A model for the practical nonlinear analysis of reinforcedconcrete frames
including joint ﬂexibility
Anna C.Birely
⇑
,Laura N.Lowes,Dawn E.Lehman
Civil and Environmental Engineering,University of Washington,Seattle,WA 98195,United States
a r t i c l e i n f o
Article history:
Received 5 January 2011
Revised 1 September 2011
Accepted 1 September 2011
Keywords:
Beam–column joints
Analytical models
Rotational springs
Ductility
a b s t r a c t
A model is developed to simulate the nonlinear response of planar reinforcedconcrete frames including
all sources of ﬂexibility.Conventional modeling approaches consider only beam and column ﬂexibility
using concentrated plasticity or springs to model this response.Although the joint may contribute the
majority of the deformation,its deformability is typically not included in practice.In part,this is because
few reliable,practical approaches for modeling all sources of frame nonlinearity are available.The
research presented herein was undertaken to develop a practical,accurate nonlinear model for reinforced
concrete frames.The model is appropriate for predicting the earthquake response of planar frames for
which the nonlinearity is controlled by yielding of beams and/or nonductile response of joints and is
compatible with the ASCE/SEI Standard 4106 nonlinear static procedure.The model was developed to
facilitate implementation in commercial software packages commonly used for this type of nonlinear
analysis.The nonlinearity is simulated by introducing a dualhinge lumpedplasticity beam element to
model the beams framing into the joint.The dualhinge comprises two rotational springs in series;
one spring simulates beamﬂexural response and one spring simulates joint response.Hinge parameters
were determined using data from45 planar frame subassemblage tests.Application of the model to sim
ulate the response of these subassemblages shows that the model provides accurate simulation of stiff
ness,strength,drift capacity and response mechanismfor frames with a wide range of design parameters.
2011 Elsevier Ltd.All rights reserved.
1.Introduction
Under seismic loading,the beams and columns in a concrete
moment frame typically experience moment reversals at the
beam–column joint.To ensure that strength is maintained under
multiple large drift cycles,design guidelines for frames in regions
of high seismicity are intended to result in ﬂexural yielding in
the beams at the face of the joint and essentially elastic response
in the columns above the base (e.g.,ACI 31808 [1]).This can result
in high joint shear demand and high bond stress demand for beam
longitudinal reinforcement anchored in the joint;design guidelines
seek to limit both to ensure that joint damage does not reduce
frame toughness.The results of experimental tests on frame
subassmblages [2–13] show that joint damage can reduce frame
strength and stiffness and,in some cases,result in premature loss
of loadcarrying capacity.
Prediction of frame response,as part of a performancebased
seismic design of a new structure or evaluation of an existing
structure,requires modeling of all sources of ﬂexibility in the
frame as well as stiffness and strength loss under earthquake load
ing.Thus,to conduct a nonlinear analysis of a reinforced concrete
momentresisting frame,engineers require not only accurate mod
els for beams and columns but also models that simulate joint re
sponse.For these models to be practical for use in the design ofﬁce,
they must (1) be compatible with commonly employed commer
cial software packages,(2) support rapid model building,(3) be
computationally efﬁcient and robust,and (4) provide acceptable
accuracy over a range of design conﬁgurations.Several practical
nonlinear modeling approaches are available for beams and col
umns (e.g.,[14,15]).These models,which have been validated by
others,are incorporated into the fame model proposed herein.
However,these models do not simulate the response of the
beam–column joint.Here,the focus is on developing an appropri
ate approach for simulating joint ﬂexibility and degradation in
frame strength due to joint response mechanisms.Although non
linear joint models are found in the literature,fewof these models
meet the requirements for widespread use by practicing structural
engineers.A model is proposed which uses conventional nonlinear
frame elements and is easily implemented in commercial struc
tural analysis software.The model is developed and validated
using experimental data to simulate the full frame response,
including the joint.
01410296/$  see front matter 2011 Elsevier Ltd.All rights reserved.
doi:10.1016/j.engstruct.2011.09.003
⇑
Corresponding author.
Email address:birely@uw.edu (A.C.Birely).
Engineering Structures 34 (2012) 455–465
Contents lists available at SciVerse ScienceDirect
Engineering Structures
j ournal homepage:www.el sevi er.com/l ocat e/engst ruct
1.1.Previously proposed beam–column joint models
The simplest approach for modeling joint response within the
context of a nonlinear frame analysis is to introduce a spring at
the intersection of the beamand column line elements.Often rigid
offsets are included in the beamand column elements to deﬁne the
physical size of the joint and ensure that the joint spring is the sole
source of simulated deformation in the joint region.The load
deformation response of the joint spring is typically deﬁned on
the basis of the expected shear stress–strain response of the joint
volume and/or the bond slip of longitudinal reinforcement within
the joint.Models of this type were developed by ElMetwally and
Chen [16],Kunnath [17],Ghobarah and Biddah [18],and Anderson
et al.[19].The advantage of this type of model is its overall sim
plicity.One disadvantage of the model is the need to deﬁne dupli
cate nodes at the center of the joint,a process that is typically not
well supported in commercial software and can hinder the model
building process.The primary disadvantage of this type of model,
however,is the challenging and timeconsuming process of cali
brating the model to provide an accurate simulation of joint re
sponse.Often,model calibration is accomplished by simplifying
the assumed primary response of the joint.Speciﬁcally,the model
parameters are speciﬁc to a limited set of design parameters,a lim
ited set of data,or both.Typically the modeling assumptions de
pend on the design parameters,and applying these assumptions
to generate the model response history typically requires signiﬁ
cant computation by the engineer.In some cases,models are cali
brated directly fromexperimental data characterizing the response
of joints in frames with design parameters and details that are sim
ilar to those in the structure of interest;in this case,the accuracy of
the simulations depends entirely on the similarity of the two
frames.
Macroelement joint models are a secondapproachfor modeling
joint behavior.Altoontashand Deierlein[20],Lowes and Altoontash
[21],and Mitra and Lowes [22] have proposed models that connect
beamandcolumncenterline elements toﬁnitevolume joint macro
elements.These models comprise a shearpanel component and
rotational springs or zerolength springs to represent bar slip and
interface shear.Other macromodels have been developed by
Youssef and Ghobarah [23],Elmorsi et al.[24],Shin and LaFave
[25],Uma and Prasad [26],and Tajiri et al.[27].Relative to a single
concentrated spring,a macroelement model typically allows for a
simpler,more objective calibration and enables simulation of joints
with a wide range of design parameters.Aprimary drawback of this
modeling approach,however,is that macroelement models are not
easily implemented in commercial software.
Given the complexity of beam–column joint response,contin
uummodeling offers the greatest potential for accurately simulat
ing the nonlinear response of joints with a range of design
parameters.Continuum modeling has been the focus of a number
of research studies [28–30].However,it has not been validated
using large data sets,is computationally demanding,requires sig
niﬁcant modelbuilding effort on the part of the engineer,and can
not be accomplished using the analysis software employed
typically by practicing structural engineers.
1.2.Proposed model
The objective of this study was to develop a model to accurately
simulate the nonlinear response (including initial stiffness,
strength,and deformation capacity) of a reinforced concrete frame.
