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Abstract
Reinforced concrete (RC) beam-column connections especially those
without transverse reinforcement in joint region can exhibit brittle
behavior when intensive damage is concentrated in the joint region
during an earthquake event. Brittle behavior in the joint region can
compromise the ductile design philosophy and the expected overall
performance of structure when subjected to seismic loading. Con-
sidering the importance of joint shear failure influences on strength,
ductility and stability of RC moment resisting frames, a finite
element modeling which focuses on joint shear behavior is present-
ed in this article. Nonlinear finite element analysis (FEA) of RC
beam-column connections is performed in order to investigate the
joint shear failure mode in terms of joint shear capacity, defor-
mations and cracking pattern. A 3D finite element model capable of
appropriately modeling the concrete stress-strain behavior, tensile
cracking and compressive damage of concrete and indirect modeling
of steel-concrete bond is used. In order to define nonlinear behavior
of concrete material, the concrete damage plasticity is applied to
the numerical model as a distributed plasticity over the whole
geometry. Finite element model is then verified against experi-
mental results of two non-ductile beam-column connections (one
exterior and one interior) which are vulnerable to joint shear fail-
ure. The comparison between experimental and numerical results
indicates that the FE model is able to simulate the performance of
the beam-column connections and is able to capture the joint shear
failure in RC beam-column connections.
Keywords
Finite element analysis, Concrete damage plasticity, RC beam-
column connection, Joint shear failure, Numerical model.
Finite Element Analysis of Reinforced Concrete Beam-Column
Connections with Governing Joint Shear Failure Mode
M.A. Najafgholipour a
S.M. Dehghan b
Amin Dooshabi c
Arsalan Niroomandi d
a Faculty of Civil and Environmental
Engineering, Shiraz University of
Technology, Shiraz, Iran,
najafgholipour@sutech.ac.ir
b Faculty of Civil and Environmental
Engineering, Shiraz University of
Technology, Shiraz, Iran,
smdehghan@sutech.ac.ir
c Graduate student of Civil and
Environmental Engineering, Shiraz
University of Technology, Shiraz, Iran
d Department of Civil and Natural
Resources, University of Canterbury,
Christchurch, New Zealand
http://dx.doi.org/10.1590/1679-78253682
Received 15.01.2017
In revised form 15.03.2017
Accepted 05.05.2017
Available online 26.05.2017
M.A. Najafgholipour et al. / Finite Element Analysis of Reinforced Concrete Beam-Column Connections with Governing Joint Shear… 1201
Latin American Journal of Solids and Structures 14 (2017) 1200-1225
1 INTRODUCTION
Beam-column connection in Reinforced Concrete (RC) frame is one of the most critical parts of the
structure which has important role in seismic performance of these buildings. When a beam-column
connection of a moment resisting frame is subjected to lateral forces, beam-column joint is prone to
joint shear failure due to high shear stress which appears in the joint panel as a result of opposite
sign moments on both sides of the joint core. This type of failure is not favorable because it has
undesirable effects on seismic performance of RC buildings, especially moment resisting frames. The
joint shear failure is a brittle type of failure which can adversely affect ductility of the RC frames.
Also, early occurrence of this failure causes the beams not to reach their ultimate flexural capacity.
Finally, the exposure of severe damage in the joint region may lead to total collapse of the building
when the structure is in the seismically active regions and designed according to non-seismic design
guidelines before the application of modern seismic design codes (ACI 318-14, ACI 352R-02, ASCE
41-13 and NZS3101). Therefore, this type of failure has undesirable effects on global seismic perfor-
mance of RC structures, especially moment resisting frames.
Considerable amount of experimental and analytical studies have been carried out to investigate
the seismic performance of RC beam-column joints subjected to lateral earthquake loading (Elsouri
and Harajli (2013), Lee et al. (2009), Masi et al. (2013) and Sasmal et al. (2013)) or development of
retrofitting techniques for vulnerable existing RC beam column joints (Eslami and Ronagh (2014),
Esmaeeli et al. (2014), Vecchio et al. (2016), Shwan and Abdul Razak (2016), Esmaeeli et al. (2015)
and Campione et al (2015)). Some of these researches were concentrated on shear resistance and
behavior of RC beam-column joints. As one of the first studies in this field Hanson and Connor
(1967) suggested a quantitative definition of joint shear. Based on their definition, joint shear was
determined from a free-body diagram, at mid-height of the joint panel. Pauley et al (1978) intro-
duced qualitative analytical shear resistance mechanisms of RC beam-column joints including a
concrete strut and a truss. The first mechanism is a diagonal compression concrete strut that trans-
fers the compression forces from the beam and column actions without the contribution of shear
reinforcement. The second mechanism is a stress mechanism that transfers bond forces from the
longitudinal bars utilizing horizontal and vertical joint shear reinforcement and concrete struts,
therefore the shear resistance in the joint region corresponding to concrete truss mechanism is at-
tributed to the bond capacity between reinforcement and concrete. These basic studies were fol-
lowed by experimental researches on RC joints aimed at evaluation of their shear behavior (De Risi
et al. (2016), Kotsovou and Mouzakis (2012) and Hakuto et al. (2000)). The tests were done on
different joint types such as exterior, interior, knee and T joints in different scales. In these experi-
mental studies the effects of different parameters such as geometrical properties of the joint panel,
concrete compressive strength, transverse joint reinforcement, beam longitudinal reinforcement,
existence of RC slab, column axial load and etc. were investigated on the joint shear strength and
load-displacement curves of the RC joints. Clyde et al (2000) and Pantelides et al (2002) carried out
seismic tests on some half-scale exterior beam-column joint specimens in order to determine perfor-
mance levels according to FEMA273 guidelines. Park and Mosalam (2012) tested four full-scale
exterior beam-column joints with orthogonal transverse beams and floor slabs under variable col-
umn axial loads. They investigated the behavior of unreinforced exterior joints subjected to cyclic
loading. Their investigation was focused on the effect of some crucial parameters, such as joint as-
1202 M.A. Najafgholipour et al. / Finite Element Analysis of Reinforced Concrete Beam-Column Connections with Governing Joint Shear…
Latin American Journal of Solids and Structures 14 (2017) 1200-1225
pect ratio and beam longitudinal reinforcement ratio on the shear strength and deformability of
exterior connection joints comparing their test results with the ASCE 41 seismic design recommen-
dations.
