ValidationofaModifiedSteelBarModel IncorporatingBond-Slipfor Seismic Assessment of Concrete Structures

Michele D’Amato1; Franco Braga2; Rosario Gigliotti3; Sashi Kunnath, F.ASCE4; and Michelangelo Laterza5

Abstract: In this paper the implementation and validation of a modified steel bar model including bond-slip of longitudinal bars that was pro-posed in a companion paper is discussed. The model is developed on the key assumption of linear slip field along the steel bar with differentconfigurations at the ends of the bar. The simplified model is capable of predicting the axial slip displacement with suitable accuracy comparedwith a refinedmodel but with considerably fewer computational steps. The proposedmodel avoids nested iterations in the context of fibermodeldiscretization of a section that requires the representation of all actions in terms of stress and strain. The model is applied to two componenttests—one with poor and another with improved reinforcing detailing. Findings from the simulations indicate that the proposed model ismore suitable for use in connections with poor detailing and pronounced slip in the plastic hinge zones. DOI: 10.1061/(ASCE)ST.1943-541X.0000588. © 2012 American Society of Civil Engineers.

CE Database subject headings: Reinforced concrete; Bonding; Slip; Nonlinear analysis; Concrete structures; Seismic effects.

Author keywords: Reinforced concrete; Bond; Slip; Modeling; Nonlinear analysis; Existing buildings.

Introduction

Experimental investigations carried out on RC assemblages sub-jected to lateral displacements have highlighted the importance ofincorporating bond-slip phenomenon between longitudinal bars andsurrounding concrete (Eligehausen et al. 1983; Hakuto et al. 2000;Braga et al. 2009). It influences the global response and results ina reduction in stiffness and hysteretic dissipation capacity. Usually,bond-slip problems are pronounced in structures with poor bondconditions, such as older concrete buildings generally reinforcedwith longitudinal smooth bars or having inadequate detailing.Bond-slip in these structures is typically amplified by an insufficientlap splice of bars, poor bond strength of concrete, and a low con-finement level.

Slippage becomes particularly significant under lateral loads atthe interface of beam-column joint panels or at the footing of col-umns. Within these regions (often termed the plastic hinging re-gions), important inelastic deformations mostly arise with: (1) anincrease in bond-slip of longitudinal bars with consecutive openingof large concrete cracks; and (2) considerable compressive stressesleading to microcrushing of concrete. Moreover, under cyclic loads,

the bond strength degrades and consequently the longitudinal barslippage is further increased.

In the companion paper (Braga et al. 2012), the formulation ofa simplified slippage model is presented considering both straightand hooked longitudinal bars. This paper aims to show the abilityof the proposed model to simulate the nonlinear response of RCconnections including bond-slip of steel bars with respect to thesurrounding concrete. The procedure to assign the derived bar re-lationship in terms of the corresponding stress-strain constitutive lawis discussed. The model may be implemented in any general pur-pose finite-element programs in which the section fiber state de-termination requires amaterial relationship in the formof (s, «). Thispaper also shows some comparisons with experimental results ofnonlinear analyses carried out on RC substructures.

Simplified Slippage Model

The model including bond-slip herein discussed assumes a linearslip field along the embedded bar in a concrete block and also de-scribes the response of the anchorage at the bar end by means of anelastic constitutive relationship. Details of the proposed model arepresented in the companion paper (Braga et al. 2012). The first stepin applying themodel is to calculate the anchorage lengthL0 [Eq. (1)]where the bar is in tension, and to compare it with the entire em-bedment length L

L0 ¼ D

ffiffiffiffiffiffiffiffiffiffiffiffi3p2

Es

Ed

rð1Þ

in which Ed is named bond stiffness given by

Ed ¼ tdu1

ðpDÞ ð2Þ

where td5 bond strength and u15 corresponding slip; Es and D5longitudinal bar elastic modulus and bar diameter, respectively.When L0 # L (Fig. 1), the simplified model yields a set of

1Research Fellow, Dept. of Structural Engineering, Univ. of Basili-cata, C. da Macchia Romana 85100 Potenza, Italy (corresponding author).E-mail: damato.mic@gmail.com

2Professor, Dept. of Structural Engineering, “Sapienza”—Univ. ofRome, via Eudossiana 18, 00184 Roma, Italy.

