allowable free end slip when α = 2
δall3 = allowable free end slip when α = 3
δ
= free end slip just after release
δAVE = average free end slip after stabilization period obtained in all
specimens of series with embedment length equal to or greater
than transfer length
δl
= stressed (loaded) end slip
δl
= stressed end slip just after release
δlAVE = average stressed end slip after stabilization period obtained in all
specimens of series with embedment length equal to or greater
than transfer length
ΔP
= force loss after stabilization period
ΔPAVE = average force loss after stabilization period obtained in all
specimens of series with embedment length equal to or greater
than transfer length
ΔP
= force loss just after release
εpi
= initial strand strain
ACI Structural Journal/JulyAugust 2007
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67.
ACI member Mary Beth D. Hueste is an Associate Professor in the Department of
Civil Engineering at Texas A&M University, College Station, Tex. She is a member of
ACI Committees 374; PerformanceBased Seismic Design of Concrete Buildings; 375,
PerformanceBased Design of Concrete Buildings for Wind Loads; E803, Faculty
Network Coordinating Committee; and Joint ACIASCE Committee 352, Joints and
Connections in Monolithic Concrete Structures. Her research interests include
earthquakeresistant design of reinforced concrete structures, structural rehabilitation
including seismic retrofitting, performancebased seismic design, and design and
evaluation of prestressed concrete bridge structures.
ACI member JoAnn Browning is an Associate Professor in the Department of Civil,
Environmental, and Architectural Engineering at the University of Kansas, Lawrence,
Kans. She is a member of ACI Committees 314, Simplified Design of Concrete Buildings;
318D, Flexure and Axial Loads; Beams, Slabs, and Columns; 341, EarthquakeResistant Concrete Bridges; 374, PerformanceBased Seismic Design of Concrete
Buildings; and 408, Bond and Development of Reinforcement. Her research interests
include the performance of reinforced concrete structures under seismic loads,
design and analysis of concrete structures, and durability of concrete structures.
Andres Lepage, FACI, is an Assistant Professor in the Department of Architectural
Engineering at Pennsylvania State University, University Park, Pa. He is a member of
ACI Committees 318H, Seismic Provisions; 335, Composite and Hybrid Structures;
369, Seismic Repair and Rehabilitation; 374, PerformanceBased Seismic Design of
Concrete Buildings; and 375, PerformanceBased Design of Concrete Buildings for
Wind Loads. His research interests include the design of concrete, steel, and hybrid
structural systems subjected to extreme events.
John W. Wallace, FACI, is a Professor of civil engineering at the University of
CaliforniaLos Angeles, Los Angeles, Calif. He is a member of ACI Committee 318H,
Seismic Provisions; 335, Composite and Hybrid Structures; 369, Seismic Repair and
Rehabilitation; 374, PerformanceBased Seismic Design of Concrete Buildings;
E803, Faculty Network Coordinating Committee; and Joint ACIASCE Committee
352, Joints and Connections in Monolithic Concrete Structures. His research interests
include response and design of buildings and bridges to earthquake actions, laboratory
and field testing of structural components and systems, and structural health monitoring
and use of sensor networks.
for punching failures; therefore, in regions of high seismic
risk, slabcolumn frames are used in conjunction with beamcolumn moment frames or shear walls. Compatibility of
lateral deformations between the slabcolumn frame and the
LFRS, however, must be considered to determine the
demands on the connections.
The seismic performance of reinforced concrete structures
with flatslab construction has demonstrated the vulnerabilities
of the system. For example, following the 1985 Mexico City
earthquake, punching shear failures were noted in a 15story
building with waffle flatplate construction (Rodriguez and
Diaz 1989). This failure was partly attributed to a high
flexibility combined with lowductility capacities of the
waffle slabtocolumn connection. In a department store
during the 1994 Northridge earthquake, discontinuous
flexural reinforcement at slabcolumn connections led to
punching failures at column drop panels (Holmes and
Somers 1996). Punching failures around shear capitals were
also noted in the posttensioned floor slabs of a fourstory
building during the same event (Hueste and Wight 1997).
CURRENT DESIGN APPROACH
General
The shear strength of slabs in the vicinity of columns is
governed by the more severe of two conditions, either beam
action or twoway action. In beam action, the slab acts as a
wide beam with the critical section for shear extending
across the entire width of the slab. This critical section is
assumed to be located at a distance d (effective slab depth)
from the face of the column or shear capital. For this condition,
conventional beam theory applies and will not be discussed
in detail herein. For the condition of twoway action, the critical
section is assumed to be located at a distance d/2 from the
perimeter of the column or shear capital, with potential diagonal
tension cracks occurring along a truncated cone or pyramid
passing through the critical section (refer to Fig. 1, where d1
ACI Structural Journal/JulyAugust 2007
Fig. 1—Critical sections for twoway shear for interior
slabcolumn connection with shear capital.
is the effective slab depth within the thickened shear capital
region and d2 is the effective slab depth).
Existing methods for calculating the shear strength of slabcolumn connections include applications of elastic plate
theory, beam analogies, truss analogies, strip design methods,
and others. The design method specified by ACI 31805
(ACI Committee 318 2005) provides acceptable estimates of
shear strength with reasonable computational effort. The
procedure is based on the results of a significant number of
experimental tests involving slabcolumn specimens.
The eccentric shear stress model is the basis of the general
design procedure embodied in ACI 318 for determining the
shear strength of slabcolumn connections transferring shear
and moment. The model was adopted by the 1971 version of
the ACI 318 and only minor modifications have been
included in subsequent versions. Recently, ACI 31805 has
incorporated special provisions related to the lateralload
capacity of slabcolumn connections in structures located in
regions of high seismic risk or structures assigned to high
seismic performance or design categories.
The design approach presented in this section of the paper is
based on the design procedures given in ACI 31805 complemented by ACI 421.1R99 (Joint ACIASCE Committee 421
1999) and 352.1R89 (Joint ACIASCE Committee 352 1989).
ACI 318 eccentric shear stress model
Slabcolumn connections experience very complex
behavior when subjected to lateral displacements or unbalanced
gravity loads. This involves transfer of flexure, shear, and
torsion in the portion of the slab around the column.
Combined flexural and diagonal cracking are coupled with
significant inplane compressive forces in the slab induced
by the restraint of the surrounding unyielding slab portions.
Relatively simple design equations have been derived by
considering the critical section to be located at d/2 away
from the face of the column and by assuming that shear stress
on the critical perimeter varies linearly with distance from
the centroidal axis. This eccentric shear stress model is based
on the work by DiStasio and Van Buren (1960) and reviewed
by Joint ACIASCE Committee 326 (1962).
For a slabcolumn connection transferring shear and
moment, the ACI 31805 design equations for limiting the
shear stresses vu are given by
vu ≤ φvn
(1)
Vu
γv Mu c
vu =  ± J
bo d
(2)
449
68.
where vu is the factored shear stress; φ is the strength reduction
factor for shear; vn is the nominal shear stress; Vu is the
factored shear force acting at the centroid of the critical
section; Mu is the factored unbalanced bending moment
acting about the centroid of the critical section; d is the
distance from the extreme compression fiber to the centroid
of the longitudinal tension reinforcement; bo is the length of
the perimeter of the critical section; c is the distance from the
centroidal axis of the critical section to the point where shear
stress is being computed; J is a property of the critical
section analogous to the polar moment of inertia; and γv is
the fraction of the unbalanced moment considered to be transferred by eccentricity of shear, defined by
1
γ v = 1 – 2 b1
1 +   3 b2
(3)
where b1 and b2 are the widths of the critical section measured
in the direction of the span for which Mu is determined
(Direction 1) and in the perpendicular direction (Direction 2).
For an interior column and a critical section of rectangular
shape, bo and J are determined by
bo = 2(b1 + b2)
3
3
(4)
2
db 1 b 1 d db 2 b 1
J =  +  + 6
6
2
(5)
The first term of Eq. (2), the shear stresses due to direct
shear, is assumed uniformly distributed on the critical
section, and the fraction γv Mu is assumed to be resisted by
linear variation of shear stresses on the critical section. The
portion of the moment not carried by eccentric shear is to be
carried by slab flexural reinforcement placed within lines 1.5h on
either side of the column (h is the slab thickness, including drop
panel, if any). This flexural reinforcement is also used to resist
slab design moments within the column strip.
The provisions of the ACI 318 specify that in absence of shear
reinforcement, the nominal shear strength (in stress units)
carried by the concrete vc in nonprestressed slabs is given by
⎧
4 f c′ (psi)
⎪
⎪
4
⎪ ⎛ 2 +  ⎞ f c′ (psi)
v c = min ⎨ ⎝
β c⎠
⎪
⎪ ⎛ αs d
⎞
⎪ ⎝  + 2⎠ f c′ (psi)
bo
⎩
For prestressed slabs without shear reinforcement, Eq. (6)
is replaced by
⎧
⎪
⎪
v c = min ⎨
⎪
⎪
⎩
Vp
3.5 f c′ (psi) + 0.3f pc + bo d
Vp
⎛ α s d + 1.5⎞ f ′ (psi) + 0.3f + pc
⎝b
⎠ c
b d
o
o
(7)
⎧
Vp
⎪
0.29 f c′ (MPa) + 0.3f pc + ⎪
bo d
v c = min ⎨
Vp
αs d
⎪
⎪ 0.083 ⎛  + 1.5⎞ f c′ (MPa) + 0.3f pc + ⎝b
⎠
bo d
⎩
o
where αs equals 40, 30, and 20 for interior, edge, and
corner columns, respectively; bo and d are defined previously;
βc is the ratio of long side to short side of column; fc′ is the
specified concrete compressive strength (psi units); fpc is the
average compressive stress in two vertical slab sections in
perpendicular directions, after allowance for all prestress
losses; and Vp is the vertical component of all effective
prestress forces crossing the critical section.
The use of Eq. (7) is restricted to cases where fc′ is less than
5000 psi (35 MPa); fpc ranges between 125 and 500 psi (0.9 and
3.5 MPa) in each direction; and no portion of the column
cross section is closer than four times the slab thickness to a
discontinuous edge. If these conditions are not satisfied, the slab
should be treated as nonprestressed and Eq. (6) applies.
When vu > φvn, the slab shear capacity can be increased by: (a)
thickening the slab in the vicinity of the column with a
column capital, shear capital, or drop panel; (b) adding
shear reinforcement; (c) increasing the specified compressive
strength of concrete; or (d) increasing the column size. In a flat
slab with shear capitals or drop panels, stresses must be checked
at all critical locations—both at the thickened portion of the slab
near the face of the column and at the section outside the shear
capital or drop panels (refer to Fig. 1).
Shear reinforcement, which can be in the form of bars or
wires and single or multipleleg stirrups properly anchored,
increases both the shear strength and the ductility of the
connection when transferring moment and shear. Shear
reinforcement consisting of structural steel shapes (shearheads)
is also effective in increasing the shear strength and ductility
of slabcolumn connections. Design procedures for shearhead reinforcement are presented in Corley and Hawkins
(1968) and are not discussed in this paper. For members with
shear reinforcement other than shearheads, the nominal
shear strength (in stress units) is calculated using
(6)
⎧
0.33 f c′ (MPa)
⎪
⎪
2
⎪ 0.17 ⎛ 1 +  ⎞ f c′ (MPa)
⎝
or v c = min ⎨
β c⎠
⎪
⎪
⎛ αs d
⎞
⎪ 0.083 ⎝  + 2⎠ f c′ (MPa)
bo
⎩
450
v c = v c + v s ≤ 6 f c′ (psi) or 0.5 f c′ (MPa)
(8)
v c = 2 f c′ (psi) or 0.17 f c′ (MPa)
(9)
A v f yv
v s = bo s
(10)
ACI Structural Journal/JulyAugust 2007
69.
where vs is the nominal shear stress provided by shear
reinforcement; Av is the area of shear reinforcement; fyv is
the specified yield strength of shear reinforcement; s is the
spacing of shear reinforcement; and vc, fc′ , and bo are defined
previously.
When lightweight aggregate concrete is used, the value of
f c′ in Eq. (6) through (9) is multiplied by 0.75 for all
lightweight concrete or by 0.85 for sandlightweight
concrete. The extent of the shearreinforced zone is
determined to ensure that punching shear failure does not occur
immediately outside this region for the design actions.
The nominal ultimate concrete shear stress along the critical
section acting with shear reinforcement is taken as
2 f c′ (psi) ( 0.17 f c′ [MPa] ) because at approximately this
stress, diagonal tension cracks begin to form and cracking is
needed to mobilize the shear reinforcement. The shear
reinforcement or shear capital must be extended for a sufficient
distance until the critical section outside the reinforced
region satisfies Eq. (9). In nonprestressed slabs, the maximum
spacing of shear reinforcement is 0.5d. In prestressed slabs,
the spacing of shear reinforcement is allowed to reach 0.75h
but not to exceed 24 in. (0.61 m).
For both prestressed and nonprestressed slabs, ACI 318
mandates continuity reinforcement to give the slab some
residual capacity following a single punching shear failure at
a single support. Thus, in nonprestressed slabs, all bottom
bars within the column strip shall be continuous and at least
two of the column strip bottom bars in each direction shall
pass through the column core (ACI Committee 318 2005,
Section 13.3.8.5). In prestressed slabs, a minimum of two
tendons shall be provided in each direction through the
critical shear section over columns (ACI Committee 318
2005, Section 18.12.4).
ACI 421.1R99 refinements
ACI 318 sets out the principles of design for slab shear
reinforcement but does not make specific reference to
mechanically anchored shear reinforcement, also referred to
as shear studs. ACI 421.1R99 (Joint ACIASCE Committee
421 1999) gives recommendations for the design of shear
reinforcement using shear studs in slabs. This report also
includes equations for calculating shear stresses on
nonrectangular critical sections.
Shear studs have proven to be effective in increasing
the strength and ductility of slabcolumn connections.
ACI 421.1R99 suggests treating a shear stud as the equivalent
of a vertical branch of a stirrup and to use higher limits on
some of the design parameters used in ACI 318. In particular,
ACI 421.1R99 suggests higher allowable values for vn, vc,
s, and fyv, as follows
v c = v c + v s ≤ 8 f c′ (psi) or 0.66 f c′ (MPa)
(11)
v c = 3 f c′ (psi) or 0.25 f c′ (MPa)
(12)
⎧
vu
⎪ 0.75d when  ≤ 6 f ′ (psi) or 0.5 f ′ (MPa)
c
c
⎪
φ
s≤⎨
(13)
vu
⎪
⎪ 0.5d when  > 6 f c′ (psi) or 0.5 f c′ (MPa)
φ
⎩
ACI Structural Journal/JulyAugust 2007
fyv ≤ 72,000 psi (500 MPa)
(14)
The justification for these higher values is mainly due to
the almost slipfree anchorage of the studs and that the
mechanical anchorage at the top and bottom of the stud is
capable of developing forces in excess of the specified yield
strength at all sections of the stud stem.
ACI 352.1R89 recommendations
ACI 352.1R89 (Joint ACIASCE Committee 352 1989)
includes recommendations for the determination of connection
proportions and details to ensure adequate performance of
monolithic, reinforced concrete slabcolumn connections. The
recommendations address connection strength, ductility, and
structural integrity for resisting gravity and lateral forces.
ACI 352.1R89 only applies to nonprestressed slabcolumn connections with fc′ less than 6000 psi (42 MPa),
with or without drop panels or shear capitals, and without
slab shear reinforcement. The provisions are limited to
connections where severe inelastic load reversals are not
expected, and do not apply to slabcolumn connections that
are part of a primary LFRS in regions of high seismic risk
because slabcolumn frames are generally considered to be
inadequate for multistory buildings in these areas.
ACI 352.1R89 classifies slabcolumn connections as one
of two types: 1) Type 1—connections not expected to
undergo deformations into the inelastic range; and 2) Type 2—
connections requiring sustained strength under moderate
deformations into the inelastic range. In structures subjected
to high winds or seismic loads, a slabcolumn connection
should be classified as Type 2 even though it is not designated
as part of the primary LFRS.
To ensure a minimal level of ductility, ACI 352.1R89
references the work by Pan and Moehle (1989) and
recommends that for all Type 2 connections—without
shear reinforcement—the direct factored shear Vu acting on
the connection, for which inelastic moment transfer is
anticipated, must satisfy
Vu ≤ 0.4Vc = 0.4vcbod
(15)
where vc is determined by either Eq. (6) or (7).
The limitation defined by Eq. (15) was based on a review
of test data that revealed that the deformation capacity of
interior connections without shear reinforcement is inversely
related to the direct shear on the connection. Connections not
complying with Eq. (15) exhibit virtually no postyield
deformation capacity under lateral loading. Pan and Moehle
(1989) found that when the stress due to direct shear
approaches 0.4vc, the connection experiences a brittle failure
for story drift ratios of approximately 1.5%. No additional
statements are made in ACI 352.1R89 regarding other
combinations of shear stress and story drift ratio. The report
states that Eq. (15) may be waived if calculations demonstrate
that the imposed displacement will not induce yield in the
slab system. For example, the use of structural walls may
adequately limit the imposed drifts on slabcolumn frames
such that yield at the slabcolumn connection may not occur.
The approach by ACI 352.1R89 suggests that the deformation
capacity of slabcolumn connections may be defined as a
function of the shear stress due to direct shear only. This
approach has been developed further by Moehle (1996) and
Megally and Ghali (2000). ACI 31805 has incorporated this
451
70.
Vu
VR = φv c b o d
Fig. 2—ACI 31805 relationship for determining adequacy
of slabcolumn connections in seismic regions.
Fig. 3—Design steps when adding shear reinforcement.
concept into a general approach for addressing the deformation
capacity of slabcolumn connections not designated as part
of the LFRS.
Requirements of ACI 31805, Section 21.11.5
Model building codes (SEI/ASCE 2005) have deformation
compatibility requirements for members that are not designated
as part of the LFRS. These members should be able to resist
the gravity loads at lateral displacements corresponding to
the design level earthquake. ACI 31805, Section 21.11.5,
has incorporated a design provision to account for the
deformation compatibility of slabcolumn connections.
Instead of calculating the induced effects under the design
displacement, ACI 31805 describes a prescriptive approach.
The connection is evaluated based on a simple relationship
between the design story drift ratio (DR) and the shear stress
due to factored gravity loads. The design DR (story drift
divided by story height) should be taken as the largest value
for the adjacent stories above and below the connection. The
maximum DR (in percent) that a slabcolumn connection can
tolerate, in the absence of shear reinforcement, is given by
the following relationship and illustrated in Fig. 2.
⎧
DR = ⎨ 3.5 – 5.0VR ( for VR < 0.6 )
( for VR ≥ 0.6 )
⎩ 0.5
where VR is the shear ratio, defined as
452
(16)
(17)
The term vc is calculated using Eq. (6) or (7). The factored
shear force Vu on the slab critical section for twoway action
is determined for the load combination 1.2D + 1.0L + 0.2S,
where D, L, and S are the dead, live, and snow loads.
If the DR exceeds the limit given by Eq. (16), shear
reinforcement must be provided (or the connection can be
redesigned). When adding shear reinforcement, ACI 31805
prescribes that the term vs, defined by Eq. (10), must exceed
3.5 f c′ (psi) ( 0.29 f c′ [MPa] ) and the shear reinforcement
must extend at least four times the slab thickness from the
face of the support. Given that this approach is relatively simple,
and that the added cost of providing shear reinforcement at
connections is not significant for structures designed for
high seismic performance categories, use of this prescriptive
approach is likely to be common. The representative design
steps are shown in Fig. 3.
If shear capitals, column capitals, or drop panels are used,
all potential critical sections must be investigated. ACI 31805
does not prescribe a minimum extension of shear capitals.
Wey and Durrani (1992), however, recommend a minimum
length equal to two times the slab thickness from the face of
the column.
ANALYTICAL MODELING
The shear stresses due to the combined factored shear and
moment transferred between the slab and the column under
the design displacement can be determined by creating an
appropriate analytical model of the slabcolumn frame and
directly assessing the potential for punching. Recommendations
by Hwang and Moehle (2000) may be used to establish the
effective stiffness of the slab and to include the impact of
cracking. Hwang and Moehle (2000) recommend that the
uncracked effective stiffness for a model with rigid joints,
for ratios of c2/c1 from 1/2 to 2 and a slab aspect ratio l2/l1
greater than 2/3, be determined using an effective beam
width represented as
l1
b int = 2c 1 + 3
(18)
where bint is the effective width for interior frame connections
(interior connections and edge connections with bending
perpendicular to the edge); c1 and l1 are the column dimension
and slab span parallel to the direction of load being considered;
and c2 and l2 correspond to the orthogonal direction. For
exterior frame connections (corner connections and edge
connections with bending parallel to the edge), half the width
defined in Eq. (18) is used. Effects of cross section changes,
such as slab openings, are to be considered. One way to
accomplish this is to vary the width of the effective beam
along the span (Hwang and Moehle 1990).