A series of researchers have collected the test results in literature and have developed empirical
and analytical relations to predict the RC joint shear capacity (Pauletta et al. (2015), Muhsen and
Umemura (2011), Park and Mosalam (2012), Lima et al. (2012), Wong and Kuang (2014), Wang et
al. (2012), Jeon et al. (2014) and Kassem (2016)). For example Kim and Lafave (2007) collected
experimental results of semi-static cyclic tests carried out by different researchers on various types
of RC beam-column connections (exterior, interior and corner joints). All specimens had a minimum
amount of joint confinement and the main objective of the study was to investigate the most influ-
ential parameters affecting the joint shear load-displacement behavior. The parameters investigated
were material property, joint panel geometry, confining reinforcement, reinforcement bond condition
and the column axial force. A design approach for RC beam-column joint was presented by
Kotsovou and Mouzakis (2012). The basic assumption of their model was that the load transferred
from the linear elements to the joint is mainly resisted by a diagonal strut mechanism. In a state of
the art article, Lima et al (2012) reviewed some of the analytical and empirical models introduced in
the last two decades for predicting shear strength of RC beam-column joints. Some simplified mod-
els are also developed for simulation of the joint shear behavior in nonlinear static and dynamic
analysis of RC frames (Sharma et al. (2011), Favvata et al. (2008) and Shayanfar et al. (2016)). For
instance, model developed by Shayanfar et al (2016) is one of the recent models for this purpose.
They have introduced a frame model to simulate the RC joints in nonlinear analysis of RC frames.
A series of numerical studies have been done on RC joints using FEM analysis. For example Ni-
roomandi et al (2014) performed numerical investigation of affecting parameters on the shear failure
of non-ductile exterior joints. According to their numerical results, two crucial parameters influenc-
ing the joint shear behavior were joint aspect ratio and beam longitudinal reinforcement ratio.
Although a majority of numerical studies have been carried out on RC beam-column connec-
tions (Parvin and Granata (2000), Mostofinejad and Talaeitaba (2006), Niroomandi et al (2010),
Mahini and Ronagh (2011), Masi et al. (2013) and Haach et al. (2008)) by FEM softwares such as
ANSYS, ABAQUS and DIANA, these studies have concentrated on simulating flexural behavior of
beams and columns adjacent to the joint region and are not focused on joint shear behavior of RC
connections, while in many cases the shear strength and behavior of joints control the overall re-
sponse of RC beam-column connections subjected to seismic actions. It should be noted that captur-
ing shear dominant failure mode in discontinuity region of connection is a very complicated numeri-
cal simulation problem.
The suggested modeling technique in this paper has been conducted by means of the commer-
cial FEA program ABAQUS and calibrated by modeling and analyzing experimentally tested exte-
rior and interior beam-column connections in which the governing failure mode during simulated
seismic actions on the specimens was the joint shear failure type. The comparison between numeri-
cal and experimental results, indicates the ability of the proposed method in simulating the govern-
ing joint shear behavior even at post peak phase.
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2 FINITE ELEMENT MODELING
Nonlinear finite element analysis of RC beam-column connections with joint shear failure as the
governing failure mode is performed using ABAQUS/Standard. Geometrical and material nonlinear-
ities are considered in order to properly simulate the behavior of connections and the joint shear
failure mode under seismic loading. In pre-processing stage, geometry and boundary conditions,
element types, material properties and nonlinear analysis solution are defined. Finite element simu-
lation of the complex behavior of concrete as a non-homogeneous and anisotropic material is a chal-
lenge in the finite element analysis of reinforced concrete structures and their components. Among
constitutive models defining concrete nonlinear behavior as a quasi-brittle material available in
ABAQUS, such as smeared and brittle cracking models, the Concrete Damage Plasticity (CDP) is
selected and introduced to the numerical model. The main and essential elements of any model
based on classical plasticity theory, which are the "yield criteria", "flow rule" and "hardening rule"
are all effectively considered in damage plasticity model. Numerical modeling of RC joint shear be-
havior calibrated by experimental results of other researchers is the main strategy of this study. To
verify the model, two RC beam-column connections with exterior and interior joint configurations
are selected and simulated. The numerical modeling is first calibrated by experimental results of a
quasi-static cyclic test on a half scale exterior beam-column connection which is previously done by
Clyde et al (2000) and then applied to an interior full scale RC beam-column connection which is
tested by Hakuto et al (2000) under simulated cyclic load. In both cases the ultimate failure of spec-
imens were attributed to the joint shear failure.