3Assistant Professor, Dept. of Structural Engineering, “Sapienza”—Univ. of Rome, via Eudossiana 18, 00184 Roma, Italy.

4Professor, Dept. of Civil and Environmental Engineering, Univ. ofCalifornia, Davis 95616, CA.

5Associate Professor, Dept. of Structural Engineering, Univ. of Basili-cata, C. da Macchia Romana 85100 Potenza, Italy.

Note. This manuscript was submitted on June 15, 2011; approved onFebruary 17, 2012; published online on February 22, 2012. Discussionperiod open until April 1, 2013; separate discussions must be submitted forindividual papers. This paper is part of the Journal of Structural Engi-neering, Vol. 138, No. 11, November 1, 2012. ©ASCE, ISSN 0733-9445/2012/11-1351e1360/$25.00.

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relationships consisting of four branches and three characteristicpoints: Point A, when the bond strength at the free end is reached(uL 5 u1) and corresponds to the upper limit of the linear branch;Point B, when the bond strength is distributed over the entire straightbar length L (u05 0); and Point C, when bond strength is reached atthe embedded end of the bar (u0 5 u1). When L0 . L (Fig. 2), thesimplified axial stress-slip relationship is described by threebranches and by only two characteristic points: Point A, when bondstrength at the free end is reached (uL 5 u1); and Point C corre-sponding to the linear slip field with u0 5 u1. Beyond Point C, bothfor L0 # L and L0 . L, the axial stress-slip relationship has a linearbehavior.Moreover, in Figs. 1 and 2 the longitudinal bar relationshipis reported regardless of the steel yielding or bar pull-out that mighttake place after Point C or somewhere else on the curve depending on

the problem boundary conditions. If the anchorage is adequate thenbar yielding will occur before the slip engages the entire embedmentlength L. Otherwise, bar yielding or pull-out will occur when theanchorage spreads along the entire bar.

Figs. 3 and 4 show different monotonic curves obtained with theproposedmodel. Fig. 3 depicts the reinforcing bar behaviorwhen theanchorage length L/D is varied. As far as the straight bar is con-cerned, the higher the L/D ratio the stiffer the axial stress-axialdisplacement (uL) relationship at the free end. Instead, when thehooked bar is applied, the longer the bar the lower the influence ofthe hook stiffness on the global bar behavior. For the case of thesufficient embedment length L/D, the monotonic relationships ofa longitudinal bar with or without a hooked end are identical (dashedcurve), whereas Fig. 4 shows different reinforcing bar relationshipsby increasing the bond strength td. The higher the bond strength td,the higher the anchorage efficiency. Moreover, in very good an-chorage condition (4td) the hook at the bar end becomes unnecessaryand also may take place in the case of a straight bar yielding.

The slip uL represents the total relative axial displacement ofa longitudinal bar with respect to the concrete block. The proposed(s, uL) law should typically be iterated for obtaining an accuratesolution in terms of equilibrium and compatibility. However, theproposed formulation without iterations provides good agreementin terms of the global axial slip uL when compared with the resultsobtained using the refined model proposed by Monti et al. (1997),as demonstrated in Figs. 5 and 6. The first set (Fig. 5) comparesthe two models for varying L/D values, whereas the second set(Fig. 6) compares the stress-slip responses for varying steel yieldstrengths.

Fig. 7 compares the predicted values of uL using the proposedsimplified model with the refined model of Monti et al. (1997).Results are presented in terms of uL for more than 50 simulations byconsidering the following values: td 5 0.5, 1.0, and 2.0 MPa;fy 5 230 and 310 MPa; L/D5 40, 80, and 160; D5 6 and 12 mm;u1 5 0.1 and 1.0 mm. The simulations considered both the case of

Fig. 1. Tensile stress-slip curve when L0 # L

Fig. 2. Tensile stress-slip curve when L0 . L

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bar pull-out as well as bar yielding. Both models provide the samefailure condition of the longitudinal bar for the range of parameters.The simplified model provide reasonable estimates of the axialdisplacement for use in nonlinear structural analysis.