To account for cracking, a stiffness reduction factor β has
been proposed by Hwang and Moehle (2000) for nonprestressed slabs and is given by
c 1
β = 4  >  l 3
(19)
ACI Structural Journal/JulyAugust 2007
71.
where c and l are the column dimension and slab span
parallel to the load direction. Kang and Wallace (2005)
recommend β = 0.5 for posttensioned floor systems with
approximate values for spantoslab thickness ratios of 40,
c1/l1 of 1/14, and precompression of 200 psi (1.4 MPa).
The analytical model of the slabcolumn frame should
capture the potential for both slab yielding and connection
failure due to punching as recommended in FEMA 356.
Figure 4 shows an approach where yielding within the slab
column strip is modeled using slabbeam elements (in this
case, an elastic slabbeam with stiffness properties defined
by the effective beam width model, and zerolength plastic
hinges on either side of the connection). Further details of
this model are described by Kang et al. (2006). Punching
failures can occur if the capacity of the connection element
is reached or if a limiting story drift ratio is reached for a
given gravity shear ratio. Hueste and Wight (1999)
suggested an approach for incorporating this behavior into a
nonlinear analysis program, where, after prediction of a
punching shear failure, the member behavior is modified to
account for the significant reduction in stiffness and
strength. Kang and Wallace (2005) suggest a direct approach
by employing a limit state model.
The FEMA 356 guidelines note that the analytical model for
a slabcolumn frame should consider all potential failures
including flexure, shear, shearmoment transfer, and reinforcement development at any section. The modeling information
mentioned previously gives a convenient and relatively
straightforward approach to modeling the behavior of slabcolumn frames for nonlinear static and dynamic analysis.
PERFORMANCEBASED DESIGN CRITERIA
A review of current practice with respect to performancebased
design is needed to provide context to the material presented
subsequently on performance objectives for slabcolumn
connections. The FEMA 356 prestandard (ASCE 2000)
provides analytical procedures and criteria for the performancebased evaluation of existing buildings and for designing
seismic rehabilitation alternatives. This prestandard includes
recommended limits for deformation capacities based on the
calculated gravity shear ratio, as well as a general framework
for creating performance levels and objectives.
In FEMA 356, performance levels describe limitations on
the maximum damage sustained during a ground motion,
while performance objectives define the target performance
level to be achieved for a particular intensity of ground
motion. Structural performance levels in FEMA 356 include
immediate occupancy, life safety, and collapse prevention.
Structures at collapse prevention are expected to remain
standing, but with little margin against collapse. Structures at
life safety may have sustained significant damage, but still
provide an appreciable margin against collapse. Structures at
immediate occupancy should have only minor damage. In
FEMA 356, the Basic Safety Objective is defined as life
safetyperformance for the basic safety earthquake 1 (BSE1)
earthquake hazard level and collapse prevention performance
for the BSE2 earthquake hazard level. BSE1 is the smaller
event corresponding to 10% probability of exceedance in
50 years (10% in 50 years) and 2/3 of the BSE2 (2% in
50 years) event.
For a given design event and a target performance level,
FEMA 356 provides acceptance criteria when using either
static or dynamic analysis based on linear and nonlinear
procedures. To evaluate acceptability using linear procedures,
ACI Structural Journal/JulyAugust 2007
Fig. 4—Modeling of slabcolumn connection (adapted from
Kang et al. 2006).
an action is classified as either deformationcontrolled or
forcecontrolled. Deformationcontrolled actions are applicable
for components that have the capacity to undergo deformations
into the inelastic range without failure. Based on the demand
to capacity ratio (DCR), calculated using the linear static or
dynamic analysis procedures, components are classified as
having low (DCR < 2), moderate (2 ≤ DCR ≤ 4), or high
(DCR > 4) ductility demands.
The acceptance criteria based on linear analysis procedures
are expressed in terms of mfactors. The factor m is intended to
provide an indirect measure of the total deformation
capacity of a structural element or component. As such,
the factor m is only used to evaluate the acceptability of
deformationcontrolled actions
mκQCE ≥ QUD
(20)
where κ is the knowledge factor used to reduce the strength of
existing components based on quality of information, QCE is the
expected strength of a component or element at the deformation
level considered, and QUD is the deformationcontrolled
design action. Equation (20) can be rearranged for direct
comparison of the DCR to m to determine acceptability
Q UD
DCR ≤ m = κQ CE
(21)
The FEMA 356 limiting values for mfactors for twoway
slabs and slabcolumn connections are provided in Table 1.
The mfactors for slabcolumn connections range from 1 to
4 and depend on several parameters: the gravityshear ratio,
the presence of continuity reinforcement through the column
cage, the development of reinforcement, and the selected
performance level. The connections must also be classified
as primary or secondary elements to determine the limits
for life safety and collapse prevention. Secondary
elements are those typically not considered to provide
resistance to earthquake effects.
For nonlinear static and dynamic analysis procedures,
FEMA 356 restricts inelastic response values determined
from the analytical model in terms of maximum plastic
rotations. Generally, plastic rotation is computed as the
453
72.
Table 1—Acceptance criteria for linear procedures—
twoway slabs and slabcolumn connections
(adapted from FEMA 356 [ASCE 2000])
mfactors by performance level*
Component type
Primary
Conditions
IO
LS
Secondary
CP
LS
1. Slab controlled by flexure and slabcolumn connections
CP
†
Vg /Vo‡
≤ 0.2
Yes
2
2
3
3
4
≥ 0.4
Yes
1
1
1
2
3
≤ 0.2
No
2
2
3
2
3
≥ 0.4
Fig. 5—Test data for interior slabcolumn connection
specimens with no shear reinforcement.
Continuity
reinforcement§
No
1
1
1
1
1
2. Slabs controlled by inadequate development or splicing along span†
—
—
—
3
4
3. Slabs controlled by inadequate embedment into slabcolumn joint†
2
2
3
3
4
*IO
= immediate occupancy; LS = life safety; and CP = collapse prevention.
When more than one of Conditions 1, 2, and 3 occurs for given component, use
minimum appropriate numerical value from table.
‡
Vg = gravity shear acting on slab critical section and Vo = direct punching shear
strength as defined by ACI 318.
§Under heading “Continuity reinforcement,” use “Yes” where at least one of the main
bottom bars in each direction is effectively continuous through column cage. Where
that slab is posttensioned, use “Yes” where at least one of posttensioning tendons in
each direction passes through column cage. Otherwise, use “No.”
†
Fig. 6—Test data for interior slabcolumn connection
specimens with shear reinforcement.
difference between the maximum rotation during analysis
and the yield rotation at the member end. Therefore, it is
critical for the nonlinear model to represent the maximum
plastic rotation for a certain level of demand. The plastic
rotation limits in FEMA 356 range from 0.0 to 0.02 radians
for primary slabcolumn connections and from 0.0 to 0.05
radians for secondary slabcolumn connections. These limits are
based on the gravityshear ratio, the presence of continuity
reinforcement through the column cage, the development
of reinforcement, and the selected performance level (immediate
occupancy, life safety, or collapse prevention).
EXPERIMENTAL DATA
Over the past 40 years, experimental studies have been
conducted by researchers at a number of universities. Much
of the earlier data has been summarized by Pan and Moehle
(1989), Megally and Ghali (1994), and Luo and Durrani
(1995). Tables 2 and 3 provide information on interior slabcolumn connection test specimens, with and without shear
reinforcement. Limited tests have been conducted for
nonductile slabcolumn connections where the bottom slab
reinforcement is discontinuous at the interior slabcolumn
connection (Durrani et al. 1995; Dovich and Wight 1996;
Robertson and Johnson 2006) and available data is included
in Table 2. The failure mode for each specimen is provided,
when available, as either: punching shear P, flexure F, or a
454
combination of flexure and punching shear (FP) where a
punching shear failure occurred at a higher drift level
following yielding of the slab reinforcement. The gravity
shear ratio and peak drift are also provided for each specimen.
The peak drift is defined as the drift corresponding to the
peak lateral load. Therefore, the maximum drift attained for a
particular specimen may be larger than the reported peak drift.
The maximum drift at which an interior connection will
fail can be estimated from the gravity shear ratio Vg/Vo (Pan
and Moehle 1989; Luo and Durrani 1995). The gravity shear
ratio represents the unfactored vertical gravity shear Vg
divided by the theoretical punching shear strength without
moment transfer Vo determined using
Vo = vc bo d
(22)
The term vc is calculated using Eq. (6) or (7). A similar
ratio can be computed for slabs with shear reinforcement by
replacing vc with vn defined by Eq. (8) through (10).
Figure 5 provides a plot of peak drift as a function of Vg/Vo for
interior slabcolumn connection specimens with no shear
reinforcement. The figure shows the direct influence of the
gravity shear ratio on the lateral drift capacity of slabcolumn connections. It may be observed that punching shear
occurs for a large range of Vg/Vo values (approximately 0.1 to
0.9), while flexural failures primarily occur for Vg/Vo values
of 0.3 or less.
Figure 6 provides a similar plot for interior slabcolumn
connection specimens with shear reinforcement. The
experimental data indicates that larger drift ratios are possible
when shear reinforcement is used. In particular, a number of
slabcolumn specimens with studshear reinforcement (SSR)
attained story drift ratios well over 3% before failure.
The data from slabcolumn connection tests, with and
without shear reinforcement, are compared in Fig. 7, along
ACI Structural Journal/JulyAugust 2007
73.
Table 2—Test data for interior slabcolumn connection test specimens with no shear reinforcement
Label
Vg/Vo
Peak drift, %
0.85
0.90
NA
I.I
0.08
5.00
F
CD 2
0.65
1.20
NA
INT1
0.43
NA
P
0.52
1.40
NA
DNY 1*
0.20
3.00
F
INT2
0.50
NA
P
DNY 2*
0.30
2.00
P
MG2A
0.58
1.17
P
*
0.24
2.00
F
MG7
0.29
3.10
FP
DNY 4*
Durrani et al. (1995)
Vg/Vo
CD 8
Dilger and Cao (1991)
Label
CD 1
Source
0.28
2.60
FP
MG8
0.42
2.30
FP
1
0.46
NA
P
MG9
0.36
2.17
FP
DNY 3
Elgabry and Ghali (1987)
Peak drift, % Mode
Source
Luo and Durrani (1995)
Megally and Ghali (2000)
Mode
1
S1
0.03
4.70
F
F
S2
0.03
2.80
F
3
0.26
3.56
P
S3
0.03
4.20
F
0.30
2.40
P
S4
0.07
4.50
F
0.31
6.00
F
S5
0.15
4.80
F
SM 1.0
0.33
2.70
FP
AP 1
0.37
1.60
FP
SM 1.5
0.30
2.70
FP
A12
0.29
NA
P
A13L
0.29
NA
P
AP 4
B16
Hanson and Hanson (1968)
F
4.04
SM 0.5
Ghali et al. (1976)
4.81
0.00
4
Farhey et al. (1993)
0.00
2
0.29
NA
P
1
2
0.35
1.50EW/0.79NS
P
3
0.22
3.10
FP
P
Morrison and Sozen (1983)
Pan and Moehle (1989)
AP 2
0.36
1.50
FP
AP 3
0.18
3.70
FP
0.19
3.50
FP
0.35
1.50
P
FP
NA
FP
0.05
5.80
F
4
0.22
3.20EW/1.75NS
0.33
3.75
P
1
0.21
2.75
F
S2
0.45
2.00
P
2C
0.22
3.50
FP
S3
0.45
2.00
P
S4
0.40
2.60
P
4 Int. Joints
0.24
4.00
1
0.25
3.67
2
0.23
3.33
P
3C
Islam and Park (1976)
3.80
0.24
S1
Hwang and Moehle (1990)
0.04
C8
Hawkins et al. (1974)
B7
C17
0.23
4.00
FP
*
Pan and Moehle (1992)
3SE
0.19
3.50
F
5SO
0.21
3.50
F
NA
6LL
0.54
0.85
P
P
7L
0.40
1.45
P
Robertson and Durrani (1990)
8I
Robertson et al. (2002)
0.18
3.50
FP
1C
0.17
3.50
P
S6
0.86
1.10
P
S7
0.81
1.00
P
ND1C
0.23
3.00 to 5.00
FP
ND4LL*
0.28
3.00
FP
*
0.47
1.50
P
SC 0
0.25
3.50
P
*
ND6HR
0.29
3.00
P
SC 2
0.18
6.00
F
NC7LR*
0.26
3.00
FP
ND5XL
Robertson and Johnson (2006)
ND8BU*
0.26
3.00
FP
Symonds et al. (1976)
Wey and Durrani (1992)
SC 4
Zee and Moehle (1984)
0.15
6.00
F
SC 6
0.15
5.00
P
INT
0.21
3.30
FP
*Bottom
slab reinforcement is discontinuous at interior connection.
Note: EW = eastwest lateral load for biaxial test; NS = northsouth lateral load for biaxial test; F = flexural failure; P = punching shear failure; and FP = flexural and punching
shear failure. NA: Not available.
with the ACI 31805 limits for assessing the need for shear
reinforcement. The line defined by ACI 31805 is a reasonable
lowerbound limit for the data corresponding to specimens
without shear reinforcement. A strength reduction factor
of φ = 1 is used when determining Vg/Vo for the test data.
PERFORMANCEBASED SEISMIC
DESIGN RECOMMENDATIONS
Research studies and past structural performance have
shown that slabcolumn frames provide lateral stiffness
contributions to the overall LFRS and, as such, they do resist
lateral loads during a seismic event even if they were
designed for gravity loads only. For this reason, compatibility of
deformations must be considered to calculate the demands at
the slabcolumn connections. Likewise, the analytical model
should include the strength and stiffness of the slabcolumn
ACI Structural Journal/JulyAugust 2007
frames to ensure an accurate representation of the overall
building stiffness and allow an evaluation of the magnitude
of the lateral load that must be resisted by the slabcolumn
frame members. The appropriate parameters that should be
included in such a model were highlighted previously (effective
slab width for equivalent beams, cracked section properties,
and hysteretic behavior for nonlinear models).
Performancebased seismic design (PBSD) criteria are
suggested in the following. The criteria are based on
experimental data of interior slabcolumn connections under
combined gravity and lateral load. The suggested criteria
reference FEMA 356 performance levels (immediate
occupancy, life safety, and collapse prevention) and seismic
design requirements for slabcolumn connections that are
adopted in ACI 31805. As noted previously, in regions of
high seismic risk, the slabcolumn connections of twoway
455
74.
Table 4—Key points for recommended PBSD
criteria for interior slabcolumn connections
Table 3—Test data for interior slabcolumn
connection test specimens with shear
reinforcement
Source
Label
Shear
Vg /Vo Peak drift, % reinforcement Mode
SJB1
0.48
5.50
SSR
S1
SJB2
0.47
5.70
SSR
S1
SJB3
SSR
0.47
7.60
SSR
5.70
SSR
7.10
SSR
0.91
3.50
SSR
4.80
SSR
0.64
5.40
SSR
0.51
5.60
SSR
0.47
NA
SSR
0.87
NA
SSR
0.85
NA
SSR
P
5
1.20
NA
SSR
P
SS1
0.49
3.50
Stirrups
C3
SS2
0.47
3.43
Stirrups
P
SS3
0.48
4.10
Stirrups
F
SS4
0.47
5.50
Stirrups
NA
SS5
0.42
4.90
Stirrups
F
4S
4.33
Bent up
P
0.23
4.17
Shear head
6CS
0.24
4.00
Stirrups
P
0.24
3.70
Stirrups
P
8CS
0.27
5.00
Stirrups
0.16
4.50
Closed hoop
F
4HS
0.15
5.00
Headed stud
F
3SL
4.50
Single leg
F
0.60
5.20
SSR
NA
MG3
0.56
5.40
SSR
NA
MG4
0.86
4.60
SSR
FP
MG5
0.31
6.50
SSR
FP
MG6
Robertson and
Durrani (1990)
0.10
MG10
Megally and
Ghali (2000)
0.59
6.00
SSR
FP
4S
0.19
3.50
Closed hoop
F
1
Note: SSR = studshear reinforcement; S = shear failure outside shear reinforced zone; S2
= shear failure in shear in zones without shear reinforcement; C3 = crushing failure at
column face without apparent punching shear failure; F = flexural failure; P = punching
shear failure; and FP = flexural and punching shear failure. NA: not available.
slabs without beams must be checked for the induced effects
caused by the lateral displacement expected for the designbasis earthquake. It is important to note the direct influence
of the gravity shear ratio on the lateral drift capacity of slabcolumn connections without shear reinforcement illustrated by
the test data in Fig. 5. As suggested by the FEMA 356 limits
for slabcolumn connections, this relationship is critical to the
development of appropriate PBSD criteria for slabcolumn
connections. The ACI 31805 seismic design limits for slabcolumn connections given in Eq. (16) also underscore the direct
relationship between these two parameters.
Linear regression analysis on the experimental data for
slabcolumn connections without shear reinforcement and
having a gravity shear ratio Vg/Vo less than 0.6, results in a
line defined by a slope of –6.95 and a zero intercept of 4.97.
456
Fig. 7—Comparison of recommended performancebased
seismic design limits with slabcolumn connection test data.
Thus, the mean for the data gives the following expression
for the maximum story drift ratio (in percent)
P
2CS
Note: IO = immediate occupancy; LS = life safety; and CP = collapse prevention.
P
7CS
Robertson et al.
(2002)
0.23
5S
Islam and Park
(1976)
0.75
P
4
0.75
0.5
P
3
0.5
0.25
NA
2
0.25
1.0
NA
CD 7
0.6
NA
CD 6
5.0
NA
0.62
3.5
S2
CD 4
CP
1.75
S2
0.49
LS
0.0
S1
0.46
IO
S2
CD 3
Hawkins et al.
(1975)
SSR
6.40
SJB9
Elgabry and
Ghali (1987)
5.00
0.43
SJB8
Dilger and Cao
(1991)
0.48
SJB4
SJB5
Dilger and
Brown (1995)
S2
Drift ratio, %, by performance level
Gravity shear
ratio (Vg /Vo)
Vg
DR = 5 – 7 Vo
(23)
The PBSD criteria suggested herein use Eq. (23) as a
reference for selecting the collapse prevention performance
level limits. The life safety performance level was initially
defined as 2/3 of the values used for collapse prevention; and
for immediate occupancy, 1/3 of the values for collapse
prevention was used. The drift limits determined using the
aforementioned parameters were the basis for finalizing the
key points of the graphed PBSD criteria. Table 4 summarizes
the key points for the recommended PBSD criteria and the
values are shown graphically in relationship to the test data
in Fig. 7.
For the suggested PBSD criteria, the drift limits for the
immediate occupancy performance level are relatively low
so that the slabcolumn frame members remain at or near the
elastic range of behavior. The suggested line for life safety
corresponds to the ACI 31805 design limits (refer to Fig. 2).
The life safety performance level includes the combination
of Vg/Vo = 0.4 and a drift of 1.5%, which is consistent with
the recommendation in ACI 352.1R89 that the gravity shear
ratio should be kept below 0.4 to ensure some minimal
ductility with the availability of approximately 1.5% drift
capacity. The collapse prevention limits correspond to approximately the mean of the experimental data for specimens
without shear reinforcement. For all performance levels, a
constant story drift ratio capacity is assigned for gravity
shear ratios in excess of 0.6.
As the approximate mean of the data for specimens
without shear reinforcement (Fig. 7), the collapse prevention
limits correspond to a 50% probability of failure (without
considering the load and resistance factors provided in the
ACI Structural Journal/JulyAugust 2007
75.
code). Assuming a normal distribution, the life safety
limits, defined as 2/3 of collapse prevention, correspond
to approximately 5% probability of failure, and the
immediate occupancy limits, defined as 1/3 of collapse
prevention, correspond to less than 1% probability of failure.
When the story drift limit corresponding to the acting
gravity shear ratio is exceeded for the performance level
considered, various options exist, including: 1) reduce the
gravity shear ratio by thickening the slab, adding shear capitals,
or adding drop panels; 2) reduce the story drift ratio to be
within the allowable limit by stiffening the lateral system; or
3) add shear reinforcement as prescribed by ACI 31805. For
Options 1 and 2, consideration must be given to increased lateral
forces resulting from the structural modification. For Option 3,
the experimental data indicates that larger drift ratios are
possible when shear reinforcement is used (refer to Fig. 6).
The data for the shear reinforced specimens are included in
Fig. 7 for comparison.
A direct comparison of the suggested PBSD criteria to the
FEMA 356 acceptance criteria is not simply accomplished
because FEMA limits are in terms of plastic rotations rather
than drift ratios. FEMA 356 is intended for assessing
existing structures and also addresses cases involving
several possible deficiencies, including: 1) inadequate
development or splicing along the slab span; 2) inadequate
embedment into the slabcolumn joint; and 3) lack of continuity
reinforcement through the column cage. In addition, a
distinction is made between primary and secondary components.
In general, the proposed PBSD limits appear to be in the range of
the corresponding FEMA 356 limits. One exception is that
FEMA 356 does not allow plastic rotation in primary
components when the gravity shear ratio is above 0.4.