This requires simulation of all important sources of deformation
including the beam,column,and joint response mechanisms.Pre
viously proposed joint models do provide accurate representation
of joint behavior,and some have been calibrated to explicitly ac
count for speciﬁc response modes,such as bond slip and shear
deformation.However,these models require the addition of a sep
arate element,typically concentrated springs or macroelements,
for every joint,making their use in commercial nonlinear struc
tural analysis software packages time consuming and cumber
some.However,modeling a frame using only standard nonlinear
beam–column elements neglects simulation of joint response and
typically produces inaccurate results.The proposed model seeks
to provide an intermediate modeling alternative between the sim
ple use of frame elements without joint representation and the
more time consuming use of detailed joint elements.
One of the most common approaches to modeling nonlinear re
sponse of reinforced concrete frames is to use lumpedplasticity
elements for the beams and columns.This approach is used here,
with the momentrotation history of the beam plastic hinges (for
the systemconsidered in this study,columns do not exhibit signif
icant nonlinearity) modiﬁed to include simulation of not only the
beam response,but also the joint response.In the modiﬁed ele
ment,the conventional singlehinge representation is replaced by
a dualhinge,which comprises two rotational springs in series.
One of the springs simulates the nonlinear ﬂexural response of
the beamand is referred to as the beamspring.The second spring
simulates the response of the joint,and is referred to as the joint
spring.Rotation limits are provided for each spring to simulate
the onset of loss of load carrying capacity.Rigid offsets are in
cluded in the beams and columns that frame into the joint to en
sure that joint ﬂexibility is deﬁned entirely by the joint spring in
the dual hinge.
The model was developed with the objective of satisfying prac
tical nonlinear modeling needs.The proposed model can be used in
existing commercial software with an effort that is equivalent to
using a conventional nonlinear beam–column element.By modify
ing a traditional method for modeling frames,the intent of the
model is to provide increased accuracy over models commonly
used in practice.Consequently,the model should not be used to
extract local response of a joint.If such information is needed,
other models,such as those identiﬁed above,are more appropriate.
2.Experimental data set
The proposed joint model was developed using data from45pla
nar frame subassemblages,tested by 11 research groups [2–13],as
listed in Table 1.The data set used in this study is a subset of that
assembled by Mitra and Lowes [22],where a detailed description
of the individual specimens canbe found.Specimens are bareframe
subassemblages without slabs.The data set does not include spec
imens constructedof lightweight concrete,veryhighstrengthcon
crete (over 16 ksi,or 110 MPa),or plain reinforcing bars.For the
specimens included in the data set,the lateral capacity was limited
by ﬂexural yielding of beams and/or damage in the joint.The pro
posed model is appropriate for use in simulating the response of
frames with these characteristics and failure modes.
The data set spans a wide range of design parameters and in
cludes specimens that are representative of newconstruction in re
gions of high seismic hazard as well as construction that predate
modern seismic detailing requirements.Fig.1 shows histograms
for select design parameters,including concrete compressive
strength (f
0
c
),beamsteel yield strength (f
y
),ratio of the sumof col
umn ﬂexural strengths to the sum of the beam ﬂexural strengths
(
R
M
nc
/
R
M
nb
),as well as bond demand (
l
) computed
l
¼
a
f
y
d
b
2h
c
ﬃﬃﬃﬃ
f
0
c
p
ð1Þ
where d
b
is the maximum diameter of the beam longitudinal rein
forcement and all other variables are previously deﬁned,joint rein
forcement ratios (
q
j
) computed
456 A.C.Birely et al./Engineering Structures 34 (2012) 455–465
q
j
¼
A
t
s
t
b
j
ð2Þ
where A
t
is the area of one layer of joint transverse reinforcement
passing through a plane normal to the axis of the beams,s
t
is the
vertical spacing of hoops in the joint region,and b
j
is the outof
plane dimension of the joint,and axial load ratio (p) computed
p ¼
P
A
g
f
0
c
ð3Þ
where P is the applied column axial load,A
g
is the gross area of the
column,and f
0
c
is the measured concrete compressive strength.
Additionally,Table 1 lists for each specimen,the following shear
stress values used in the current study:
Designshear stress demand,
s
design
,computedusing anapproach
that is consistent with ACI Com352 [31] recommendations:
s
design
¼
1
h
c
b
j
a
ðf
y
ðA
top
s
þA
bottom
s
Þ V
n
Þ ð4Þ
where h
c
is the height of the column and b
j
is the outofplane
dimension of the joint.The column shear,V
n
,corresponds to
the average nominal moment strength of the beam.The vari
ables A
top
s
and A
top
s
are the areas of steel in the top and bottom
of the beam,respectively,and f
y
is the measured strength of
the reinforcement.To account for hardening of the steel under
earthquake loading and overstrength in the nominal value,f
y
is typically multiplied by a factor of 1.25 [1].However,as the ac
tual strength of the steel was available for all specimens in the
data set,the 1.25 factor was reduced by 1.1 as recommended
by ACI Com.352 [31].Thus,
a
= 1.25/1.1 was used in Eq.(4).
Maximum measured shear stress demand,
s
max
s
max
¼
1
h
c
b
j
M
L
þM
R
jd
V
max
ð5Þ
Table 1
Ductility classiﬁcation data.
Test program Test specimen
s
design
ﬃﬃﬃ
f
0
c
p
(MPa/
p
MPa)
a
s
meas
max
ﬃﬃﬃ
f
0
c
p
(MPa/
p
MPa)
a
s
model
max
ﬃﬃﬃ
f
0
c
p
(MPa/
p
MPa)
a
l
D
b
Ductility
Test Model
Durrani and Wight [7] DWX1 11.10 1.27 1.06 4.7 D D
DWX2 1.12 1.35 1.07 5.5 D D
DWX3 0.87 1.12 0.85 6.4 D D
Otani et al.[3] OKAJ1 1.22 0.97 1.01 5.3 D D
OKAJ2 1.26 1.03 1.04 5.6 D D
OKAJ3 1.26 1.15 1.04 6.7 D D
OKAJ4 1.22 0.95 1.01 5.1 D D
OKAJ5 1.15 0.90 0.96 4.4 D D
Meinheit and Jirsa [10] MJ1 2.24 1.32 1.70 – B B
MJ2 1.78 1.52 1.57 – B B
MJ3 2.23 1.47 1.70 – B B
MJ5 1.92 1.57 1.60 – B B
MJ6 1.89 1.69 1.59 – B B
MJ12 1.93 2.04 1.62 3.5 LD B
MJ13 1.79 1.49 1.57 – B B
Alire [5] and Walker [6] PEER14 0.87 0.94 0.89 3.6 LD D
PEER22 1.55 1.17 1.49 – B B
PEER0850 0.61 0.70 0.64 4.3 (E) D D
PEER0995 0.93 1.04 0.95 5.3 D D
PEER4150 3.40 1.90 1.97 – B B
Park and Ruitong [12] PR1 0.46 0.51 0.47 29.5 (E) D D
PR2 0.72 0.81 0.72 6.8 D D
PR3 0.51 0.53 0.52 13.2 (E) D D
PR4 0.69 0.73 0.69 7.1 D D
Noguchi and Kashawazaki [11] NKOKJ1 1.88 1.52 1.67 – B B
NKOKJ3 1.90 1.54 1.63 – B B
NKOKJ4 1.88 1.60 1.67 – B B
NKOKJ5 2.37 1.59 1.88 – B B
NKOKJ6 2.02 1.61 1.75 – B B
Oka and Shiohara [13] OSJ1 1.45 1.44 1.34 3.0 LD B
OSJ2 3.60 1.55 1.84 – B B
OSJ4 1.55 1.55 1.39 3.8 LD B
OSJ5 2.05 1.75 1.72 – B B
OSJ6 1.57 1.55 1.41 3.0 LD B
OSJ7 1.16 1.24 1.17 5.0 (E) D D
OSJ8 1.95 1.76 1.59 2.9 LD B
OSJ10 2.35 1.60 1.80 – B B
OSJ11 2.83 1.84 1.87 – B B
Kitayama et al.[9] KOAC1 0.91 1.01 0.87 116 (E) D D
KOAC3 0.91 0.97 0.87 42.3 (E) D D
Park and Milburn [4] PM1 1.25 1.15 1.22 3.1 (E) LD D
Endoh et al.[8] HC 1.02 1.10 0.96 8.2 (E) D D
A1 2.24 1.45 1.64 – B B
Beckingsale [2] B11 0.76 0.83 0.78 No Loss D D
B12 0.76 0.86 0.79 No Loss D D
a
Conversion factor:1 MPa = 145 psi;1 MPa/
p
MPa = 12.04 psi/
p
psi.