Reinforcing Steel Model Including Bond-Slip

As discussed in the companion paper (Braga et al. 2012), froma computational perspective, one can distinguish different categoriesof modeling approaches with an increasing level of complexity andrefinement in describing the slippage phenomenon. The family offrame finite elements offers the best compromise between com-plexity of mathematical formulation and accuracy in simulating theglobal response of RC structures. In this approach, the overallstructure is regarded as an assemblage of interconnectedmembers inwhich nonlinearities may be described in two ways: by lumpinginelastic behavior at expected locations of yielding (typically theends of a member for seismic loading); or by assigning distributednonlinearities along each element. The latter approach assumes thateach beam/column consists of an aggregate of fibers forming theelement cross section. Lumped plastic hingesmay bemodeled eitheras fiber hinges or as discrete hinges with specified moment-rotationcharacteristics. At the fiber level, constitutive relationships toproperly model the behavior of concrete and reinforcing steel arenecessary in the form of uniaxial stress- strain relationships.

Element response is described referring to certain representativesections whose location depends on the adopted integration scheme.The chosen sections are the sampling integration points, and theelement response is obtained as theweighted sum of these individualsections.ThesectionweightwIP represents part of the element lengthL on which the element response is constant. Thus, Li is specified as

Fig. 3. Monotonic tensile stress-axial displacement at the free end ofa reinforcing bar by varying the anchorage length L/D (D 5 16 mm,fy 5 300 MPa, Ea 5 206 GPa, td 5 0.68 MPa, u1 5 0.1 m)

Fig. 4.Monotonic tensile stress-axial displacement of a reinforcing barby varying the bond strength td (D5 16 mm, fy 5 300 MPa, Ea 5 206GPa, L/D 5 40, u1 5 0.1 mm)

Fig. 5. Comparisons between the simplified model and the modelproposed by Monti et al. (1997); D5 12 mm, fy 5 230 MPa, Ea 5 206GPa, td 5 0.5 MPa, u1 5 0.1 mm: (a) L/D 5 40; (b) L/D 5 160

Fig. 6. Comparisons between the simplified model and the modelproposed byMonti et al. (1997);D5 12 mm, L/D5 40, Ea5 206 GPa,td 5 1.0 MPa, u1 5 0.1 mm: (a) fy 5 230 MPa; (b) fy 5 310 MPa

Fig. 7. Scatter diagram from over 50 simulations of the axial dis-placement uL comparing the proposed model with the model proposedby Monti et al. (1997)

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Li ¼ wIPL ð3Þ

where Li 5 part of the element in which the sectional response isassumed constant.

In the force-based formulation, Gauss-Lobatto quadrature iscommonly adopted inwhich the referring integration scheme includesthe element end sections where inelastic excursions occur under lat-eral loads. Accordingly, the proposed model may be implementedin a general purpose finite-element program on the condition that theaxial displacementuL represents the integrated axial displacement ofa longitudinal bar including both steel deformation and bond-slipalong the weighted length Li of the element ends. In doing this, thelength Li as the plastic hinging region Lpl is assumed. Under theseconditions, the uniaxial strain is defined as

« ¼ uL; totLpl

ð4Þ

where uL,tot 5 total relative axial displacement of a longitudinal barcalculated with the proposed model, and Lpl 5 plastic hinge length.Eq. (4) permits the expression of the (s, uL) relationship as a (s, «)relationship of the longitudinal steel, thereby making it convenientduring the fiber-section state determination, which is based ona section strain distribution.

The so-derived stress-strain relationship indirectly incorporatesthe bond-slip phenomenon of the longitudinal bar, because it pro-vides the reinforcing bar constitutive relationship including bothsteel deformation and anchorage slip. For this reason, it can beconsidered a pseudostress-strain relationship of the reinforcing steelwithin the plastic hinge length, where the term pseudo implies thefact that the (s, «) relationship is an abstraction of the (s, uL) lawnecessary to implement the model in a fiber-section framework.