The aforementioned PBSD criteria are intended primarily
for new construction. The criteria, however, could also be
applied to existing structures that contain subpar seismic
details where a moderate seismic demand is expected. For
assessing the expected performance of a structure, the value
of Vo should be computed with φ = 1.0, whereas for new
building design, Vo should include a strength reduction
factor for shear, currently φ = 0.75 in ACI 31805.
SUMMARY AND CONCLUSIONS
This paper focuses on the behavior and design of interior
slabcolumn connections under combined gravity and lateral
loading and serves to review current design procedures,
performancebased seismic design (PBSD) approaches, and
relevant experimental data. Practical recommendations are
provided for PBSD of slabcolumn connections under
seismic loading conditions that can be readily implemented
into design practice.
An assessment of the experimental data versus the ACI 31805
recommendations for slabcolumn connections indicate
that the limits for determining the necessity of slab shear
reinforcement are a reasonable lower bound of the test data.
Very few reports for slabcolumn connection specimens
include plastic rotation data. FEMA 356, however, provides
limits in terms of plastic rotations for nonlinear analysis
procedures that are determined in part by the gravity shear
ratio at the slabcolumn connections. The recommended
PBSD criteria in this paper use two key parameters for
assessing slabcolumn connections: the gravity shear ratio at
the connection and the maximum story drift ratio. The use of
story drift ratio allows a direct comparison to the experimental
data and is readily available when conducting a structural
ACI Structural Journal/JulyAugust 2007
analysis. A relationship between drift capacity and gravity
shear ratio is provided in Eq. (23), representing an average of
the collected experimental data. Three performance levels
are used to match those in FEMA 356: immediate occupancy,
life safety, and collapse prevention. The proposed limits
correlate well with the ACI 31805 seismic design provisions
for slabcolumn connections and provide a practical approach
for conducting PBSD for slabcolumn connections.
ACKNOWLEDGMENTS
The authors wish to thank the members of ACI Committee 374, PerformanceBased Seismic Design of Concrete Buildings, for their input. The contribution of
Y.H. Kim, a graduate student at Texas A&M University, College Station,
Tex., is also appreciated.
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ACI Structural Journal/JulyAugust 2007
78.
Juan F. Correal is an Assistant Professor of civil and environmental engineering at
the University of Los Andes, Colombia, where he received his BS and MSCE. He
received his PhD in 2004 from the University of NevadaReno, Reno, Nev. His
research interests include the seismic design of bridges and applications of innovative
materials for design, repair, and rehabilitation of structures.
M. Saiid Saiidi, FACI, is a Professor of civil and environmental engineering and is the
Director of the Office of Undergraduate Research at the University of NevadaReno. He
is a Past Chair and a member of ACI Committee 341, EarthquakeResistant Concrete
Bridges, and is member of ACI Committees 342, Bridge Evaluation; E803, Faculty
Network Coordinating Committee; and Joint ACIASCE Committee 352, Joints and
Connections in Monolithic Concrete Structures. His research interests include analysis
and shake table studies of reinforced concrete bridges and application of innovative
materials.
David Sanders, FACI, is an Associate Professor of civil and environmental
engineering at the University of NevadaReno. He is Chair of ACI Committee 445,
Shear and Torsion, is Past Chair of ACI Committee 341, EarthquakeResistant
Concrete Bridges, and is a member of the ACI Technical Activities Committee; ACI
Committees 318, Structural Concrete Building Code; 369, Seismic Repair and
Rehabilitation; 544, Fiber Reinforced Concrete; E803, Faculty Network Coordinating
Committee; E804, Educational Awards Nomination Committee; and Joint ACIASCE
Committee 423, Prestressed Concrete. His research interests include shake table
studies of reinforced concrete bridges.
Table 1—Test variables
Steel reinforcement
Specimen
ISL1.0
ISL1.5
Scale
factor
0.25
Shear
index
Aspect
ratio
di (× R)
ρl , %
ρs ,* %
3.0
3.3
1.0
2.0
1.1
3.0
3.6
1.5
2.0
1.1
ISH1.0
7.0
2.0
1.0
2.9
0.6
ISH1.25
7.0
2.0
1.25
2.8
0.9
7.0
2.1
1.5
2.9
0.9
7.0
2.1
1.5
2.9
0.9*
ISH1.5
ISH1.5T
0.2
*
Steel ratio from additional crossties is not included.
Note: ρl = ratio of longitudinal reinforcement and ρs = ratio of transversal reinforcement.
Saad EIAzazy is the Seismic Research Program Manager at the California Department
of Transportation (Caltrans). He received his BS from Cairo University, Giza, Egypt,
and his MS and PhD from Ohio State University, Columbus, Ohio. His research
interests include bridge seismic retrofit and performance of new bridges.
80% of the gross area (SDC).2 A shear stress index was
defined as the average shear stress divided by 0.083√f ′c
(MPa) (√f ′c [psi]). This index represents the level of shear in
the column. In this study, two levels of shear were selected:
low index equal to 3 and high index equal to 7. These indexes
represent column shear stresses in real bridges. Actual
bridge columns are designed to be ductile and the load
capacity is controlled by flexure, although shear damage is
expected to increase as the shear index increases.
The Caltrans BDS1 states that when more than one cage is
used to confine an oblong column core, the spirals must be
interlocked or the pier must be designed as though it consists
of multiple single columns. A maximum limit of 1.5 times
the radius of the spirals, R, (where R is measured to the
outside edge of the spiral) for the horizontal distance of the
spirals, measured centertocenter of the spirals, di, is
specified. A minimum distance of 1.0R is recommended to
avoid overlaps of more than two spirals in multiple spiral
cases. Of the six models used in this study, two were
designed with a di of 1.0R, one with a di of 1.25R, and three
with a di of 1.5R.
Three alphabetical characters followed by a number were
used to identify the test specimens. The initials I and S
represented interlocking and spirals, respectively. The third
initial L or H was for the shear index of low or high,
respectively. A numeral indicated the fraction of R used for
di. In one specimen an initial T was added at the end of the
specimen, designation to indicate the presence of
supplementary crossties (Fig. 1).
The experimental program was developed to use one of
the shake tables at the LargeScale Structures Laboratory at
the University of NevadaReno. Scale factors of 1/4 for the
specimens with low shear and 1/5 for the columns with high
shear were selected. These were the largest scales that could
be used without exceeding shake table capacity. The
displacementbased design procedure in the SDC2 was used
for a target displacement ductility capacity of 5. In the SDC,2
the displacement ductility is defined as the displacement
divided by the effective yield displacement excluding bond
slip and shear deformations. Typical steel ratios of 2.0% and
394
Fig. 1—Test specimens cross sections.
2.8% were selected for the longitudinal reinforcement. The
transverse steel ratio was designed to provide sufficient
confinement for the columns to reach the target displacement
ductility capacity. Additional crossties with the same bar
size as the spirals and spacing of two times the spacing of the
spirals were used based on a design recommendation
described in Reference 6. An axial load index, defined as the
axial load divided by the product of the gross crosssectional
area and the specified concrete compressive strength of 10%,
was used to represent the axial load level in real bridge columns.
The details of the cross section and the elevations of the
specimens are shown in Fig. 1 and 2, respectively. The
spirals were continuous with constant pitch throughout the
height of the specimens. The spirals were extended along the
height of the footing and the top loading head. The
longitudinal reinforcement was continuous with 90degree
standard hooks at the ends. In the specimens with low shear,
the height was taken from the top of the footing to the center
of the lateral loading head because these columns were
tested in single curvature cantilever mode. The height for
others was taken as the clear distance between the top of the
footing and the bottom of the loading head because these
columns were tested in double curvature.
The specified concrete compressive strength was 34.5 MPa
(5000 psi) with 9.52 mm (3/8 in.) maximum aggregate size. The
average measured concrete strength of the standard cylindrical
ACI Structural Journal/JulyAugust 2007
79.
samples on the day of testing was 36.8 MPa (5339 psi) for
Specimens ISL1.0 and ISL1.5, 31.1 MPa (4514 psi) for
Specimens ISH1.0 and ISH1.5, and 45.1 MPa (6542 psi) for
Specimens ISH1.25 and ISH1.5T. The specified yield stress for
all the reinforcement was 420 MPa (60 ksi). The average
measured yield stress of the steel samples was 462 MPa (67 ksi)
for Specimens ISL1.0 and ISL1.5, 443 MPa (64 ksi) for
Specimens ISH1.0 and ISH1.5, and 431 MPa (63 ksi) for
Specimens ISH1.25 and ISH1.5T.
Test setup, instrumentation, and testing procedure
Figure 3 shows the shake table setup for the high shear
specimens. The test setup for the lowshear specimens was
similar but with only one link between the mass rig and the
column to achieve singlecurvature testing. All specimens
were tested in the strong direction. The lateral load was
applied through an inertial mass system off the table for better
stability. Two sets of swiveled links were used to connect the
inertial mass system to the specimens. One set consisted of
one link connected at the column loading head to test the
specimens as a cantilever member with single curvature. The
other set consisted of two links connected at the top of the
column, allowing the specimens to be tested in double
curvature. The doublelink system was designed to prevent
rotation of the loading head. The specimens with low shear
(ISL1.0 and ISL1.5) were tested in single curvature whereas
the specimens with high shear (ISH1.0, ISH1.25, ISH1.5, and
ISH1.5T) were tested in double curvature. The total equivalent
weight of the inertia mass was 445 and 356 kN (100 and
80 kips) for specimens tested in single and double curvature,
respectively. The axial load was applied through a steel
spreader beam by prestressed bars connected to hydraulic jacks
and an accumulator to limit axial load fluctuation. Electrical
strain gauges were attached to the longitudinal and transverse
steel to measure strain variation. A series of curvature
measurement instruments were installed in the plastic
hinge zones. Additional displacement transducers forming
panels were placed along the height of the columns with high
shear. Load cells were used to measure both the axial and
lateral forces. The acceleration at the top of the columns was
measured using an accelerometer placed on the link
connecting the mass rig to the specimens. Wire potentiometers
were used to measure the lateral displacements of the columns.
Preliminary momentcurvature analysis was performed to
estimate the lateral load and displacement capacities of the
specimens. Once the capacity was estimated, a series of
dynamic analyses were conducted to select the input motion
to be simulated in the shake table tests. The 1994 Northridge
earthquake, recorded at the Sylmar Hospital (0.606g peak
ground acceleration [PGA]) was selected as the input motion
based on the maximum displacement ductility demand
placed on the columns without exceeding the shake table
capacity. The earthquake record is referred to as “Sylmar”
hereafter. The time axis of the input record was compressed
to account for the scale effect and the minor differences
between the axial load and the effective mass.
Each column was subjected to multiple simulated
earthquakes, each referred to as a “run.” The amplitude of
the motions was increased in subsequent runs. Small
increments of the Sylmar record (10 to 20% of the full
Sylmar amplitude) were initially applied to the specimens to
determine the initial stiffness, the elastic response, and the
effective yield point. Once the effective yield was reached,
the amplitude of the input record was increased until failure.
ACI Structural Journal/JulyAugust 2007
Fig. 2—Test specimens elevations.
Fig. 3—Double cantilever test setup.
Intermittent free vibration tests were conducted to measure
the change in frequency and damping ratio of the columns.
EXPERIMENTAL RESULTS
Important aspects of the seismic performance of the test
columns were evaluated. The observed damage progression,
loaddisplacement response, and strains were used to judge
the behavior of the columns. Additional response parameters,
the curvature and plastic hinge length, were computed based
on the measured data and used in performance evaluation.
Observed response
Specimens with low shear—The observed performance was
correlated with the displacement ductility μd, which represents
the displacement divided by the effective measured yield
displacement. Only flexural cracks were observed during the
first three runs (displacement ductility of up to 0.8) in
Specimen ISL1.0 and during the first six runs (μd of up to 1.5)
in Specimen ISL1.5. Most of these cracks were located in the
lower 1/3 of the column height. First spalling and shear cracks
were seen in Specimen ISL1.0 after 0.5 × Sylmar (μd = 1.5) and
in Specimen ISL1.5 after 1.25 × Sylmar (μd = 2.4). The shear
cracks were located in the interlocking region in the lower 1/3
of the height of the column and were extensions of the flexural
cracks. Considerable spalling at the bottom of the column, as
well as propagation of flexural and shear cracks, was observed
after 1.25 × Sylmar (μd = 2.8) in Specimen ISL1.0 and 1.5 ×
Sylmar (μd = 3.1) in Specimen ISL1.5. Spirals were visible
395
80.
Fig. 4—Specimen ISL1.0 after failure.
Fig. 6—Specimen ISH1.25 after failure.
Fig. 5—Vertical crack (µd = 0.7) Specimen ISH1.5.
Fig. 7—Specimen ISH1.5T after failure.
after 1.5 × Sylmar (μd = 4.1) and longitudinal bars were
exposed after 1.75 × Sylmar (μd = 5.6) in Specimen ISL1.0.
The spirals were visible in Specimen ISL1.5 after 1.75 ×
Sylmar (μd = 4.5) and became exposed over a large area after
2.0 × Slymar (μd = 7.5). There was no visible core damage in
either specimen. Specimens ISL1.0 and ISL1.5 failed during
2.0 × Sylmar (1.21g PGA and μd = 9.6) and 2.125 × Sylmar
(1.29g PGA and μd = 10.4), respectively. Figure 4 shows the
damage after failure in Specimen ISL1.0. The failure in both
columns was similar and was due to rupture of the spirals and
buckling of the longitudinal bars at the bottom of the column in
the plastic hinge zone.
Specimens with high shear—Even though these columns
had a relatively high shear index, they were flexural members
and, hence, only flexural cracks were formed during the initial
three or four runs. The measured displacement ductilities
associated with initial flexural cracks were 0.4, 0.6, 0.7, and
0.6 in Specimens ISH1.0, ISH1.25, ISH1.5, and ISH1.5T,
respectively. The flexural cracks were located in the plastic
hinge zones near the top and bottom of the columns. These
cracks were concentrated mainly at the top and bottom 1/3 of
the column height. A vertical crack in the interlocking region
extending from the top of the column to the midheight was
observed after 0.4 × Slymar (μd = 0.7) in Specimen ISH1.5
(Fig. 5). Diagonal cracks were formed in the interlocking
region in the plastic hinge zones of all the specimens. These
cracks began to form starting with 0.5 × Sylmar (μd = 0.6) and
became noticeable under 0.75 × Sylmar (μd = 0.9) in Specimen
ISH1.0, and 1.0 × Sylmar (μd = 1.4) in Specimen ISH1.25. In
Specimen ISH1.5, shear cracks were visible starting with
0.75 × Sylmar (μd = 1.0) and in Specimen ISH1.5T under
1.0 × Sylmar (μd = 1.2). Localized small vertical cracks were
observed in Specimen ISH1.5T under 1.0 × Sylmar. After
1.0 × Sylmar (μd = 1.4), first spalling at the top and bottom of
the column was observed in Specimens ISH1.0 and ISH1.5,
whereas in Specimens ISH1.25 and ISH1.5T, the first spalling
was observed during 1.25 × Sylmar (μd = 1.6 in Specimen
ISH1.25 and 1.7 in Specimen ISH1.5T). Flexural and shear
cracks propagated and more concrete spalled during 1.5 ×
Sylmar (μd = 2.5) in Specimen ISH1.0, 1.75 × Sylmar (μd = 2.2)
in Specimen ISH1.25, 1.25 × Sylmar (μd = 1.7) in Specimen
ISH1.5, and 1.75 × Sylmar (μd = 2.5) in Specimen ISH1.5T.
The spirals were visible at the top and bottom of the column
after 2.125 × Sylmar (μd = 2.9) in Specimen ISH1.25. The
longitudinal bars were exposed after 1.75 × Sylmar (μd = 3.6)
in Specimen ISH1.0, 2.25 × Sylmar (μd = 3.7) in Specimen
ISH1.25, 1.5 × Sylmar (μd = 2.2) in Specimen ISH1.5, and
2.0 × Sylmar (μd = 2.8) in Specimen ISH1.5T. Specimens
ISH1.0 and ISH1.25 (Fig. 6) failed in flexure/shear during
2.0 × Sylmar (μd = 4.7) near the bottom and 2.375 × Sylmar
(μd = 4.7) near the top, respectively.
Damage in the core was observed in Specimen ISH1.5 after
2.125 × Sylmar (μd = 4.7) and in Specimen ISH1.5T after
2.25 × Sylmar (μd = 3.0). The longitudinal bars buckled at the
bottom of the column during 2.25 × Sylmar (μd = 3.4) in
Specimen ISH1.5 and 2.5 × Sylmar (μd = 3.4) in Specimen
ISH1.5T Specimens ISH1.5 and ISH.5T (Fig. 7) failed
during 2.375 × Sylmar (μd = 4.0) and 2.625 × Sylmar (μd =
3.8), respectively. Failure in Specimen ISH1.5, was due to
fracture of the spirals and buckling of the longitudinal bars,
whereas in Specimen ISH1.5T, failure was due to fracture of
the spirals and one of the longitudinal bars.
396
ACI Structural Journal/JulyAugust 2007
81.
Fig. 8—Hysteretic curves and envelopes for lowshear specimens.
Fig. 9—Hysteretic curves and envelopes for highshear specimens.
Forcedisplacement relationships
The accumulated measured hysteresis curves for the ISL
and ISH groups are plotted in Fig. 8 and 9, respectively. For
each column, a backbone forcedisplacement envelope was
developed based on the peak forces with corresponding
displacements for all the motions before failure. The failure
point in the backbone curve was assumed either at the point
of maximum displacement or at a point with 80% of the peak
force with the corresponding displacement. The latter was
used when the force at the maximum displacement dropped
more than 20% of the pick force (Fig. 8 and 9). The
backbone curves for the predominant direction of the motion
were idealized by elastoplastic curves to quantify the
ductility capacity. The force corresponding to the first
reinforcement yield and the corresponding displacement on
the measured envelope was used to define the elastic portion
of the idealized curve. Once the elastic portion was defined,
the yield level was established by equalizing the area
between the measured backbone and the idealized curves.
Figures 8 and 9 show the idealized curves for specimens with
ACI Structural Journal/JulyAugust 2007
low and high shear, respectively. Based on the elastoplastic
curves, displacement ductility capacities of 9.5 and 10.4 were
obtained for Specimens ISL1.0 and ISL1.5, respectively. In
Specimens ISH1.0, ISH1.25, ISH1.5, and ISH1.5T, the
measured displacement ductility capacities were 4.7, 5.0,
4.0, and 3.8, respectively.
The column section total depths were different within each
specimen group due to different distances between the spiral
sets. As a result, the lateral load capacity varied among the
columns. To compare the performance of the specimens,
forces were normalized with respect to the effective yield
force of each specimen and the normalized forcedisplacement envelopes were compared (Fig. 10). The effect
of a large distance between the spiral sets in lowshear
columns can be seen in Fig. 10(a). The overall ductility
capacity of the two lowshear specimens was comparable.
The strength of the specimen with di of 1.5R (Specimen ISL1.5),
however, degraded starting with displacement ductility of
7.4, whereas the strength of the column with di of 1.0R
397
82.
Fig. 10—Normalized lateral forcedisplacement envelopes.
Fig. 11—Measured displacement ductility capacity versus
average shear stress index.
Fig. 12—Measured curvature for ISL group.
(Specimen ISL1.0) did not drop until failure. At a
displacement ductility of 9, the strength degradation in the
column with di of 1.5R was 10% whereas it was 4% when di
was 1.0R. Nevertheless, degradation started at a relatively
high ductility and hence is not of concern. Note that the
target design displacement ductility for the columns was 5.
In specimens with high shear, the displacement ductility
capacity was comparable in the two columns with di of 1.0R
or 1.25R. The ductility capacity dropped by approximately
20% when di was increased to 1.5R. The slightly lower
ductility of Specimen ISH1.5T versus Specimen ISH1.5 (3.8
versus 4) suggests that the addition of crossties had little
effect on the ductility capacity. A comparison of Fig. 9(c)
and (d) indicates that the response of Specimen ISH1.5
contained limited excursions into the negative displacement
range, whereas the Specimen ISH1.5T response was somewhat symmetric. Variations of concrete strength properties,
column stiffness, and the shake table response are attributed
398
to the difference in the column responses. Symmetric cyclic
displacements tend to place higher demands on reinforced
concrete members. It is hence concluded that, had the
displacements in the two columns been identical, Specimen
ISH1.5T would have shown a higher ductility capacity.
Nonetheless, the ductility capacity of approximately 4
measured in Specimens ISH1.5 and ISH1.5T was considered
to be satisfactory. The displacement ductility at which
strength degradation began in columns with di of 1.0R and
1.25R was approximately 3.7, and in those with di of 1.5R
was approximately 3. The larger spread of the spirals clearly
shows some effect on the overall loaddisplacement
response. The addition of the crossties reduced the slope of
the degradation part of the responses (Fig. 10(b)).