b
(E):Displacement ductility at 10% strength loss extrapolated;(–):brittle joint,no displacement ductility.
A.C.Birely et al./Engineering Structures 34 (2012) 455–465
457
where V
max
is the maximumexperimental column shear,M
L
and
M
R
are the corresponding moments in the left and right beamat
the joint face,respectively.The termjd is the moment lever arm
for the beam,where d is the depth from the extreme compres
sion ﬁber to the tension reinforcement in the beam,assumed
to be 0.9h
b
if unknown,and j is an empirically derived parameter
typically taken equal to 0.85 [32].
Fig.2 shows a typical test specimen and setup.Test specimens
represent a subassemblage froma twodimensional building frame,
comprising a segment of a continuous beam extending from mid
spanof one frame bay tomidspanof the next,a segment of a contin
uous column extending frommidheight of one story to midheight
of the next,and the beam–column joint at the intersection of these
two members.The midlength points are assumed to approximate
points of inﬂection in an actual frame and therefore are a suitable
location to apply the shear forces and sustain the reactions.Frame
subassemblages were subjected to reversed cyclic lateral loading
under displacement control in the laboratory,and load was applied
either as a shear load at the top of the column or as a pair of equal
andopposite shear loads at the beamends.Some subassemblages
were also subjected to simulated gravity load applied as a constant
axial load to the top of the column.
2.1.Force–displacement data
The primary data used from the experimental tests were the
column shear versus lateral displacement envelope for each spec
imen.In some cases,data characterizing the response of the indi
vidual frame components (beams,columns and joints) were
available,in other cases they were not.Therefore individual com
ponent responses were not directly considered;instead,the overall
subassemblage response was used.This is consistent with previ
ous research addressing performancebased seismic design of rein
forced concrete frames that has shown frame drift to be the most
practical engineering demand parameter [33].
The response envelope for each specimen was determined from
the experimental cyclic load–displacement history.Points on the
envelope correspond to the following load–displacement pairs:
(i) theoretical initial ﬂexural yielding of beams,(V
y
,
D
y
),(ii) maxi
mumcolumn shear,(V
max
,
D
V
max
),(iii) 20% loss of strength follow
ing maximum load,(V
80%
,
D
80%V
),and (iv) any additional points
required to accurately represent the shape of the load–displace
ment envelope.The theoretical yield load,V
y
,was computed using
a moment–curvature analysis of a ﬁber crosssection of the beams
with ﬂexural yield strength of the beams deﬁned by initial tensile
yielding of the beam longitudinal reinforcement.Fig.3 shows an
example of the cyclic response and the response envelope for Spec
imen PEER0995 [5–6].
2.2.Ductility classiﬁcation
To support the model calibration effort,specimens were classi
ﬁed as brittle,ductile or limitedductility.Brittle specimens exhibit
a maximum strength that was less than that required to yield all
beam longitudinal reinforcement in tension.Ductile specimens
reach the yield shear force and exhibit displacement ductility,
l
D
,greater than four.Limitedductility specimens are specimens
not classiﬁed as brittle or ductile.Displacement ductility,
l
D
,was
deﬁned as
l
D
¼
D
90%
D
y
ð6Þ
20
40
60
80
100
120
0
10
20
30
(a) f’
c
(MPa)
# Specimen
250
500
750
1000
1250
1500
0
10
20
30
(b) f
y
(MPa)
# Specimen
0.5
0.75
1
1.25
1.5
1.75
2
2.25
0
5
10
15
20
(c) Σ M
nc
/Σ M
nb
# Specimen
10
20
30
40
50
60
0
5
10
15
(d) μ
# Specimen
0
0.5
1
1.5
2
2.5
3
3.5
4
0
5
10
15
(e) ρ
j
(%)
# Specimen
0
0.1
0.2
0.3
0.4
0.5
0
5
10
15
20
(f) P/(A
g
f’
c
)
# Specimen
Fig.1.Histogram of basic beam–column joint subassemblage properties.
Fig.2.Experimental setup for Park and Milburn [4] and Alire [5] subassemblages
(equal and opposite forces applied to beams).
0
40
80
120
0
150
300
450
Dis
p
lacement, [mm]
Column Shear, [kN]
Experimental
Envelope
Second Beam Yield
90% Maximum Shear
Δ
Y
V
Y
Δ
Vmax
V
max
Δ
80%Vmax
V
80%
(V
fy
, Δ
fy
)
(V
90%
, Δ
90%Vmax
)
Fig.3.Positive force–displacement history for specimen PEER 0995 with envelope.
458 A.C.Birely et al./Engineering Structures 34 (2012) 455–465
where
D
90%
is the displacement at which 10% strength loss
occurred,as determined fromthe column lateral load–displacement
envelope,and
D
y
is as deﬁned previously.A solid circle in Fig.3
shows
D
90%
and the corresponding column shear force for Specimen
PEER0995.The ductility classiﬁcation of each specimen is provided
in Table 1.For the complete data set,18 specimens were classiﬁed
as brittle,20 as ductile,and 7 as limitedductility.
3.Model deﬁnition
The proposed dualhinge model is incorporated in a lumped
plasticity beam–column element,where two rotational springs
are combined in series to form the dualhinge that represents the
inelastic deformations of the beam and the joint.One of the rota
tional springs simulates the beam response.The second spring
simulates the joint response,including that caused by joint shear
deformation and bond slip.
Fig.4 shows a model of a typical frame subassemblage in
which the proposed dualhinge beam–column element is used to
model the beams.Beams are modeled as elastic outside the hinge
region.Columns are modeled using elastic beam–column ele
ments;elastic elements were considered adequate because,for
the subassemblages in the data set,columns did not yield.The
effective elastic stiffness values recommended in ASCE/SEI Stan
dard 4106 [34] were used for beams and columns.Rigid offsets
equal to the joint dimensions were included at the ends of the
beamand column elements to deﬁne the joint volume and ensure
that the joint spring was the only source of joint deformation.