The scheme in Fig. 8 shows the entire relative slip of a longitu-dinal bar that occurs within a concrete crack, for example, at theinterface of beam-column joint panels or at the footing of columns.By referring to this scheme, the total relative slip uL,tot in Eq.(4) maybe calculated as

uL; tot ¼ uL;A þ uL;B ð5Þ

where uL,A and uL,B 5 relative axial displacements including thebond-slip related to concrete blocks A and B, respectively.

To obtain uL,tot, one may refer to two different embedmentschemes for each part of a reinforcing bar, as represented in Fig. 9.Thes5s (uL,tot) relationshipmay be calculated by combining, as ina series system, the two relationships s5 s (uL,A) and s5 s (uL,B)of anchorage Schemes A and B, respectively.

For the sake of clarity, Fig. 10 depicts an example of the lon-gitudinal reinforcing bar relationship including the total bond-slipbetween two concrete embedment blocks. In this case, bars witha diameter of D 5 20 mm with hooked ends comprise the longi-tudinal reinforcement of a concrete column, as shown in Fig. 10(a).When inelastic action under lateral load occurs at the base, eachlongitudinal bar may be represented as a series system made of: (a)a hooked bar with an embedment length in the basement of (L/D)A540 (Scheme A); and a straight bar with an embedment length withinthe column of (L/D)B 5 150 (Scheme B). The monotonic tensilestress-axial displacement relationships based on both schemes andthe resulting relationship (s, uL,tot) (thicker line) are plotted inFig. 10(b). The results shown also compare the relationship forSchemes A and B by doubling the axial displacement.

Fig. 8. Total relative slip at the concrete crack

Fig. 9. Two anchorage schemes considered

Fig. 10. (a) Bar anchorage lengths; (b) steel relationships including slippage between the two concrete embedment blocks (D5 20mm, fy5 300MPa,Ea 5 206 GPa, td 5 0.68 MPa, u1 5 0.1 mm)

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In Fig. 10(b), when an adequate anchorage length is available,both embedment schemes provide the same relationship

�s; uL;A

� ¼ �s; uL;B

� ¼ ðs; uLÞ ð6Þ

and thus it is possible to obtain the resulting steel relationship (s,uL,tot) by simply doubling the total displacement uL. Therefore

�s; uL;tot

� ¼ ðs; 2uLÞ ð7Þ

Numerical Simulations

To implement the constitutive model for longitudinal steel fibers ina general-purpose finite-element program, the derived steel stress-strain response may be transformed from a nonlinear curve toa multilinear relationship based on the energy principle (Fig. 11).This procedure may be very useful, because existing linearizedmaterial models already available in many finite-element programs,which also include cyclic hysteretic behavior for unloading andreloading behavior, can be applied in simulating the slippagephenomenon with the proposed model.

In this section, comparisons of analytical predictions with ex-perimental results using the proposed model are presented. Com-parisons involve two different RC specimens tested under cyclicloads: an internal beam column joint (Braga et al. 2009) with poorconfinement and reinforced with smooth longitudinal bars; anda well-confined RC cantilever with deformed bars (Saatcioglu andOzcebe 1989). Analyses have been carried out by using theOpenSees software frame work (OpenSees 2009). The confinementeffects within the section core have been calculated using the BGLmodel (Braga et al. 2006; D’Amato 2009).

Specimen C11 (Brega et al. 2009)

To investigate failure mechanisms and their interaction in existingRC existing buildings made up with inadequate detailing, tests onRC subassemblages were conducted by Braga et al. (2009).Specimens were reinforced with smooth longitudinal bars andconsisted of internal and external joints (named C-joints andT-joints, respectively) and were subjected to lateral cyclic dis-placements at the top column in a quasi-static form. More details onthe experimental program and results can be found elsewhere(Gigliotti 2002; Braga et al. 2009).

