nonlinear finite element analysis of non-seismically detailed

ScienceAsia 34 (2008): 049-058

Nonlinear Finite Element Analysis of Non-SeismicallyDetailed Interior Reinforced concrete Beam-Column

Connection under Reversed Cyclic LoadTeeraphot Supaviriyakita, Amorn Pimanmasa* and Pennung Warnitchaib

a School of Civil Engineering and Technology, Sirindhorn International Institute of Technology, ThammasatUniversity, Thailand.

b Asian Institute of Technology, Thailand.

* Corresponding author, E-mail: [email protected]

Received 2 Mar 2007Accepted 19 Oct 2007

ABSTRACT: This paper presents a nonlinear finite element analysis of non-seismically detailed reinforced concrete(RC) beam-column connections under reversed cyclic load. The test of half-scale nonductile reinforcedconcrete beam-column joints was conducted. The tested specimens represented those of the actual mid-risereinforced concrete frame buildings designed according to the non-seismic provisions of the AmericanConcrete building code (ACI). The test results show that the specimens representing small and mediumcolumn tributary area failed in brittle joint shear while the specimen representing large column tributaryarea failed by ductile flexure though no ductile reinforcement details were provided. The nonlinear finiteelement analysis was applied to simulate the behavior of the specimens. The finite element analysis employedthe smeared crack approach for modeling beam, column and joint, and employed the discrete crack approachfor modeling the interface between beam and joint face. The nonlinear constitutive models of reinforcedconcrete elements consisted of coupled tension-compression model to model normal force orthogonal andparalleled to the crack and shear transfer model to capture the shear sliding mechanism. The finite elementmodel (FEM) shows good comparison with test results in terms of load-displacement relations, hystereticloops, cracking process and the failure mode of the tested specimens. The finite element analysis clarified thatthe joint shear failure was caused by the collapse of principal diagonal concrete strut.

KEYWORDS: beam-column connection, nonlinear analysis, joint shear failure, diagonal compressive strut,reinforced concrete plate element.

doi: 10.2306/scienceasia1513-1874.2008.34.049

INTRODUCTION

Although many cities in South East Asia areconsidered as being located in low to moderate seismiczone due to a relatively long distance from active earthfaults, they are not absolutely exempt from seismichazard. Bangkok, for example, is founded on a softbasin of marine clay of several ten-meter depths. Thissoil characteristic has a potential to amplify the seismicwave up to 3-4 times1. The soft ground condition isquite similar to that of Mexico City, which was devastatedby 1985 Mexico earthquake with almost 10,000 deathtoll. Recently, the 2004 Sumatra earthquake in theAndaman Sea recorded at a magnitude of 9.3 on RichterScale, caused violent shaking of many highrises inBangkok though the epicenter was more than 800kilometers away. The quake has prompted a seriouspublic concern on seismic safety of buildings. Many tallbuildings, though not designed for seismic loads, usuallyhave shear walls to resist wind forces, and consequentlymay not be vulnerable to the overall collapse under theearthquake2-3. On the contrary, many low-rise and mid-

rise buildings of up to 10 stories are rigid frame withoutshear walls. The frame structures mainly resist lateralforces through bending of beams and columns. Mostof these frame structures were designed for gravityload only according to the American Concrete buildingcode (ACI) in Thailand, and British code (BS) in Singaporeand Malaysia4.

As a result of the lack of seismic consideration instructural design, the reinforcement details of theseframe buildings are usually weak against earthquakeloading. As shown in Fig. 1, the deficiencies ofreinforcement details are characterized by (a) little or noconfining reinforcement in beam-column joint, (b) lapsplice of longitudinal column bars immediately abovefloor level, and (c) widely-spaced column ties and beamstirrups. Under lateral force, the joint has to carry a largehorizontal shear force (Fig. 2(a)) in order to equilibratemoments acting on the joint in the same direction byframing beams. Concurrently, the longitudinal beam barin the joint is also subject to a large bond stress. As a resultof these forces, the joint commonly fails by either jointshear or bar pull-out failure.