2.1 Elements
3D 8-noded hexahedral (brick) elements having 3 degrees of freedom in each node (translations in
X, Y and Z directions) are utilized for modeling concrete elements with reduced integration
(C3D8R) to prevent the shear locking effect. In order to model reinforcements, 2-noded truss ele-
ments (T3D2) having 3 degrees of freedom in each node (translations in X, Y and Z directions of
global coordinates system) are used. The embedded method with perfect bond between reinforce-
ment and surrounding concrete is adopted to properly simulate the reinforcement-concrete bonding
interaction. It is notable that the effects usually associated with reinforcement-concrete interface,
such as bond slip and dowel action are modeled indirectly by defining "tension stiffening" into the
reinforced concrete model to approximately simulate load transfer across cracks through the rebar
(ABAQUS user’s manual (2014)).
2.2 Analysis Approach
Finite Element Analysis (FEA) of the connection joint specimens is performed in a nonlinear static
analysis format and the analysis procedure considers both material and geometric nonlinearities. In
a nonlinear analysis, the total specified loads acting on a finite element body will be divided into a
number of load increments. At the end of each increment the structure is in approximate equilibri-
um and the stiffness matrix of structure will be modified in order to reflect nonlinear changes in
structure's stiffness.
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ABAQUS/Standard uses the Newton-Raphson method to obtain solutions for nonlinear prob-
lems. Newton–Raphson equilibrium iterations provide convergence at the end of each load incre-
ment within tolerance limits for all degrees of freedom in the model. Residual load vector, which is
the difference between the internal forces (the loads corresponding to the element stresses) and the
external applied loads are analyzed again by Newton–Raphson approach. Subsequently, the pro-
gram carries out a linear solution using residual loads and considering initial stiffness of structure, in
order to check the convergence criteria. If the residual load vector is less than the current tolerance
value, the external and internal forces are in equilibrium (this tolerance value is set to 0.5% of an
average force in the structure, averaged over time). If convergence criteria are not satisfied, the
stiffness matrix is updated, the residual load vector is re-calculated and a new solution is achieved.
ABAQUS/Standard also checks that the displacement correction is small relative to the total in-
cremental displacement and it is regularized in such a way that both convergence checks (loads and
displacements) must be satisfied before a solution considered to be converged for that load incre-
ment.
2.3 Concrete Damage Plasticity Model
Concrete damage plasticity is used as the governing concrete material plasticity model over the
whole geometry of the specimens. The model is a plasticity-based model which is developed using
concepts of continuum damage mechanics and the application of scalar damaged elasticity in com-
bination with isotropic tensile and compressive plasticity to properly represent the inelastic behavior
of concrete (Lubliner et al. (1989)). The main two failure mechanisms of the concrete material are
tensile cracking and compressive crushing according to fundamental assumptions of damage plastici-
ty model. The evolution of the yield (or failure) surface and the degradation of elastic stiffness in
damage plasticity model are controlled by two hardening variables which are tensile and compres-
sive equivalent plastic strains (
p
l
t
and
p
l
c
). Increasing values of the hardening variables leads to
the initiation of micro-cracking and progressive propagation of cracks or the occurrence of crushing
in the concrete material. The mentioned yield surface was then modified by Lee and Fenves (1998)
in order to take into account the different evolution of concrete tensile and compressive strengths.
The current yield surface is defined in the form of effective stresses according to Eq. (1):
max max
1
,3
1
pl pl pl
cc
Fqp
(1)
Where ()
2
x
x
x
denotes the Macaulay bracket function. In Eq. 1, p is the effective hydro-
static pressure stress and q is the Mises equivalent effective stress. These two parameters are stress
invariants of the effective stress tensor which are used by the yield surface and plastic flow potential
function (Lubliner et al (1989)). max
ˆ
is the algebraically maximum eigenvalue of the deviatory part
of effective stress tensor (
).The parameters
,
and
in Eq. 1 are dimensionless material con-
stants. Further details on how these parameters affect the damage plasticity model have been dis-
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cussed by Lubliner et al (1989) and a summary of their definitions is presented in subsequent para-
graphs. The function
is defined as:
11
pl
cc
pl
pl
tt
(2)
Here
p
l
cc
σε
and
p
l
tt
σε
are effective compressive and tensile cohesion stresses, respectively. In
biaxial compression with the maximum principal effective stress equal to zero ( max
ˆ0
), the yield
surface function of Eq.1 reduces to Drucker-Prager yield condition in which the only parameter
needed to define the yield surface is
. The parameter
then can be obtained by comparing the
initial equibiaxial and uniaxial compressive yield stresses 0b
and 0c
according to Eq.3:
0
0
0
0
1
21
b
c
b
c
(3)
Typical values of the ratio
00bc
for concrete based on experimental results are reported
in the range from 1.10 to 1.16, yielding values of
between 0.08 and 0.12 (Lubliner et al (1989)).
Finally, the parameter
determines the shape of the yield surface in the plasticity model and en-
ters the yield function only for stress states of triaxial compression, when
0. The
coeffi-
cient can be determined by comparing the yield conditions along the tensile and compressive merid-
ians. This coefficient is obtained according to Eq.4:
31
21
c
c
K
K
(4)
The coefficient Kc is the ratio of the hydrostatic effective stress in tensile meridian to that on
the compressive meridian when the maximum principal stress is negative. This coefficient defines
the shape of yield surface in the deviatory plane. The shape of deviatory plane was first assumed to
be circular as in the classic Drucker-Prager strength hypothesis ( 1
c
K
). The CDP model suggests
to assume default value of 23
c
K
based on triaxial stress test results. Figure 1 shows a plane
stress cross section of the yield surface in principal stress space together with the yield surface cross
section on the deviatory plane corresponding to two values of c
K
representing different strength
criteria.