In this simulation the internal beam-column joint C11 was beenchosen for comparison. During the test, a constant vertical load of270 kN (axial load ratio of 16%) was applied at the top of the uppercolumn by a hydraulic actuator without a P-Delta effect. Concretecompressive strength of f9c0 5 22.5 MPa and steel yield strengthfy 5 340.5 MPa was measured. Each column had a flexural heightof 1,250 mm and the gross section was 300 3 300 mm withfour longitudinal bars of 18-mm diameter, resulting in 1.13%reinforcement ratio. The longer and the shorter beam had a section

of 300 3 500 mm and longitudinal bars consisting of 4f12 plusa reinforcement of 18-mm bars near the joint panel. The elementshad inadequate volumetric transverse reinforcement ratio (comparedwith typical amounts in existing concrete buildings) provided bysimple hoops: 0.23% and 0.3% in beams and columns, respectively.The test was performed by applying the lateral displacement bothin the positive and negative direction at the top of the uppercolumn. At each displacement level three cycles were applied.During the tests a failure mechanism corresponding to weakcolumn-strong beam behavior was observed. Flexural hingingof each column because of one primary concrete crack at thejoint panel interface was observed. No damage in the joint paneloccurred.

For numerical simulations, beams and columns have beenmodeled in the OpenSees software (OpenSees 2009) with theBeamWithHinges element (Scott et al. 2006), which assumes thatplasticity is concentrated in a hinge length localized at both elementends. In the middle region of each element linear-elastic propertiesare assigned; whereas, over the hinge length fiber sections areadopted. The hinge length Lpl was assumed equal to h/3, where h isthe section depth. This value represents, as experimentallymeasuredby linear variable differential transformers placed along the ele-ments, the zone in which the lumped crack takes place because ofsignificant bond-slip of the longitudinal bars (Braga et al. 2012). Thepanel zone region has been modeled through rigid elements whileelastic trusses for reproducing the test apparatus have been in-troduced. The loss of compressive strength of the bars crossing thecolumns has been considered by assigning only tensile strength tothe steel constitutive relationship. This modeling criterion simulatesthe effective equilibrium conditions when the through bars are stillin tension within the contiguous column because of the inadequateanchorage conditions inside the joint panel (Hakuto et al. 1999;Calvi et al. 2002). This assumption makes the model particularlyappropriate when the level of displacement becomes significant.Fig. 12 shows the beam column joint model used.

The analytical stress-strain relationships of materials are plottedin Figs. 13, 14, and 15. Fig. 13 reports the constitutive relationshipassigned to concrete fibers. The confinement effects have been takeninto account by using the BGL model (Braga et al. 2006), and noultimate compressive strain has been defined. Because the tensilestrength influences only the early cycles and cracks occur fairly earlyin the response, no tensile strength for concrete fibers has beenconsidered. Whereas, in Figs. 14 and 15, steel stress-strain rela-tionships accounting for bond-slip behavior are depicted for columnand beam reinforcement, respectively. The bond-slip response hasbeen calculated by applying the proposed model, and the associatedstress-strain relationship has been calculated, as previously de-scribed. For the sake of completeness, Figs. 14 and 15 also displaythe steel stress-strain relationship by considering the classical fullbond assumption (dashed line). The difference in stiffness wasbecause the steel stress strain includes both the steel elongation andthe slippage phenomenon. In the cyclic analyses, the obtained steelrelationship has been assigned through the uniaxial material model

Fig. 11. Conversion of a nonlinear curve to a bilinear and trilinear relationship derived from the energy principle

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available in the OpenSees software (2009), having hysteretic be-havior for an unloading/reloading branch without any pinching. Nodegradation and pinching for stress and strain during the reloadingbranch have been considered. The required points for defining thesteel envelope curve have been calculated by applying the energyprinciple discussed previously.