Fig.5 shows the moment–curvature response of the two com
ponents of the dualhinge.The beamresponse is similar to a tradi
tional nonlinear beam–column hinge,but has a rotation limit,
h
fail
beam
,beyond which a loss of strength occurs.The joint spring
has a bilinear response,with stiffness K
1
,to the yield moment of
the beam,and K
2
beyond yield.For the joint spring,strength loss
initiates at a rotation demand of h
fail
joint
.The following sections dis
cuss the calibration of these values.
3.1.Rotational spring simulating nonlinear beam ﬂexural response
Typically,the momentrotation response for a lumpedplasticity
beam or column element is determined from the moment–
curvature response of the member cross section and a plastic
hinge length.As described previously,the proposed dualhinge
model uses two springs in series.The rotational spring represent
ing the nonlinear ﬂexural response of the beam was calibrated to
simulate the nonlinear response of the beam plasticend zone
including a rotation limit deﬁned to simulate the onset of ﬂexural
strength degradation.The beam ﬂexural momentrotation
response prior to strength loss was determined by performing a
moment–curvature analysis (for both positive and negative
bending) using a ﬁbertype model of the beam section.The beam
depth was discretized into 32 concrete ﬁbers;ﬁber thickness was
approximately onehalf inch (13 mm).Concrete was assumed to
have no tensile strength.A parabolic compressive stress–strain
response was assumed for unconﬁned concrete,with maximum
strength corresponded to a strain of 0.002 mm/mm.The modiﬁed
Kent–Park model [35] was used to deﬁne the compressive
response of conﬁned concrete.A bilinear stress–strain response
was assumed for reinforcing steel in tension and compression;
the hardening stiffness was deﬁned to be 0.01% of the initial
stiffness,essentially creating an elastic–plastic model.
The resulting moment–curvature response was transformed to
a momentrotation response by multiplying curvatures by a plastic
hinge length equal to onehalf the beam depth.This hingelength
model was proposed by Corley [36] and has been found by several
research groups [16–19] to provide good estimates of member dis
placement.Fig.5a shows the momentrotation relation for the
beam spring,including the proposed rotation limit.Calibration of
the rotation limit is discussed in Section 3.3.
3.2.Rotational spring simulating nonlinear joint response
In the proposed model,joint nonlinearity was explicitly mod
eled using a second rotational spring in series with the beamspring
discussed in Section 3.1.The properties of the joint spring were
Rigid Offsets
h
b
h
c
L
c
L
b
BeamSpring
Joint Spring
Dualhinge
Lumpedplasticity
beamelement
Fig.4.Proposed model of subassemblage using rigid offsets and dualhinge at
beamjoint interface.
(a)
(b)
M
hinge
θ
bea
m
M
pos
yield
M
neg
yield
θ
fail
beam
−
θ
fail
beam
M
hinge
θ
joint
M
min
yield
−
M
min
yield
θ
fail
joint
−
θ
fail
joint
K
1
K
2
K
1
K
2
Fig.5.Dualhinge components:(a) beam spring and (b) joint spring.
A.C.Birely et al./Engineering Structures 34 (2012) 455–465
459
determined assuming that joint response may be deﬁned by a joint
shear stress versus shear strain model and related to a moment
rotation response using frame geometry.This approach to model
calibration is similar to that used in previous studies [16–19]
employing a single joint hinge.The joint spring in the dualhinge
includes a rotation limit beyond which the strength drops signiﬁ
cantly.Discussion of the spring follows,with calibrated values
listed in Table 2.Calibration of the rotation limit is discussed in
Section 3.3.
To calibrate the spring stiffness,equations were derived relating
joint shear strain to rotation at the beamjoint interface and joint
shear stress to the moment at the beamjoint interface for a frame
subassemblage.Detailed derivations are provided in Appendix A.
To determine the relationship between joint shear strain and hinge
rotation,the story drift resulting froma joint shear strain of
c
j
,was
constrained to be equal to that resulting from equal but opposite
rotations,h
js
,in the hinges at the beamjoint interface.Imposing
this constraint results in
h
js
¼
v
c
c
j
ð7Þ
where
v
c
¼ 1
h
b
L
c
1
h
c
L
b
ð8Þ
L
c
and L
b
are the lengths of the column and beam (Fig.4),respec
tively,and h
c
and h
b
are the depth of the column and beam,respec
tively.Similarly,a relationship between joint shear stress,
s
j
,and
moment in the hinge,M
hinge
,was determined by relating the beam
moment at the joint face for a particular column shear to the joint
shear stress for the same column shear.For a particular column
shear load,the moment at the beamjoint interface is assumed to
transfer into the joint as a tension–compression couple,with ten
sion and compression forces determined assuming a moment arm
of jd,where d is the depth in the beam crosssection from the ex
treme compression ﬁber to the tension reinforcement (assumed
to be 0.9h
b
if unknown) and j is an empirically derived parameter
taken equal to 0.85 [32].The joint shear,V
j
,is deﬁned equal to
the sumof the couple forces (one tension and one compression) less
the column shear acting on the top half of the joint.The joint shear
stress is the joint shear force divided by the crosssectional area of
the joint,A
j
.The resulting relationship between joint moment and
shear stress is
M
hinge
¼
v
s
s
j
ð9Þ
where
v
s
¼
A
j
1
h
c
L
b
jd
2 1
jd
L
c
h
c
L
b
ð10Þ
and all parameters in Eq.(10) are as previously deﬁned.
The joint shear stress is related to the shear strain by an effec
tive shear modulus,
j
G:
j
G ¼
s
c
ð11Þ
where G is the shear modulus of the concrete.Using Eqs.(7)–(11)
the stiffness of the joint rotational spring,K,can be written:
K ¼
M
hinge
h
js
¼
j
G
v
s
v
c
ð12Þ
A bilinear shear stress–shear strain response,with a change in stiff
ness occurring at initial beam yielding,was used.Prior research
shows that this is an approximate yet accurate model of the nonlin
ear response of the joint [5].The initial joint spring stiffness,K
1
,was
established using the relationship (Eq.(13)) to
j
1
,the effective
shear stress–strain stiffness parameter.
K
1
¼
j
1
v
s
v
c
G ð13Þ
The parameter
j
1
,was computed as that which minimized the
cumulative error in the predicted initial yield displacement for
the data set.The Matlab (www.mathworks.com) fminbnd function
was used for the optimization,and the cumulative error was
deﬁned
e
yield
¼
X
N
i¼1
D
y
D
analytical
y
D
y
!
2
i
ð14Þ
where
D
analytical
y
is the simulated initial yield displacement corre
sponding to the initial yield load,V
y
,
D
y
is the measured yield dis
placement,and only specimens that reached the yield load,V
y
,
were included in the summation.The optimumeffective shear stiff
ness factor,
j
1
,is listed in Table 2.
The postyield tangent stiffness of the joint spring was deﬁned
as:
K
2
¼
j
2
v
s
v
c
G ð15Þ
The parameter
j
2
was determined using the simulated,
D
analytical
fy
,
and measured,
D
fy
,displacements at the column shear correspond
ing to full yield,V
fy
.Full yield was deﬁned as the point at which both
beams had yielded,with yielding for a beamdeﬁned by initial yield
ing of longitudinal reinforcement as determined from the moment
curvature analysis of the beamsection.The experimental displace
ment at full yield,
D
fy
,was taken fromthe subassemblage response
envelope.For most subassemblages,the full yield force is larger
than the initial yield force due to differences in conﬁguration and
strength of the reinforcement in the top and bottom of the beams.