1206 M.A. Najafgholipour et al. / Finite Element Analysis of Reinforced Concrete Beam-Column Connections with Governing Joint Shear…
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Figure 1: The yield surface in: a) plane stress cross section; b) deviatory plane (Lubliner et al (1989)).
The flow rule in concrete damage plasticity is introduced to the model as an essential element of
plasticity theory utilizing non-associated flow potential function
G
. The CDP model uses Druck-
er-Prager hyperbolic function as non-associated flow potential function according to Eq.5.
2
2
0tan tan
t
Gqp
(5)
The equation involves three material parameters. The first one is dilation angle
which is a
concrete performance characterizing parameter when it is subjected to triaxial compound stress
state. The next parameter ∈ is eccentricity which adjusts the shape of hyperbola in the plastic po-
tential flow function. The eccentricity parameter is a small positive value and can be estimated as
the ratio of concrete tensile strength to its compressive strength. The default value of eccentricity
parameter is considered to be equal to 0.1. Finally, the last parameter of plastic flow potential func-
tion is the concrete initial uniaxial tensile strength 0t
.
Damage in CDP model is associated with the failure mechanisms consist of concrete cracking
and crushing, therefore the occurrence of damage leads to reduction in elastic stiffness. To introduce
damage to the model Eq.6 is applied.
0
1 1 : pl
ddE
(6)
Where 0
E is the initial (undamaged) elastic stiffness of the material and the operator (:) denotes
the product of the related tensors. Based on the scalar-damage theory, the stiffness degradation is
isotropic and defined by a degradation variable d. Damage is defined in both tensile and compres-
sive states t
dand c
das functions of plastic strains. Damage parameter can take values in the range
from zero (corresponding to undamaged material) to one (corresponding to fully damaged material).
Visco-plastic regularization of the concrete constitutive equations is defined in the numerical
model using a generalization of Duvaut-Lions regularization. As a result, the plastic strain tensor
M.A. Najafgholipour et al. / Finite Element Analysis of Reinforced Concrete Beam-Column Connections with Governing Joint Shear… 1207
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and consequently the stiffness degradation variable were modified. Visco-plastic strain rate tensor
(
p
l
) is defined according to Eq.7:
1
μ
p
lplpl
(7)
Here
p
l
is the evaluated plastic strain and the visco-plastic stiffness degradation variable d
is
defined according to Eq.8:
1
μ
ddd
(8)
Where d is the evaluated degradation variable. The stress-strain relation based on viscoplastic
model is defined according to Eq.9:
0
1 : pl
dE
(9)
2.4 Material Model
2.4.1 Concrete Material Modeling
Uniaxial stress-strain behavior of concrete is simulated utilizing Hognestad type parabola (Hognes-
tad (1951)). The uniaxial stress-strain behavior of concrete can be categorized into three main do-
mains. The first one represents the linear-elastic branch which continues to reach the stress level of
co
that is taken as 0.4
co c
f
. The second stage shows the hardening part of the concrete uni-
axial compressive stress-strain behavior which describes the ascending branch of the stress–strain
relationship reaching to the peak load at the corresponding strain level 02cc
f
E
. The last part
of concrete uniaxial compressive stress-strain relationship attributes to the post-peak softening be-
havior and therefore represents the initiation and progression of compressive damage in the concrete
material until the ultimate compressive strain u
. Figure 2 shows the compressive stress-strain be-
havior of concrete which is introduced to the presented numerical model.
Figure 2: Concrete uniaxial compressive stress-strain diagram.
1208 M.A. Najafgholipour et al. / Finite Element Analysis of Reinforced Concrete Beam-Column Connections with Governing Joint Shear…
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The uniaxial stress-strain behavior of concrete in tension consists of two parts. As shown in
Figure 3, the first part is a linear elastic behavior up to concrete tensile strength 0t
. The second
phase initiates together with crack occurrence and its propagation in concrete material under ten-
sion demonstrating a descending branch in the uniaxial tensile stress-strain diagram. The behavior
in this phase is modeled by a softening procedure which can be modeled using linear, bilinear or
nonlinear stress-strain relationships (Belarbi and Thomas (1994)). According to analysis assump-
tions in this study the linear behavior is applied to the model. Figure 3 demonstrates details of ten-
sile softening assumptions in the presented model.
Figure 3: Concrete uniaxial tensile stress-strain behavior and its softening branch assumptions.
The ultimate tensile strength of concrete is estimated by Eq.10 (Genikomsou and Polak (2015)
and Wang and Vecchio (2006)).
0.33 (MPa )
tc
ff
(10)
Damage is defined both for uniaxial tension and compression during softening procedure in con-
crete damage plasticity model. Damage in compression occurs just after reaching to the maximum
uniaxial compressive strength corresponding to strain level 0
. The degradation of elastic stiffness in
softening regime is characterized by two damage variables, dt and dc corresponding to tensile and
compressive damage, respectively, which are assumed to be functions of the plastic strains. Tensile
and compressive damage in concrete damage plasticity model in the presented numerical model is
assumed to be according to equations and diagrams of Figure 4.
2.4.2 Steel Reinforcement Modeling
The uniaxial tensile stress–strain behavior of reinforcement was assumed to be elastic with conven-
tional Young’s modulus and Poisson’s ratio. The plastic behavior is also modeled including yield
stress and corresponding plastic strain. Properties of plastic phase is defined to the model using
bilinear behavior.
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Figure 4: Definition of damage parameter in CDP model:
a) Uniaxial tensile damage; b) Uniaxial compressive damage.