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Recently, the authors have conducted a survey ofexisting reinforced concrete (RC) buildings in Thailand5.The building database was compiled for mid-rise RCbuildings having the number of stories in the region of5 to 15. All buildings were beam-column rigid framewithout shear wall. They were designed according tothe non-seismic provisions of ACI building codeconsidering only gravity load. The buildings weregrouped into 3 categories based on their columntributary area as shown in Fig. 2(b). The beam-columnconnections in each category were studied in terms ofstructural designs and reinforcement details. Threehalf-scale beam-column specimens that representedeach building category were prepared and tested in thelaboratory under reversed cyclic load. Since buildingswere designed in accordance with the ACI buildingcode, the experimental findings are supposedlyapplicable to not only buildings in Thailand, but alsobuildings designed and detailed according to the ACIcode in other countries.

The experiment showed a sudden shear failure inthe joint region of specimen representing connectionof buildings with small and medium column tributaryarea (9-30 m2) and a ductile flexural failure of beamsections near the joint in the specimen representingconnection of building with large column tributary

area (40-48 m2). In this paper, the authors presentedthe nonlinear finite element method as a tool to simulatereversed cyclic behaviors of these specimens. Thebeam-column joint specimens are modeled with two-dimensional plane stress elements in line with thesmeared crack approach. The interface between beamand column is modeled with one-dimensional discretejoint element to simulate reinforcement pull-out. Thefinite element method is used to investigate the crackingprocess, the failure mode, the global load-displacementbehavior and the principal load resistant mechanism.

NONLINEAR FINITE ELEMENT ANALYSIS OFREINFORCED CONCRETE (RC) INTERIOR BEAM-COLUMN CONNECTION

Finite Element Model (FEM)Fig. 3(a) illustrates a finite element model of a typical

reinforced concrete (RC) interior beam-columnconnection of a rigid frame. The cruciform-shapedspecimen is normally preferred for laboratory testing.It consists of the beam-column joint core with beamsand columns extending horizontally and vertically totheir mid-length where moments are assumed to bezero, i.e., point of inflection. The beam, column andbeam-column regions are modeled with two-dimensional 8-node plane stress RC elements as shownin Fig. 3(b). The interfaces between beam and columnare modeled with one-dimensional 6-node jointelement as shown in Fig. 3(c).

The two-dimensional reinforced concrete elementis constructed by combining the constitutive laws ofconcrete and reinforcing bar. Here, cracks andreinforcing bars are assumed to be distributed orsmeared over the entire element. The bond betweenconcrete and reinforcement is considered to be perfect,that is, concrete and reinforcement are subject toidentical strains throughout the entire loading. Thetotal stress carried by the reinforced concrete element

Fig 1. Typical non-seismic reinforcing detail of RC framedesigned for gravity load only.

Fig 2. Horizontal shear force in joint core (a) and definition of column tributary area (b).

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is the sum of concrete and reinforcement stresses. Inthe smeared crack approach, the constitutive modelsof reinforcement and concrete are formulated in termsof relations between average stress and average strainas follows,

{ } { } { }, , ,T rc c rc r rcσ σ σ= + (1)

{ } { }{ } { }

, , ,

( , , ) ( , , )

T rc x y xy

rc x y xy rc x y xyf g

σ σ σ τ

ε ε γ ε ε γ

=

= + (2)

where {σT,rc

}, {σc,rc

}, and {σr,rc

} are total stresses,concrete stresses and reinforcement stresses of RCplate element respectively, {σ

x, σ

y, τ

xy} and {ε

x, ε

y, γ

xy} are

global stress and strain respectively; and frc and g

rc

represent constitutive laws of concrete andreinforcement, respectively. The joint element is one-dimensional with zero thickness. It represents theinterface between two elements with different sectionalstiffness where local discontinuities, such asreinforcement pull-out may occur. Similar to two-dimensional RC plate element, the total stresses of thejoint element are the sum of concrete and reinforcementstresses. However, the constitutive laws of the jointelement are formulated in terms of average stressesand relative displacements, i.e., crack opening andsliding.

