Engineering Structures 33 (2011) 273–286

Contents lists available at ScienceDirect

Engineering Structures

journal homepage: www.elsevier.com/locate/engstruct

Review article

Numerical modelling of the cyclic behaviour of RC elements built with plainreinforcing bars

José Melo a, Catarina Fernandes a, Humberto Varum a,∗, Hugo Rodrigues a, Aníbal Costa a, António Arêde b

a Civil Engineering Department, University of Aveiro, Aveiro, Portugalb Civil Engineering Department, Faculty of Engineering, University of Porto, Porto, Portugal

a r t i c l e i n f o a b s t r a c t

Article history:Received 16 February 2010Received in revised form16 September 2010Accepted 2 November 2010Available online 30 November 2010

Keywords:Concrete–steel bondPlain reinforcing barsCyclic behaviourNumerical modelling

The bond–slip mechanism is one of the features that significantly controls the response as well asdamage evolution of reinforced concrete structures when subjected to severe cyclic loadings, such asthose induced by earthquakes. Its effect is particularly important in structures built with plain reinforcingbars. For a rigorous simulation of the response of existing RC structures, built mainly with plain bars, thebond–slip mechanism should be considered. However, the majority of the available concrete–steel bondnumericalmodelswere developed and calibrated for elementswith deformed reinforcing bars. Moreover,the available experimental data on the cyclic behaviour of RC elements built with plain bars is still limited.In this framework, the objective of the present paper is to calibrate a numerical model based on results ofa cyclic test performed on a two-span RC beam built with plain bars, whichwas collected from an existingstructure. The numerical modelling was carried out with the nonlinear OpenSees software platform.Particular awareness was devoted to the bond–slip mechanism. The numerical results obtained with thecalibrated nonlinearmodel are presented and comparedwith the experimental results. The considerationof the bond–slip effect in the numerical model was fundamental to achieve a good agreement betweenthe numerical simulation and the test results.

© 2010 Elsevier Ltd. All rights reserved.

Contents

1. Introduction........................................................................................................................................................................................................................2742. Overview on OpenSees platform and models ..................................................................................................................................................................274

2.1. Structural elements’ modelling ............................................................................................................................................................................2752.1.1. BeamWithHinges element .....................................................................................................................................................................2752.1.2. Zero-length section element..................................................................................................................................................................275

2.2. Concrete and steel material models .....................................................................................................................................................................2752.3. Force–slip model....................................................................................................................................................................................................276

3. Numerical modelling of a two-span RC beam built with plain reinforcing bars ...........................................................................................................2763.1. Brief description of the tested beam and cyclic test procedure..........................................................................................................................2763.2. Numerical model ...................................................................................................................................................................................................277

3.2.1. Modelling strategy..................................................................................................................................................................................2773.2.2. Materials..................................................................................................................................................................................................2783.2.3. Imposed displacement history ..............................................................................................................................................................279

3.3. Numerical results...................................................................................................................................................................................................2793.3.1. Force–displacement diagrams ...............................................................................................................................................................2793.3.2. Damage evolution...................................................................................................................................................................................2793.3.3. Energy dissipation evolution .................................................................................................................................................................2803.3.4. Bending moment evolution ...................................................................................................................................................................281

∗ Corresponding author. Tel.: +351 234 370938; fax: +351 234 370094.E-mail addresses: josemelo@ua.pt (J. Melo), cfernandes@ua.pt (C. Fernandes), hvarum@ua.pt (H. Varum), hrodrigues@ua.pt (H. Rodrigues), agc@ua.pt (A. Costa),

aarede@fe.up.pt (A. Arêde).

0141-0296/$ – see front matter© 2010 Elsevier Ltd. All rights reserved.doi:10.1016/j.engstruct.2010.11.005

274 J. Melo et al. / Engineering Structures 33 (2011) 273–286

3.3.5. Moment–curvature diagrams ................................................................................................................................................................2823.3.6. Evolution of neutral axis position..........................................................................................................................................................2823.3.7. Stress–strain diagrams ...........................................................................................................................................................................282

3.4. Bond–slip influence in the beam response ..........................................................................................................................................................2834. Main conclusions and final comments .............................................................................................................................................................................283

Acknowledgements............................................................................................................................................................................................................285References...........................................................................................................................................................................................................................285

1. Introduction

A significant number of existing reinforced concrete (RC) build-ing structureswere constructed before the 70s,with plain reinforc-ing bars, prior to the enforcement of the modern seismic-orienteddesign philosophies. As a consequence of poor reinforcement de-tailing andof the absence of capacity designprinciples, a significantlack of ductility, at both the local and global levels, is expected forthese structures, resulting in inadequate structural performanceeven under moderate seismic excitations [1,2].

The hysteretic behaviour of RC structures is highly dependenton the interaction between steel and concrete. Cyclic load reversals(like the ones induced by earthquakes) result in accelerated bonddegradation, which leads to significant bar slippage. The bond–slipmechanism is reported to be one of the most common causesof damage and collapse of existing RC structures subjected toearthquake loading, assuming particular importance in structureswith plain reinforcing bars.

In elements with plain reinforcing bars, low bond capacitiesdirectly influence the three main deformations mechanisms [3]:bending, shear and fixed-end rotation. Crack development alongthe element is affected by poor bond between the longitudinalreinforcing bars and the surrounding concrete. Low numberof cracks is developed, influencing the shear and bendingcontributions to the element deformation. Shear deformabilityis reduced and bending deformability is increased. In particular,poor bond has a strong influence on fixed-end rotations, greatlyincreasing its contribution to the total element deformation. Thiscontributionmay represent up to 80%–90% of the element’s overalldeformability [3–5].

EC8 [6] evaluates the ultimate deformation capacity of RCelements under cyclic loads in terms of ultimate rotation capacity.The formulas presented for determining the ultimate rotationcapacity were developed for elements with deformed bars. Forelements with plain reinforcing bars, correction coefficients, basedon experimental data, must be used. Though, experimental dataconcerning elements with plain reinforcing bars is scarce incomparison with the data for elements with deformed bars.But the need for a reliable assessment of seismic capacity ofexisting structures has been producing an increasing number ofexperimental campaigns aimed at the study of behaviour of non-conforming elements [3]. Reports of recent experiments made onelements with plain reinforcing bars can be found, for example,in [7] (beams), [5,8] (columns), [2,9,10] (beam–column joints)and [11–13] (frames).