Figs. 16 and 17 compare the experimental response and theanalytical simulations without (full bond) and including bond-slipbehavior, respectively. The differences between the two compar-isons are evident. Under the classical assumption of full bond be-tween longitudinal rebars and surrounding concrete fuller, cycles arepredicted with an overestimation of stiffness and strength with re-spect to the experimental ones. Moreover, in this case equal strengthin compressive and tensile branch has been assigned to the rein-forcing bars through the joint panel. Whereas, when bond-slip wasconsidered the analytical prediction provided a global responsecloser to the experimentally observed response, especially at larger

Fig. 12. Analytical model adopted for the internal joint C11

Fig. 13. Concrete relationships assigned to concrete fibers (generatedfrom material properties reported in Braga et al. 2006)

Fig.14. Stress-strain relationship assigned to columns longitudinal barsaccounting for bond-slip (D5 18mm, fy5 340.51MPa,Ea5 206GPa,L/D 5 50, td 5 0.71 MPa, u1 5 0.1 mm, Lpl 5 100 mm)

Fig. 15. Stress-strain relationship assigned to beams longitudinal barsaccounting for bond-slip (D5 12mm, fy5 340.51MPa,Ea5 206GPa,L/D 5 58, td 5 0.71 MPa, u1 5 0.1 mm, Lpl 5 167 mm)

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displacement amplitudes. The model resistance sharply drops(positive and negative cycles) after the spalling of the cover con-crete, which is particularly notable in this case because of the loss ofthe compressive strength of the longitudinal bars in the column.Slippage provides an additional deformation of the system anddelays the steel yielding of longitudinal bars. A better agreementalso in terms of energy dissipation capacity was obtained, asdemonstrated by comparing force-deformation cycles in groups(Fig. 18) at equal amplitudes. In the early cycles a slightly higherresistance and stiffness than the experimental observation werepredicted.

Specimen U4 (Saatcioglu and Ozcebe 1989)

The second comparison considers a well-detailed RC cantileverspecimen subjected to a constant axial load and a lateral displacementat the top (Saatcioglu and Ozcebe 1989). The section was square withside of 350 mm, while the flexural length is 1,000 mm (aspect ratio:2.86). Longitudinal reinforcement consisted of eight deformed bars25 mm in diameter (reinforcement ratio: 3.22%) with a yield strengthof 438MPa. Theunconfined strength f9c0 of concretewas 32MPa, anda constant axial load equal to the 18%of the axial capacitywithout theP-Delta was imposed on the column. Transverse reinforcementconsisted of single closed hoops of 10-mm diameter placed at 50-mmspacing along the column length (volumetric transverse reinforcementratio: 2.54%). The column failed in flexure.

The cantilever has been modeled with a single BeamWithHingeselement, whose properties have been summarized in Fig. 19. In thenumerical simulations the hinge length at the base has been esti-mated equal to the section depth (Lpl 5 500 mm). This value is ingood agreement with many empirical formulations published inthe literature (Park and Paulay 1975; Paulay and Priestley 1992)for evaluating the plastic hinge length. The confined concrete

Fig. 16. Comparison between experimental response and analyticalprediction by considering full-bond for longitudinal bars

Fig. 17. Comparison between experimental response and analytical pre-diction by considering proposed bond-slip modeling of longitudinal bars

Fig.18.Cycle-by-cycle (groupedbyamplitude) comparisonof experimental responseandanalytical predictionbyconsideringbond-slip of longitudinal bars

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relationship was calculated with the BGL model and is plotted inFig. 20; whereas, the steel relationship is depicted in Fig. 21. Thelongitudinal steel stress-strain has been calculated by extendingthe proposedmodel for deformed bars. In doing so, the bond stress-slip relationship was defined by using the values proposed in theModel Code (1990, 1991) for ribbed reinforcing steel. Theassigned bond strength was associated with the residual bondstrength under monotonic loading. The steel stress-strain re-lationship has the same resistance both in tension and compres-sion. The uniaxial Giuffrè-Menegotto-Pinto steel material hasbeen used without any Bauschinger effect under cyclic loading.Details on the cyclic behavior of this material are reported else-where (OpenSees 2009).