{ } { } { }, , ,T jo c jo r joσ σ σ= + (3)

{ } { } { } { }, , ( , ) ( , )T jo n jo jof gσ σ τ ω δ ω δ= = + (4)

where {σT,j0

}, {σc,j0

}, and {σr,j0

} are total stresses,concrete and reinforcement stresses of RC jointelement, respectively; {σ

n, τ} is the vector of joint

normal and shear stress, {ω, δ} is the vector of jointopening and sliding; and, f

j0 and g

j0 represent

constitutive laws of concrete and reinforcement ofjoint element, respectively.

Essentially, the constitutive laws for plate and jointelement describe the same physical behaviors ofconcrete and reinforcement. However, the kinematicvariables used in the constitutive formulations aredifferent as previously explained.

The constitutive laws of concrete consist of thecompressive stress model under crack closure, thetensile stress model upon crack opening and the shearstress model upon crack sliding (Fig. 4). The constitutivelaw of reinforcement includes the elastic-hardeningmodel under tensile yielding, and the elasto-plasticmodel under compressive yielding. These lawsrepresent major non-linearity of reinforced concrete.The combined nonlinearity of all elements subject todifferent stress states constitutes the entire nonlinearbehavior of the specimen. In the next section, only theaverage stress-average strain constitutive laws of theRC plate element are presented.

Constitutive Models of Concrete and Reinforcing BarsCracked concreteThe local strains of cracked concrete are defined

with respect to the crack axis. They consist of strainnormal to a crack, strain parallel to a crack and theshear strain along a crack face as shown in Fig. 4. Inputvariables for constitutive laws are local strains definedabove as well as relevant path-dependent parameters.The constitutive laws are formulated for both envelopeand hysteretic loops. Under cyclic load, two sets ofdiagonal cracks are generated and alternate betweenopening and closing in accordance with the loaddirection. At any load step, however, only a set ofcracks is active and principally contributes to thenonlinearity of the element. The active crack is defined

Fig 3. Finite element model of RC interior beam-column connection and Types of RC element.


as the one that has the largest normal tensile strain. Theconstitutive laws are expressed with respect to theactive crack. In the following, the three previouslymentioned constitutive laws are described.

(a) Combined compression-tension model for normalstress orthogonal and parallel to a crack.

The constitutive law for computing normal stressorthogonal and parallel to a crack is shown in Fig. 5. Onthe tensile side, the model covers both tension softeningand tension stiffening behavior. After crack occurs, thetensile stress sharply drops in the un-reinforced elementwhile it gradually decreases in the reinforced elementas a result of bond between concrete and steel bar. Torepresent both behaviors under the same formulation,the following equation is adopted for concrete tensilestress. c

tut t

t

fε

σε

⎛ ⎞= ⎜ ⎟

⎝ ⎠ (5)

where σt and ε

t is the tensile stress and strain

orthogonal to a crack, respectively, ft is the concrete

tensile strength, εtu is the cracking strain, and c is a

softening/stiffening parameter. By varying theparameter c, the above model can cover both softeningand stiffening behavior. In this paper, the parameter cis set to 2.0 and 0.4 for un-reinforced and reinforcedelements to represent softening and stiffening behaviorsrespectively.

To compute the compressive stress parallel to acrack, the following elasto-plastic fracture model 6 isused

0 0( )c c c pK Eσ ω ε ε= − (6)

where σc and ε

c is the compressive stress and strain

parallel to a crack, respectively, ω is transverse tensilestrain factor, K

0 is the fracture parameter, E

c0 is the

initial stiffness, and εp is the compressive plastic strain.

The fracture parameter (K0)

and compressive strain

(εp)

are the key parameters of the model. They are

empirically formulated as 6.

0 exp 0.73 1 exp 1.25c cKε εε ε

⎛ ⎞⎛ ⎞⎛ ⎞= − − −⎜ ⎟⎜ ⎟⎜ ⎟′ ′⎝ ⎠⎝ ⎠⎝ ⎠ (7)

202 1 exp 0.35

7c c

p

ε εε ε

ε ε⎛ ⎞⎛ ⎞⎛ ⎞′= − − −⎜ ⎟⎜ ⎟⎜ ⎟′ ′⎝ ⎠⎝ ⎠⎝ ⎠

(8)

where ε′ is the concrete strain at peak compressive

strength.The model combines the nonlinearity caused by

plasticity through plastic strain and fracturing damagethrough fracture parameter. These parameters accountfor the permanent deformation and irrecoverable lossof elastic strain energy, respectively. In addition, theconcrete compressive strength is reduced owing to thepresence of transverse tensile strain6-8. This is takeninto account by an additional damage factor (ω) inequation 6. The lower limit of ω depends on thecompressive strength of concrete. It varies linearlyfrom 0.4 to 0.6 as concrete compressive strengthincreases from 20 to 100 MPa. The relation between ωand transverse tensile strain is shown in Fig. 5.