Some considerations about the EC8 prescriptions for theevaluation of the ultimate rotation capacity of elements withplain reinforcing bars can be found in [3]. Also, a proposal forthe calibration of the correction coefficient is presented, basedon recent experimental data concerning RC columns, highlightingthe conservativeness of the EC8 prescription. The experimentalevidence reported in [3] shows that the ultimate rotation ofelements with plain reinforcing bars is higher compared withelements with deformed bars, on average by 35%, given equal thestructural characteristics and details.

Bond behaviour is usually described in terms of a bondstress–slip relationship. Though the procedure for deducing

a bond–slip relationship starting from experimental data isnot univocally defined [14], pull-out tests and beam testsare the common experimental procedures for assessing bondperformance. Existing literature regarding bond performance ofdeformed bars offers many experimental data and constitutiverelationships (see [15]), monotonic and hysteretic, taking intoaccount the main parameters that influence bond capacities [14].Conversely, there is a lack of experimental data concerningspecimens with plain reinforcing bars, namely for the cyclicfield. Also, the influence of several parameters affecting bondperformance (concrete strength, embedment length, bar diameter,among others) is not fully established. Reports on monotonicpull-out tests on specimens with plain bars can be foundin [16,17]. Recentmonotonic and cyclic pull-out tests are describedin [18]. Lack of experiments made on this type of specimenstranslates into a scarce number of reliable models for describingbond performance. For example, analytical approaches proposedby Feldman and Bartlett [17] and CEB-217 [19] are limited tothe monotonic phase. A recent proposal for a bond stress–sliprelationship and relative hysteretic rules, based on experimentaldata from [16], are presented in [14].

Bond–slip effects should be included in the numerical modelsof structural analysis for achieving a better simulation of the realbehaviour of RC structures [13,20–22]. Though, perfect bond isusually assumed in the analysis of RC structures, implying fullcompatibility between concrete and reinforcement strains at eachstructural member point.

Considerable research work (see [23]) has been developedaiming the implementation of bond–slip effects in the analyses ofRC elements. In terms of available numerical tools, only a few (forexample, CASTEM 2000 [24], OpenSees [25] and ATENA [26]) allowconsidering bond–slip in the structural analysis of RC structures.

In this content, the present paper describes the numericalmodel adopted for reproducing the cyclic response of a two-spanRC beam built with plain reinforcing bars that was cyclically testedup to collapse. The numericalmodelwas developed using the OpenSystem for Earthquake Engineering (OpenSees) platform [25]. Theresults from the cyclic test were used to calibrate the numericalmodel taking into account the bond–slip effects. The modellingstrategy and parameters adopted, and the main numerical resultsare presented and discussed. The obtained numerical results arepresented in straight comparison with the experimental resultsfor a better observation of the model accuracy in reproducingthe cyclic behaviour of RC elements representative of existingstructures. For better understanding how the consideration of thebond–slip mechanism affects the cyclic response of the beam,additional numerical analyses were performed with the calibratedmodel, butwithout considering slippage. Themost relevant resultsfrom this analysis are also compared with the results of the cyclictest of the RC beam.

2. Overview on OpenSees platform and models

The OpenSees platform is an open source software frameworkdeveloped to simulate the response of structural and geotechnicalsystems subjected to earthquakes. For frame structural membersOpenSees performs fibre-based analysis. The flexural member is

J. Melo et al. / Engineering Structures 33 (2011) 273–286 275

Fig. 1. Linear element and zero-length section element.

represented by unidirectional steel and concrete fibres which areassumed to be characterized by the selected material stress–strainrelationship. The member stiffness and forces are obtained bynumerically integrating the stiffness and forces of sections alongthe member length. The section deformation is used to obtainthe strain in each fibre, based on the plane section assumption.The fibre stress and stiffness are updated according to thecorresponding material models, followed by upgrading of thesection force resultant and the corresponding stiffness [27].

OpenSees has available several models for describing thebehaviour of materials (steel and concrete) and the global cyclicbehaviour of RC structural elements. It has also available modelsfor simulating the bond–slip effects on the elements’ behaviour.Although the models were originally developed for RC elementsbuilt with deformed bars and good anchorage conditions, themodification of the proper input parameters allows using of thesemodels for elements with plain bars. In this section reference ismade only to themodels and tools used in the numericalmodellingof the RC beam, later described in detail in Section 3.

2.1. Structural elements’ modelling

2.1.1. BeamWithHinges elementThe BeamWithHinges element, available in OpenSees, was used

to simulate the response of the RC beam under study. Plasticityis concentrated over specified hinge lengths at the element endsand each BeamWithHinges element is divided into three parts:two hinges at the ends and a linear-elastic region in the middle(see Fig. 1). By contrast with other types of distributed plasticityelements in which the Gauss integration points are distributedalong the element length, the BeamWithHinges element localizesthe integration points in the hinges’ regions.

In RC elements built with plain reinforcing bars the plastichinges (located at the element end zone) generally have smalllengths. Therefore, the BeamWithHinges element is considered tobe very suitable for representing elements with plain reinforcingbars.

A zero-length section element at the end of a beam–columnelement as shown in Fig. 3 can incorporate the fixed-end rotationcaused by strain penetration to the beam–column element.

Fig. 3. Hysteretic response of steel bars with the Steel02 model [28].

2.1.2. Zero-length section elementThe zero-length section element in OpenSees is assumed to have

a unit-length such that the element deformations (i.e., elongationand rotation) are equal to the section deformations (i.e., axial strainand curvature). This element at the end of a beam–column element(for this case the BeamWithHinges element as shown in Fig. 1) canincorporate the fixed-end rotation caused by strain penetrationto the beam–column element [28]. In order to incorporate a zero-length section element in the structural model, a double node is re-quired (twonodeswith the same coordinates). The relative transla-tional degree-of-freedom of these nodes should be constrained toeach other, avoiding the beam–column element (between nodesj and k of the Fig. 1) sliding under lateral loads, because the shearbehaviour control is not included in the zero-length section element.For each BeamWithHinges element and zero-length section element,the section is idealized through fibre modelling, assigning to eachfibre the corresponding material properties and cyclic behaviourrules (for confined or unconfined concrete or longitudinal steel).Because of the unit-length assumption, the material model for thesteel fibres in the zero-length section element represents the bar slipinstead of strain for a given bar stress [27].