The full yield point,rather than the point of maximum load,was
used to determine
j
2
because the full yield point was found to pro
vide a good representation of the postyield stiffness while the low
tangent stiffness of subassemblages in the vicinity of maximum
load was found to result in a high level of variation and poor repre
sentation of the postyield stiffness.Only tests in which the V
fy
shear force level was achieved or exceeded were used to calibrate
the postyield stiffness ratio.Fig.3 shows the full yield point for
specimen PEER0995.
Initially,postyield stiffness parameter
j
2
was calibrated using
the same errorminimization approach that was used to calibrate
j
1
;however,this resulted in an unreasonable value for
j
2
.The fail
ure of this approach was attributed to two factors.First,the post
yield response of the dualhinge was dominated by the response of
the beamspring and,as a result,the simulated response was fairly
insensitive to changes in the postyield stiffness of the joint hinge.
Second,because the calibrated initial stiffness of the joint spring
underestimated the stiffness of some of the specimens,application
of the model resulted in underprediction of stiffness and over
prediction of displacement at full yield of the beams for these
specimens.
Table 2
Summary of model parameters.
Parameter Calibrated value
j
1
0.14
j
2
0.038
c
fail
joint
0.0069
/
fail
beam
0.0056
460 A.C.Birely et al./Engineering Structures 34 (2012) 455–465
Ultimately,
j
2
was determined by averaging the
j
i
2
values com
puted for each subassemblage,i,with an imposed upperbound
limit on
j
i
2
equal to the initial stiffness coefﬁcient
j
1
.The following
error function was minimized for each specimen to ﬁnd the
j
i
2
values
e
i
postyield
¼
D
i
fy
D
i analytical
fy
D
i
fy
!
ð16Þ
where
D
i
fy
and
D
i analytical
fy
are as deﬁned previously.The optimal value
for
j
2
is listed in Table 2.
3.3.Spring rotation limits to determine onset of strength degradation
To simulate strength loss,rotation limits were included in both
components of the dualhinge;beyond the rotation limit,the
strength of the springs was reduced.The characteristics of strength
loss and simulation of postpeak response were not considered,
and postpeak response is shown in Fig.5 as a dashed line.ASCE/
SEI Standard 4106 [34] provides recommendations for simulation
of strength degradation.To determine appropriate rotation limits
for use in the model,a pushover analysis was performed for each
specimen using the dualhinge model (without rotation limits).
The beam and joint spring rotations corresponding to the drift at
onset of degradation in lateral loadcarrying capacity were re
corded.Ductile and limitedductility specimens were considered
to reach this limit state when 10% of the specimen’s peak strength
was lost,
D
90%
;brittle specimens were considered to reach this lim
it state at maximum strength,
D
V
max
.Joint spring rotations were
converted to equivalent shear strains using the relationship pro
vided by Eq.(7).Beam spring rotations were converted to equiva
lent curvatures using an assumed hinge length (l
p
) equal to
onehalf the beam depth.
To determine an appropriate approach for deﬁning rotation lim
its for response prediction,the relationships between specimen de
sign parameters and spring rotation limits were explored.
However,no signiﬁcant correlation was found.Comparing the
equivalent strains and curvatures at the onset of strength loss for
brittle,ductile and limitedductility specimens did show correla
tion between these values and subassemblage ductility.Fig.6
shows the equivalent shear strain and curvature values plotted
versus specimen number with different markers used to indicate
the ductility classiﬁcation of each specimen.Fig.6a shows that
the equivalent shear strains at strength loss were signiﬁcantly lar
ger for brittle specimens than for ductile specimens;Fig.6b shows
that the equivalent beam curvatures were signiﬁcantly larger for
ductile specimens than for brittle specimens.Thus,by deﬁning
constant rotation limits for each spring in the dualhinge model,
the onset of strength loss can be predicted without prior knowl
edge or prediction of the response mechanism.Strength degrada
tion is triggered by the joint spring component of the dualhinge
for a brittle specimen and by the beam spring for a ductile
specimen.
Beam and joint spring rotation limits were deﬁned as the con
stant values that minimized the sum of the error squared in the
simulated drift at strength loss.Equivalent beam curvatures
(/
fail
beam
) and equivalent joint shear strains (
c
fail
joint
) at initiation of
strength loss were calibrated using data from ductile and brittle
specimens,respectively.These limits were incorporated into the
dualhinge using the relationships deﬁned by Eqs.(17) and (18).
A constant value for each parameter was found that minimized
the sumof the error squared in the simulated drift at strength loss.
The calibrated values are provided in Table 2.
h
fail
beam
¼/
fail
beam
l
p
ð17Þ
h
fail
joint
¼
c
fail
joint
v
c
ð18Þ
4.Application of the model
The proposed model is intended for use with the commercial
structural analysis software used commonly in design ofﬁces,
and can be implemented using the following procedure.
1.Determine the properties of the beam spring.Compute the
moment–curvature response for each beam cross section and
convert beam curvatures to rotations using a plastic hinge
length,l
p
,equal to onehalf of the beam depth.Using the com
puted crosssectional response,identify yield moments for use
in creating the joint spring momentrotation history.Compute
the beam rotation limits (positive and negative bending) for
each section,using Eq.(17) with
u
fail
beam
deﬁned in Table 2.
2.Determine the joint spring response history.Determine the ini
tial and postyield stiffness values using Eqs.(13) and (15),with
j
1
and
j
2
deﬁned in Table 2.The joint shear stress at which
stiffness change occurs is the joint shear stress corresponding
to initial beam yielding.The strength of the joint spring
degrades at an absolute rotation demand given by Eq.(16),with
c
fail
joint
deﬁned in Table 2.
3.Create a centerline model of the structure with elastic beam–
column elements used for the columns,lumpedplasticity
beam–column elements used for the beams,rigid offsets in
the beam and column elements deﬁning the physical volume
of the joint,and dualhinges introduced in the lumpedplasticity
beamelements at the joint faces.The dual hinges consist of the
zerolength beamand joint springs placed in series.
4.Perform the desired analysis of the structure.
0
10
20
30
40
50
0
0.005
0.01
0.015
Joint Number
(b)
φbeam
pos
0
10
20
30
40
50
0
0.005
0.01
0.015
Joint Number
(a)
γjoint
pos
Brittle
Ductile
Limited Ductility
Fig.6.Engineering parameter values corresponding to strength drop for all data:
(a) joint spring shear strain limits and (b) beamspring curvature limits for all data.
A.C.Birely et al./Engineering Structures 34 (2012) 455–465
461
5.Comparison of simulated and observed response
For each of the subassemblages in the data set,a nonlinear
model was created using the steps described above and a push
over analysis was performed.