Most finite element studies of RC structures do not consider bond-slip of reinforcing steel and
the inherent interaction between reinforcement and concrete in RC members. The post-failure be-
havior for direct straining is modeled with tension stiffening in CDP model, which helps to define
the strain-softening behavior for cracked concrete, therefore in order to consider concrete-
reinforcement interaction such as bond slip, an indirect approach which defines "tension stiffening"
into the reinforced concrete finite element model is applied as mentioned before in this study. Two
different approaches to introduce tension stiffening to the numerical model are the application of
post-failure stress-strain relation and the definition and application of fracture energy cracking crite-
rion. The presented numerical model utilizes the first method (post-failure stress-strain relation) to
appropriately introduce the tension stiffening behavior of reinforced concrete, the method in which
post-failure properties of reinforced concrete is defined by giving the post-failure stress as a function
of cracking strain ( ck
t
). Cracking strain is defined as the total strain minus the elastic strain corre-
sponding to the undamaged material according to Eq.11 (Lubliner et al. (1989))
ck el
ttot
(11)
The elastic strain of undamaged material ( el
ot
) is defined according to Eq.12:
0
el
ot t E
(12)
When unloading data are available the tension softening data can be provided in terms of ten-
sile damage-cracking strain relationship. A linear tension stiffening data based on uniaxial tensile
damage definition in CDP model is introduced to the numerical model of the beam-column connec-
tions according to the diagram of Figure 5.
1210 M.A. Najafgholipour et al. / Finite Element Analysis of Reinforced Concrete Beam-Column Connections with Governing Joint Shear…
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Figure 5: Tension stiffening data introduced to the numerical model in terms of
tensile damage-cracking strain linear relationship.
3 VERIFICATION OF THE FINITE ELEMENT MODELING
An exterior half-scale beam-column connection namely test#2 conducted by Clyde et al (2000) and
a full scale interior beam-column connection namely unit-01 tested by Hakuto et al (2000) are cho-
sen to validate the numerical model. Both specimens were designed to have joint shear failure as
their governing failure mode. To enforce shear failure in the joint, the beam longitudinal reinforce-
ment of exterior beam-column connection was increased to prevent yielding and early degradation
of the beam. To represent non-ductile detail of the joint, no transverse reinforcement was consid-
ered within the joint panel in both cases (exterior and interior beam-column connections). It is
worth mentioning that the interior beam-column connection tested by Hakuto et al (2000) is identi-
cal to an interior beam- column connection of an existing RC building constructed in New Zealand
before mid-1960s.
3.1 Boundary Conditions, Loading, Dimensions and Details
Geometry and reinforcement details of the exterior and interior joints are shown in Figure 6. A
schematic representation of the loading apparatus regarding both test connections is shown in Fig-
ure 7. According to boundary conditions applied to the exterior beam-column connection specimen,
column was simply supported at both ends and the compressive axial load was applied to the col-
umn’s end using a hydraulic cylinder. The column axial compressive load is applied to the top of
column corresponding to 10% of the compressive strength of concrete in the first step. The lateral
load was applied in second step to the beam tip through a loading collar in a quasi-static cyclic
manner. The interior beam-column connection had a different boundary condition and loading pro-
cedure. The pinned connection at the bottom of the column restrained its lateral displacement, but
the column was allowed to rotate and elongate. The ends of the beams were connected by pin-ended
steel members to the reaction floor, so that the ends of the beams were free to rotate and to trans-
late horizontally but restrained vertically. No axial load was applied to the column during the test
since according to Hakuto et al (2000) the column axial pressure may contribute to the load bearing
capacity of the joint making an unfavorable condition. Loading was applied to the interior beam-
M.A. Najafgholipour et al. / Finite Element Analysis of Reinforced Concrete Beam-Column Connections with Governing Joint Shear… 1211
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column connection in one step only (lateral load). Cyclic horizontal loading was applied to the top
end of the columns of the interior test unit using a double acting hydraulic jack.
Figure 6: Specimen dimensions and details: a) Exterior beam-column connection
(Clyde et al (2000)); b) Interior beam-column connection (Hakuto et al (2000)).
Figure 7: Test setup of beam-column connection specimens: a) Exterior beam-column connection
(Clyde et al (2000)); b) Interior beam-column connection (Hakuto et al (2000)).
1212 M.A. Najafgholipour et al. / Finite Element Analysis of Reinforced Concrete Beam-Column Connections with Governing Joint Shear…
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3.2 Material Properties of the Test Specimens
The measured uniaxial concrete compressive strength of the test specimens, the yield strength and
the ultimate tensile strength of reinforcement used in the tests, are reported in Table 1.
Connection
specimen
Concrete
compressive
strength
c
f
(MPa)
Reinforcement
type
Bar
size
Yield
strength
y
F(MPa)
Ultimate tensile
strength
u
F(MPa)
Exterior
connection
(Test #2)
46.2
Beam
longitudinal #9(28.65mm) 454.4 746
Column
longitudinal #7(22.23mm) 469.5 741.9
Stirrups #3(9.53mm) 427.5 654.3
Interior
connection
(Test 01)
41
Beam and
Column
longitudinal
D24(24mm) 325 Not
reported
Stirrups R6(19mm) 339 Not
reported
Table 1: Material properties of the test specimens.
4 FINITE ELEMENT ANALYSIS
4.1 Elements
A uniform mesh size of 40 mm is chosen for the concrete elements over the whole geometry in both
exterior and interior connection specimens as shown in Figure 8. The same size for reinforcement
mesh is also adopted for steel bars. With this configuration, the exterior RC connection specimen
has 11144 elements and 13414 nodes and the interior connection has 17414 elements and 21088
nodes. Details regarding element types are presented in Table 2.