2.2. Concrete and steel material models

OpenSees has available several material models for describingthe monotonic and cyclic response, based on hysteretic rulesof concrete and steel reinforcement fibres. In the numericalmodelling of the RC beam presented in this work the followingmaterial models were used: Concrete01 and Concrete02 for theconcrete (Fig. 2) and Steel02 for the steel (Fig. 3). Both concretemodels take into account the confinement effect due to the stirrupsand are based on the law proposed by Hognestad [29] adapted byGuedes [30]. The Concrete02model takes into account the concretetensile strength which is not account for in Concrete01 model.

a b

Fig. 2. Material parameters for the concrete models’ definition [28]: (a) Concrete01; (b) Concrete02.

276 J. Melo et al. / Engineering Structures 33 (2011) 273–286

a b

Fig. 4. Force–slip model [27] — bar stress versus loaded end slip for reinforcing bars fully anchored into footings: (a) under monotonic loading; (b) hysteric response.

The Steel02 model is based on the Giuffré–Pinto formulation,implemented later byMenegotto and Pinto [31]. The nomenclatureof the parameters of the concrete models, shown in Fig. 2, aredescribed in detail in Section 3.2 (Table 4).

2.3. Force–slip model

For simulating the bond–slip mechanism, OpenSees providesthe Bond_SP01model. Thismodel is to be usedwith the zero-lengthsection element, referred in Section 2.1.2. Because of the unit-lengthassumption, the material model for the steel fibres in the sectionelement represents the bar slip instead of strain for a given barstress [27,28]. Therefore, Zhao and Sritharan [27] proposed thebar stress–slip relationship model, represented in Fig. 4(a). Thisgeneric model was developed based on the pull-out tests resultsof deformed steel reinforcing bars anchored in concrete footingswith sufficient embedment length, loaded at its free end zone,namely on the measured bar stress and loaded end slip evolutions.Fig. 4(b) shows the corresponding hysteretic response schemesubjacent to the mentioned model. The hysteretic rules wereestablished using experimental data on the cyclic response ofwell-anchoredbars andobserved cyclic response of columns. Thismodelwas calibrated with the cyclic test results of RC elements withdeformed reinforcing bars, namely a short rectangular column,a tall circular column and bridge T-joint system [27]. A betternumerical approach to the element’s real behaviour was achievedconsidering this model.

Regarding the models presented in Fig. 4(a), the slip valuescorresponding to yielding (sy) and ultimate strengths (su) arecomputed using Eqs. (1) and (2), respectively.

sy (mm) = 0.4

db (mm)

4·

fy (MPa)f ′c (MPa)

· (2α + 1)

1/α

+ 0.34 (1)

su (mm) = 30 ∼ 40 · sy (2)

where, db is the bar diameter, fy is the yielding strength of thesteel bar, f ′

c is the concrete compressive strength and α is atuning parameter used for adjusting the local bond stress–sliprelationship. In the formulation [27] α is taken equal to 0.4, whichis the value given for deformed bars in accordance with [32,33].The parameter b shown in Fig. 4(a) is a stiffness reduction factor.

Since the force–slip model was developed considering de-formed bars, for the beam numerical modelling addressed herein,some of the force–slipmodel parameters had to be adjusted for theplain reinforcing bar situation. In particular, the parameter α wasincreased to 0.5 according to the CEB-217 [19] recommendations,the maximum recommended value was adopted for ultimate slip(su = 40sy) and the minimum value was considered for the stiff-ness reduction factor (b = 0.3).

Fig. 5 illustrates the bond stress–slip relationship for plainbars presented in [19]. This curve is expressed by Eq. (3) andconsists on a nonlinear curve increasing up to the s1 slip value

Fig. 5. Bond stress–slip relationship for plain reinforcing bars [13,19].

Table 1Parameters for the bond stress–slip relationship for plain bars [7,18].

Mean valuesBond conditionsCold drawn wire Hot rolled barsGood All other cases Good All other cases

s1 0.01 mm 0.1 mmα 0.5 0.5τmax (N/mm2) 0.1

√fck 0.05

√fck 0.3

√fck 0.15

√fck

and followed by a plateau. The model parameters of the bondstress–slip relationship are presented in Table 1 and depend onthe roughness of the bar surface, bond conditions and concretestrength. The model and associated parameters are valid for bothconfined and unconfined concrete [13]. According to [13,19], themaximum bond stress for plain bars can be about 12% of themaximum bond stress for deformed bars on equivalent loadingconditions.

τ = τmax · (s/s1)α0 ≤ s ≤ s1. (3)

Regarding the models for the concrete fibres surrounding steelin the zero-length section element, it is suggested by otherauthors [27,28] that they can follow one of the concrete modelsavailable in OpenSees (for example, Concrete01).

3. Numerical modelling of a two-span RC beam built with plainreinforcing bars

3.1. Brief description of the tested beam and cyclic test procedure

A two-span RC beam built with plain reinforcing bars, with0.18 × 0.22 (m2) cross-section dimensions and a total length ofabout 8 m (Fig. 6), was tested under cyclic loading conditions. Thebeam was collected in 2007 from an existing building of the SantaJoanaMuseum, located in Aveiro, Portugal, during its rehabilitationworks.

J. Melo et al. / Engineering Structures 33 (2011) 273–286 277

a

b

Fig. 6. Cyclic testing of the RC beam: (a) testing setup, beam dimensions and reinforcement detailing; (b) loading history.

No information was available regarding the materials’ mechan-ical properties. Cylindrical concrete samples were extracted afterthe beam test and subjected to compression tests. Amean strengthequal to 19 MPa was obtained and the characteristic compressivestrength was estimated to be equal to 16.8 MPa. According to theEurocode 2 (EC2, [34]) classification, the corresponding concreteclass is C16/20. Steel bar samples could not be obtained for per-forming tensile strength tests. For that reason, the steelmechanicalproperties were not estimated. The number and position of longi-tudinal and transversal (U-shaped stirrups) reinforcing bars wasestimated based on the measurement made with a rebar detectorand confirmed after the beam test.