The displacement ductility capacity versus the average
shear stress index is shown in Fig. 11. The measured
concrete compressive strengths were used in this graph. In
general, the displacement ductility capacity decreased when
the average shear stress index increased. This was because
columns subjected to high shear failed in shear/flexural
mode, whereas those with low shear failed in flexure with no
significant shear damage.
Measured curvatures
Displacement transducers were used to measure curvature
in the plastic hinge regions at the bottom of the ISL group
and at the top and bottom of the ISH group. The strain on
each side of the column was calculated from the vertical
displacement measured by each external transducer divided
by the gauge length. The average curvature over the gauge
length was computed as the difference between the strains on
the opposite sides of the column divided by the horizontal
distance between the instruments. This procedure assumes
that sections remained plane.
The curvature profiles for the predominant direction of
motion are shown in Fig. 12 and 13 for specimens with low
and high shear, respectively. High values of curvature were
measured in the plastic hinges, as expected. The curvatures
at the ends are influenced by the localized longitudinal
reinforcement bond slip and are not purely due to flexural
deformation of the plastic hinge.
The maximum ultimate curvatures in Specimens ISL1.0
and ISL1.5 were comparable, indicating that the change in
distance of the spiral sets did not affect the curvature
performance. This observation was in agreement with the
displacement ductility capacities of the two models. The
maximum curvatures in the columns with high shear were
also comparable within the group, but were approximately 2/3
of the curvatures of the ISL group. The lower curvatures are
consistent with the smaller displacement ductility capacities
that were observed for this group. The peak top and bottom
curvatures in Specimens ISH1.0 and ISH1.25 were comparable,
confirming that the loading mechanism to bend the columns
in doublecurvature fixedfixed mode was successful. In
Specimens ISH1.5 and ISH1.5T, the peak top curvatures
were 20 to 25% lower than the bottom curvature due to slight
rotation of the loading head that occurred under high loads
and prevented fully fixed response at the top.
Measured strains
The strain gauges on the longitudinal reinforcement were
placed at the potential plastic hinge regions of all the
columns and the footings, and in the loading heads of the
ISH group. In all specimens, the longitudinal bars yielded
ACI Structural Journal/JulyAugust 2007
83.
Fig. 13—Measured curvature for ISH group.
extensively and flexural deformations dominated the
response. Higher strains were measured at or near the base of
all the columns and also at the top of the ISH group. Because
the response in all the specimens was dominated by flexure,
the test variables did not significantly affect the trends in the
longitudinal and spiral bar strains except as noted in the
following.
The correlation between the apparent damage and the longitudinal bar strains was studied. Five damage states were
selected representing an increasing level of damage: 1) flexural
cracks; 2) first spalling and shear cracks; 3) extensive cracking
and spalling; 4) visible spirals and longitudinal bars; and 5)
imminent failure. The fifth damage state refers to the case where
core damage is observed or is about to occur and some of the
longitudinal bars show signs of bending that might lead to
buckling and failure in subsequent runs. This damage state
corresponded to the run before the failure run in the shake table
tests. Figure 14 shows the average of the highest three strain
data in the longitudinal bars versus the damage states in each
model. The average data for three gauges, rather than the
maximum strain, were used because local bar strains are
influenced by cracks and present erratic patterns. The data for
all specimens were averaged and shown on the graph.
It can be seen in Fig. 14 that, within each damage state, the
longitudinal bar strains were generally higher in the ISL
group. This is because the moment gradient in the highshear
columns is relatively high, making the strain more localized
and the average strains lower. The larger distance between
the spiral sets in Specimen ISL1.5 led to higher bar strains in
the first three damage states. Within the ISH group, the bar
strains did not seem to be sensitive to the distance between
the spiral sets.
The average bar strains in all specimens increased
especially during the first three damage states. Average
strains of approximately 3.5 times the yield strain were
recorded when flexural cracks were observed in the
columns. When first spalling and shear cracks were visible,
the strain in the longitudinal bars increased to approximately
7.5 times the yield strain. An average strain of 14.5 times the
ACI Structural Journal/JulyAugust 2007
Fig. 14—Longitudinal bars strain versus observed damage.
yield strain was recorded when extensive cracking and
spalling was observed in the columns. Average strains of 18
and 19 times the yield strain were recorded for the last two
damage states.
The correlation between the spiral bar strains and different
damage states was also reviewed. It was found that spiral bar
strains remain small (generally less than 2/3 of the yield
strain) until the run before failure. These data are presented
and discussed in more detail in Reference 7. It was
determined that it would be more useful if the trends in spiral
bar strains are studied as a function of displacement
ductilities. The average of peak spiral strains is plotted
against displacement ductilities in Fig. 15. It can be seen that
average strain was below yield until higher ductilities were
reached. The larger distance between the spiral sets in
Specimen ISL1.5 led to higher strains than those of
Specimen ISL1.0 under large ductilities. The higher spiral
strains are attributed to the slight degradation of the load
capacity (Fig. 10) observed in Specimen ISL1.5. In addition,
Fig. 15 shows slightly smaller strains in Specimen ISH1.0
compared with the rest of the highshear specimens until the
last motion. The average maximum spiral strains in
399
84.
Fig. 15—Maximum average strain in the spirals.
Table 2—Data for plastic hinge length
Specimen
Variables
ISL1.0
ISL1.5
ISH1.0 ISH1.25 ISH1.5 ISH1.5T
φp , Rad/mm
(Rad/in.)
0.204
(0.008)
0.159
(0.006)
0.124
0.116
(0.005) (0.005)
0.101
(0.004)
0.074
(0.003)
Δy, mm (in.)
16.901
(0.67)
18.172
(0.72)
21.1
(0.83)
21.1
(0.83)
32.1
(1.26)
26.7
(1.05)
Δu, mm (in.)
161
(6.34)
188
(7.4202)
98.6
(3.88)
105
(4.15)
127
(5.02)
102
(4.00)
L, mm (in.)
1473
(58)
1828
(72)
1473
(58)
1600
(63)
1753
(69)
1753
(69)
lp , mm (in.)
351
(13.8)
428
(16.84)
363
(14.3)
384
(15.1)
480
(18.9)
541
(21.3)
Specimens ISH1.25 and ISH1.5T were nearly the same, and
the average maximum spiral strain in Specimen ISH1.5 was
the highest until a displacement ductility of approximately
1.6 was reached.
Plastic hinge length
The plastic hinge length (PHL) is used to estimate postyield lateral displacements based on the moment curvature
properties of the plastic hinge while empirically taking into
account displacements due to bond slip and shear
deformation. To determine the sensitivity of PHL to the
spiral set distance and the level of shear, the PHL for each
column was estimated using the measured plastic curvatures
and displacements. The moment area method was used to
relate displacements and curvatures assuming that the plastic
rotation θp over the equivalent PHL, lp, is defined by
θ p = ( φ u – φ y )l p
(1)
where φu equals the ultimate curvature capacity, and φy
equals the idealized yield curvature.
The center of rotation was assumed to be at the center of
the plastic hinge. Equation (2) was assumed to relate plastic
rotation and plastic displacements. The PHL was determined
using this equation. In the ISH group, two plastic hinges
were formed and, hence, the average measured curvatures at
the top and bottom were used.
lp
Δ p = θ p ⎛ L – ⎞
⎝
2⎠
(2)
where L equals the distance from point of maximum moment
to the point of contraflexure.
400
In Eq. (1), the average of the measured curvatures over the
extreme two gauge lengths (203 mm [8 in.] in lowshear
columns and 254 mm [10 in.] in highshear columns) was used
because most of the plastic deformation was concentrated over
that region according to the measured curvatures and strains.
Table 2 lists the data used to determine the measured lp for
Specimens ISL1.0 and ISL 1.5. The values of lp of 0.75 and
0.83 times the total depth of the column were found for
Specimens ISL1.0 and ISL1.5, respectively. It can be seen that
the larger spiral distance in Specimen ISL1.5 led to an increase
in the ratio of the PHL over the column depth by
approximately 10%. The aspect ratio (column height divided
by the column section depth in the loading direction) of
Specimen ISL1.5 was approximately 10% larger than the
Specimen ISL1.0 aspect ratio. Under equal conditions,
Specimen ISL1.5 would experience a smaller shear
deformation. The larger spread of the spirals in Specimen
ISL1.5, however, appear to have led to higher shear
deformations that necessitated a larger PHL to match the
measured displacement.
The values of lp of 0.98, 0.96, 1.12, and 1.27 times the total
depth of the columns were found for Specimens ISH1.0,
ISH1.25, ISH1.5, and ISH1.5T, respectively. The aspect
ratios for these columns were nearly the same. In highshear
columns, the increase in the distance between the spirals
from 1.0R to 1.5R appears to have increased displacement
due to shear, thus increasing the apparent plastic hinge
length by approximately 20%.
CONCLUSIONS
Based on the observations and the experimental results of
this study, the following conclusions are drawn:
1. The seismic performance of columns with relatively
low shear with spiral distance di of 1.0R and 1.5R was similar
and satisfactory with displacement ductility capacities of
near 10. The strength degradation was slightly larger when di
was 1.5R. This degradation began at a displacement ductility
of 7.4, however, which exceeded the target design
displacement ductility of 5;
2. Because the lowshear column with di of 1.5R did not
experience significant shear cracking, and based on the
satisfactory displacement ductility capacity, it appears that
the Caltrans provision of allowing a di value of up to 1.5R is
appropriate for columns with low shear;
3. The seismic performance of column models with di of
1.0R and 1.25R subjected to high shear was similar and
satisfactory. Even though the columns failed in shear/flexure
mode, they were ductile and achieved the design
displacement ductility capacity of 5;
4. Vertical cracks in the interlocking region were observed
under small earthquakes in the column with high shear and
di of 1.5R. The large area of plain concrete in the interlocking
zone is susceptible to cracking when di is 1.5R and the
column shear is relatively high. The addition of horizontal
crossties connecting the interlocking hoops not only reduced
and delayed vertical cracks in the interlocking region, but
also reduced the strength degradation;
5. The measured displacement ductility capacity was
approximately 4 in columns with high shear and a di of 1.5R.
Even though the desired ductility capacity was 5, the column
is considered to be sufficiently ductile for most applications.
Crossties are recommended to reduce premature vertical
cracking in these columns; and
ACI Structural Journal/JulyAugust 2007
85.
6. The plastic hinge length to match the measured plastic
lateral displacement increased as the distance of the spirals
sets increased from 1.0R to 1.5R by 10 to 20%, depending on
the level of shear.
ACKNOWLEDGMENTS
The research presented in this paper was sponsored by the California
Department of Transportation. The dedicated assistance of P. Laplace, J.
Pedroarena, and P. Lucas of the University of NevadaReno bridge
laboratory is gratefully acknowledged. Specials thanks are expressed to N.
Wehbe of South Dakota State University for developing a momentcurvature
analysis program for interlocking spiral columns.
REFERENCES
1. California Department of Transportation, “Bridge Design Specifications,”
Engineering Service Center, Earthquake Engineering Branch, Calif.,
July 2000, 250 pp.
2. California Department of Transportation, “Seismic Design Criteria
ACI Structural Journal/JulyAugust 2007
Version 1.2,” Engineering Service Center, Earthquake Engineering Branch,
Calif., Dec. 2001, 133 pp.
3. Tanaka, H., and Park, R., “Seismic Design and Behavior of Reinforced
Concrete Columns with Interlocking Spirals,” ACI Structural Journal, V. 90,
No. 2, Mar.Apr. 1993, pp. 192203.
4. Buckingham, G. C., “Seismic Performance of Bridge Columns with
Interlocking Spirals Reinforcement,” MS thesis, Washington State University,
Pullman, Wash., 1992, 146 pp.
5. Benzoni, G.; Priestley, M. J. N.; and Seible, F., “Seismic Shear Strength
of Columns with Interlocking Spiral Reinforcement,” 12th World Conference
on Earthquake Engineering, Auckland, New Zealand, 2000, 8 pp.
6. Correal, J.; Saiidi, M.; and Sanders, D., “Seismic Performance of RC
Bridge Columns Reinforced with Two Interlocking Spirals,” Report No.
CCEER046, Center for Civil Engineering Earthquake Research, Department
of Civil Engineering, University of NevadaReno, Reno, Nev., Aug. 2004,
438 pp.
7. Correal, J., and Saiidi, M., “Lessons Learned from Shake Table Testing of
RC Columns in Relation to Health Monitoring,” IMACXXIII—A Conference
& Exposition on Structural Dynamics—Structural Health Monitoring,
Orlando, Fla., 2005, 9 pp.
401
86.
DISCUSSION
Disc. 103S67/From the Sept.Oct. 2006 ACI Structural Journal, p. 656
Shear Strength of Reinforced Concrete TBeams without Transverse Reinforcement. Paper by A. Koray
Tureyen, Tyler S. Wolf, and Robert J. Frosch
Discussion by Himat Solanki
Professional Engineer, Building Dept., Sarasota County Government, Sarasota, Fla.
The authors have presented an interesting paper on the
shear strength of reinforced concrete Tbeams without
transverse reinforcement. However, the discusser would like
to offer the following comments:
1. The authors have mentioned the basic outline of a
derivation of Eq. (1), but Eq. (1) appears to be based on the
neutral axis (NA) located at the center of the beam in a
typical homogeneous rectangular concrete beam. The
authors’ Eq. (2) was a simplification of Eq. (1) based on the
experimental database of reinforced concrete beams, which
is inconsistent with the Rankine’s failure criteria of a plain
homogeneous concrete beam.
Based on ft = 6 f c′ (or 0.1fc′ ) and assuming the Rankine’s
failure criteria of plain concrete, and by considering various
strength ratios of flexural stress σm versus concrete compressive
stress fc′ (σm/f ′c = 14.2%,10 the flexural stress σm of a plain
homogeneous concrete beam equals to 114.2%10 of the tensile
strength of plain concrete ft . Based on the aforementioned
assumptions, the discusser arrived at the authors’ Eq. (2)
without considering the experimental database of reinforced
concrete beams.
Another simplified approach is that Eq. (2) can also be
derived from the current ACI Building Code9 (that is,
authors’ Eq. (5)) by assuming an average depth of NA equals
0.4d11,12 and by substituting c = 0.4d in the authors’ Eq. (5),
which would result in authors’ Eq. (2). Based on the aforementioned two approaches, the discusser believes that there
is no need to have a reinforced concrete beam database, that
is, Fig. 1 and 2. Is this consistent with the shear strength of
reinforced concrete Tbeams without transverse reinforcement plain concrete?
2. The authors’ concept on shear funnel (Fig. 8 and 10) is
somewhat unclear. Please note that there is no reinforcement
within the compression and/or flange area. Based on Fig. 8,
considering a simplified approach, a portion of the crosssectional area above NA in the Tbeam could be converted
into an equivalent rectangular section, but not the entire
section of the Tbeam when a shear force is computed. The
discusser has computed over 100 specimens of Tbeams
from Reference 1 by assuming the flange depth as one unit
and the overall depth and web width were transferred into the
flange depth units with varying depths of NA (that is, NA
was assumed within the flange and within the web of the
Tbeams) and found that approximately 20% of the crosssectional area increases above NA as compared with its
equivalent rectangular section and approximately 10% of the
crosssectional area increases to its equivalent rectangular
section, if the entire beam was compared with the rectangular
section. These values are somewhat inconsistent in the
authors’ Table 2.
ACI Structural Journal/JulyAugust 2007
REFERENCES
10. Kato, K., Concrete Engineering Data Book, Nihon University,
KoriyamaCity, Fukushima Prefecture, Japan, 2000.
11. Eurocode No. 2, “Design of Concrete Structures, Part 1: General
Rules and Rules of Buildings,” ENV 199211, Commission of the European
Communities,1991.
12. British Standard Institution, “Structural Use of Concrete, Part 1:
Code of Practice for Design and Construction,” BS 8110:Part 1:1997, London,
UK, 1997.
AUTHORS’ CLOSURE
The authors thank the discusser for his interest in this
paper. The comments are addressed in the same order as
presented by the discusser.
The detailed derivation of Eq. (1) and its simplification
into Eq. (2) are presented in Reference 2 of the paper. This
derivation was not based on the neutral axis located at the
center of a beam, but rather based on the location of the
neutral axis as calculated based on a cracked section analysis.
The discusser is referred to Reference 2 for further clarification.
As noted in Reference 2, Eq. (1) was derived considering
that failure initiates when the principal stress in the compression
zone reaches the tensile strength of concrete ft. It was shown
that this equation could be simplified for an assumed tensile
strength (6 f c′ ) and considering the flexural stress σm. The
experimental results, however, were considered so that a
complete perspective of the performance of the simplified
expression could be accessed.
The discusser notes that Eq. (2) can be derived from the
ACI code. It appears that the discusser is referring to ACI
Eq. (113) rather than (115). Perhaps a better view is that
Eq. (113) is a subset of Eq. (2). For k = 0.4, Eq. (2) simplifies
it to 2 f c′ bwd. The neutral axis depth, c = kd varies according
to the flexural reinforcement ratio ρ and the modular ratio n.
Therefore, Eq. (2) accounts for the reinforcement ratio and
the concrete compressive strength, whereas ACI 318 Eq. (113)
is insensitive except with respect to its inclusion in the term f c′ .
Unfortunately, the discusser’s question “Is this consistent
with the shear strength of reinforced concrete Tbeams
without transverse reinforcement plain concrete?” is unclear
and cannot be addressed.
The results presented in Fig. 10 are based on an angled
approach using a 45degree angle. Simplification can be
achieved using an effective flange width approach. Based on
the area achieved from the 45degree shear funnel, an effective
overhanging flange width of 0.5 times the flange depth on
each side of the web can be considered for shear. If the
neutral axis falls within the thickness of the flange, this
effective width approach is conservative. It should be noted
that in either the shear funnel or equivalent flange width
approach, the neutral axis depth should be computed using
an effective flange width that is based on flexural behavior
503
87.
and that is different from the flange width considered effective
for shear.
Table 2 presents a statistical comparison of the performance
of the various design methods considering the ratio of Vtest/
Vcalc. Therefore, it is unclear what inconsistencies the
discusser is referring to. However, as emphasized in the
paper, for the evaluation of the shear area when the flanges
were ignored, the neutral axis depth was calculated ignoring
the flanges while the shear funnel approach computed the
neutral axis depth with the flanges considered. This may
explain the perceived inconsistencies in the discusser’s analysis if he was directly comparing the results provided in
Table 2. Regardless, the main premise is that additional
shear area beyond that bounded by the web can be considered
as effective in shear transfer. The percentage of flange area
considered will vary depending on the section considered and
the location of the neutral axis.
Disc. 103S71/From the Sept.Oct. 2006 ACI Structural Journal, p. 693
Shear Strength of Reinforced Concrete TBeams. Paper by Ionanis P. Zararis, Maria K. Karaveziroglou, and
Prodromos D. Zararis
Discussion by Himat Solanki
Professional Engineer, Building Dept., Sarasota County Government, Sarasota, Fla.
The authors have presented an interesting concept in their
paper on shear strength of reinforced concrete Tbeams.
However, the discusser would like to offer the following
comments:
1. The authors have considered εco = 0.002 and fct =
0.30f ′ 2/3 (Eq. (4)) based on Reference 10, but no consideration
c
was given to the depth of compression zone equals 0.8c
value as suggested in Reference 10. Also, the authors have
not thoroughly explained the assumption of 0.667c value (in
the Appendix) other than the test result values versus their
theoretical values. Please note that BS 8110:Part 1:199715
considers the depth of compression zone equal a value of
0.9c.
2. Based on Eq. (7) and Fig. 7, the authors assumption for
a 45degree projection angle from web to flange appears to
be inconsistent with Fig. 6(b) and other researchers. The
45degree projection angle was a simplified assumption
based on the depth of compression equals the depth of
flange, that is, the neutral axis (NA) is located at the interface
of the bottom of flange and the top of web.
3. In conclusion, the authors’ statement “An increase of
stirrups does not give any advantage to Tbeams over the
rectangular beams” is a little confusing without thorough
explanation, because the authors have converted a Tbeam
into a rectangular beam with bef web width in lieu of bw web
width. Let’s consider beam pairs from Table 1: Beam Pair
TA11TA12 of Reference 2; Beam Pair T2T3 and Beam
Pair T15T16 of Reference 4; Beam Pair T3aT3b of
Reference 5; and Beam Pair A00A75 of Reference 7. These
beam pairs all have test parameters such as concrete strength
fc′ , longitudinal reinforcement ρ%, and shear spantodepth ratio a/d approximately identical, except for the
shear reinforcement ρv fvy; but the shear strength increases
with an increase in shear reinforcement ρv fvy. This means
the shear reinforcement ρv fvy does have some influence on
the shear strength.
4. The authors’ Eq. (9) and the calculated values of A s of
′
the depth of compression block in Fig. A (of the authors’
Appendix) are unclear. It appears that the authors have
considered a routine rectangular beam with compression
reinforcement but have not considered the reinforcement
within the flange width when a Tbeam section was
converted into a rectangular beam section above NA. The
504
reinforcement in the flange would improve the value of c
(depth of NA) as well as the value of Vcr in Eq. (8) and Vu in
Eq. (10).