(b) Shear stress transfer modelFor computing shear stress transmitted along a

crack face, the contact density model 6 is adopted asshown in Fig. 6. The equation of the shear envelopemay be expressed as,

2

21cr stfβτβ

=+ (9)

where fst is interface shear strength along the crack

and may be expressed as,

( )1 / 33.8st cf f ′= , MPa (10)

β is the normalized shear strain defined as,

cr

t

γβ

ε= (11)

Model of reinforcing barsLocal strains of reinforcing bar are defined with

respect to the bar axis. For reinforcing steel, local

Fig 4. Local stresses and strains of cracked concrete.

Fig 5. Combined compression-tension model for normal stressparallel and orthogonal to a crack.


Numerical Simulation of RC Beam-ColumnConnection

Test of Non-Seismically Detailed Interior Beam-Column Connection

The authors conducted the reversed cyclic test ofthree half scale interior beam-column joint specimens.These specimens represented those of actual buildingsdesigned and detailed according to the non-seismicprovisions of ACI building code. The database of tenreinforced concrete mid-rise buildings in Bangkok wascompiled. These buildings have 5-15 stories and wereconstructed as reinforced concrete frame without shearwall. The types of buildings cover university, school,apartment and hospital. The buildings were groupedinto three categories based on the sizes of the columntributary area (Fig. 2(b)), as buildings with large, mediumand small column tributary area. Based on the collecteddata, the area range is set to 40-48 m2, 20-30 m2 and 9-18 m2 for the large, medium and small categories,respectively. In order to characterize the structuralbehavior of the beam-column connection, thestructural indices were defined for the beam, columnand beam-column joint. The structural indices of thebeam included tension and compression reinforcementratio, shear span-to-depth ratio, flexural capacity-to-shear capacity ratio, and transverse reinforcement ratio.The structural indices of the column included axialforce ratio, shear span-to-depth ratio, flexural capacity-to-shear capacity ratio, longitudinal reinforcement ratio,and transverse reinforcement ratio. The structuralindices of the beam-column joint included columndepth-to-bar diameter ratio, column width-to-beamwidth ratio, column depth-to-beam depth ratio,confinement reinforcement index, column flexuralcapacity-to-beam flexural capacity ratio, and jointshear-to-joint shear strength ratio.

These structural indices were calculated for eachbuilding in each category. Three half-scale beam-columnspecimens, namely, JL, JM and JS were constructed torepresent those of each building category. The structuralindices of these specimens were designed to be as closeas possible to the mean values calculated from the actualbuilding in each category. The reinforcement details andthe construction methods were also as similar as possibleto the actual construction. The specimen dimensionsand reinforcement details are illustrated in Figs. 8 and9. The 12-mm diameter reinforcing bar was used aslongitudinal reinforcements in the beam and column.The average tested yield and tensile strengths of the barwere 499 and 615 MPa, respectively. The 3-mm diameterplain mild steel was used as transverse reinforcementsin the beam and column. The tested yield and tensilestrengths of the bar were 291 and 339 MPa, respectively.The average tested cylindrical compressive strength ofconcrete was 26.7 MPa.

yielding at the vicinity of cracks has to be taken intoaccount in the constitutive formulation. Sincereinforcement sections close to cracks are subject toyielding first whereas others are still elastic, allreinforcement sections are not simultaneously yieldingthroughout the entire length. This causes the apparentreduction in the average yield strength of reinforcingbars. The average yield strength of reinforcing barsembedded in concrete may be expressed as 9,

2t

y y

ff f

ρ= − (12)

where fy is an average yield strength of reinforcing

bar embedded in concrete, fy is the yield strength of

bare bars, ft is the tensile strength of concrete and ρ is

the reinforcement ratio.The model of reinforcing bar for both envelope and

hysteretic loops6 is shown in Fig. 7. The figure alsoshows that of bare bar for comparison.