At the museum, the beam was located at the building’sroof, loaded by two vertical elements, one at each middle-span,supporting the roof structure and its covering. This was a solutioncommonly adopted in the period between 1950 and 1980, insouthern European countries in the rehabilitation works of oldmasonry buildings’ roofs. So, for simulating the loading conditions,considering the dead load and the vertical component of theseismic action, the beam was cyclically loaded by two verticalforces, symmetrically positioned at the mid-span of each bay(Fig. 6(a)).

The cyclic forces were applied under force-controlled con-ditions, following the loading history shown in Fig. 6(b). Theforces are always descending, describing series of three load-ing–unloading cycles of increasing amplitude, up to a maximumforce of about 25 kN, when the beam collapse due to flexural fail-ure was observed.

Measures of vertical displacementswere provided by drawwiredisplacement transducers placed along the two spans.

From the experimental data only the global behaviour ofthe beam could be evaluated. The main results are shown inFig. 7, namely the force–displacement diagrams and deformedshape evolution. The beam deformed shape remained roughly

symmetrical until the beginning of cracking at the left andright mid-span section. Afterwards, larger displacements wereregistered for the left span. By the end of the cyclic test, the leftand right spans exhibited a deflection equal to 0.12 m and 0.03m, respectively. The force–displacement diagrams show that thetwo spans have similar stiffness and that a slightly higher resistantcapacity is displayed by the right span. Though being subjected tosymmetric loading conditions, the beambehaviourwas not strictlysymmetric.

Fig. 8 shows the location of the three plastic hinges formedduring the test, with indication of the corresponding length,estimated based only on visual observation and made equalto the length of the zone where significant damage occurred.Damage concentrates within the plastic hinges, with almost nocrack spreading outside the plastic hinges region. Instead, thewidth of the existing cracks was increased significantly during thetest. The small length of the plastic hinges and the insignificantcrack distribution is mainly justified by the slippage between thelongitudinal reinforcing bars and the surrounding concrete.

The experimental results of the beam testwere used to calibratethe adopted numerical model, as described in Section 3.2. Thenumerical analysis results are described and compared against theexperimental results in Section 3.3.

3.2. Numerical model

3.2.1. Modelling strategyFig. 9 illustrates the RC beam numerical model built-up and

implemented within the OpenSees platform. The element lengthsrepresented in themodel scheme shown in Fig. 9 correspond to theeffective distances between the support points. The model of thebeamwas built with BeamWithHinges elements. Zero-length sectionelements were used for the simulation of the bond–slip effect and

278 J. Melo et al. / Engineering Structures 33 (2011) 273–286

a

b

Fig. 7. Experimental results: (a) force–displacement diagrams; (b) deformed shape.

Fig. 8. Plastic hinges’ location and length.

a b

Fig. 9. Model of the RC beam at: (a) beam element dimensions, zero-length section element location; (b) fibre modelling at section level.

located at the plastic hinges’ sections as observed in the beam test.The zero-length section elements concentrate the representation ofthe slippage mechanism within the plastic hinge region and at itsvicinity. The adopted lengths in the numericalmodel for the plastichinges corresponds to the values measured in the test, i.e.: 0.14m for the hinge located at the left mid-span section; 0.05 m forthe hinge located at the middle support, and 0.15 m for the hingelocated at the right mid-span section.

Next section provides the description of the assumed values forthe material properties assigned to the BeamWithHinges elementand zero-length section element in the numerical analyses.

3.2.2. Materials

3.2.2.1. Steel. As previously stated, no specific information wasavailable about the steel mechanical properties and it was not

J. Melo et al. / Engineering Structures 33 (2011) 273–286 279

Table 2Average steel mechanical properties (S235NL).

Properties Value

Yielding strength 260 MPaUltimate strength 360 MPaElastic modulus 200 GPa

Table 3Steel02 and Bond_SP01 model parameters considered in the beam numericalmodelling.

Material model Parameter

Steel02 Hardening modulus ≈ 0R0 16.5

Bond_SP01

α 0.55b 0.30sy 0.44 (mm)su 40sy (mm)R 0.80

possible to take out steel bar samples to perform tensile strengthtests. The specifications given by the National design code RBA [35]in force at the time of the beam erection, in terms of reinforcementsteel mechanical properties, impose as minimum strength thecorresponding to the S235NL steel class. In fact, this was themost common steel used in the past in RC building structuresconstruction. Its average mechanical properties were consideredfor the steel material properties in the beam model, as listedin Table 2. As observed in Section 3.3, the excellent agreementbetween the numerical and experimental results supports thisassumption for the steel properties.

For the beam modelling, two different material models hadto be adopted for the steel. For the BeamWithHinges elementsthe Steel02 material model was assigned while to the zero-lengthsection elements was assigned the Bond_SP01 model. As previouslystated, the bondmodel was developed for deformed bars and goodanchorage conditions. The beam has good anchorage conditionsand plain reinforcing bars. The modification of the properinput parameters allowed using this model. Accordingly, Table 3summarizes the values adopted for each parameter required tocompletely define both Steel02 and Bond_SP01models, beyond thenominal mechanical properties presented in Table 2. Regardingthe Steel02 model, the hardening modulus was considered to beapproximately zero, which represents the typical yielding plateauwith large deformation for hot rolled steel. The value adoptedfor the radius of the initial nonlinear branch was lower than therecommended value R0 = 18.50 [28] for deformed reinforcingbars. Consequently, the energy dissipation under cyclic loading isreduced,which is consistentwith the behaviour of the RC elementswith plain reinforcing bars.3.2.2.2. Concrete

Table 4 summarizes the parameters for the concretemechanicalmodels adopted in the numerical analyses. The residual strengthfor the unconfined and confined concrete (fcum) was consideredequal to 50% and 60% of the maximum strength obtained fromthe compression tests, respectively, in order to apply the [29]formulation for the concrete compressive behaviour. Concrete01model was used for the concrete fibres of the zero-lengthsection elements while Concrete02 model was adopted for theBeamWithHinges elements. The behaviour of the unconfined andconfined concrete is not much different in terms of maximumstrength and corresponding strain, as well as for the post-peakbehaviour, evidencing poor concrete confinement. In fact, in theNational design code RBA [35] in force at the time of the beamerection, the transversal reinforcement was designed only toguarantee the shear resistance. No confinement concerns wereconsidered at that time.