5. The discusser has calculated all Tbeams except
Beam ET1, which is a rectangular beam from References 1 and
2, as outlined in the authors’ Table 1, by considering the
reinforcement in the flange width and by using authors’
Eq. (10) for a calculation of NA, c, and then Vcr and Vu were
calculated. Based on this concept, a mean value of Vu,exp /Vu,th
of 1.006 and a standard deviation value of 0.05 were found.
It was also noticed from Table 11,2,48 that the thinner web
width with higher reinforcement ratios (both longitudinal
and shear reinforcement ratios) do not have any advantage over
wider web width with lower reinforcement ratio in Tbeams.
REFERENCES
15. British Standard Institution, “Structural Use of Concrete, Part 1:
Code of Practice for Design and Construction,” BS 8110:Part 1:1997,
London, UK, 1997.
AUTHORS’ CLOSURE
The authors would like to thank the discusser for his
interest in the paper and his kind comments. The authors
would like to reply to his comments in the order they are asked.
In the case of rectangular or Tsection beams, the true
distribution of stresses in the compressive zone is usually
replaced for simplification by an equivalent rectangular
stress block. In the ultimate limit state, that is, when the
compressive strain in concrete at extreme fiber is εc = 0.0035,
the true distribution of stresses in the compressive zone
follows a parabolarectangular diagram. Then, the compressive
force of concrete, as a resultant of stresses, is Fc = 0.81bcfc′ .
Thus, the equivalent rectangular stress distribution has an
approximate height equal to 0.8c. In this case, however, the
authors choose a state where the strain of concrete at extreme
fiber is εco = 0.002. This strain corresponds in a true, exactly
parabolic distribution of stresses in the compressive zone. In
this case, the corresponding compressive force of concrete is
Fc = 0.667bcfc′ . Thus, the equivalent rectangular stress
distribution (shown in Fig. A) has a height equal to 0.667c.
There has never been made a 45degree projection angle by
the authors. As it is written in the text of the paper, the failure
occurs due to a splitting of concrete that takes place in the
compression zone of the Tbeam. Taking into account Fig. 2
ACI Structural Journal/JulyAugust 2007
88.
and 6, the splitting takes place in an inclined area, the projection
of which, on a cross section of the beam, is approximately
defined from the shaded part of the section in Fig. 7.
Equation (7), giving the effective width, results simply from
the area of this shaded part of cross section of the Tbeam.
This statement means that the contribution of stirrups in the
shear strength is the same for Tbeams and rectangular beams,
as it results from the second part of Eq. (10). The increase in
the strength of the beams that the discusser has mentioned is
due to an increase of the first part of Eq. (10).
The compression reinforcement As′ within the flange
width has been considered and takes part in Eq. (9) with the
ratio ρ′ = As′ bwd. Nevertheless, an increase of As′ does not
increase the shear strength of a beam; on the contrary, it
decreases the shear strength, exactly because the reinforcement
As′ improves the value of c. Equations (8) and (10) show that
a decrease of the depth c decreases the strength. This has
been observed both in Tbeams and rectangular test beams.
The compression reinforcement A s has not been considered
′
in the calculations of Table 1, because of the lack of data
regarding this reinforcement for all the test beams. As it
results from the discusser’s calculations, however, the small
ratios of ρ′ have only a small effect on the shear strength.
Disc. 103S76/From the Sept.Oct. 2006 ACI Structural Journal, p. 736
Effect of Reinforced Concrete Members Prone to Shear Deformations: Part I—Effect of Confinement.
Paper by Suraphong Powanusorn and Joseph M. Bracci
Discussion by Himat Solanki
Professional Engineer, Building Dept., Sarasota County Government, Sarasota, Fla.
Though the authors have presented an interesting concept
on shear deformations in their paper, they have not fully
explained all necessary assumptions other than the use of
Mander et al.’s methodology. Also, the authors have not
provided the details as outlined by Mander et al. (1988).
Without a detailed explanation and information, particularly
of the test specimens supplemented by the associated
assumptions, it is very difficult to verify the author’s results
as well as published results available elsewhere; therefore, the
discusser has the following comments:
1. The discusser has tried to understand the authors’
methodology, and has described the authors’ methodology
to the best as follows. In the following concept, there are
several assumptions that were neither mentioned by Mander
et al. (1988), nor by the authors.
The authors’ Eq. (12) is expressed as
0.0024 + (0.0024 + 0.002) cot2 35 degrees = 0.0114. Now,
εcc = εco[1 + R((f ′cc/f ′co) – 1)].
Based on the test results of Mander et al. (1988) and
Scott et al. (1980), f ′cc/f ′co ≈ 1.75 and εco ≈ 0.002 (Richart
et al. 1928).
In the previous equation, the R value varies from 3 to 6
(Park and Paulay 1990). Based on the authors’ Fig. 1 and 2,
the transverse reinforcement details with respect to the longitudinal reinforcement, R = 5, as suggested by the authors in
their Eq. (14) appears to be on the low side. Therefore, R
= 6 was appropriate and was assumed in the aforementioned
equation by the discusser. That is, εcc = 0.002 [1 + 6((1.75) – 1)]
= 0.011.
Based on the Mander et al. (1988) and Scott et al. (1980)
test results, εcc ≈ 0.0115.
Based on an average value of εcc = 0.01125
βf ′cc xr
σ c = r
r–1+x
1
β =  ≤ 1
0.8 + 0.34 ( ε 1 ⁄ ε cc )
where
1
β =  ≤ 1
0.8 + 0.34 ( ε 1 ⁄ ε cc )
in which
Also, based on an average value of εcc = 0.01125 and εc ≈
0.0048 was chosen due to lateral expansion (biaxial tensioncompression)
x = εc/εcc = 0.0048/0.01125 = 0.425
Esec = f ′co/εcc = 1.75f ′co /3.52εco
ε1 = εs + (εs + 0.002)cot2α
Ferguson (1964) suggested that the stress in steel develops
from 1.15fy to 1.20fy. Therefore, an average value of 1.175fy
was considered. That is, εs = 1.175fy/Es, where Es = 29,000 ksi.
Furthermore, it was assumed that the tensile strain is
causing approximately a 35degree skew angle crack to the
strut’s axis. The 35 degrees falls within the range from 25 to
45 degrees, and this angle is consistent with Cusson and
Paultre (1994) and Fig. 5 and 13 of Ferguson (1964): ε1 =
ACI Structural Journal/JulyAugust 2007
≈ 0.5Ec
Ec
Ec
Now, r =  =  = 2.0Ec
E c – E sec E c – 0.5E c
βf ′ xr
cc
Now σc = r
r–1+x
505
89.
Substituting β, x, and r values in the previous equation
( 0.8737 )f ′cc ( 0.425 ) ( 2.0 )
σ c =  = 0.629f ′cc
2.0 – 1 + 0.425 ( 2.0 )
Because f ′cc ≈ 1.75fc′
σc = 1.1008fc′ or ≈ 1.10fc′
This means approximately 10% compressive stress
increases due to the confinement. This value is consistent
with Vecchio’s (1992) concept as well as the authors’ tests
results as shown in Tables 1 through 3.
Based on Vecchio’s study (Vecchio 1992), an average
stress in shear panels was increased by approximately 5.6%,
while an average stress in shearwalls was increased by
approximately 13.4%, that is, an overall average value
increased in stress would be 9.5%. Is this consistent with the
methodology/concept/logic used in this paper?
2. Based on Fig. 1(a), the authors have considered a
symmetrical loading case, but the symmetrical loading case
may not be the case for all structures in the practice. Because
asymmetrical loading conditions would create unbalanced
loading, it would require some additional reinforcement per
truss analogy in the dark area, as shown by the authors in
Fig. 9(a) and (b), depending on the unbalanced load due to
the asymmetrical loading condition.
3. It is unclear how the theoretical values stated in Tables 1
through 3 were calculated. Was any correction for variable
depth considered? Or was a uniform depth considered?
Though the authors stated the advantage of overlapping stirrups
versus single stirrups, the effectiveness of stirrups as
compared with the longitudinal reinforcement was unclear
from Table 1 through 3.
4. The discusser would like to point out that because the
shear strength and shear deformations relate to the strength
of concrete, a simplified method proposed by Muttoni
(2003) could be extended to the authors’ specimens.
5. Using the aforementioned concept outlined in this
discussion and Muttoni’s (2003) methodology, the discusser
has also analyzed other test specimens available in the literature
elsewhere (Rodrigues and Muttoni 2004; Fukui et al. 2001;
Ferguson 1964). The results are found to be in good agreement
with the test results. Due to brevity, the results are not
included in the discussion.
ACKNOWLEDGMENTS
The discusser gratefully appreciates S. Unjoh, Leader, Earthquake
Engineering Team, Public Works Research Institute, Tokyo, Japan; A.
Muttoni, Institut de Structures, Ecole Polytechnique Fédérale de Lausanne,
Lausanne, Switzerland; and N. Pippin and A. Wards, TTI, Texas A&M
University, College Station, Tex., for providing publications related to the
shear strength of beams.
REFERENCES
Fukui, J.; Shirato, M.; and Umebara, T., 2001, “Study of Shear Capacity
of Deep Beams and Footing,” Technical Memorandum No. 3841, Public
Works Research Institute, Tokyo, Japan. (in Japanese)
Cusson, D., and Paultre, P., 1994, “High Strength Concrete Columns
Confined by Rectangular Ties,” Journal of Structural Engineering, ASCE,
V. 120, No. 3, Mar., pp. 783804.
Mander, J. B.; Priestley, M. J. N.; and Park, R., 1988, “Observed Stress
and Strain Behavior of Confined Concrete,” Journal of Structural Engineering,
ASCE, V. 114, No. 8, Aug., pp. 18271849.
Muttoni, A., 2003, “Schubfestigkeit und Durchstanzen von Platten ohne
Querkraftbewehrung,” Beton und Stahlbetonbau, V. 98, No. 2, Feb., pp. 7484.
Park, R., and Paulay, T., 1990, “Bridge Design and Research Seminar:
506
V. I, Strength and Ductility of Concrete Substructures of Bridges,” RR
Bulletin 84, Transit New Zealand, Wellington, New Zealand.
Richart, F. E.; Brandtzaeg, A.; and Brown, R. L., 1928, “A Study of Failure
of Concrete under Combined Compressive Stresses,” Bulletin 185, University
of Illinois Engineering Experimental Station, Champaign, Ill.
Rodrigues, R. V., and Muttoni, A., 2004, “Influence des Déformations
Plastiques de l’Armature de Flexion sur la Résistance a l’Effort Trenchant
des Pouters sans étriers: Rappart d’essai,” Laboratoire de Construction en
Béton (ISBETON), Istitut de Structures, Ecole Polytechnique Fédérale de
Lausanne, Oct.
Scott, B. D.; Park, R.; and Priestley, M. J. N., 1980, “StressStrain
Relationships for Confined Concrete: Rectangular Sections,” Research
Report 806, Department of Civil Engineering, University of Canterbury,
Christchurch, New Zealand, Feb.
AUTHORS’ CLOSURE
The authors would like to express a sincere gratitude to the
discusser for comments that give the authors an opportunity
to clarify certain issues in the article. The authors’ response
to the discusser is as follows:
General
The purpose of the article under discussion was to present
an alternative method that incorporates the effects of
confinement into the constitutive equations of the Modified
Compression Field Theory (MCFT), first proposed by
Vecchio and Collins (1986). In essence, the extension of the
MCFT proposed by the authors is based on twodimensional
stress and strain analysis. All necessary assumptions were
stated at the beginning of the article under the section
Proposed analytical model.
Response to discusser comments
1. The discusser demonstrates the application of Eq. (12)
on the constitutive relationship of concrete in compression
taken into account the effect of confinement given in the
paper with assumptions on a few parameters shown in the
equation. It was concluded that the results from applying
Eq. (12) led to an approximate 10% increase in compressive
strength of concrete, which was compared with a study by
Vecchio (1992) on shearwalls and panels and also by the
authors’ reinforced concrete (RC) bent cap tests. From the
authors’ point of view, however, the application of Eq. (12)
alone to obtain an increase in strength is only part of the
comparative study. It is the forcedeformation behavior that
is important for comparative purposes, especially for
members prone to shear deformations near ultimate loading.
MCFT is generally developed on the basis of: 1) twodimensional states of stress and strain; 2) the superposition
of stresses in the concrete and reinforcing steel as shown in
Eq. (1); and 3) the compatibility of strains in the concrete and
reinforcing steel as shown in Eq. (2). The model can be
categorized into the socalled rotating crack model to maintain
the coaxiality between the concrete principal stresses and
principal directions. For twodimensional states of stress and
strain, three components of stresses and strains, which are εx,
εy, and γxy and σx, σy, and τxy, are required to define a state
of stress and strain at a given point within the member. The
constitutive relationships under the context of MCFT,
however, have been defined in the principal stress and strain
components (σ1, σ2) and (ε1, ε2). The general state of stress
and strain, εx, εy, and γxy and σx, σy, and τxy, are related to the
principal stress and strain components (σ1, σ2) and (ε1, ε2)
using Mohr’s circle of stress and strain. The concrete
constitutive equation in compression defined in the principal
stress and strain directions are given in Eq. (4) through (8)
and (11) through (13). The special emphasis of the article is
ACI Structural Journal/JulyAugust 2007
90.
on the incorporation of the beneficial effects of lateral
confinement of the transverse reinforcement on the concrete
stressstrain relationship in the principal compressive direction
using an approach adopted by Mander et al. (1988) using the
fiveparameter failure surface derived by Willam and
Warnke (1974). Due to space limitations, the authors did not
include the complete development of fiveparameter failure
surface in the article. Interested readers should consult the
original paper by Willam and Warnke (1974) or books by
Chen (1982), Chen and Han (1988), and Chen and Saleeb
(1982) for further details.
Regarding the discusser’s comments on the R value for
determining the peak strain corresponding to the peak
concrete stress, additional studies by the authors have shown
that the use of R = 6 led to only a marginal change in the
strength prediction.
2. The MCFT was formulated on the basis of three
fundamental principles of structural mechanics, which are:
1) equilibrium; 2) compatibility; and 3) material constitutive
relationships. The rationality and generality of the MCFT
should make the theory applicable to any loading pattern.
The case of unsymmetric loading, however, was not considered
ACI Structural Journal/JulyAugust 2007
in this work and would require further experimental and
analytical research to justify recommendations.
3. To justify the proposed model, the authors implemented
the proposed model into a finite element code using a userdefined material subroutine. It is the results from FEM analysis
that are summarized in Tables 1 through 3.
4 and 5. The authors agree with the discusser that the shear
strength and deformation are related to the compressive
strength of concrete and would like to look into further
details on the article by Muttoni (2003).
REFERENCES
Chen, W.F., 1982, Plasticity in Reinforced Concrete, McGrawHill, New
York, 474 pp.
Chen, W.F., and Han, D. J., 1988, Plasticity for Structural Engineers,
SpringerVerlag, New York, 606 pp.
Chen, W.F., and Saleeb, A. F., 1982, “Constitutive Equations for Engineering
Materials,” Elasticity and Modeling, V. 1, John Wiley & Sons, New York.
Mander, J. B.; Priestley, M. J. N.; and Park, R., 1988, “Theoretical StressStrain Model for Confined Concrete,” Journal of Structural Engineering,
ASCE, V. 114, No. 8, pp. 18041826.
Willam, K. J., and Warnke, E. P., 1974, “Constitutive Model for the Triaxial
Behavior of Concrete,” Concrete Structures Subjected to Triaxial Stresses,
Paper III1, International Association of Bridge and Structural Engineers
Seminar, Bergamo, Italy, pp. 130.
507
92.
ACI member KyoungKyu Choi is a Research Assistant Professor at the University of
New Mexico, Albuquerque, N. Mex. He received his BE, MS, and PhD in architecture
from Seoul National University, Seoul, Korea. He is an associate member of ACI
Committees 440, Fiber Reinforced Polymer Reinforcement; 548, Polymers in
Concrete; and Joint ACIASCE Committee 445, Shear and Torsion. His research
interests include shear strength and seismic design of reinforced concrete structures
and application of artificial intelligence in structural engineering.
ACI member Mahmoud M. Reda Taha is an Assistant Professor in the Department
of Civil Engineering at the University of New Mexico. He received BSc and MSc from
Ain Shams University, Cairo, Egypt, and his PhD from the University of Calgary,
Calgary, Alberta, Canada, in 2000. He is a member of ACI Committees 209, Creep
and Shrinkage in Concrete; 235, Electronic Data Exchange; 440, Fiber Reinforced Polymer
Reinforcement; 548, Polymers in Concrete; and E803, Faculty Network Coordinating
Committee. His research interests include structural monitoring, using artificial intelligence
in structural modeling, and fiberreinforced polymers.
ACI member Alaa G. Sherif is an Associate Professor in the Civil Engineering
Department, Helwan University, MatariaCairo, Egypt. He received his BSc from
Cairo University, Cairo, Egypt, and his MSc and PhD from the University of Calgary.
He is an associate member of Joint ACIASCE Committee 352, Joints and Connections in
Monolithic Concrete Structures. His research interests include the behavior and serviceability of reinforced concrete structures and systems for multispan cablestayed bridges.
approach and targets predicting the punching shear strength
of the slabcolumn connections based on various geometric
and material parameters. The proposed fuzzybased model
presented in a simple form respects the failure mechanics of
punching shear by learning its rules from the experimental
database with the ability to address the interaction between
the modeling variables and the uncertainty in these variables.
The fuzzybased model shows high accuracy in predicting
punching shear strength.
Fig. 1—Current design codes for punching shear.
FUZZY LEARNING OF PUNCHING
SHEAR DATABANK
Fuzzy systems have been widely used in the last decade
for modeling complex engineering systems (for example,
modeling robots16 and in assessing concrete durability17)
and their feasibility as universal approximators has been
proven.18 The capability of the fuzzy systems to model
complex systems is attributed to their inherent ability to
accommodate a tolerance for uncertainty in the modeling
parameters.19,20 While probabilistic empirical models are
limited to random uncertainties, fuzzy systems have the ability
to consider random and nonrandom types of uncertainties
that arise due to vagueness and/or ambiguity in the modeling
parameters/process.1820
The fundamental concept in modeling complex phenomena
using fuzzy systems is to establish a fuzzy rulebase that is
capable of describing the relationship between the input
parameters and the output parameters while considering
uncertainty bounds.19 This fuzzy rulebase captures individual
and group relationships that distinguish the internal complex
relations between the system parameters.20 As such, system
nonlinearity is not recognized by using nonlinear equations
but through establishing a number of fuzzy rules (that could
use linear relations) such that the fuzzy system becomes
capable of describing the phenomena to a prespecified level
of accuracy.20 A group of successful techniques to establish a
fuzzy rulebase using exemplar observations was recently
developed.20,21
Here, the use of the fuzzy set theory to model the punching
shear strength of a slabcolumn connection is demonstrated.
Preliminary investigations using Bayesian analysis of
significance22 have been performed to identify the most
primary input parameters that have a significant influence on
the punching shear strength. Possible parameters included
concrete compressive strength, slab thickness and effective
depth, span length, column geometry, punching shear
perimeter, and compression and tension reinforcement
ratios. Assuming the geometry of punching shear perimeter
to be known a priori, the Bayesian analysis showed that for
circular and square columns (c1/c2 ratio equals to 1.0), the
most significant parameters that affect the punching shear
strength are concrete compressive strength fc′, slab thickness
h, and tension reinforcement ratio ρ. The assumption of the
punching shear perimeter to be known a priori is based on the
fact that the punching shear databank does not include
detailed information about the failure pattern and the
punching shear perimeter. This hinders the ability to learn
the failure patterns of slabcolumn connections as part of the
new model. It is also noted that the results of Bayesian analysis
showed that the compression reinforcement does not have a
significant effect on the maximum punching shear strength.
This finding is in agreement with the literature8,23 showing
that the primary effect of compression reinforcement is on
postpunching behavior providing a membrane action.
Hereafter, these three parameters have been used as input
parameters to the fuzzybased model for predicting the
punching shear strength. By considering these three parameters,
the fuzzybased model considers the major criteria on punching
shear examined by many researchers.2428 These include shear
strength and cracking capacity conventionally represented
by the cubical or square root of the compressive strength,6,24,27
size effect related to slab thickness,9 and membrane effect28
represented by the flexural reinforcement ratio.
While the ratio of the column dimensions of rectangular
columns and the perimetertodepth ratio (bo/d) have been
reported to affect the punching shear strength of slabcolumn
connections,24,29 the experimental database for rectangular
columns or for slabs with significantly large perimeterto
ACI Structural Journal/JulyAugust 2007
439
93.