Connection specimen Element type Element shape Geometrical order Number of
elements
Exterior
connection
(Test#2)
C3D8R Hexahedral Linear 9536
T3D2 Line Linear 1608
Interior
connection
(Test 01)
C3D8R Hexahedral Linear 15328
T3D2 Line Linear 2086
Table 2: The number and type of elements in the finite element model.
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Figure 8: Modeled specimens; a) Concrete element mesh of exterior connection; b) Reinforcement details of exterior
connection; c) Concrete element mesh of interior connection and d) Reinforcement details of interior connection.
4.2 Model Geometry and Boundary Conditions
Restraints were defined at both top and bottom surfaces of the specimen's column in the exterior
test specimen according to boundary conditions addressed in the test setup. Restraints are also de-
fined in the interior connection specimen according to the test setup boundary conditions both on
beam and column tip surfaces. Details regarding to the geometry and boundary conditions of the
RC exterior and interior beam-column connections which are applied to the finite element models
are illustrated in Figure 9.
Loading is introduced to the model in two separate steps in exterior beam-column connection.
The column compressive axial load is applied to the column top surface in the first step which re-
mained constant during the analysis procedure.
The second step corresponds to the monotonic lateral loading of the specimen by applying the
lateral displacement at the beam's end surface. However for the interior connection, only one step of
load that is related to the lateral loading of the specimen is applied at column's top surface.
1214 M.A. Najafgholipour et al. / Finite Element Analysis of Reinforced Concrete Beam-Column Connections with Governing Joint Shear…
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Figure 9: Simulated boundary conditions and loading of the specimens:
a) Exterior beam-column connection; b) Interior beam-column connection.
4.2 Material Parameters and Calibration of the Model
The numerical model is calibrated on the experimental test results of exterior connection specimen
as a reference experiment. The calibrated numerical model is then utilized to conduct nonlinear
finite element analysis on the interior connection specimen (Test01).
Concrete material parameters used in the presented analysis consists of concrete Young’s modu-
lus of elasticity (E0), Poisson’s ratio (ν) and concrete compressive and tensile strengths. The poi-
son’s ratio value for concrete material is considered to be equal to 0.2. The concrete damage plastic-
ity input parameters which were discussed in section 2.3 are considered in the plasticity model as
presented in Table.3.
Plasticity Parameters Notation Parameter’s Value
considered in the model
Dilation angle 35° (calibrated)
Shape factor 0.667 (default value)
Stress ratio
1.16 (default value)
Eccentricity 0.1 (default value)
Table 3: Concrete damage plasticity input parameters.
The uniaxial tensile stress–strain behavior of reinforcement was assumed to be elastic with
Young’s modulus ( 5
210
s
E
MPa ) and Poisson’s ratio ( 0.3
s
). The plastic behavior is also
modeled including yield stress and corresponding plastic strain. Properties of plastic phase is defined
to the model using bilinear behavior. Figure 10 illustrates the typical stress-strain relationship of
reinforcement introduced to the numerical model. Typical reinforcement properties are also present-
ed in Table 4 for both exterior and interior beam-column connections.
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Figure 10: Typical uniaxial stress-strain behavior of reinforcements introduced to the numerical model.
Connection
specimen
Reinforcement
type
Yield
stress
Yield
strain
Ultimate
strength
Ultimate
strain
Exterior
connection
(Test#2)
Beam
Longitudinal bar 454.4 0.00227 746 0.09
Column
longitudinal bar 469.5 0.00235 741.9 0.09
Stirrups 427.5 0.00214 654.3 0.11
Interior
connection
(Test01)
Beam & Column
longitudinal bar 325 0.001625 396.5 0.10
Stirrups 339 0.001695 413.6 0.11
Table 4: Typical stress-strain properties of steel reinforcements.
In order to investigate the role of parameters in constitutive equations of damage plasticity
model and to calibrate the model, effects of different logical values of the dilation angle and the
viscosity parameter and mesh sensitivity analysis of the model are presented.
The dilatancy of concrete material represents the occurrence of volume expansion when the ma-
terial is subjected to triaxial stress state and the consequent inelastic strain. Internal actions on the
connection joint specimens under simulated lateral loads induce high shear stress in the joint core,
an example of which compound stress state leads to considerable dilatancy sensitiveness of the ana-
lytical model. A sensitivity analysis on dilation angle as a concrete material parameter is performed
to investigate the parameter influence on lateral load-displacement response of the reference speci-
men. Figure 11 shows that by increasing the value of dilation angle, the displacement capacity and
the ultimate failure load of the connection are increased subsequently.
A range between 31° to 42° of the dilation angle parameter is recommended for concrete materi-
al according to series of fundamental studies performed by different authors (Lee and Fenves
(1988), Wu et al. (2006) and Voyiadjis et al. (2009)). As shown in Figure 11 a value of 35° can rea-
sonably capture the lateral load-deformation curve and the failure mode.
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Figure 11: Lateral load-displacement response of FEA for different values of dilation angle.
The influence of visco-plastic regularization on the CDP constitutive equations through the in-
troduction of viscosity parameter was another source of calibration process of the finite element
analysis in this study. Defining visco-plastic regularization to the numerical model was first pro-
posed as a method to improve the rate-dependent plastic damage model that brings uniqueness to
the incremental stress field in the constitutive equations. It was shown that while the appropriate
choice of viscosity parameter and defining it to the model provides relaxation time for visco-plastic
system of equations to overcome convergence difficulties in the softening regime, the influence of
considering different values of viscosity parameter in monotonic loading cases do not impose consid-
erable amount of changes in the behavior of concrete material under uniaxial tension and compres-
sion (Lee (1996)). It was shown that best results are obtained by small values of viscosity parameter
compared to the pseudo time of analysis in the study performed by Lee (1996). Series of guidelines
and recommendations are suggested by various researchers in order to consider best values of vis-
cosity parameter in the CDP model depending on the type of predominant internal actions which
are active and the degree of nonlinearity that involved in different problems (Wosatko et al (2015),
Genikomsou and Polak (2015), Wei Ren et al (2015)).