Table 4Concrete mechanical properties adopted in the numerical model (unconfined andconfined).

Concrete E(GPa)

fcm(MPa)

ε0(h)

fcum(MPa)

εu(h)

fctm(MPa)

ε0t (h)

Unconfined 18.5 19.0 2.0 9.5 7.0 1.2 0.12Confined 17.9 19.2 2.1 11.5 33.0 1.2 0.12

Fig. 10. Imposed vertical displacements at mid-span sections.

3.2.3. Imposed displacement historyAlthough the vertical forces imposed in the beam test were

symmetric, the response observed was not strictly symmetric,namely after major damage begun in the left span of the beam.In fact, small differences in the geometrical characteristics, sup-port conditions, material properties, etc., imply non-symmetricaldistribution of stiffness and strength, which is consistent with thedamage concentration on one span. Therefore, in the numericalanalysis, to the beam model was imposed the displacement his-tories recorded during the beam cyclic test. The imposed displace-ment laws at themid-span sections of the beam, left mid-span sec-tion (dl) and right mid-span section (dr ), are represented in Fig. 10.

3.3. Numerical results

This section addresses the presentation and discussion ofthe results obtained with the numerical analyses performedfor the cyclic response of the beam, comparing them with theexperimental results, namely in terms of force–displacementdiagrams, damage evolution and energy dissipation. Bendingmoment and curvature evolutions, and stress and strain evolution,computed based on the numerical model, are also presented.

3.3.1. Force–displacement diagramsFig. 11 shows the vertical force versus vertical displacement

diagrams computed for the left and right mid-span sections of thebeam. In general terms, a good agreement was found between thenumerical and experimental results, in terms of maximum force aswell as force and displacement evolutions.

Plotting together the force–displacement diagrams numericallyobtained for both the left and right mid-span sections, as shownin Fig. 12, it can be observed that both spans reached a similarmaximum force, of about 25 kN. Since during the test developmentlarger displacements are imposed to the left span, the damageappears first in this span. Therefore, for a given displacement level,the unloading stiffness for the right span is larger than for the leftspan. Note that Fig. 13 shows the results for the left span only until3 cm displacement in order to facilitate the comparison.

3.3.2. Damage evolutionFor better understanding the damage evolution and its

influence in the global response of the beam, the concrete andsteel stress–displacement diagramswere computed and compared

280 J. Melo et al. / Engineering Structures 33 (2011) 273–286

a b

Fig. 11. Force–displacement diagrams: (a) left mid-span section; (b) right mid-span section.

Fig. 12. Numerical results: force–displacement relations for the left and right mid-span sections.

with the force–displacement diagrams previously shown in Fig. 11.Based on this evaluation, the points corresponding to the beginningof cracking, steel yielding and concrete crushing were determinedand are represented in Fig. 14. The sequence of the maincalculated damage occurrences is totally in accordance with thedamage evolution observed during the beam test. The first cracksappeared at the middle support and followed by the cracking atthe left and right mid-span sections. This early cracking stage,associated to the low tensile strength of the concrete, does nothave an important influence in the global force–displacementrelations. After the cracking onset, the longitudinal reinforcingsteel bars yielding developed: first on the top bars at the middlesupport, then in the bottom bars at the left mid-span section(which strongly influenced the remaining test as evidenced in theforce–displacement diagrams) and finally on the bottombars at theright mid-span section. For the next damage steps, compressivecrushing of the unconfined concrete took place at the middlesupport, which was followed by the concrete crushing at theleft and right mid-span sections. Finally, the confined concretecrushing occurred at the middle support and then at the left andright mid-span sections.

The estimated value for the serviceability loads, approximatelyequal to 17 kN, corresponding to 68% of the maximum loadimposed at the mid-spans, is shown in Fig. 13. According to thenumerical results, for this load value, in situ, cracking at the mid-spans and middle support, and yielding at the middle support,had already occurred. In fact, cracking was observed at the plastichinges’ sections before the beam test.

In Fig. 13 is also identified the ultimate rotation capacitycomputed using Eq. (4), according to EC8 [6]. The resulting valuewas multiplied by the correction coefficient prescribed by thecode for elements with plain reinforcing bars and without lappingof longitudinal bars, equal to 0.575. The beam is considered asprimary seismic element, according to the EC8 [6] classification.Therefore, in Eq. (4) the parameter γel assumes value of 1.5. Thecomputation of the other parameters was made considering thegeometrical and mechanical properties of the beam and the rulesgiven at the EC8 [6]. The ultimate rotation capacity given by EC8is equal to 0.023 rad. The ultimate rotation of the beam wasalso evaluated from the experimental results, for the maximumimposed displacement, and is 4.8 times and 2.8 times the EC8 valuefor the left and right mid-spans, respectively. Verderame et al. [3]have also studied the EC8 provisions for the estimation of theultimate rotation capacity of elements with plain reinforcing bars.

θum =1γel

· 0.016 · (0.3ν) ·

[max(0.01, ω′)

max(0.01, ω)fc

]0.225·

LVh

0.35

· 25αρsx

fywfc

· (1.25100ρd). (4)

The maximum displacement at the left mid-span section is equalto 0.12 m, which is about four times the maximum displacementrecorded at the rightmid-span section (0.028m). Even a very smalldifference in terms of strength capacity between the two spansjustifies the response observed. Since the testwas force-controlled,once the right span raises its maximum strength capacity theforce cannot increase in any of the two loading points. Despitethe significant deformation of the left span, with larger sectionrotations, the reinforcement steel did not break at any point dueto the bar slippage and to the larger ductility of this type of steel.