Fig 6. Shear stress transfer model.

Fig 7. Modeling of reinforcing bar.


The test set-up and boundary conditions are shownin Fig. 10(a). The lateral forced displacement was appliedat the top of the column through a 500 kN hydraulicactuator. The ends of the beam were supported byrollers that allowed free horizontal movement tosimulate lateral drift. The bottom end of the columnwas supported by a hinge which allowed no movementin all directions. The axial load of 12.5% of column axialcapacity was applied to the column by means of verticalprestressing. The column was pushed forward andpulled backward in a reversed cyclic pattern with thetarget lateral drifts of 0.25%, 0.50%, 0.75%… as shownin Figs. 10(b) and 10(c). The target loop was repeatedtwice for each drift level. The load was continued untiland beyond the peak load to trace the post-peakbehavior. The experimental results shall be presentedand discussed in conjunction with the numericalanalysis in the next section.

Finite Element AnalysisThe typical finite element mesh of the specimen is

shown in Fig. 3. For each specimen, the mesh iscomposed of 733 nodes, 216 two-dimensional RC plateelements and 12 one-dimensional joint elements. Loadsacting on specimens include self-weight, column axialload and horizontal forced displacement. Thecomparison between experiment and analysis is givenin Table 1. The failure is defined as the point where theapplied load drops more than 80%.

Fig 8. Geometry, dimension and reinforcement of allspecimens (unit: mm).

Fig 9. Cross section of beams and columns (unit: mm).

Table 1. Comparison between analytical (FEM) andexperimental force and displacement (drift)relations.

SpecimenSpecimenSpecimenSpecimenSpecimen Force / DriftForce / DriftForce / DriftForce / DriftForce / Drift FEM analysisFEM analysisFEM analysisFEM analysisFEM analysis ExperimentExperimentExperimentExperimentExperiment

JL Maximum force (kN) 90 92Drift (%) 4.6 4.6

JM Maximum force (kN) 73 72Drift (%) 2.0 1.75

JS Maximum force (kN) 70 68Drift (%) 1.5 1.5

The analytical deformed shape and crackingpattern of specimen JL are compared with theexperimental results in Fig. 11. The FEMdemonstrates an extensive deformation at the endsof beam framing into the joints, indicating a flexuralfailure. The evident opening of the discrete jointelement indicates pull-out of the beam reinforcingbars. The FEM also shows concentration of verticalcracks near the ends of the beams. This crack patternindicates that bending moments act on the beam


ends in the same direction. On the contrary, the FEMpredicts little distortion and cracking in the elementsof the joint region. Hence the failure does not originatefrom the joint. Moreover, no major cracks are computedin the column adjacent to the joint core. The failure istherefore categorized as ductile strong column-weakbeam. In a building with a large column tributary area,the size of the columns is usually large, whereas thespan of the beams is long. This configuration leads toa large column flexural capacity, a high joint shearstrength and a low beam flexural strength.

The comparison between analytical and experimentload versus applied displacement is shown in Fig. 12(a).A reasonable match is obtained for both envelope andhysteretic loops. The hysteretic loops are wide,indicating large energy dissipation. In the experiment,concrete crushing and subsequent spalling occurredin the beam and consequently exposed longitudinaland transverse reinforcements. This was followed bybuckling of compression reinforcement. The spallingand reinforcement buckling are not included in thepresent constitutive models, hence the numerical loopsare relatively wider than the experimental ones. TheFEM predicts ductile yielding with long yield plateau upto 4.6% drift which is the same as the experimental

value. It shall be noticed that the sub-standard beam-column joint in buildings with large column tributaryarea is quite ductile though it was designed and detailedwithout considering seismic loads in accordance withthe ACI building code. The ductility can be enhancedby providing closed stirrups near the beam’s end toprevent concrete spalling and reinforcement buckling.