Fig.7 illustrates the impact the joint spring has on the simu
lated response by showing envelopes for (i) the experimental data,
(ii) an analytical model that includes a conventional beam hinge/
spring,and (iii) the proposed dualhinge model,including termina
tion when the rotation limits are reached.Envelopes are shown for
(a) PEER 0850,for which specimen drift is due primarily to beam
ﬂexure and (b) PEER 4150,for which specimen drift is due primar
ily to joint shear.The addition of the joint spring to the model has a
greater impact on the response of the subassemblage with greater
measured shear deformation (PEER 4150).The proposed rotation
limits for the beamand joint springs result in the model accurately
simulating (a) the drift capacity of the ductile PEER 0850 specimen
and (b) the peak strength of the brittle PEER 4150 specimen.
Table 1 presents the ductility classiﬁcation for each specimen as
observed experimentally and as predicted using the model.Fig.8
shows the best and worst simulated versus observed response his
tories for the ductile,limitedductility and brittle specimens.
For the full data set,the model was evaluated ﬁrst by comparing
the simulated and experimentally determined ductility of each
specimen.Section 2.2 described classiﬁcation of specimen ductility
using experimental data.The simulated response of the specimen
was considered ductile if beam rotation demand exceeded the
beam spring rotation limit but not the joint spring rotation limit
and brittle if joint rotation demand exceeded the joint spring
rotation limit but not the beamspring rotation limit.Brittle behav
ior was simulated for all 18 specimens experimentally classiﬁed as
brittle.Ductile behavior was simulated for all 20 specimens exper
imentally classiﬁed as ductile.Of the seven limitedductility spec
imens,the simulated response of six was controlled by joint
failure;for these six specimens the rotation demand imposed on
the beamﬂexural spring was only 7% of the rotation limit.Of the
limitedductility specimens,specimen PEER14 was the only speci
men for which simulated strength loss was due to activation of the
beamspring rotation limit;this occurred at a joint spring rotation
demand equal to 45% of the rotation limit.Thus,the model was
considered to provide accurate simulation of specimen ductility.
To further quantify the accuracy and precision of the model,the
error in the simulated displacement/drift and load at critical points
of the response history was computed.The critical points consid
ered were:initial yield of beam longitudinal steel (
D
y
,V
y
),maxi
mum strength (
D
max
,V
max
),and 10% loss of lateral load carrying
capacity (
D
90%
,V
90%
).For each specimen,the error in displace
ment/drift was computed
e
D
¼
D
measured
D
simulated
D
measured
ð19Þ
and the error in load was computed
e
V
¼
V
measured
V
simulated
V
measured
ð20Þ
Error data are reported in Table 3.
The data in Fig.8 and Table 3 indicate the following capabilities
of the model to simulate the ductile and limitedductility
specimens.
1.Initial stiffness is simulated accurately for both the ductile spec
imens (average displacement error of 8% at initial yield) and the
limitedductility specimens (average error of 4%).
2.The standard deviation of the error for simulation of initial stiff
ness for ductile and limitedductility specimens is approxi
mately 25%.This is considered to be acceptable for nonlinear
analysis of concrete components and is consistent with other
recently developed models for performancebased seismic
design and evaluation (e.g.,Berry et al.[37]).
3.The strength of ductile specimens is accurately and precisely
simulated with an average error of 8% and a standard deviation
of 7%.It should be noted that strength is determined by the
computed moment–curvature response history for the beam
section.
4.Displacements at maximum strength are not reported for the
nonductile specimens.Specimens typically exhibit very low
stiffness once beam(s) reach full yield strength;thus,the dis
placement at which maximum strength is achieved is not
meaningful.
5.For ductile specimens,the displacement at strength loss is sim
ulated with a high level of accuracy and moderate precision
(average error of 8%;standard deviation of 35%).For limited
ductility specimens,the accuracy of the model for simulation
of strength loss is much worse (average displacement error at
strength loss of 36%).This inaccuracy could be expected given
that the model is intended to simulate the onset of strength loss
in ductile specimens and was calibrated using data for these
specimens.
For brittle specimens,the data in Fig.8 and Table 3 support the
following conclusions.
1.Initial stiffness (same as stiffness at maximum load) for brittle
joints is,on average,fairly accurate (8% average error) but
lacks precision (55% standard deviation).
0
1
2
3
4
5
6
0
50
100
150
200
250
% Drift
(a)
Column Shear [kN]
Experimental
Conventional Hinge
Dual−Hinge
Dual−Hinge Limit
0
1
2
3
4
5
6
0
200
400
600
800
1000
% Drift
(b)
Column Shear [kN]
Experimental
Conventional Hinge
Dual−Hinge
Dual−Hinge Limit
Fig.7.Comparison of experimental response to simulated response using a
conventional beam hinge and the proposed dualhinge for specimens (a) PEER
0850 and (b) PEER 4150.
462 A.C.Birely et al./Engineering Structures 34 (2012) 455–465
2.The model provides an accurate and precise simulation of max
imumstrength (average error of 5%,standard deviation of 9%)
as well as accurate simulation of displacement at maximum
strength (average error of 8%).
3.The simulated displacement at maximum strength has a rela
tively high level of uncertainty,with a standard deviation of
53%;this was attributed to the simplicity of the model.
As previously discussed,the model provides accurate simula
tion of frame ductility and,thus,may be used in evaluation of
existing reinforced concrete frames to assess frame ductility under
seismic loading.To further explore the parameters that control
frame ductility,the computed joint shear stress demands for duc
tile and brittle joints were compared.The simulated maximum
joint shear stress,
s
model
max
,was computed as follows:
s
model
max
¼
M
max
L
þM
max
R
2
v
s
ð21Þ
where M
max
L
and M
max
R
are the maximummoments developed in the
left and the right beams,respectively.Values are provided in Ta
ble 1.The simulated maximum shear stresses are similar to those
computed directly from experimental data using Eq.(5);the aver
age difference between simulated and experimental stresses was
approximately 10%.Joint failure controlled the response of brittle
specimens;thus,for these specimens,the stress computed using
Eq.(21) is the joint shear strength (capacity).For brittle specimens,
simulated joint shear stress demands at failure ranged from
17.9
ﬃﬃﬃﬃ
f
0
c
p
psi to 23.7
ﬃﬃﬃﬃ
f
0
c
p
psi,with an average value of 20.5
ﬃﬃﬃﬃ
f
0
c
p
psi
and a coefﬁcient of variation of 7%.For ductile specimens,specimen
response was controlled by beam yielding and the joint shear de
mand,computed using Eq.(21),is less than the joint shear strength.
For ductile specimens,shear stress demand ranged from5.7
ﬃﬃﬃﬃ
f
0
c
p
psi
to 14.1
ﬃﬃﬃﬃ
f
0
c
p
psi with an average of 10.6
ﬃﬃﬃﬃ
f
0
c
p
psi and coefﬁcient of var
iation of 22%.Because there is no overlap in the ranges of simulated
shear stress for the ductile and brittle specimens,and given the rel
ative low coefﬁcient of variation on shear strength for brittle spec
imen,a strengthbased limit model for the joint spring could be
expected to provide reasonably accurate prediction of strength.
However,the strainbased model simulates the range of strengths
observed in the lab,provides accurate (5% error) and precise (9%
standard deviation) prediction of strength and provides accurate
prediction of the drop at maximum strength (8% error).