3.3.3. Energy dissipation evolutionFig. 14 shows the energy dissipation evolution for the beam,

corresponding to the cumulative sum of the energy dissipationcomputed for each cycle for both spans. On its turn, the energydissipation for each cycle was estimated as the integral of thefunction that defines the force–displacement diagram in thecorresponding displacement range. The left span presents largerdisplacement demands than the right span and, consequently, hasa more significant contribution to the total energy dissipation. Theenergy dissipation evolution obtained with the numerical modelshows a good agreementwith the corresponding evolution directlyobtained from the experimental results, featuring a maximumdifference of about 1.8%.

J. Melo et al. / Engineering Structures 33 (2011) 273–286 281

a

b

Fig. 13. Damage evolution at: (a) left mid-span section; (b) right mid-span section.

Fig. 14. Total energy dissipation evolution.

3.3.4. Bending moment evolutionFig. 15 represents the bending moments’ evolution computed

at the middle support section and at the left and right mid-spansections. Since the amount of longitudinal steel reinforcement isnot the same at the top and bottom of the beam cross-section,the beam moment strength depends on the moments’ sense. Thearea of the bottom steel bars is about twice the area of thecorresponding top steel bars. Thus, for the beam under analysisthe moment strength for sagging bending moments is larger thanfor hogging. As evidenced in Fig. 15(b), for increasing load, when

Table 5Comparison between the moment strength computed using the EC2 [34] rules andthe numerical model.

Sagging momentstrength (kN m)

Hogging moment strength(kN m)

EC2 17.1 11.1Numerical 17.2 11.2Difference (%) 0.5% 0.9%

the moment capacity of the middle support section is achieved(hogging moment strength), the bending moment at the left andright mid-span sections do not reach the corresponding saggingmoment strength, and therefore larger positive bending momentscan still be installed at these sections.

Table 5 lists the sagging and hogging moment strengthobtainedwith the numerical analysis and the corresponding valuescomputed according to EC2 [34], by adopting the mean values forthe material properties and unitary safety factors. The numericalvalues are very close to the strength estimated with the EC2standard, which underlines the adequacy of the adopted numericalmodel.

Finally, Fig. 16 shows the evolution of the bending moments’diagrams calculated for the loading steps corresponding to theoccurrence of the main damages referred to in Section 3.3.2. Thehogging moment strength is achieved when the top steel bars at

282 J. Melo et al. / Engineering Structures 33 (2011) 273–286

a b

Fig. 15. Evolution of computed bending moments at the middle support and left and right mid-span sections: (a) complete test; (b) detail.

Fig. 16. Evolution of the numerical and elastic bending moment diagrams (in kN m).

the middle support section reach the yielding stage. In Fig. 16 isalso shown the bending moments obtained from a linear-elasticanalysis, demonstrating the bending moment redistribution alongthe beam, that is more relevant after yielding of the top steelreinforcement at the middle support occurs. From this point themoment–curvature relationship becomes nonlinear.

3.3.5. Moment–curvature diagramsFig. 17 presents the moment–curvature diagrams plotted for

the sections corresponding to the plastic hinges’ location, namelyleft mid-span section, right mid-span section and middle support(at the left and right sides of the support). It was verified thatbefore reaching yielding of the top steel reinforcement at themiddle support, the moment–curvature relationship is linear andequal for the three analysed sections. Regarding the unloading andreloading branches of the cyclic response, the moment–curvaturerelationships are linear with approximately the same stiffness ofthe initial branch. When the sagging moment strength is reached,curvature values of 6.7 × 10−4 m−1 and 2.2 × 10−3 m−1 areestimated at the left mid-span and middle support sections,respectively.

3.3.6. Evolution of neutral axis positionThe evolution of the sections’ neutral axis positions in relation

to its geometric gravity centre are depicted in Fig. 18, for the threesections in which the plastic hinges were formed. The analysisof these diagrams allow understanding the part of the concretesection under compression for each step of the cyclic loading,and also which type of damages are associated to the abruptchanges of the neutral axis position. Because the beam cross-section has a non-symmetric distribution of the longitudinal steelreinforcement, before the cracking onset the neutral axis is located

Fig. 17. Moment–curvature diagrams.

slightly below the sections geometric centre. For the unloadingand reloading branches, the maximum variation of the neutralaxis position is of about 6 mm. After the concrete crushing at theleft and right mid-span sections, the neutral axis stabilises at aposition close to the gravity centre of the longitudinal reinforcingbars under compression and to the confined–unconfined concreteinterface.

3.3.7. Stress–strain diagramsStress–strain diagrams for the steel and concrete, at represen-

tative points, were computed showing that for the cyclic load-ing considered the steel under tension yields but do not achievesfailure. This is in accordance with the visual damage observationmade after the cyclic testing of the beam. Regarding the confinedand unconfined concrete, at the two control points, both reachedtheir maximum compressive capacity. In comparison to the con-trol point for the confined concrete, the unconfined concrete pointtends to have significantly larger maximum strain demands, due

J. Melo et al. / Engineering Structures 33 (2011) 273–286 283

Fig. 18. Evolution of the neutral axis’ position.

a b

Fig. 19. Relative partial contribution of the slip for the mid-span displacement diagrams: (a) left mid-span section; (b) right mid-span section.

to the particular position of the neutral axis for the last loadingcycles. Accordingly, Table 6 shows the evolution of the stress andstrain profiles during the increasing cyclic loading path, computedfor the left mid-span section.

Before the cracking onset, the stress diagram is approximatelysymmetric.When cracking starts, stress at the bottom longitudinalreinforcing bars increases significantly. Until cracking starts, themaximum compressive stress at the unconfined concrete regionis larger than the maximum stress at the confined concrete. Thelinearity of the stress diagrams is in accordance with the planesection assumption, for the response in the elastic branch.

3.4. Bond–slip influence in the beam response

Fig. 19 shows the relative partial contributions of the slip andbending for the mid-span displacements, computed based on thenumerical results with consideration of the bond–slip mechanism.The shear influence is insignificant and is included in the bendingcontribution. The slip contribution becomes more significant asthe mid-span displacements increase. Initially the slip parcel isabout 10%. In the end is equal to 85%. This value is in agreementwith Verderame et al. [3,4] which verified that 80%–90% of theoverall deformability of elementswith plain reinforcing bars can berelated to the bond–slip effect. Comparing the diagrams of Fig. 19with the force–displacement diagrams in Fig. 11, it is possible toobserve that the larger variation of the slip contribution occurs inthe yielding load region.