The influence of considering additional relaxation time for the visco-plastic system on the analy-
sis results using different values of viscosity parameter is shown in Figure 12.
According to the diagram of figure 12, the differences in the responses appear only at the soften-
ing and to some extent at the hardening regions. For constant element mesh size, when the viscosi-
ty parameter was taken as a relative small value of 0.007985, the calculation procedure of the nu-
merical results gave us the most accurate response in comparison to the experimental results. Also
the concrete cracking pattern and the nature of joint shear failure were consistent with the test
results.
In order to analyze mesh size sensitivity of the numerical model in this study, the finite element
analysis results of the exterior connection specimen with 40 mm and 25 mm element mesh sizes are
presented in terms of lateral load-displacement curve in Figure 13. Although effects associated with
mesh size and strain localization in the presented numerical model are within the margins of error
expected for most numerical simulations based on plasticity models, slight differences are observed
in the peak lateral loads and displacements.
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Figure 12: Lateral load-displacement response of FEA for different values of the viscosity parameter.
Slight mesh size dependency is available in most distributed plasticity models which consider
strain softening phenomenon in the constitutive equations (Genikomsou and Polak (2015)). It
should be investigated that the mesh size dependency does not considerably affect the overall re-
sponse of the specimen in FEA. It is notable that best results are obtained when the coarse mesh
(40 mm) is used which may be due to the fact that most damage processes, causing concrete crack-
ing propagation usually involve length scales in the order of two to three dominant aggregate sizes
of the base concrete material (Bazant (1986)).
Figure 13: Lateral load-displacement response of RC exterior beam-column
connection for 40mm and 25mm mesh sizes.
Among the various methods to reduce the mesh size dependency due to the strain-softening lo-
calization in limited number of elements, one method is to introduce the characteristic internal
crack length at the softening branch of the stress-strain relationship into the constitutive model.
The other method is the introduction of concrete visco-plastic regularization to the numerical mod-
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el. The second approach is used in this study but the model is still slightly mesh size dependent
however the influence of mesh size dependency on pushover lateral load-displacement behavior of
the studied beam-column connections is within the acceptance limits for numerical simulations.
5 FINITE ELEMENT ANALYSIS RESULTS
The FEA results of the RC beam-column connections subjected to lateral loading is presented in
terms of force-displacement curves, ultimate loads and displacements and cracking pattern which
are monitored at different key points of shear behavior of the connection joints. Comparison be-
tween force-displacement curves predicted by simulation and experimental results of exterior and
interior connection specimens are presented in Figure 14 and Figure 15, respectively. Figure 14a,
and Figure 15a, show the finite element analysis results of force-displacement curves which are
compared with the envelope curve of the cyclic loading response shown in Figure 14b and Figure
15b which are obtained by experimental works of Clyde et al(2000) and Hakuto et al(2000) for ex-
terior and interior connection test specimens respectively.
Figure 14: Lateral load-displacement diagrams of exterior Test#2 specimen: a) Comparison between FEA and test;
b) Test results of exterior specimen (Test#2) obtained by Clyde et al(2000) experimental study.
Figure 15: Lateral load-displacement diagrams of interior Test unit 01: a) Comparison between FEA and test;
b) Test results of interior specimen (Test unit 01) obtained by Hakuto et al (2000) experimental study.
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Peak lateral loads and displacements predicted by the numerical simulation and reported by
experimental test results are presented in Table 5.
Connection
specimen
Experimental results Finite element analysis results
Peak lateral
load (kN)
Displacement at peak
lateral load (mm)
Peak lateral
load (kN)
Displacement at peak
lateral load (mm)
Exterior connection
(Test#2) 267 23.77 268.82 24.5
Interior connection
(Test 01) 89 60.5 87.38 58.97
Table 5: Ultimate lateral loads and displacements obtained from FEA and experiments.
The finite element analysis shows that the initial response is slightly stiffer than the test results.
This may be due to effects of some assumed variables such as the choice of concrete tensile and
compressive properties, or the uncertainties often involved with experimental efforts such as proba-
ble existence of material deficiencies and also the inherent differences which exist between response
obtained by cyclic loading of the experiment and monotonic loading in presented analysis may also
contribute in limited stiffer initial response of FEA comparing to testing.
As shown in Figure 14 and Figure 15, three key points of shear behavior that are specified on
the FEA resultant lateral load-displacement diagrams for both exterior and interior connection
specimens, reasonably match with reported experimental results by Clyde et al (2000) and Hakuto
et al (2000), respectively. Point A displays measurable beam flexural and joint shear cracking in
exterior specimen and first diagonal shear cracking of the joint core in the interior specimen. Point
B displays the ultimate strength of both exterior and interior specimens and finally point C is relat-
ed to the end of FEA procedure attributed to 25% drop of ultimate strength in exterior connection
specimen test and 30% drop of ultimate strength in the interior connection specimen test.