6.Summary and conclusions
A dualhinge lumpedplasticity beamelement was developed to
provide a practical model capable of simulating the nonlinear re
sponse of planar concrete frames.The dualhinge consists of two
0
2
4
6
0
100
200
300
400
% Drift
(f)
Column Shear [kN]
0
2
4
6
8
0
50
100
150
% Drift
(e)
Column Shear [kN]
0
2
4
6
0
100
200
300
400
500
% Drift
(d)
Column Shear [kN]
Experimental
Dual−Hinge
Dual−Hinge Limit
Experimental
Dual−Hinge
Dual−Hinge Limit
Experimental
Dual−Hinge
Dual−Hinge Limit
0
2
4
6
0
100
200
300
% Drift
(c)
Column Shear [kN]
0
2
4
6
0
20
40
60
80
% Drift
(b)
Column Shear [kN]
0
2
4
6
0
50
100
150
% Drift
(a)
Column Shear [kN]
Experimental
Dual−Hinge
Dual−Hinge Limit
Experimental
Dual−Hinge
Dual−Hinge Limit
Experimental
Dual−Hinge
Dual−Hinge Limit
Fig.8.Experimental (open circles) and model (thick line) envelopes for best and worst predictions.Best (a) brittle,MJ3,(b) ductile,PR3,and (c) limitedductility,OSJ.Worst
(d) brittle,PEER22,(e) ductile,HC,and (f) limitedductility,OSJ8.
Table 3
Error in proposed model for evaluation of experimental frame subassemblages.
Data subsets Normalized error Initial yield Max.load 10% loss
Stiffness Disp.Load Disp.Load
Ductile Average 8% – 8% 8% 2%
Stand.dev.25% – 7% 35% 9%
Limited ductility Average 4% – 11% 36% 5%
Stand.dev.24% – 9% 41% 11%
Brittle Average
1
8% 8% 5% – –
Stand.dev.
1
55% 55% 9% – –
1
For brittle joints,the stiffness at yield is the same as that at maximum load
A.C.Birely et al./Engineering Structures 34 (2012) 455–465
463
rotational springs in series to simulate the nonlinear ﬂexural re
sponse of the beam and the nonlinear response of the joint.The
beam spring response is determined from the moment–curvature
response of the beam cross section and an assumed plastic hinge
length.A rotation limit for the beamspring,which deﬁnes the on
set of strength loss,was determined using laboratory data from
frame tests in which beams yielded in ﬂexure.The rotation limit
was represented as an equivalent curvature value.The joint spring
response is deﬁned by a bilinear shear stress–strain relationship,
which is converted to a momentrotation response using geomet
ric transformations.Joint spring stiffnesses were determined using
laboratory data from frame subassemablage tests.Experimental
data from frame subassemblages exhibiting joint failure prior to
beamyielding were used to determine a joint spring rotation limit
at which strength loss initiates.This rotation limit was represented
as an equivalent shear strain value.
Frame models were constructed using standard beam column
elements and the dualhinge spring model to simulate the re
sponse of 45 experimental subassemblages.The interaction of
the beam and joint springs and the introduction of rotation limits
in both springs resulted in a model that accurately simulated the
ductile (beamcontrolled) or brittle (jointcontrolled) response ob
served in the laboratory.Results show that the model provides
accurate simulation of initial stiffness,strength and displacement
at strength loss for ductile specimens as well as accurate simula
tion of stiffness and strength for brittle specimens.
Acknowledgements
Support of this work was provided primarily by the Earthquake
Engineering Research Centers Program of the National Science
Foundation,under Award Number EEC97015668 through the
Paciﬁc Earthquake Engineering Research Center (PEER).Any
opinions,ﬁndings,and conclusions or recommendations expressed
in this material are those of the authors and do not necessarily
reﬂect those of the National Science Foundation.
Appendix A
A.1.Derivation of shear force transformation coefﬁcient
Fig.A.1 shows the reaction forces of a frame subassemblage
subject to a column shear V
c
.The forces acting on the joint are
shown in Fig.A.2.For a moment M
jf
at the face of the joint,the col
umn shear V
c
is:
V
c
¼
2M
jf
L
c
1
h
c
L
b
ðA:1Þ
and the moment produces the tension/compression couples:
T ¼ C ¼
M
jf
jd
ðA:2Þ
where jd is the beammoment armwith d equal to the distance from
the extreme compression ﬁber to the extreme tension ﬁber and j is
taken as 0.85.Fromthe column shear and tension/compression cou
ples,the joint shear force can be calculated as:
V
j
¼ C þT V
c
ðA:3Þ
Because the joint shear force V
j
is equal to the shear stress times the
area of the joint,the shear stress can be written as a function of the
moment M
jf
:
s
¼
V
j
A
j
¼
2M
jf
A
j
1
jd
1
L
c
1
h
c
L
b
0
@
1
A
ðA:4Þ
The moment at the joint face can then be expressed as a function of
the shear stress and joint geometry:
M
jf
¼
v
s
s
ðA:5Þ
v
s
¼
A
j
2
1
1
jd
1
L
c
1
h
c
L
b
0
B
B
@
1
C
C
A
ðA:6Þ
The transformation coefﬁcient
v
s
can be simpliﬁed to:
v
s
¼
A
j
jd 1
h
c
L
b
2 1
jd
L
c
h
c
L
b
ðA:7Þ
A.2.Derivation of shear strain transformation coefﬁcient
Fig.A.3 shows an idealization of a frame subassemblage with a
ﬁnite volume joint and centerline elements for the beams and col
umns framing into the joint.The joint is subject to a shear strain
c
,
resulting in displacements
D
b
and
D
c
of the beams and columns,
respectively.
D
b
¼
c
2
L
c
2
h
b
ðA:8Þ
D
c
¼
c
2
L
b
2
h
c
ðA:9Þ
In the proposed model,shown in Fig.A.4,the column is modeled
such that the column extends to the center of the joint,rather than
the face of the joint.The displacement
D
c
can be replicated using a
rotation of h
c
,measured from the vertical axis,at the center of the
joint:
L
c
1/2 (L
b
h
c
)
L
b
V
b
V
c
V
c
V
b
M
jf
Fig.A.1.Subassemblage forces and reactions.
V
c
V
j
C
T
T
C
h
c
h
b
jd
M
jf
M
jf
Fig.A.2.Forces acting on joint.
464 A.C.Birely et al./Engineering Structures 34 (2012) 455–465
h
c
¼
c
2
1
2h
b
L
c
ðA:10Þ
This rotation causes a displacement in the beams of:
D
h
c
b
¼ h
c
L
b
2
¼
c
2
L
b
2
h
b
L
b
L
c
ðA:11Þ
The hinge at the beam–joint interface must account for the differ
ence between the beam displacement of the physical system,
D
b
,
and the beamdisplacement due to the column rotation of the mod
eled system,
D
hc
c
.This is represented in equation form as:
D
b
D
h
c
b
¼
1
2
ðL
b
h
c
Þh
b
ðA:12Þ
Inserting Eqs.(A.9) and (A.11) into Eq.(A.12) and solving for h
b
pro
vides a relationship between the spring rotation h
b
and the shear
strain
c
that will ensure that the boundary conditions of the model
are satisﬁed.
h
b
¼
c
L
b
h
b
L
b
L
c
h
c
1
L
b
h
c
ðA:13Þ
h
b
¼
c
1
h
b
L
c
h
c
L
b
0
@
1
A
ðA:14Þ
References
[1] ACI Committee 318.Building code requirements for structural concrete (ACI
31808) and commentary (ACI 318R08).Farmington Hills,Michigan:
American Concrete Institute;2008.