Fig. 2—Cross section of slabcolumn connection showing
critical section at distance d/2 from column face to intersect
most plausible failure planes (angle θ ranges between 30
and 45 degrees). Choice of d/2 allows obtaining good estimate
of average ultimate punching shear strength vc.
failure as observed by many researchers.69 The choice of the
critical perimeter to be considered at a distance d/2 from the
column face is attributed to the possible use of this location
to estimate the average ultimate shear strength vc for usually
intersecting most plausible failure planes, as shown in Fig. 2,
which is similar to the value (h/2) proposed in Nielsen.7
The modeling is started by defining N fuzzy sets A over the
domain of each input parameter x. This definition˜ provides
each value of the parameter x with N membership values
representing its level of belonging to the N fuzzy set A . The
˜
concept of membership or degree of belonging represents the
18,20,21 The
basis in the formulation of fuzzy set theory.
membership denoted μ A(x) ranges between 0.0 and 1.0.
˜
μ A(x) does not express probability of x but characterizes the
˜
extent to which x belongs to fuzzy set A .20 Several methods
for establishing membership functions˜with different levels
of complexity exist. While simplified methods can be used
according to expert opinion, complex automated methods
using artificial neural networks or inductive reasoning are
usually considered to be efficient for modeling complex
phenomena.20,30 A technique is adopted herein that is based
on providing an initial definition of the fuzzy sets using
kmeans clustering31 followed by the automated update of
the fuzzy sets during the learning process.20,21
The modeling process depends on fuzzifying all three
input domains and constructing a fuzzy rulebase, which
describes the relationship between the fuzzy sets defined on
the input domains and the punching shear strength using a
group of linear equations. Exemplar rule in the fuzzy rulebase can be defined as
k
Fig. 3—Pictorial representation of bellshaped membership
function used to represent fuzzy sets defined over input
domains.
depth ratio (bo/d > 15) is insufficient to develop the knowledge
rule base that is necessary for the fuzzybased model to
consider both effects on the punching shear strength. Therefore, first, the fuzzybased model is trained by using the
experimental data with square and circular columns only and
with perimeter to depth ratio (bo/d) < 15. Based on this fact,
prediction of the fuzzybased model will be modified to
consider the effect of rectangularity of columns or high
perimetertodepth ratios in excess of that used in the training
(bo/d > 15) as shown in the Results and discussion section.
In the present study, the punching shear failure load of
slabcolumn connections without shear reinforcement Vc is
defined as
k
k
If f ′ ∈ A f , h ∈ A h , and ρ∈ A p ,
c
˜
˜
˜
then vi = ai f c′ + bih + ciρ + di
k
k
(2)
k
where A f , A h , and A p are the kth fuzzy set (k = 1, 2, … Nj)
˜
defined˜ on ˜ fuzzy domains of compressive strength f ′c,
the
slab thickness h, and tension reinforcement ratio ρ, respectively.
The value of Nj is the total number of fuzzy sets defined over
the jth input parameter. In the present study, ρ is defined
with respect to effective depth. Equation (2) represents the ith
rule in the fuzzy rulebase. The values ai, bi, ci, and di are
known as the consequent coefficients that define the output
side of the ith rule in the fuzzy rulebase.
A bellshape membership function is employed to represent
the fuzzy sets defined on the input domains. The use of other
membership functions (for example, gaussian and triangular) is
possible, but constrained by having a differentiable membership
function.21 The bellshape membership function to represent
the kth fuzzy set of the jth input parameter xj can be
k
described as μ A (x).
˜
Vc = vcbod
(1)
where Vc equals the punching failure load and bo equals the
critical perimeter at a distance d/2 from the column face;
bo = (2c1 + 2c2 + 4d) for a square column and bo = π(D + d)
for a circular column. The values c1 and c2 equal the short
and long sizes of a rectangular column, D equals the diameter
of a circular column, and vc represents the average ultimate
punching shear strength, which is defined with respect to
defective depth. Equation (1), although simplified, has been
adopted by almost all current design codes and respects the
fundamental mechanics governing the slabcolumn punching
440
k
1
μ A ( x j ) = k
˜
k 2q j
x j – x cj
1 + k
wj
(3)
k
where x cj , w jk, and q k represents the center, the top width,
j
and the shape parameters of the membership function
defining the kth fuzzy set defined over the jth input parameter.
A pictorial representation of the bellshaped membership
function is shown in Fig. 3. By considering the Tnorm
(product) operator (Π) to capture the influence of the interaction
ACI Structural Journal/JulyAugust 2007
94.
between the input parameters32 on the output, the weight of
the ith rule (λi) in the fuzzy rulebase can be computed as
T
1
Π j =1 k
k 2q
x j – x cj j
1 + k
wj
λ i =  for i = 1...R
R
T
1
Σ i =1 Π j =1 k
k 2q j
x j – x cj
1 + k
wj
Table 1—Dimensions and properties of specimens
Investigator*†
No. of specimens
f ′,
c
Training Verification MPa h, mm ρ, %
Hallgren and Kinnunen
(1993a), Hallgren and
Kinuunen (1993b),
Hallgren (1996)
3
3
79.5 to 239 to 0.6 to
108.8 245
1.2
Tomaszewicz (1993)
7
6
64.3 to 120 to 1.5 to
119.0 320
2.6
Ramdane (1996),
Regan et al. (1993)
4
4
28.9 to
74.2
Marzouk and Hussein (1991)
6
8
30.0 to 90 to 0.4 to
80.0
150
2.1
Lovrovich and McLean (1990)
2
2
(4)
Factors affecting the choice of the implication operator are
discussed in the following. The value T represents the total
number of input parameters (herein, T = 3). The number of
fuzzy rules R is a function of the number of input variables T and
the number of fuzzy sets Nj defined over each input domain.
The punching shear strength vc can then be computed as
⎛
⎞
vc = ⎜
λ i v i⎟
⎝i = 1
⎠
R
∑
⎛
⎞
⁄⎜
λ i⎟
⎝i = 1 ⎠
1.7
Tolf (1988)
4
3
Regan (1986)
11
11
8.4 to
37.5
80 to 0.8 to
250
2.4
Swamy and Ali (1982)
1
1
37.4 to
40.1
125
Marti et al. (1977),
Pralong et al. (1979)
1
1
23.1 to 180 to 1.2 to
30.4
191
1.5
Schaefers (1984)
1
1
23.1 to 143 to 0.6 to
23.3
200
0.8
Ladner et al. (1977),
Schaeidt et al. (1970),
Ladner (1973)
2
3
24.6 to 110 to 1.2 to
29.5
280
1.8
Corley and Hawkins (1968)
1
1
44.4
146
1.0 to
1.5
Bernaert and Puech (1996)
9
9
14.0 to
41.4
140
1.0 to
1.9
Manterola (1966)
4
4
24.2 to
39.7
125
0.5 to
1.4
Yitzhaki (1966)
5
6
8.6 to
19.0
102
0.7 to
2.0
Moe (1961)
7
7
20.5 to
35.2
152
1.1 to
2.6
Kinnunen and Nylander
(1960)
6
6
21.6 to 149 to 0.5 to
27.7
158
2.1
Elstner and Hognestad
(1956)
8
9
9.0 to
35.6
Hawkins et al.34
0
6
25.9 to 138 to 0.77 to
32.0
142 1.12
Teng et al.29
0
4
33.0 to
40.2
Criswell35
0
1
96
(5)
where vi is the output of the ith rule in the fuzzy rulebase
and λi represents the weight of the ith rule in the fuzzy rulebase as computed using Eq. (4).
The process for learning from example aims at extracting
a knowledge rulebase from a group of inputoutput datasets.
This knowledge rulebase can be used later to model the
behavior of the system (herein the punching shear of slabcolumn connections) for input datasets not used in the
training process. While other techniques capable of building
similar learning systems were reported in the literature (for
example, artificial neural networks), the advantage of fuzzy
systems is being able to consider nonrandom uncertainty in the
modeling process and thus yields robust modeling systems.20
The learning process starts by initializing the premise
parameters (parameters describing the membership functions
x k , w k , and q k ) using the kmeans clustering technique.31
cj
j
j
This is followed by computing the consequence coefficients
(ai, bi, ci, and di) using least square techniques33 such that the
root mean square prediction error E of the punching shear
strength does not exceed a target root mean square prediction
error, herein 1.0 × 10–5. The root mean square prediction
error E is defined as
100
1.0 to
1.3
20.1 to 120 to 0.4 to
25.1
240
0.8
R
∑
39.3
125
Total
82
152
0.6 to
0.7
1.2 to
3.7
150
1.24
35.4
146
1.24
8.4 to
119.0
80 to 0.4 to
320
3.7
*
Reference to investigators work, unless otherwise noted, can be found in Reference 3.
Properties and dimensions of these test specimens were collected from fib Bulletin 12.3
Note: 1 MPa = 0.145 ksi; 1 mm = 0.04 in.
†
Nd
∑ ( vpn – vdbn )
E =
2
n=1

Nd
(6)
where vpn is the predicted punching shear strength for the nth
dataset, vdbn is the punching shear of the nth dataset from
the database, and Nd is the total number of training datasets.
As the target mean square prediction error will not be
achieved from the first learning trial (using the initial fuzzy
sets and consequence coefficients), the premise parameters
describing the fuzzy sets can be updated using the gradient
descent method as
k
k
∂E ( m )x cj ( m ) = x cj ( m – 1 ) + η ∂x cj ( m )
ACI Structural Journal/JulyAugust 2007
(7)
k
k
∂E ( m )w j ( m ) = w j ( m – 1 ) + η ∂w j ( m )
(8)
k
k
∂E ( m )
q j ( m ) = q j ( m – 1 ) + η ∂q j ( m )
(9)
where x k (m), w k (m), and q k (m) are the center, the top width,
j
j
j
and the shape of the membership function, respectively,
defining the kth fuzzy set defined over the jth input parameter
in the mth learning epoch (trial). The values x k (m – 1),
j
w k (m – 1), and q k (m – 1) are the center, the top width, and
j
j
the shape of the membership function, respectively, defining
the kth fuzzy set defined over the jth input parameter in the
(m – 1) learning epoch. The value η is the learning rate and
∂E(m)/∂xj(m), ∂E(m)/∂wj(m), and ∂E(m)/∂qj(m) are components
441
95.
Fig. 4—Fuzzy sets used to describe concrete compressive
strength, slab thickness, and tension reinforcement ratio.
Before training (left) and after training (right): MF1
(Membership Function 1), MF2 (Membership Function 2),
and MF3 (Membership Function 3).
Table 2—Parameters describing premise
parameters (membership functions)*
Compressive strength f ′
c
xc, MPa (ksi)
1
Af
˜
2
Af
˜
w, MPa (ksi)
q
–23.83 (–3.40)
29.9 (4.34)
1.98
78.30 (11.40)
78.2 (11.30)
2.02
Slab thickness h
xc, mm (in.)
w, mm (in.)
q
1
42.05 (1.66)
68.0 (2.68)
1.982
2
127.07 (5.00)
89.4 (3.52)
2.011
3
272.58 (10.73)
126.7 (4.99)
1.994
Ah
˜
Ah
˜
Ah
˜
Tension reinforcement ratio ρ
xc
w
q
1
Aρ
0.001
0.012
1.997
2
0.035
0.018
2.005
˜
Aρ
˜
*For
compressive strength f c , slab thickness h, and reinforcement ratio ρ.
′
of the gradient vector of the mean square prediction error
with respect to the premise parameters of the jth input
parameter evaluated at the mth learning epoch. The updated
premise parameters are then used to recompute a new set of
consequence parameters and a new root mean square prediction
error. The process continues and the fuzzy rulebase parameters
(premise and consequent parameters) are updated in each
training epoch until the target root mean square prediction error
442
or a maximum number of training epochs is reached. The
update process therefore allows the fuzzybased model to
reduce the root mean square prediction error and thus learn
from examples in a much more robust manner compared with
any other empirical techniques.
For training and testing of the fuzzybased model, 178 test
specimens performed by 21 researchers as reported in the fib
bulletin3 and other reports in the literature29,34,35 were used.
Only specimens that were reported to fail in pure punching
shear (no flexural shear failure) were considered. A specimen
reported by Lovrovich and McLean36 was excluded in this
study because its span length was extremely short (l1/c1 = 2).
Also, six specimens by Yitzchaki,4 Elstner and Hognestad,37
and Tolf38 were also excluded because their tension reinforcement ratios were extremely beyond practical design
range (ρ ≥ 6.9%). The specimens had two types of boundary
geometries (circular and rectangular flat plates) and two types of
column shapes (circular and square columns). The dimensions
and properties of the specimens are summarized in Table 1.
The test specimens had a broad range of design parameters:
8.4 ≤ f′c ≤ 119.0 MPa (1.2 ≤ f ′c ≤ 17.3 ksi), 80 ≤ h ≤ 320 mm
(3.1 ≤ h ≤ 15.6 in.), 0.4 ≤ ρ ≤ 3.7%, and 5.5 ≤ bo/d ≤ 24.
These data cover a wide range of the material and geometric
properties of slabcolumn connections. Eightytwo specimens
were used for training of the fuzzybased model while 96
specimens were used for testing the model. All specimens used in
the testing were not used in training the fuzzybased model.
All modeling parameters were normalized to their maximum
values determined from the database (178 data sets). The
normalization process is necessary to avoid the influence of
numerical weights on the learning process.39 The fuzzy rulebase that achieved the lowest root mean square error during
training was used for testing and verification of the model capability to predict punching shear strength in slabcolumn connections. The optimum number of fuzzy sets for each modeling
parameter was developed using the kmeans clustering technique.31 The number of membership functions defined on the
domain of any variable x can be used to indicate the sensitivity
of the model to this variable x. The higher the sensitivity of the
model to the variable x, the larger the number of membership
functions used to describe the variable x. It is worth noting,
however, that increasing the number of membership functions
does not guarantee enhancing the model accuracy.20,21
It was found that the best learning represented by the
lowest root mean square prediction error was achieved while
using two fuzzy sets to represent the compressive strength
and the tension reinforcement ratio. Three fuzzy sets were
necessary for describing the slab thickness (N1 = N3 = 2,
N2 = 3). The initial and final fuzzy sets, as established by the
learning algorithm, are shown in Fig. 4 and Table 2. The total
number of rules in the rulebase can be computed by
multiplying the number of membership functions of the
three variables as R = N1N2N3. Thus, 12 rules (R = 12) were
needed to describe the relationship between the input
parameters: concrete compressive strength, slab thickness,
tension reinforcement ratio, and the punching shear strength.
While reduction of the total number of rules in the fuzzy
rulebase is possible for limiting combinatorial explosion,20
researchers showed that the efficient reduction of the number
of rules shall be performed considering both accuracy and
robustness of the model. Exemplar methods for rule reduction
in the fuzzy rulebase include the Combs and Andrews40
method and the method suggested by Lucero41 but are
beyond the scope of this work.
ACI Structural Journal/JulyAugust 2007
96.
RESULTS AND DISCUSSION
The fuzzybased model was trained using test results with
specific geometrical limits: circular and square columns and
slabs with perimetertoslabdepth ratio (bo/d) ranging between
5.8 and 14.9. Therefore, the punching shear strength of any
slabcolumn connection within the geometrical limitations
listed previously can be computed using Eq. (10) to (12).
Equation (10) can be used to compute the weight λ for each
rule in the rulebase using the premise parameters listed in
Table 1. Equation (11) presents the 12 rules forming the
fuzzy knowledge rulebase.
3
1
Π j =1 k
k 2q
x j – x cj j
1 + k
wj
λ i =  for i = 1...12
12
3
1
Σ i =1 Π j =1 k
k 2q j
x j – x cj
1 + k
wj
1
1
1
h
2
h
0.83 to
1.00
0.86 to
0.97
0.93 to
1.02
0.96 to
1.07
1.39 to
1.64
0.80 to
1.17
0.94 to
1.29
1.41 to
1.64
0.70 to
1.23
Ramdane (1996),
Regan et al.
(1993)
1.46 to
1.66
1.15 to
1.31
1.20 to
1.37
1.27 to
1.47
1.25 to
1.41
Marzouk and
Hussein (1991)
0.71 to
1.61
1.13 to
1.84
0.97 to
1.64
0.63 to
1.40
0.91 to
1.42
1.18 to
1.26
0.73 to
0.78
0.79 to
0.85
1.02 to
1.10
0.87 to
0.94
Tolf (1988)
0.88 to
1.21
0.92 to
1.34
0.82 to
1.15
0.77 to
1.05
0.91 to
1.02
Regan (1986)
1.17 to
1.78
0.97 to
1.47
1.04 to
1.47
1.02 to
1.54
0.60 to
1.29
Swamy and Ali
(1982)
1.10
1.19
1.12
0.96
1.00
Marti et al. (1977),
Pralong et al.
(1979)
1.32
0.97
1.00
1.15
0.77
Schaefers (1984)
1.19
1.14
1.05
1.04
1.00
Ladner et al.
(1977), Schaeidt
et al. (1970),
Ladner (1973)
1.48 to
1.79
1.22 to
1.34
1.26 to
1.47
1.29 to
1.56
0.89 to
1.26
Corley and
Hawkins (1968)
0.87
0.85
0.85
0.75
0.72
Bernaert and
Puech (1996)
0.88 to
1.93
0.80 to
1.28
0.81 to
1.43
0.76 to
1.68
0.70 to
1.45
Manterola (1966)
0.88 to
1.36
0.81 to
0.96
0.85 to
0.98
0.76 to
1.18
0.65 to
0.92
Yitzhaki (1966)
1.51 to
1.98
1.01 to
1.54
1.01 to
1.53
1.31 to
1.72
0.80 to
1.16
Moe (1961)
1.24 to
1.65
0.70 to
1.38
0.83 to
1.40
1.07 to
1.43
0.68 to
1.12
Kinnunen and
Nylander (1960)
0.83 to
1.75
0.93 to
1.23
0.92 to
1.23
0.72 to
1.52
0.85 to
1.36
Elstner and
Hognestad (1956)
1.19 to
2.23
0.88 to
1.20
1.05 to
1.30
1.03 to
1.94
0.79 to
1.27
Hawkins et al.34
0.90 to
1.05
0.87 to
1.10
0.89 to
1.05
0.78 to
0.91
0.88 to
1.19
Teng et al.29
1
ρ
0.88 to
0.98
Lovrovich and
McLean (1990)
(10)
Hallgren and
Kinnunen
(1993a), Hallgren
and Kinuunen
(1993b), Hallgren
(1996)
0.88 to
1.15
0.89 to
1.15
0.92 to
1.19
0.76 to
1.00
1.02 to
1.49
R = 1: if f c ′ ∈ A f , h ∈ A h, and ρ ∈ A ρ
˜
˜
˜
then v 1 = 0.247f c ′ + 0.008h + 153.7ρ + 4.90
R = 2: if f c ′ ∈ A , h ∈ A , and ρ ∈ A
˜
˜
˜
then v 2 = – 0.506f c ′ + 0.026h + 835.4ρ – 11.42
R = 3: if f c ′ ∈ A , h ∈ A , and ρ ∈ A
˜
˜
˜
then v 3 = 0.174f c ′ + 0.028h + 63.9ρ – 8.12
1
2
2
1
3
1
R = 4: if f c ′ ∈ A f , h ∈ A h, and ρ ∈ A ρ
˜
˜
˜
then v 4 = 0.149f c ′ + 0.031h – 136.65ρ – 3.49
R = 5: if f c ′ ∈ A f , h ∈ A h, and ρ ∈ A ρ
˜
˜
˜
then v 5 = – 0.248 f c ′ + 0.001h – 236.32ρ + 3.91
1
3
2
2
1
1
2
f
1
h
2
ρ
2
2
1
R = 6: if f c ′ ∈ A f , h ∈ A h, and ρ ∈ A ρ
˜
˜
˜
then v 6 = 0.243f c ′ – 0.006h – 53.35ρ + 3.16
(11)
R = 7: if f c ′ ∈ A f , h ∈ A h, and ρ ∈ A ρ
˜
˜
˜
then v 7 = – 0.005 f c ′ – 0.031h – 84.38ρ + 3.14
R = 8: if f c ′ ∈ A , h ∈ A , and ρ ∈ A
˜
˜
˜
then v 8 = 0.006f c ′ + 0.116h – 136.57ρ – 2.67
R = 9: if f c ′ ∈ A f , h ∈ A h, and ρ ∈ A ρ
˜
˜
˜
then v 9 = 0.006f c ′ – 0.002h – 30.73ρ + 0.05
2
2
2
2
f
3
h
1
ρ
2
3
FuzzyCSA
based
ACI 31805 CEBFIP Eurocode 2 A23.304 model
VTest/
VTest/
VTest/
VTest/
VTest/
Vpredicted† Vpredicted† Vpredicted† Vpredicted† Vpredicted†
Tomaszewicz
(1993)
2
ρ
1
f
Investigator*
1
1
f
Table 3—Testing to predicted punching shear
strength ratio using existing design codes and
fuzzybased model
2
R = 10: if f c ′ ∈ A f , h ∈ A h, and ρ ∈ A ρ
˜
˜
˜
then v 10 = 0.021f c ′ + 0.043h + 19.19ρ – 7.58
R = 11: if f c ′ ∈ A , h ∈ A , and ρ ∈ A
˜
˜
˜
then v 11 = 0.001f c ′ – 0.006h + 49.86ρ + 1.96
Criswell35
⎛ 12
⎞ ⎛ 12 ⎞
λ i⎟
v cf = ⎜
λ i v i⎟ ⁄ ⎜
⎝i = 1
⎠ ⎝i = 1 ⎠
∑
∑
(12)
It is important to emphasize the fact that several implication
operators exist.42 The selection of the implication operator is
governed by three main issues: the needed logical implication
ACI Structural Journal/JulyAugust 2007
0.89
0.96
0.82
0.86
1.375
1.098
1.139
1.219
1.019
Standard deviation
where vi, f ′c, and h are in MPa, MPa, and mm, respectively.