For a better comprehension of the influence of not consid-ering the bond–slip mechanism in the beam response, an addi-tional numerical analysis was performed, with the same model

adopted to reproduce the test results and subjected to the sameimposed displacement history, but now without considering theslippage. Thus, Fig. 20 represents the numerical response ofthe beam in terms of force–displacement diagrams, in compar-ison with the experimental results, for the model without thebond–slipmechanism. Additionally, Fig. 21 shows the correspond-ing evolution of the dissipated energy. Regarding the numericalforce–displacement results, even if themaximum force is achievedwhether considering or not the slippage mechanism, a larger stiff-ness than the real is obtained when the bond–slip effects are ne-glected. In terms of energy dissipation evolution, when slippage isnot considered in the numerical results, larger energy dissipationis found than that for the experimental results, with a maximumdifference of 10%. Therefore, in general terms, by considering thebond–slip in the numerical model, a much better simulation of theexperimental beam response is achieved.

4. Main conclusions and final comments

A numerical model was developed with OpenSees for thesimulation of the cyclic response of a continuous two-span RCbeam built with plain bars which was collected from an existingbuilding structure. The available results from the cyclic test of thebeam were used to calibrate the numerical model. The Bond_SP01model, available in OpenSees, was used to simulate the bond–slipmechanism.

Themain numerical results were presented in comparisonwiththe test results, so that the model accuracy in reproducing the

284 J. Melo et al. / Engineering Structures 33 (2011) 273–286

Table 6Evolution of the stress and strain profiles at the left mid-span section.

Damage Strain distribution (h) Stress distribution (MPa)Concrete Steel

0 — Before cracking (no damage)

A — Concrete cracking onset at the middlesupport

B — Concrete cracking onset at the leftmid-span section

C — Yielding of the top steel bars at themiddle support

D — Yielding of the bottom steel bars at theleft mid-span section

D — Yielding of the bottom steel bars at theright mid-span section

E — Compressive crushing of the unconfinedconcrete at the middle support section

F — Compressive crushing of the unconfinedconcrete at the left mid-span section

F — Compressive crushing of the unconfinedconcrete at the right mid-span section

G — Compressive crushing of the confinedconcrete at the middle support

cyclic behaviour of the beam could be better addressed and dis-cussed. The good agreement between the numerical and exper-imental results in terms of force–displacement diagrams (initialstiffness, maximum strength, ductility, unloading–reloading inter-nal cycles), damage evolution (in terms of type of damage andlocation), aswell as dissipated energy evolution, shows that the be-haviour of the beam is well represented by the numerical model,

at global and local levels. Regarding the energy dissipation, a max-imum difference of about 1.8% was found between the experimen-tal and the numerical results. The numerical results show that theslip contribution to the deformation of the beam becomes moresignificant as the mid-span displacements increase. In the end ofthe test, the slip represents 85% of the total deformation of the leftspan and 45% of the right span.

J. Melo et al. / Engineering Structures 33 (2011) 273–286 285

a b

Fig. 20. Force–displacement results (experimental and numerical without the bond–slip mechanism): (a) left mid-span section; (b) right mid-span section.

Fig. 21. Dissipated energy evolution (numerical results without consideringslippage phenomena).

Considering the impossibility to install a more refined in-strumentation setup for the beam tested, the calibrated numer-ical model was used to calculate some other local responseresults, namely: stress and strain profiles, stress–strain andmoment–curvature diagrams.

For better understanding how the consideration of the bond–slip mechanism affects the global response of the beam, addi-tional numerical analyses were made without considering thebond–slip mechanism. The comparison between the correspond-ing numerical results (in terms of force–displacement relation-ships and energy dissipation evolution) and the test resultsallowed proving that numerical simulation of the cyclic re-sponse of the beam without considering the bond–slip mecha-nism leads to inadequate numerical results, since the real beambehaviourwith plain reinforcing bars cannot be suitably simulated.A force–displacement numerical relationship with much largerstiffness than the experimental is obtained when the effects of barslippage are neglected. Moreover, larger values of dissipated en-ergy are obtained, in comparison with the energy dissipation com-puted from the experimental results, with a maximum differenceof 10%.

Experimental data is essential for the development andcalibration of reliable numerical models for the assessment of theseismic capacity of existing RC structures. However, data on theresponse of RC elements with plain reinforcing bars are still scarce.Additional experimental work needs to be developed for coveringall aspects and variables that influence the cyclic behaviour of thistype of elements and structures.

Acknowledgements

This paper reports research developed under financial sup-port provided by ‘‘FCT — Fundação para a Ciência e Tecnologia’’,

Portugal, namely through the Ph.D. grants of the first, sec-ond and fourth authors, with references SFRH/BD/62110/2009,SFRH/BD/27406/2006 and SFRH/BD/63032/2009, respectively; andthrough the sabbatical leave grant of the third author with ref-erence SFRH/BSAB/939/2009. The authors would like to acknowl-edge: (i) Eng. Alexandre Costa, Mr. Valdemar Luís and Mr. AndréMartins, from the Laboratory of Seismic and Structural Engineeringof the Faculty of Engineering of the University of Porto (Portugal),and Eng. Romeu Vicente, Eng. Henrique Pereira and Eng. Elsa Neto,from the Civil Engineering Department of the University of Aveiro(Portugal), for their collaboration in the execution of the tests;(ii) Civilria Construções, Silva Tavares & Bastos Almeida, Lda. andArlindo Correia & Filhos S.A. for help in the construction of the testsetup; and, (iii) the Santa Joana Museum, Aveiro, for giving accessto the building where the beam specimen was collected.

References

[1] Hertanto E. Seismic assessment of pre-1970s reinforced concrete structures.Ph.D. thesis. New Zealand: University of Canterbury; 2005.

[2] Pampanim S, Calvi GM, Moratti M. Seismic behaviour of RC beam–columnjoints designed for gravity loads. In: Proceedings of the 12th Europeanconference on earthquake engineering, EAEE. London: Elsevier Science Ltd.;2002. Paper no. 726.