[2] Beckingsale CW.Post elastic behavior of reinforced concrete beam–column
joints.Ph.D.thesis,Department of Civil Engineering,University of Canterbury,
Christchurch,New Zealand;1980.
[3] Otani S,Kobayashi Y,Aoyama H.Reinforced concrete interior beam–column
joints under simulated earthquake loading.US–New Zealand–Japan seminar
on design of reinforced concrete beam–column joints.California:Monterey;
1984.
[4] Park R,Milburn JR.Comparison of recent New Zealand and United States
seismic design provisions for reinforced concrete beam–column joints and test
results from four units designed according to the New Zealand Code.New
Zealand Nat Soc Earthquake Eng Bull 1983;16(1):3–24.
[5] Alire D.Seismic evaluation of existing unconﬁned reinforced concrete beam–
column joint.MSCE,Department of Civil Engineering,University of
Washington;2002.
[6] Walker S.Seismic performance of existing reinforced concrete beam–column
joints.MSCE,Department of Civil Engineering,University of Washington;
2001.
[7] Durrani AJ,Wight JK.Experimental and analytical study of beam to column
connections subjected to reverse cyclic loading.Technical report,Department
of Civil Engineering,University of Michigan;1982.
[8] Endoh Y,Kamura T,Otani S,Aoyama H.Behavior of RC beam–column
connections using lightweight concrete.Trans Jpn Concr Inst 1991;13:
319–26.
[9] Kitayama K,Otani S,Aoyama H.Earthquake resistant design criteria for
reinforced concrete interior beam–column joints.In:Paciﬁc conference on
earthquake engineering,Wairakei,New Zealand;1987.p.315–26.
[10] Meinheit DF,Jirsa JO.The shear strength of reinforced concrete beam–column
joints.Technical report,University of Texas at Austin;1977.
[11] Noguchi H,Kashiwazaki T.Experimental studies on shear performances of RC
interior column–beam joints.In:Tenth world conference on earthquake
engineering,Madrid,Spain;1992.p.3163–8.
[12] Park R,Ruitong D.A comparison of the behavior of reinforced concrete beam
column joints designed for ductility and limited ductility.Bull NewZealand
Nat Soc Earthquake Eng 1998;21(4):255–78.
[13] Oka H,Shiohara H.Tests on highstrength concrete interior beam–column
joints subassemblages.In:Tenth world conference on earthquake
engineering,Madrid,Spain;1992.p.3211–7.
[14] Scott M,Fenves G.Plastic–hinge integration methods for forcebased beam–
column elements.J Struct Eng,ASCE 2006;132(2):244–52.
[15] Neuenhofer A,Filippou FC.Geometrically nonlinear ﬂexibilitybased frame
ﬁnite element.J Struct Eng,ASCE 1998;124(6):704–11.
[16] ElMetwally S,Chen WF.Momentrotation modeling of reinforced concrete
beam–column connections.ACI Struct J 1988;85(4):384–94.
[17] Kunnath SK.Macromodelbased nonlinear analysis of reinforced concrete
structures.Structural engineering worldwide.Paper no.T1015.Oxford,
England:Elsevier Science,Ltd.;1998.
[18] Ghobarah A,Biddah A.Dynamic analysis of reinforced concrete frames
including joint shear deformation.Eng Struct 1999;21(11):971–87.
[19] Anderson M,Lehman D,Stanton J.A cyclic shear stress–strain model for joints
without transverse reinforcement.Eng Struct 2007;30(4):941–54.
[20] Altoontash A,Deierlein GD.A versatile model for beam–column joints.In:
ASCE structures congress,Seattle,WA;2003.
[21] Lowes LN,Altoontash A.Modeling the response of reinforced concrete beam–
column joints.J Struct Eng,ASCE 2003;129(12):1686–97.
[22] Mitra N,Lowes LN.Evaluation,calibration,and veriﬁcation of a reinforced
concrete beam–column joint.J Struct Eng 2007;133(1):105.
[23] Youssef M,Ghobarah A.Strength deterioration due to bond slip and concrete
crushing in modeling of reinforced concrete members.ACI Struct J
1999;96(6):956.
[24] Elmorsi M,Kianoush MR,Tso WK.Modeling bondslip deformations in
reinforced concrete beam–column joints.Can J Civil Eng 2000;27(3):490–505.
[25] Shin M,Lafave JM.Modeling of cyclic joint shear deformation contributions in
RC beam–column connections to overall frame behavior.Struct Eng Mech
2004;18(5):645–69.
[26] Uma SR,Prasad AM.Seismic evaluation of R/C moment resisting frame
structures considering joint ﬂexibility.In:13th World conference on
earthquake engineering conference proceedings;2004.
[27] Tajiri S,Shiohara H,Kusuhara F.A new macroelement of reinforced concrete
beam–column joint for elastoplastic plane frame analysis.In:Eighth national
conference of earthquake engineering,San Francisco,California;2006.
[28] Will GT,Uzumeri SM,Sinha SK.Application of ﬁnite element method to
analysis of reinforced concrete beam–column joints.In:Proceedings of
specialty conference of ﬁnite element method in civil engineering,Canada;
1972.
[29] Noguchi H.Nonlinear ﬁnite elements analysis of reinforced concrete beam–
column joints.Final report of IABSE colloquium;1981.p.639–53.
[30] Pantazoupoulou S,Bonacci J.On earthquakeresistant reinforced concrete
frame connections.Can J Civ Eng 1994;21(2):307.
[31] ACIASCE Joint Committee 352.Recommendations for design of beam–column
connections in monolithic reinforced concrete structures.Farmington Hills,
Michigan:American Concrete Institute;2005.
[32] Wight JK,MacGregor JG.Reinforced concrete:mechanics and design.Upper
Saddle River,NJ:Prentice Hall;2005.
[33] Pagni CA,Lowes LN.Fragility functions for older reinforced concrete beam–
column joints.Earthquake Spectra 2006;22(1):215–38.
[34] ASCE,Seismic rehabilitation of existing buildings (ASCE/SEI 4106).American
Society of Civil Engineers;2007.
[35] Park R,Priestly MNJ,Gill WD.Ductility of squareconﬁned concrete columns.J
Struct Div,ASCE 1982;108(ST4):135–7.
[36] Corley GW.Rotational capacity of reinforced concrete beams.J Struct Div,
ASCE 1966;92(ST5):121–46.
[37] Berry MP,Lehman DE,Lowes LN.Lumpedplasticity models for performance
simulation of bridge columns.ACI Struct J 2008;105(3):270–9.
Fig.A.3.Deformations of subassemblage due to joint shear strain.
Fig.A.4.Displacements and rotations of proposed model.
A.C.Birely et al./Engineering Structures 34 (2012) 455–465
465
Enter the password to open this PDF file:
File name:

File size:

Title:

Author:

Subject:

Keywords:

Creation Date:

Modification Date:

Creator:

PDF Producer:

PDF Version:

Page Count:

Preparing document for printing…
0%
Σχόλια 0
Συνδεθείτε για να κοινοποιήσετε σχόλιο