The analytical deformed shape and cracking patternof specimen JM and JS are compared with experimentalresults in Fig. 11, respectively. In contrast to specimenJL, the mesh of JM and JS is extremely distorted, showinga large shear deformation in the joint region. Theanalytical cracks are mostly inclined and concentratedin the joint zone. Two sets of intersecting diagonalcracks alternate opening and closing in accordancewith the change in the load direction. These cracksform the dominant X-shape pattern. The analyticaljoint damage and development of diagonal cracksclosely follow experimental results. Some small flexuralcracks are computed in the beams and columns but donot cause the failure. The failure originates from thejoint zone as a result of excessive joint shear force.

The comparison between experimental andnumerical load versus applied displacement relation isshown in Figs. 12(b) and 12(c) for specimen JM and JS,

Fig 10. Laboratory test set-up (a) and Displacement history of each specimen (b and c).


respectively. The maximum experimental drifts at peakload are 1.75% and 1.5% for specimens JM and JS,respectively. The corresponding computed drifts are2.0% and 1.5 %, respectively. The hysteretic loops arenarrow and pinched towards the origin, indicating smallenergy dissipation. This may be associated with slidingof diagonal cracks in the joint. However, the analyticalloops are wider than experimental ones. This issupposedly caused by the concrete spalling in the jointarea which is not included in the present analyticalmodel. It is also observed that the attained lateral drifttends to be smaller for a substandard beam-columnjoint in RC frame with a smaller tributary column. Thisis probably because the frame is likely to have smallercolumn dimensions, just sufficient to carry gravity load.

The joint shear failure is undesirable because it isbrittle and catastrophic. The tendency of brittleness ismore evident for the beam-column connection with asmaller column tributary area. To analytically identifythe failure mechanism of the joint region, the principalcompressive stresses of concrete are computed fromthe FEM stress outputs. The development ofcompressive struts is compared among the threeanalyzed specimens in Fig. 13.

Before peak, the formation of compressive strut isdominant in all specimens. As the lateral drift increases,

the compressive stress rapidly intensifies in specimenJS. The collapse of compressive struts generallycoincides with the failure point of the specimens. As forspecimen JL, the compressive strut is maintainedthroughout the entire loading. At 4.6% drift when failuretook place, the strut remains stable, indicating that thejoint is sound and does not cause failure. Thisobservation agrees with the experimental result. Thisanalytical results show that the diagonal strut formationis the principal load resistant mechanism for the jointof specimens JM and JS and the beam flexural yieldingis the main load resistant mechanism for specimen JL.

CONCLUSION

The paper presents the non-linear finite elementanalysis as a tool to evaluate the reversed cyclic behaviorof the sub-standard RC beam-column connections inmid-rise frame building designed and detailedaccording to the non-seismic provisions of ACI buildingcode in a low to moderate seismic region. The finiteelement analysis includes the nonlinearity of crackedconcrete in tension, compression and shear, and yieldedreinforcement in line with the smeared crack approach.It is shown that the FEM can satisfactorily reproducethe load-displacement envelope as well as hysteretic

Fig 11. Comparison among Analytical deformation (a), Analytical cracking pattern (b), and experimental failure of each specimen (c).


Fig 12. Relationship between column shear force and drift ratio of each specimen.

Fig 13. Principal compressive stresses (unit: MPa) of each specimen


loops of the tested specimens. As for the failure mode,the finite element analysis predicts a ductile beamflexural failure for specimen JL representing actualframe with a large column tributary area, and brittlejoint shear failure for specimens JM and JS representingactual frame with a small and medium column tributaryarea, respectively. The collapse of compressive diagonalstrut is verified to be the principal failure mechanism.The analytical failure modes, damage zones andcracking process compare well with the experimentalresults. However, the analytical hysteretic loops arecomparatively wider than experimental ones since thenonlinearity due to concrete spalling and reinforcementbuckling has not been taken into account in the presentanalysis.

ACKNOWLEDGEMENTS

The authors are very grateful to Thailand ResearchFund (TRF) for providing the research fundRMU4880022 to carry out the research.

REFERENCES

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6. Maekawa K, Pimanmas A, Okamura H (2003) Nonlinearmechanics of reinforced concrete. Spon Press.

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