[3] Verderame GM, Ricci P, Manfredi G, Cozenza E. Ultimate chord rotation ofRC columns with smooth bars: some considerations about EC8 prescriptions.Bulletin of Earthquake Engineering. ISSN 1570-761X (Print) 1573-1456(Online), doi:10.1007/s10518-010-9190-x, published online 3 June 2010.

[4] Verderame GM, Fabbrocino G, Manfredi G. Seismic response of RC columnswith smooth reinforcement. Part I: monotonic tests. Engineering Structures2008;30:2277–88.

[5] Verderame GM, Fabbrocino G, Manfredi G. Seismic response of R.C. columnswith smooth reinforcement. Part II: cyclic tests. Engineering Structures 2008;30:2289–300.

[6] CEN. Eurocode 8: design of structures for earthquake resistance — part3: strengthening and repair of buildings. Brussels (Belgium): EuropeanCommittee for Standardization; 2005.

[7] Marefat MS, Shirazi SMH, Rostamshirazi R, Khanmohammadi M. Cyclicresponse of concrete beams reinforced by plain bars. Journal of EarthquakeEngineering 2009;13:463–81.

[8] Fabbrocino G, Verderame GM, Manfredi G. Rotational capacity of old typeR.C. columns. In: fib symposium keep concrete atractive. Budapest (Hungary);2005.

[9] Liu A, Park R. Seismic behaviour and retrofit of pre-1970’s as-built exteriorbeam–column joints reinforced by plain round bars. Bulletin of the NewZealand Society for Earthquake Engineering 2001;34(1).

[10] Calvi GM, Magenes G, Pampanin S. Relevance of beam–column joint damageand collapse in RC frame assessment. Journal of Earthquake Engineering 2002;6:75–100. [special issue 1].

[11] Pinto AV, Verzeletti G, Molina J, Varum H, Pinho R, Coelho E. Pseudo-dynamictests on non-seismic resisting RC frames (bare and selective retrofit frames).EUR Report No. 20244 EN, ELSA, JRC, EC. Ispra (Italy); 1999.

[12] Pinto AV, VarumH,Molina J. Experimental assessment and retrofit of full-scalemodels of existing RC frames. In: Proceedings of the 12th European conferenceon earthquake engineering, EAEE. London: Elsevier Science Ltd.; 2002. Paperno. 855.

286 J. Melo et al. / Engineering Structures 33 (2011) 273–286

[13] Varum H. Seismic assessment, strengthening and repair of existing buildings.Ph.D. thesis. Aveiro (Portugal): University of Aveiro; 2003.

[14] Verderame GM, Carlo GD, Ricci P, Fabbrocino G. Cyclic bond behaviour ofplain bars. Part II: analytical investigation. Construction and BuildingMaterials2009;23:3512–22.

[15] CEB. RC elements under cyclic loading. State-of-the-art report. Thomas TelfordPublications (Editor); 1996.

[16] Fabbrocino G, Verderame GM, Manfredi G. Experimental behaviour ofanchored smooth rebars in old type reinforced concrete buildings. EngineeringStructures 2005;27:1575–85.

[17] Feldman LR, Bartlett FM. Bond strength variability in pullout specimens withplain reinforcement. ACI Structural Journal 2005;102(6):860–7.

[18] Verderame GM, Ricci P, Carlo GD, Manfredi G. Cyclic bond behaviour of plainbars. Part I: experimental investigation. Construction and Building Materials2009;23:3499–511.

[19] CEB-217. Bulletin d’Information N. 217 — selected justification notes. ComitéEuro-International du Béton; 1993.

[20] Kwak HG. Improved numerical approach for the bond–slip behavior undercyclic loads. Structural Engineering and Mechanics 1997;5(5):663–77.

[21] Manfredi G, Pecce M. A refined R.C. beam element including bond–sliprelationship for the analysis of continuous beams. Computers and Structures1998;69:53–62.

[22] Youssef M, Ghobarah A. Strength deterioration due to bond slip and concretecrushing in modeling of reinforced concrete members. ACI Structural Journal1999;96(6):956–66.

[23] Limkatanyu S, Spacone E. Reinforced concrete frame element with bondinterfaces. I: displacement-based, force-based and mixed formulations.Journal of Structural Engineering 2002;128(3):346–55.

[24] Millard A. CASTEM 2000 — guide d’utilisation. Rapport CEA 93/007. Saclay(France); 1993.

[25] Open system for earthquake engineering simulation (OpenSees).http://opensees.berkeley.edu/.

[26] Cervenka J, Jendele L. Atenauser’smanual—part 1–7. Prague (CzechRepublic):Cervenka Consulting Ltd; 2000–2002.

[27] Zhao J, Sritharan S. Modelling of strain penetration effects in fibre-basedanalysis of reinforced concrete structures. ACI Structural Journal 2007;104(2):133–41.

[28] Mazzoni S, McKenna F, Scott MH, Fenves GL. OpenSees command languagemanual. Berkeley (USA): Pacific Earthquake Engineering Research Center,University of California; 2007.

[29] Hognestad E. A study of combined bending and axial load in reinforcedconcrete. Bulletin series 339. Illinois (USA): Univ. of Illinois Exp. Sta.; 1951.

[30] Guedes JM. Seismic behaviour of reinforced concrete bridges. Modelling, nu-merical analysis and experimental assessment. Ph.D. thesis. Porto (Portugal):Faculty of Engineering of the University of Porto; 1997.

[31] Menegotto M, Pinto P. Method of analysis for cyclically loaded reinforcedconcrete plane frames including changes in geometry and non-elasticbehaviour of elements under combined normal force and bending. In: IABSEsymposium: resistance and ultimate deformability of structures acted on bywell defined repeated loads, Final Report. Lisbon (Portugal); 1973.

[32] CEB-FIPMC-90.Model code for concrete structures. Comité Euro-Internationaldu Béton; 1990.

[33] FIB. Task Group 2.5: bond models. bond of reinforcement in concrete. State-of-the-art report. FIB Bulletin 10. Lausanne; 2000.

[34] CEN. Eurocode 2: design of concrete structures — part 1–1: generalrules and rules for buildings. Brussels (Belgium): European Committee forStandardization; 2004.

[35] RBA. Regulamento do betão armado. Decreto no. 25948 de 16 de Outubro de1935. Imprensa Nacional de Lisboa; 1935.