Figs. 22 and 23 compare the experimental response by assumingeither full bond assumption or slippage between longitudinal barsand surrounding concrete, respectively. While the additional deg-radation because of bond-slip was evident in the cyclic response, thesimulated response was unable to reproduce the full strength deg-radation observed in the testing. There is evidence that bond-slip ofthe longitudinal bars delays the peak resistance (positive and neg-ative) and also results in the reduction of the column stiffness duringunloading and reloading. With exception of the final set of cycles,

Fig. 19. Cantilever model considered in numerical simulations

Fig. 20. Concrete relationships assigned to concrete fibers (generatedfrom material properties reported in Saatcioglu and Ozcebe 1989)

Fig. 21. Stress-strain relationships obtained with the proposedmodel assigned to longitudinal steel fibers (D 5 25 mm, fy 5 438MPa, Ea 5 206 GPa, L/D5 20, td 5 2.83 MPa, u1 5 1.0 mm, Lpl 5350 mm)

Fig. 22. Comparison between experimental response and analyticalprediction by considering using full-bond for longitudinal bars

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a better prediction of the energy dissipated was obtained, as depictedin Fig. 24, for each group of cycles. Differences between experi-mental and numerical results in this case are probably because thesignificant strength reduction was not a consequence of bond-slipalone. This leads to the conclusion that the proposed bar model wasmore suitable for cases where the response was primarily controlledby pronounced bond-slip.

Conclusions

The classical assumption of full bond leads to an overestimation ofglobal response in terms of stiffness, strength, and thus of hystereticdissipation capacitywhen the bond-slip phenomenonwas pronounced

in a concrete structure. The implementation in a general purposenonlinear structural analysis program of the bond-slip model pro-posed in the companion paper (Braga et al. 2012) has been pre-sented. The model has been developed on the simplifyingassumption of linear slip field along the anchorage length of the barand was capable of incorporating bond-slip in the flexural hingeregions because of concrete cracks opening at the beam/columninterface or at column footings.

The model was particularly suitable for simulating the responseof concrete buildings in which slippage of longitudinal bars wassignificant under lateral loads. This arises in the structures with pooranchorage conditions and becomes more marked when smooth barsand inadequate lap splices are present. These conditions are alsocommon in existing RC structures designed only for vertical loadsand without any seismic detailing.

To simulate the response of RC components in a structuralsoftware program with nonlinear fiber elements, the total axial slipuL provided by the proposedmodelwas assumed distributed over theelement hinge length Lpl. In this way, the resulting steel relationship(s, «) including both material elongation and anchorage slip, can beassigned to the longitudinal bars in accordance with the fiber-sectionstate determination procedure. The resulting constitutive lawmay beconsidered as an equivalent stress-strain steel relationship, because italso includes the slippage contribution of longitudinal bars. The twonumerical simulations carried out show good agreement with exper-imental results, especially when system deformations become signif-icant and the slippage phenomenon dominates the global response.Finally, the proposed approach enables implementation without theneed to resort to an iterative procedure. The model formulationovercomes issues related to numerical convergence and drasticallyreduces the number of computational steps in simulating the responseof entire RC structures.

Fig. 23. Comparison between experimental response and analytical pre-diction by considering proposed bond-slip modeling of longitudinal bars

Fig. 24. Cycle-by-cycle comparison between experimental response and analytical prediction by considering bond-slip of longitudinal bars

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Notation

The following symbols are used in this paper:D 5 longitudinal bar diameter;Es 5 steel modulus;kh 5 end secant stiffness of a bend or hook;

kh* 5 end secant stiffness of a bend or hook divided bythe bar area;

L 5 bar embedment length;L/D 5 embedment length ratio;Lpl 5 plastic hinge length;L0 5 anchorage length is equal to the portion of L where

the bond-slip occurs;R 5 hook radius;s 5 abscissa along the hooked end;

uL 5 axial slip at the free end of an embedded bar in aconcrete block;

uL,TOT 5 total axial slip between two concrete blocks;u0 5 axial slip at the embedded end;u1 5 bond slip where the residual bond strength was

attained;wIP 5 integration point weight;x 5 abscissa along the straight bar;« 5 axial strain along the embedded bar;m 5 friction coefficient between concrete and steel;s 5 axial stress along the embedded bar; andtd 5 residual bond strength between the longitudinal bar

and surrounding concrete.

References

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Braga, F., Gigliotti, R., Laterza, M., D’Amato, M., and Kunnath, S.(2012). “Modified steel bar model incorporating bond-slip for seismicassessment of concrete structures.” J. Struct. Eng., 138(11), 1342e1350.

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