Accuracy of the numerical model in capturing joint shear behavior is monitored through four
output parameters which effectively define overall joint shear behavior including concrete tensile
and compressive damage, reinforcement output stress and joint cracking pattern. The CDP model
simulates the nonlinear behavior of concrete both in tension and compression appropriately. A sig-
nificant and challenging aspect of damage plasticity model application is the appropriate definition
of damage in the constitutive model. The value of maximum principal plastic strain is the main
indicator of cracking initiation in concrete damage plasticity model. Cracks initiate when maximum
principal plastic strain is positive and the orientation of cracks is considered to be perpendicular to
the maximum principal plastic strains, therefore in order to visualize the direction of cracking, the
maximum principal plastic strains output is investigated. Figure 16 and Figure 17 display concrete
damage both in tension and compression together with reinforcement Von-Mises stress output and
joint cracking pattern at three introduced key points of joint shear behavior related to exterior
test#2 and interior test unit-01, respectively.
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Figure 16: FEA outputs of the exterior beam-column connection specimen
at key points of shear behavior: a) Point A; b) Point B; c) Point C.
Exterior beam-column connection specimen experienced beam tensile damage at point A as
shown in Figure 16a. The concrete tensile damage extended to the vicinity of the joint and the
beam longitudinal bars were placed under tension. No yielding of longitudinal beam bars and also
no compressive damage in the joint core were observed at this stage. Measurable flexural cracks in
the beam and limited shear cracks in the joint were the main source of slight stiffness change at
point A which confirms reported experimental observations at this stage. Limited shear cracking of
the joint rapidly spread to the whole joint at point B as shown in Figure 16b. Concrete tensile
damage is developed to the joint region followed by yielding of the beam longitudinal reinforcement.
The post peak behavior of the connection specimen begins at this point due to compressive damag-
ing of the concrete diagonal strut.
Finally at the end of FEA procedure (point C) corresponding to 25% drop in ultimate strength,
the failure of the connection specimen was occurred by crushing of the concrete diagonal strut in
the joint region. Figure 16c shows the ultimate joint cracking pattern at point C. The ultimate fail-
ure of the beam-column connection specimen is attributed to the joint shear failure as in the exper-
iments. Cracking pattern in the joint region obtained by FEA matches the cracking propagation
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pattern observed in the test. The diagonal cracks in the joint region are propagated to vicinity of
the column.
Figure 17: FEA outputs of the interior beam-column connection specimen
at key points of shear behavior: a) Point A; b) Point B; c) Point C.
Joint shear behavior described for the exterior beam-column connection specimen is also ob-
served in the numerical output of the finite element analysis performed on the interior beam-column
connection. Interior beam-column connection specimen experienced beam tensile damage at beam
tension sides together with diagonal concrete tensile damage in the joint core at point A as shown
in Figure 17a. Consequently, top of left side beam, bottom of right side beam and longitudinal bars
of column on tension side were subjected to tension. No yielding of longitudinal beam bars and also
no compressive damage in the joint core were observed similar to the exterior joint numerical out-
put at this stage. Limited flexural cracks in the column and adjacent beams with diagonal tension
cracking of the joint core were the main source of stiffness change at point A which is reported in
experimental observations and obtained by the FEA outputs. Diagonal tension cracking of the joint
rapidly spread to the vicinity of the column at point B as shown in Figure 17b. Concrete tensile
damage is developed, and then is followed by yielding of the column and beam longitudinal rein-
1222 M.A. Najafgholipour et al. / Finite Element Analysis of Reinforced Concrete Beam-Column Connections with Governing Joint Shear…
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forcement in the joint panel zone. The post peak behavior of the connection specimen also begins at
this point due to the initiation of compressive damage in the concrete diagonal compressive strut.
Finally at the end of FEA procedure of the interior specimen (point C) corresponding to 30%
drop in its ultimate strength, the failure of the connection specimen was occurred by crushing of the
concrete diagonal strut in the joint region. As shown in Figure 17c the ultimate failure of the interi-
or beam-column specimen is attributed to the joint shear failure as happened in the experiments.
The numerical model shows good agreement between cracking pattern in the joint region obtained
by FEA and the cracking propagation pattern observed in the test. Figure 18 presents a comparison
between the cracking patterns of the joint region at ultimate strength (point B) for both exterior
and interior specimens obtained by the FEA output and observed in the test results.
Figure 18: Comparison between joint shear cracking pattern obtained by the FEA output and observed in the test
results at point B in: a) Exterior connection specimen; b) Interior connection specimen.
6 CONCLUSIONS
The study presented herein has endeavored to propose a suitable numerical model that describes
the nonlinear shear behavior of reinforced concrete beam–column connections including poorly de-
signed and detailed interior and exterior joints. Nonlinear finite element analysis of two RC beam-
column connections with exterior and interior configurations is performed in order to capture the
joint shear failure as the connection's governing failure mode when subjected to quasi static type
lateral load. Concrete damage plasticity material model is applied to the numerical procedure as a
distributed plasticity over the whole geometry of the specimens to appropriately simulate material
nonlinearity. Two typical beam-column connections in reinforced concrete frame buildings built
before the mid-1960s which were tested by other researchers under quasi-static cyclic loading were
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chosen as target model for this research. There was no transverse reinforcement considered in the
joint panel zone.
The configuration of numerical model is implemented in finite element code ABAQUS. The fi-
nite element model is validated with experimental results. Numerical results presented in terms of
joint shear capacity, deformations and cracking pattern at introduced key points of joint shear be-
havior conform appropriately to experimental results. The finite element analysis results confirms
the capability of the developed finite element model in this study to predict RC beam-column joint
shear behavior and can be further investigated and validated for different types of joints. The model
can be used as an efficient and powerful numerical tool for further studies on different RC beam-
column joints.
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