The punching shear strength vcf can be computed using Eq. (11)
and (12)
0.94
Mean
R = 12: if f c ′ ∈ A f , h ∈ A h, and ρ ∈ A ρ
˜
˜
˜
then v 12 = 0.004f c ′ + 0.018h + 36.19ρ – 6.36
0.314
0.207
0.198
0.280
0.189
*
Reference to investigators work, unless otherwise noted, can be found in Reference 3.
†
Strength ratio (= VTest/Vpredicted), where VTest equals actual strengths (test results),
and Vpredicted equals predicted strengths by current design methods (ACI 31805,
CEBFIP, Eurocode 2, and CSA A23.304) or fuzzybased model, respectively.
of information, the influence of the fused output on the
model prediction, and the effect of the fusion method on the
computational efficiency of the learning algorithm. The
product implication Π was selected herein for three reasons.
First, to perform the fuzzy and operation as indicated by
Eq. (4). Second, the product implication tends to dilute the
influence of joint membership values that are small and
443
97.
Fig. 5—Strength prediction by current design method and fuzzybased model. (Note: 1 MPa =
0.145 ksi; 1 mm = 0.04 in.)
therefore magnify the contribution of the rules associated
with high membership values in computing the shear
strength (Eq. (4) and (5)). This fact promoted the use of the
444
product operator in artificial neural networks as an efficient
Hebbiantype learning algorithm.20 Finally, the choice of the
product implication was also controlled by the need to
ACI Structural Journal/JulyAugust 2007
98.
produce a continuous and differentiable error function (Eq. (6))
to enable efficient computation of the error gradients during
the learning process.
Table 3 presents a summary of punching shear strength of
the specimens predicted by the fuzzybased model. In the
verification, the 96 specimens, which were not used in the
learning process, were used. Figure 5(a) shows the ratios
between the actual test to the fuzzybased model predicted
strength (Vtest /Vpredicted) to have a mean value 1.019 and a
standard deviation of 18.9%. Figures 5(b) to (e) show the ratios
between actual to predicted strength (Vtest /Vpredicted) using
the CEBFIP MC 90,12 the Eurocode 2,13 ACI 31805,14 and
CSA A23.304,15 to have mean values of 1.098, 1.139,
1.375, and 1.219, respectively, with standard deviations of
20.7, 19.8, 31.4, and 28.0%, respectively (refer to Table 3).
The results show that the fuzzybased model can be used to
predict the punching shear strength of slabcolumn connections
with various slab thicknesses, reinforcement ratios, and
circular and square columns. Moreover, higher prediction
accuracy of the fuzzybased model can be observed compared
with predication accuracies for all existing design codes.
It is interesting to note that, except for Eurocode 2,13 current
design methods show a considerable scatter represented by
high standard deviations of testprediction ratios. Moreover,
observing Fig. 5(c), the CEBFIP MC 90 code underestimates
the punching shear strength of specimens with low tension
reinforcement ratios while it overestimates the punching
shear strength of specimens with high tension reinforcement
ratios. The Eurocode 213 shows good accuracy in predicting
the punching shear strength at different reinforcement ratios.
Finally, ACI 3180514 and CSA A23.30415 underestimate the
punching shear strength of specimens with high reinforcement
ratios while they overestimate the punching shear strength of
specimens with low reinforcement ratios. This is attributed
to the fact that ACI 31805 and CSA A23.304 codes do not
account for the effect of the tension reinforcement ratio on the
punching shear strength. It is also evident from Fig. 5(a) that
the fuzzybased model predicts punching shear strength at
both low and high reinforcement ratios with consistent accuracy. It is worth noting that the slab thickness and the tension
reinforcement ratio in addition to the compressive strength are
found to have a significant influence on modeling punching
shear strength using the fuzzybased model. These parameters
have also been promoted by other researchers before because of
their influence on the size effect43 and their possible role in
developing shear friction.44
To consider other rectangularity ratios c2/c1 (>1) and high
perimeter to depth ratios bo/d (>15.0), a design approach
based on the fuzzybased model is proposed as
⎧
v cf
⎪
⎪
1
⎛ 0.5 +  ⎞ v
⎪
n⎠ cf
⎝
v c = min ⎨
βc
⎪
⎪ ⎛
10 ⎛  ⎞ n⎞
⎪ ⎝ 0.5 + ⎝ b ⁄ d⎠ ⎠ v cf
o
⎩
(13)
where βc = c2/c1, c1 and c2 equal the short and long sizes of
rectangular columns, vcf is the fuzzybased shear strength
estimated using Eq. (12), and n is a power coefficient.
Equation (13) is modeled in a format similar to that of the
ACI equation for predicting the punching shear strength. If
ACI Structural Journal/JulyAugust 2007
Fig. 6—Variation of strengthprediction by fuzzybased
model according to bo /d.
Fig. 7—Variation of strengthprediction by fuzzybased
model according to c2/c1 higher than 1.
n = 1.0 as similar to the ACI equation is used, the model will
significantly overestimate the punching slabcolumn
connections with rectangular columns and with bo/d higher
than 15. A mean value and a standard deviation of the
strengthprediction ratios (Vtest /Vpredicted) of the specimens
(Table 3) using n = 1 are 0.977 and 0.193, respectively, while
those using n = 2 are 1.019 and 0.189. Therefore, the authors
recommend the use of n = 2. The model prediction with n = 2
for a wide range of bo/d and for rectangular columns are
shown in Fig. 6 and 7. The choice of n = 2 for the second and
third components of Eq. (13) was based on examining each
component separately. It has become evident that refinement
in the value of n for each part would not yield any enhancement
in the prediction accuracy of the model.
Figure 6 demonstrates the fact that the modified fuzzybased model using a modification factor (Eq. (13)) can
accurately predict the punching shear strength of slabcolumn connections with various bo/d (5.8 ≤ bo/d ≤ 24.0)
even though the fuzzybased model (Eq. (12)) was developed
within the geometrical limits (5.8 ≤ bo/d ≤ 14.9) due to the
lack of test data. This is attributed to the fact that the fuzzybased model was developed by using the average ultimate
shear strength vc considering bo and d (Eq. (1)). It is evident
that the modified fuzzybased model can properly consider
the interaction between bo/d and vc in its strength equation
(Eq. (13)). In Fig. 7, the fuzzybased model also accurately
predicts the punching shear strength of slabcolumn connections
with rectangular columns (c2/c1 > 1). From this result, it is
noted that the modified fuzzybased model properly considers
the effect of rectangularity of columns in practical design range
(1 ≤ c2/c1 ≤ 5). It is worth noting that, if enough experimental
data with high bo/d ratios and rectangular columns were
available in the literature, the use of modification factors for
445
99.
Fig. 8—Design chart for punching shear strength using
fuzzybased model. (Note: 1 MPa = 0.145 ksi; 1 mm = 0.04 in.)
Fig. 9—Strength variation according to primary design
parameters.25,37,4447
addressing these issues can be completely omitted. This
indicates the fact that a refined fuzzybased model would
always be possible to develop, once experimental data
beyond these geometrical limitations becomes available.
PROPOSED DESIGN CHART
For design purposes, the direct use of the fuzzybased
model as an empirical method using Eq. (10) to (13) and the
premise parameters from Table 2 might not be feasible for
designers. To avoid such complexity and to make use of the
demonstrated ability and relative high accuracy of the fuzzybased model in design of slabcolumn connections without
shear reinforcement, the authors suggest a simplified design
model that is developed based on a set of design charts that
are developed using the fuzzybased model. Following a
format similar to that used in ACI 31805, the design
strength for punching shear of slabcolumn connections is
defined as
φVn = φvcbod
(14)
where vc is calculated according to Eq. (13) using n = 2, and
φ is the strength reduction factor taken equal to 0.6. The
punching shear strength vcf can be estimated using Fig. 8.
Figures 8(a) to (d) show a group of design charts to estimate
the punching shear strength vcf of slabcolumn connections
using the fuzzybased model. The design charts are developed
446
for a wide range of primary design parameters: 20 ≤ f ′c ≤
100 MPa (2.9 ≤ f ′c ≤ 14.5 ksi), 100 ≤ h ≤ 300 mm (3.9 ≤ h ≤
11.8 in.), and 0.8 ≤ ρ ≤ 2.0%. For space limitations, only four
design charts are developed herein covering the aforementioned range of parameters. Additional design charts can be
developed using the model equations described previously.
The φ factor of 0.6 corresponds conservatively to the lowest
bound shown in Fig. 5(a). Obtaining a refined shear strength
reduction factor (higher than 0.6) can be done using principles
of load and resistance factor design (LRFD),45 but is beyond
the scope of this study.
It can be observed from Fig. 8(a) to (d) that the punching
shear strength decreases as slab thickness increases, which
respects previous findings of the size effect by Bažant and Cao9
and Eurocode 2.13 In cases with high reinforcement
ratios, however, this size effect is disturbed by the combined
effect of size and membrane force generated by the tension
reinforcement. As observed in Fig. 8(c) and (d), for high
tension reinforcement ratios and low concrete compressive
strength, the punching shear strength increases as the slab
thickness increases. This can be attributed to the possibility
that the increase in the slab thickness with high reinforcement
ratios results in an increase in the axial membrane
force,24,26,27 which contributes to punching shear strength
due to the increase in the shear friction effect.44 This possible
shear friction contribution to the punching shear strength has
been argued by other researchers in shear analysis.44,46
This phenomenon is due to the combined effect of the
primary parameters (compressive strength, slab thickness,
and tension reinforcement ratio) and can be also observed in
previous test results from the punching shear database.3
Figure 9 shows the punching shear strength reported in
existing test results. For this study, Elstner and Hognestad,37
Shaeidt el al.,47 Regan,48 Marzouk and Hussein,25 Hallgren
and Kinnunen,49 and Tomaszewicz’s50 specimens were
used. Each data set itself has similar dimension and property.
The dimensions and properties of the specimens are
summarized in Table 1. As expected, for all data sets with
high concrete compressive strength, the punching shear
strength of thick slabs is always less than that of thin slabs
due to the size effect24,38,43 (see Fig. 9(a)). In Fig. 9(b),
however, for low concrete compressive strength and high
reinforcement ratios (ρ ≥ 0.012), the punching shear strength
of thick slabs may be greater than that of thin slabs, which
indicates the tradeoff between size effect and shear friction
effect. These combined effects can be successfully described
by the fuzzybased model.
CONCLUSIONS
A new alternative design method and a set of design charts
based on fuzzy learning from examples are proposed. The
new method can accurately predict the punching shear
strength of simply supported interior slabcolumn connections
without shear reinforcement. One hundred and seventy eight
test specimens from the punching shear databank were used
for training and testing the proposed model (82 for training
and 96 for testing). The training and testing data sets cover a
wide range of the material and geometric properties. The
testing data set was not used in the training process. Investigations for developing a model with good accuracy showed
that concrete compressive strength, slab thickness, and
tension reinforcement ratio are the primary parameters that
dominate the punching behavior of slabcolumn connections.
This finding is limited to circular and rectangular columns
ACI Structural Journal/JulyAugust 2007
100.
and slabs with perimetertoslabdepth ratios (bo/d) ranging
between 5.8 and 24.0 and column size ratios (c2/c1) ranging
between 1.0 and 5.0. The fuzzybased model demonstrates
higher prediction accuracy compared with all current design
codes including ACI 31805, Eurocode 2, CEBFIP MC 90,
and CSA A23.304 in predicting the punching shear strength
of slabcolumn connections. The proposed model, while
addressing uncertainty and interactions between modeling
parameters, was shown to respect the fundamental mechanics
of punching shear as described by many researchers.
ACKNOWLEDGMENTS
The financial support by the Defense Threat Reduction Agency (DTRA)
University Strategic Partnership to the University of New Mexico is greatly
appreciated.
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447
102.
ACI member Nam Ho Lee is a Senior Research Engineer in the Civil Engineering
Department of the Korea Power Engineering Co. He received his BS from Seoul
National University and his MS and PhD from the Korea Advanced Institute of
Science & Technology. He is a member of ACI Committees 349, Concrete Nuclear
Structures, and 355, Anchorage to Concrete, and Joint ACIASME Committee 359,
Concrete Components for Nuclear Reactors. His research interests include the
nonlinear behavior of concrete structures and anchorage to concrete.
Kang Sik Kim is a Senior Researcher, Environment and Structure Laboratory, Korea
Electric Power Research Institute, Daejeon, Korea. His research interests include the
behavior of concretefilled steel plate structures and anchorage to concrete.
Chang Joon Bang is a Project Engineer at Korea Hydro & Nuclear Power Co. Ltd.,
Seoul, Korea. He is currently a Graduate Student of civil engineering, Lehigh
University, Bethlehem, Pa.
Kwang Ryeon Park is a Research Engineer, Civil Engineering Department, Korea
Power Engineering Co.
were performed giving 20 specimens in total. The test
program is summarized in Table 1. The test specimens are
shown in Fig. 1. All anchors were fabricated of ASTM A540
Gr. B23 Class 2 steel (equivalent to ASME SA 549 Gr. B23
Class 2 used in Korean nuclear power plants) with fy = 140 ksi
(980 MPa) and fu = 155 ksi (1085 MPa). The anchor head
consisted of a round thick plate which was fixed to the bolt
by clamping nuts (Fig. 2). The diameter of the round plate
was dh = 6 in. (152.4 mm) (db = 2.75 in. [69.9 mm]), dh =
8.5 in. (215.9 mm) (db = 3.75 in. [95.3 mm]), and dh = 10 in.
(254.0 mm) (db = 4.25 in. [108.0 mm]). The size of the
concrete test block was large enough to avoid splitting
failure. The concrete volume (width/length/depth) available
for each anchor is shown in Table 1. Furthermore, to
minimize the width of eventual shrinkage cracks, the top and
bottom of the test member were reinforced in both directions
with No. 10 bars at 16, 10, and 10 in. (406.4, 254, and 254 mm)
spacing for Specimens T1, T2, and T3, respectively. This
surface reinforcement does not significantly influence the
concrete breakout load. As shown in Fig. 1, wooden and steel
frames were constructed to suspend the castinplace
anchors in the correct position and at the correct embedment
depth. The concrete mixture for the test specimens is shown
in Table 2(a). The concrete used in the test specimens was
comparable to the concrete used in the Korean Nuclear Plant,
except that 20% by weight of the Type I cement was substituted
by fly ash and 1 in. (25 mm.) crushed aggregate was used
instead of 3/4 in. (19 mm). The target concrete strength at
42 days was fc′ = 5500 psi (37.9 MPa). The actual concrete
strength at the time of testing is given in Table 2(b). The
concrete for the specimens of one test series was placed from
one batch. Whereas in test Series T1 to T3, no special reinforcement was used to resist the applied tension load, in test Series T4
and T5, supplementary reinforcement (refer to Fig. 3) was
used to increase the ultimate load. The supplementary
reinforcements consisted of vertical stirrups (eight No. 8 bars
and 16 No. 8 bars for test Series T4 and T5, respectively), as
shown in Fig. 3.
Table 1—Description of tension test specimens
Specimen
Concrete
Diameter
volume
Anchor of anchor Effective available for
diameter, head, embedment each anchor
hef , in. (width/length/
dh , in.
Reinforce db, in.
ment
(mm)
(mm)
(mm)
depth)
T1A,B,C,D
None
2.75
(69.9)
6.0
(152.4)
25
(635)
5.9hef /5.0hef /
2.9hef
T2A,B,C,D
None
3.75
(95.3)
8.5
(215.9)
35
(889)
5.4hef/4.7hef /
2.0hef
T3A,B,C,D
None
4.25
(108.0)
10.0
(254.0)
45
(1143)
5.0hef /3.6hef /
2.0hef
T4A,B,C,D Supp. No. 1
2.75
(69.9)
6.0
(152.4)
25
(635)
5.9hef /5.0hef /
2.9hef
T5A,B,C,D Supp. No. 2
2.75
(69.9)
6.0
(152.4)
25
(635)
5.9hef /5.0hef /
2.9hef
Fig. 1—Tension test Specimens T1, T2, T3, T4, and T5.
Table 2(a)—Concrete mixture proportioning
Nominal
strength,
psi, at
W/
S/a, W,
FA,
WRA,* AEA,†
mL
42 days (C + FA) % lb C, lb lb S, lb G, lb mL
5500
0.44
44 525 514 128 1257 1617
474
26
*Waterreducing
admixture.
†
Airentraining admixture.
Table 2(b)—Concrete strength at time of testing
Test specimen
58/50/44/42
5771 (39.8)/5630 (38.8)/
5508 (38.0)/5464 (37.7)
T2A/B/C/D
41/45/47/49
5177 (35.7)/5248 (36.2) /
5291 (36.5)/5320 (36.7)
T3A/B/C/D
61/56/54/50
5448 (37.6)/5348 (36.9) /
5305 (36.6)/5220 (36.0)
T4A/B/C/D
480
Compressive strength, psi (MPa)
T1A/B/C/D
Fig. 2—Details of anchor head.
Curing ages, days
57/55/54/50
5945 (41.0)/5917 (40.8)/
5903 (40.7)/5817 (40.1)
T5A/B/C/D
71/70/69/68
6144 (42.4)/6130 (42.3)/
6130 (42.3)/6116 (42.2)
ACI Structural Journal/JulyAugust 2007
103.
Test setup
The test setup consisted of a loading frame, loading plate,
jack assembly, load cell, and other items, as shown in the
schematic and photo in Fig. 4. The load was applied to the
anchor under forcecontrol in an increment of approximately
3.5% of ultimate steel strength of the anchor bolt (Fu = 925,
1683, and 2192 kips [4114.6, 7486.4, and 9750.5 kN], for
bolts with a diameter of 2.75, 3.25, and 4.25 in. [69.90,
82.55, and 107.95 mm], respectively), that is, 30, 60, 77,
68, and 48 kips (133.4, 266.9, 342.5, 302.5, and 213.5 kN)
for Series T1, T2, T3, T4, and T5, respectively. It was
reacted in two directions by a stiff frame to minimize the
bending moment in the test specimen. The clear distance
between the supports was 4.0 hef for Specimens T1 through
T5, thus allowing for an unrestricted formation of a concrete
cone. The applied load was measured by a load cell. Additionally,
the strain along the embedment length of the anchor bolt was
measured (Fig. 5). Furthermore, the displacement of the top
end of the anchor was measured by LVDTs (Fig. 5).
surface varied from α = 20 to 30 degrees, following the
typical crack profiles shown in Fig. 6(b).
In general, test Specimens T4 and T5, with supplementary
reinforcement (Fig. 3), were not tested to failure. At the
applied peak load, the measured steel strains exceeded the
yield strain and because of safety concerns a sudden rupture
of the bolt was avoided. Only Specimen T4A was tested to
TEST RESULTS
Failure loads, failure modes and load
displacement behavior
The average failure loads are summarized in Table 3(a)
(Series T1 to T3) and Table 3(b) (Series T4 and T5). The
values given in the tables are normalized to fc′ = 5500 psi
(37.9 MPa) by multiplying the measured peak load of each
test with the factor (5500/fc,test)0.5. In test Series T1 to T3,
failure was caused by concrete cone breakout well below the
anchor bolt steel capacities (Fu = 925, 1683, and 2192 kips
[4114.6, 7486.4, and 9750.5 kN] for bolts with diameters of
2.75, 3.25, and 4.25 in. [69.90, 82.55, and 107.95 mm],
respectively). The cracking patterns in the specimen after the
test are depicted in Fig. 6(a). Generally, one major longitudinal
crack was observed, centered approximately on the sides of
the block, in combination with a horizontal crack and some
transverse cracks. On the top surface of the block, the cracks
formed a circular pattern around the anchor. To identify the
internal crack propagation defining the roughly conical breakout
body, one replicate of each specimen type was selected, and
the concrete was cored on two orthogonal planes whose
intersection coincided with the axis of the anchor. The cores
confirmed a breakout cone whose angle with the concrete
Fig. 4—Tension test setup: (a) schematic; and (b) photo.
Fig. 3—Supplementary reinforcement in Specimens T4 and T5.
ACI Structural Journal/JulyAugust 2007
Fig. 5—Location of LVDTs and strain gauges (Specimen T1).
481
104.
Table 3(a)—Tension test results and predictions
for unreinforced Specimens T1, T2, and T3
Table 3(b)—Tension test results and predictions
for reinforced Specimens T4 and T5
Concrete breakout capacities, kips (kN),
by embedment
Classification
Reference
Concrete breakout capacities, kips (kN),
by embedment
Specimen T1 Specimen T2, Specimen T3,
25 in.
