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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

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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.

M.A. Najafgholipour et al. / Finite Element Analysis of Reinforced Concrete Beam-Column Connections with Governing Joint Shear… 1203

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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.

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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

Latin American Journal of Solids and Structures 14 (2017) 1200-1225

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

Latin American Journal of Solids and Structures 14 (2017) 1200-1225

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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