35 in.
45 in.
(635 mm)
(889 mm)
(1143 mm)
Specimen T4 Specimen T5, Specimen T1,
25 in.
25 in.
25 in.
(635 mm)
(635 mm)
(635 mm)
ACI 34997,
Eq. (5)
1305 (5804)
ACI 31805,
Eq. (4)
320 (1423)
CCD method
with
1.5
h ef Eq. (1)
Reference
2138 (9510)
ACI 34997,
Eq. (5)
676 (3006)
676 (3006)
676 (3006)
562 (2499)
855 (3803)
ACI 31805,
Eq. (4)
320 (1423)
320 (1423)
320 (1423)
371 (1650)
614 (2731)
895 (3981)
CCD method
with
1.5
h ef Eq. (1)
371 (1650)
371 (1650)
371 (1650)
CCD method
with
428 (1903)
1.67
h ef Eq. (3)
Predictions
676 (3006)
Classification
750 (3336)
1142 (5079)
CCD method
with
1.67
h ef Eq. (3)
428 (1903)
428 (1903)
428 (1903)
Mean
733 (3260)
725 (3224)
509 (2264)
COV, %
1.7
3.5
5.8
5%fractile
685 (3047)
625 (2780)
393 (1748)
5%fractile/
mean
0.93
0.86
0.77
Mean
744 (3309)
1242 (5524)
COV, %
Tests
509 (2264)
5.8
2.8
6.1
5%fractile
393 (1748)
5%fractile/
mean
662 (2944)
0.77
944 (4199)
0.89
Ratios of observed to predicted capacities
Mean
of test
results
(I)
Nu,5%/
Eq. (5)
0.58
0.51
0.44
0.51
(II)
Nu,5%/
Eq. (4)
1.24
1.19
1.12
1.18
(III)
Mean/
Eq. (1)
1.37
1.21
1.39
(IV)
Mean/
Eq. (3)
1.19
0.99
1.09
Symbol in
Classification Fig. 9
5% fractile of
test results
Mean of test
results
Ratio of observed to predictions
(hef = 25 in. [635 mm])
Comparison
T4
T5
T1
T4/T1
(I)
Nu,5%/Eq. (5)
1.01
0.92
0.58
1.74
(II)
Nu,5%/Eq. (5)
2.16
1.97
1.24
1.74
(III)
Mean/Eq. (1)
1.98
1.96
1.37
1.45
(IV)
Mean/Eq. (3)
1.71
1.70
1.19
1.44
1.32
1.09
failure. Failure of this specimen was caused by forming a
concrete cone. From the loaddisplacement curves (Fig. 7), it
can be concluded that in test Series T4, the applied maximum
loads were almost identical with the failure loads. In test
Series T5, however, the failure load of the anchors was not
reached. Because Specimens T4 and T5 showed no cracking
at the concrete surface, no cores were taken to check whether
a cone had begun to form.
The loaddisplacement curves for Specimens T1, T2, T3,
T4, and T5 are shown in Fig. 7(a) through 7(e), using the
displacement measured at the top of each anchor. The
loaddisplacement relationship for each test replicate varied
based on the concrete strength at the time of testing. The
projecting lengths of the anchor shafts from the concrete
surface to the top of the anchor for Specimens T1, T2, T3,
T4, and T5 were 41.7, 48.6, 53.1, 41.7, and 41.7 in. (1059,
1234, 1348, 1059, and 1059 mm), respectively. Because the
measured displacements shown in Fig. 7 include the steel
elongation of the projecting anchor length, the actual anchor
displacements at the top of the concrete surface, which are
accumulated along the embedded portion of the anchor, are
much smaller than shown in Fig. 7. In Fig. 8, the relationship
between load and anchor displacement at the surface of the
concrete (calculated from the displacements measured at the
anchor top end subtracting the steel elongation of the
projecting length) are plotted for test Series T1 to T5. In
some tests, the calculated displacements at the concrete
surface are negative for low loads. It is believed that this is
482
Tests
0.76
SymSpecimen T1 Specimen T2, Specimen T3,
Classi bol in Com25 in.
35 in.
45 in.
fication Fig. 9 parison (635 mm)
(889 mm)
(1143 mm) Mean
5%
fractile
of test
results
Predictions
(a)
(b)
Fig. 6—(a) Cracking pattern for four test replicates (A, B,
C, and D) of Specimens T1, T2, and T3; and (b) typical
internal crack profile in Specimen T1.
ACI Structural Journal/JulyAugust 2007
105.
Fig. 7—Measured loaddisplacement relationships.
caused by bending of the anchors if they were not installed
perfectly perpendicular to the concrete surface. It can be seen
that the anchor displacements at peak load of Specimens T1
to T3 (concrete cone failure) are rather small. This can be
explained by the rather large anchor heads that, due to the
low concrete stresses, did not slip much. For head sizes
allowed by ACI 31805, Appendix D, the breakout failure
loads increase approximately proportional to hef1.5. With
much larger heads, the power on the embedment depths is
greater than 1.5.7 In the present tests, at failure, the related
pressure under the head was on average p/fc′ = 4.37, 3.36,
and 5.31 for test Series T1, T2, and T3. It was much smaller
than the pressure allowed by ACI 31805 for uncracked
concrete (pn = 10fc′ ).
Comparison of predicted and tested tensile
breakout capacities
In Table 3(a), tension test results for unreinforced
Specimens T1, T2, and T3, and results in Table 3(b) for
reinforced Specimens T4 and T5, are compared with predicted
capacities. The measured mean failure loads are compared
with the predicted mean capacities according to Eq. (1) and
(3), respectively, and the 5%fractiles of the measured
failure loads calculated by assuming an unknown standard
deviation are compared with the values according to Eq. (4)
and (5). In Fig. 9, the ratios of measured capacities to
predicted values are plotted. Figure 10 shows the measured
failure loads of each test compared with the values predicted
according to Eq. (5), Fig. 10(a); Eq. (1), Fig. 10(b); and Eq. (3),
Fig. 10(c), as a function of the embedment depth. In Fig. 11,
the measured concrete breakout loads, as well as the failure
loads according to best fit equations using the current test
results and Eq. (1), (2), (3), and (5), are plotted as a function
of the embedment depth.
ACI Structural Journal/JulyAugust 2007
Fig. 8—Relation between load and anchor displacement at
concrete surface.
EVALUATION OF TEST RESULTS FOR
UNREINFORCED SPECIMENS T1, T2, AND T3
According to the 45degree cone model (Eq. (5)), the
2
breakout capacities increase in proportion to hef . The
predicted capacities Nu,calc are much higher than the
measure values Nu,test and the ratio Nu,test /Nu,calc decreases
with increasing embedment depth (Fig. 10(a)). On average,
the 5%fractiles of the observed capacities are approximately
half the capacities predicted by ACI 34997 (Table 3(a)).
This demonstrates that the 45degree cone model is
unconservative for deep anchors. This agrees with the findings
by Fuchs et al.3 and Shirvani et. al.8 In contrast, the predictions
according to the CCD method are conservative. The
measured average breakout loads are approximately 30%
higher than the values predicted according to Eq. (1) (Nu
proportional to hef1.5) with no significant influence of the
embedment depth (Fig. 10(b)). On average, the ratio of
measured failure loads to the values predicted by Eq. (3) (Nu
1.67
proportional to hef ) is 1.09 (Table 3(a)). It decreases
slightly with increasing embedment depth (Fig. 10(c)).
In Fig. 10(d) to 10(f), the breakout failure loads of headed
anchors with an embedment depth hef ≥ 8 in. (200 mm)
measured in the present tests and taken from other sources3,8
are compared with values predicted by the CCD method.
According to Fig. 10(d), the prediction according to Eq. (1)
is conservative for large embedment depths. The failure
loads predicted by Eq. (3) agree quite well with the measured
values (Fig. 10(e)). Figure 10(f) shows that the CCD method
changing the exponent on hef from 1.5 to 1.67 at an effective
embedment depth of 10 in. (250 mm) predicts the failure
loads of anchors with hef ≥ 8 in. (200 mm) best. Only two
483
106.
test points at hef = 8 in. (200 mm) fall below the assumed
5%fractile, which is equal to 75% of the average value.
The 5%fractiles of the capacities observed in the present
tests average approximately 120% of the values predicted by
ACI 31805, Appendix D (Eq. (4)) (refer to Table 3(a)). The
higher ratio Nu,test /Nu,calc when comparing the 5% fractiles
with each other instead of the average values is due to the
rather low scatter of test results. On average, the coefficient
of variation (COV) was approximately 5%. This results in an
average ratio Nu,5%/Nu,m of 0.81, whereas in ACI 34901,
a ratio of 0.75 is assumed. In actual structures, the concrete
strength, and thus the concrete cone resistance, might vary more
than in the present test specimens. Therefore, the ratio
Nu,5%/Nu,m assumed in ACI 31805, Appendix D, should
be maintained.
Numerical investigations by Ozbolt et al.7 using a
sophisticated threedimensional nonlinear finite element
model demonstrates that the concrete breakout capacity of
headed anchors is influenced by the head size, that is, the
pressure under the head, related to the concrete compressive
strength as described previously.
Based on the previous evaluations, it is recommended to
predict the nominal concrete breakout capacities of anchors
with an embedment depth hef ≥ 10 in. (250 mm) in uncracked
concrete by Eq. (4). Equation (4) is valid, however, only if
the head size is large so that the pressure under the head at
the nominal capacity is pn ≤ 3fc′ . This limiting value is
deduced from the results of the test Series T1 to T3. In these
tests, the pressure under the head was pn/fc′ = 3.4 to 5.3, on
average 4.3. The nominal capacity is approximately 75% of
the mean capacity (compare Eq. (4) with Eq. (3)). When
applying this reduction factor, one gets pn/fc′ = 3.2 ~ 3.0.
This limiting value is supported also by the numerical analysis
results.7 For smaller heads, for which the nominal pressure
under the head is pn > 3fc′ , the breakout capacities in uncracked
concrete should be predicted by Eq. (2).
In cracked reinforced concrete, lower breakout capacities
than in uncracked concrete are observed.9 Therefore, ACI
31805, Appendix D, reduces the nominal breakout capacities of
headed anchors in cracked reinforced concrete by a factor
0.8 compared with uncracked concrete. Therefore, in
cracked concrete Eq. (4) with hef1.67, multiplied by the factor
0.8, should only be used for deep anchors if the pressure
under the head is pn ≤ 2.4fc′.
Fig. 10—Ratios of observed to predicted concrete tensile
breakout capacities as function of embedment depth.
Fig. 9—Ratios of test results (5% fractile and mean) to
predicted capacities; compare with Table 3.
484
Fig. 11—Test results and comparison with predicted capacities.
ACI Structural Journal/JulyAugust 2007
107.
EFFECT OF SUPPLEMENTARY REINFORCEMENT
Reinforced Specimen T4
Test Specimens T4, with supplementary reinforcement,
are shown in Fig. 3. The mean tested failure load (733 kips
[3260 kN]) is close to the sum (806 kips [3585 kN]) of the
calculated reinforcement strength (378 kips [1681 kN]) and
the unreinforced concrete strength (428 kips [1904 kN]) by
Eq. (3). It can be inferred that the adopted reinforcement
pattern effectively acted in the anchorage system to resist
tension load.
The tested breakout strength of the unreinforced test
Specimen T1 with the same embedment depth as Specimen T4
was 509 kips (2264 kN). Comparison of the mean tested
strengths of Specimens T1 and T4 shows that the effective
increase in capacity due to supplementary reinforcement is
roughly 224 kips (996 kN), or approximately 60% of the
calculated yield strength of the supplementary reinforcement.
The loading on Specimen T4A was increased to the
expected total yield force of the supplementary reinforcement so
that the load distribution to each of the two reinforcement
groups could be estimated. The load resisted by the
supplementary reinforcement in the inner concentric
circle (4.2 in. [106 mm] from the axis of the anchor) was
2.2 times the load resisted by the equal area of supplementary
reinforcement in the outer concentric circle (8.5 in. [216 mm]
from the axis of the anchor).
According to the measured strains in the strain gauges
attached to reinforcing bars, the reinforcing bars close to the
anchor were more effective in increasing the tensile capacity
and their maximum stress was measured close to the
anchor head.
Reinforced Specimen T5
The mean tested capacity (725 kips [3225 kN]) of the four
replicates of test Specimen T5, with supplementary reinforcement as shown in Fig. 3 was much smaller than the sum
(1129 kips [5021 kN]) of the calculated reinforcement
strength, 16 x 60 ksi x 0.79 in.2 = 758 kips (3371 kN) and
concrete breakout strength per the CCD method given by
Eq. (1), 371 kips (1650 kN). These test results indicate that
this layout of supplementary reinforcement contributes with
a low level of effectiveness to the capacity of the anchor.
This conclusion is corroborated by measured strains in the
gauges attached to the reinforcing bars, which indicates little
strain in the reinforcement. As noted previously, however,
Specimen T5 were not fully loaded up to failure due to safety
concerns. As a consequence, the results of Series T5 are judged
to not be useful in verifying the absolute effectiveness of the
supplementary reinforcement. By comparing results from
Specimens T4 with those of Specimens T5, however, it is still
possible to judge the relative effectiveness of the different
supplementary reinforcement patterns. For a given applied load,
stresses in the supplementary reinforcement of Specimens T5
along the outer circles are less than half of those along the inner
circle. The relative trends of stress distribution are similar for
each reinforcement in both Series T4 and T5. Therefore, it can
be inferred that the increase in tensile capacity is approximately
proportional to the amount of supplementary reinforcement.
The loaddisplacement curves of Series T4 show that the
peak load was nearly reached in the tests. In Series T5, the
load could still be increased. In Series T4, the supplementary
reinforcement was not strong enough to resist the concrete
breakout load. In Series T5, the loading was stopped before
the supplementary reinforcement could be fully activated.
ACI Structural Journal/JulyAugust 2007
Therefore, it is not possible to formulate a general model
from the test results. The results, however, show that with
supplementary reinforcement arranged as in Specimens T4
and dimensioned for about 80 to 100% of the expected
ultimate concrete breakout capacity, the failure load was
increased by approximately 50% over the unreinforced case.
This result can reasonably be used in the calculation of
ultimate strength.
SUMMARY AND CONCLUSIONS
Tensile loaddisplacement behavior of large
anchors without supplementary reinforcement
The test results show that ACI 34997 (Eq. (5)) significantly
overestimates the tensile breakout capacity of large anchors.
The ratio Nu,test /Nu, calc decreases with increasing embedment
depth (Fig. 10(a)). Furthermore, the slope of the concrete
cone was much flatter than 45 degrees. Therefore, the
overestimation of the failure loads would be even larger for
anchors at an edge or for anchor groups. For these reasons,
this formula in ACI 34997 should not be used in design.
1.5
The CCD method with hef (Eq. (1)) is conservative for
large anchors (Fig. 10(b)). This is probably due to the fact
that this method is based on linear fracture mechanics, which
is valid only for anchors with high bearing pressure, that is,
anchors with small heads. The tested anchors, however, had
rather large heads. The test results can best be predicted by
the CCD method with (Eq. (3)) (refer to Fig. 9 and 10(e)). On
average, the measured failure loads are approximately 10%
higher than the predicted values. If all available results are
taken into account (refer to Fig. 10(f)), however, a change of
Eq. (3) seems not to be justified.
It is proposed to calculate the characteristic resistance of
single anchor bolts with hef ≥ 10 in. (250 mm) and low
bearing pressure (pressure under the head at nominal
breakout load pn ≤ 3fc′ [uncracked concrete] or pn ≤ 2.4fc′
[cracked concrete]) according to ACI 31805, Appendix D,
or ACI 34901, Appendix B, using the equation with hef1.67).
According to the test results, however, the average cone
angle was not 35 degrees (as assumed in the CCD method)
but only approximately 25 to 30 degrees. Therefore, the
characteristic spacing scr,N and characteristic edge distance
ccr,N are probably larger than scr,N = 2ccr,N = 3hef as assumed
in ACI 31805. Therefore, it seems prudent to calculate the
resistance of anchorages at an edge or corner, or of group
anchorages, according to ACI 31805, but with scr,N = 4.0
hef instead of scr,N = 3.0 hef as given in ACI 31805.
Tensile loaddisplacement behavior of large
anchors with supplementary reinforcement
In Series T4, the supplementary reinforcement was not
strong enough to resist the applied load. Even in Test T4A,
in which the supplementary reinforcement yielded, only
approximately 1/3 (246/759 ≈ 0.33) of the applied peak load
was resisted by the reinforcement. In Series T5, which had a
stronger reinforcement, the tests had to be stopped because
of tensile yielding of the anchors before the supplementary
reinforcement had been fully mobilized. Therefore, the
results of these tests cannot be used to develop a general design
model for anchors with supplementary reinforcement.
Nevertheless, the results of test Series T4 showed that the
peak load could be increased by approximately 50%
compared with the results from test Series T1 without
supplementary reinforcement. Therefore, it is proposed to
increase the concrete breakout resistance calculated as described
485
108.
previously by a factor of 1.5 if supplementary reinforcement
is present around each anchor of an anchor group. The
supplementary reinforcement must be arranged as in Tests T4
(four Ushaped stirrups at a distance ≤ 4 in. (100 mm) or
≤ 0.15hef from the anchor) and dimensioned for the characteristic concrete breakout resistance according to Eq. (4)).
In a more general model, the supplementary reinforcement
should be dimensioned to take up 100% of the applied load,
thus neglecting the contribution of the concrete. The
supplementary reinforcement should be designed using a
strutandtie model. The characteristic resistance of the
supplementary reinforcement is given by the bond capacity
of the supplementary reinforcement in the anticipated
concrete cone, which should be assumed to radiate from the
head of the anchor at an angle of 35 degrees. The bond
capacity should be calculated according to codes of practice
(for example, ACI 318051 or Eurocode 210). The design
strength is limited by the yield capacity of the bars. This
model is described in detail in References 11 and 12.
ACKNOWLEDGMENTS
The authors would like to acknowledge the financial and technical help of
Korea Hydro & Nuclear Power Co. Ltd. and Korea Electric Power Research
Institute for financing this research work and several ongoing research
projects related to the capacity of anchorage to concrete structures. The
authors are also grateful for the valuable advice of R. Eligehausen, University
of Stuttgart, Stuttgart, Germany; R. Klingner, University of Texas at Austin,
Austin, Tex.; and members of ACI Committee 355, Anchorage to Concrete.
486
REFERENCES
1. ACI Committee 318, “Building Code Requirements for Structural
Concrete (ACI 31805) and Commentary (318R05),” American Concrete
Institute, Farmington Hills, Mich., 2005, 430 pp.
2. ACI Committee 349, “Code Requirements for Nuclear SafetyRelated
Concrete Structures (ACI 34901),” American Concrete Institute, Farmington
Hills, Mich., 2001, 134 pp.
3. Fuchs, W.; Eligehausen, R.; and Breen, J. E., “Concrete Capacity
Design (CCD) Approach for Fastening to Concrete,” ACI Structural
Journal, V. 92, No. 1., Jan.Feb. 1995, pp. 7394.
4. Rehm, G.; Eligehausen, R.; and Mallée, R., “Befestigungstechnik”
(Fastening Technique), Betonkalender 1995, Ernst & Sohn, Berlin,
Germany, 1995.
5. Comité EuroInternational du Beton, “Fastening to Reinforced
Concrete and Masonry Structures,” StateoftheArt Report, CEB, Thomas
Telford, London, 1991, pp. 205210.
6. ACI Committee 349, “Code Requirements for Nuclear Safety Related
Concrete Structures (ACI 34997),” American Concrete Institute, Farmington
Hills, Mich., 1997, 123 pp.
7. Ozbolt, J.; Eligehausen, R.; Periskic, G.; and Mayer, U., “3D FE
Analysis of Anchor Bolts with Large Embedment Depths,” Fracture Mechanics
of Concrete Structures, V. 2, No. 5, Apr. 2004, Vail, Colo., pp. 845852.
8. Shirvani, M.; Klingner, R. E.; and Graves III, H. L., “Behavior of Tensile
Anchors in Concrete: Statistical Analysis and Design Recommendations,”
ACI Structural Journal, V. 101, No. 6, Nov.Dec. 2004, pp. 812820.
9. Eligehausen, R., and Balogh, T., “Behavior of Fasteners Loaded in
Tension in Cracked Reinforced Concrete,” ACI Structural Journal, V. 92,
No. 3, MayJune 1995, pp. 365379.
10. Eurocode 2, “Design of Concrete Structures, Part 1: General Rules
and Rules for Buildings,” 2004.
11. Technical Committee CEN/TC 250, “Design of Fastening for
Use in Concrete, Part 2: Headed Fasteners,” Final Draft, CEN Technical
Specifications, 2004.
12. Comité EuroInternational du Beton (CEB), Design Guide for
Anchorages to Concrete, Thomas Telford, London, 1997.
ACI Structural Journal/JulyAugust 2007
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