Tailor Made Concrete Structures – Walraven & Stoelhorst (eds)© 2008 Taylor & Francis Group, London, ISBN 978-0-415-47535-8

Influence of bond-slip on the behaviour of reinforced concrete beam tocolumn joints

M.L. Beconcini, P. Croce & P. FormichiDepartment of Structural Engineering, University of Pisa, Pisa, Italy

ABSTRACT: The bond-slip correlation proposed by CEB is the most commonly adopted to predict slip dis-placements at the interface between steel bars and concrete.As known, the CEB model is based upon experimentalresults, obtained from pull-out tests performed on bars anchored by concrete for a limited length (usually 5 φ).In real beams the steel bar is usually bonded for a much longer length, and even in presence of concrete cracks,the maximum slip does not generally reach the values predicted by the CEB curve. An evidence of the above hasbeen observed analysing experimental data, registered during a wide campaign for testing full scale r.c. beamto column joints, carried out by the Authors. Consequently, it has been calibrated a modified τ − s curve andan innovative test arrangement has been proposed, able to investigate bond-slip correlation for rebars in actualanchoring conditions.

In the paper they are illustrated some promising preliminary results, obtained with the proposed testing setup, confirming the experimental evidence about the significant modification of the bond-slip curve with respectto the CEB one.

1 INTRODUCTION

In the framework of a research programme aimedto investigate the cyclic behaviour of r.c. beam tocolumns joints (Bartelletti et al. 2004), a specific studyhas been addressed to the definition of an analyticalprocedure able to predict the experimental bendingto rotation diagram of the beam, taking into accountslip-bond effects.

In the first phase of this study, it has been inves-tigated only the first loading cycle, leading to theformation of a plastic hinge in the beam, at theconnection with the column.

It must be stressed that the theoretical analysis ofr.c. elements in the cracked state, under the hypoth-esis of full bond condition, even considering nakedsteel bars, leads to a significant overestimation, up to400% and more, of the stiffness of the actual r.c. ele-ments, therefore the stiffness loss, caused by localisedcracking and steel-concrete relative slip, cannot bedisregarded.

In order to obtain a satisfactory fit of experimentalmoment to rotation curves, a suitable numerical iter-ative algorithm, described in the following, has beenset-up. Starting from an appropriate bond-slip model,τ − s, the algorithm allows to localise the cracks inthe beam as well as to calculate stresses both in the

concrete and in the rebars, during the whole loadingprocess.

The correlation between bond, τ, and slip, s, hasbeen widely investigated in the past (e.g. Eligehausenet al. 1999). Results of these investigations are thebackground of the commonly adopted slip-bond CEBmodel (Bulletin, 2000), to which reference has beenmade at the beginning of the present research in theimplementation of the algorithm.

Actually, numerical results derived using the CEBmodel do not fit the experimental ones. The dis-crepancies can be mainly ascribed to the differentbond conditions of the rebar in the standard pull-out test or in the beam-test, rather than in the actualconcrete beam, where the rebar itself extends for asufficient length to develop full anchorage, allow-ing for much lower slips than those foreseen by theCEB model.

The numerical algorithm has so been parametricallycalibrated, varying the τ − s model, in such a way toachieve an acceptable fit of the experimental bendingto rotation diagrams.

These results shifted the attention of the Authors tothe study of an improved test set up, allowing for moreaccurate measurement of the bond stresses in a fullyanchored rebar, better simulating its actual restraintconditions.

533

Figure 1. Test set up.

In the paper the proposed test set up and somepreliminary promising results are illustrated.

2 R.C. JOINTS BEHAVIOUR

2.1 The experimental campaign

The experimental campaign (Bartelletti et al. 2004) on22 full scale r.c. joints was carried out at the Depart-ment of Structural Engineering of the University ofPisa in 2003–04.

The main scope of the research was the investigationof the influences of concrete class and of reinforce-ment detailing in seismic behaviour of beam to columnjoints, designed according to EC8. The results anal-ysed in the present paper refer to the 11 specimensprepared with concrete class C50/60.

During each test, performed according to the typicaltest arrangement illustrated in figure 1, beside the mea-surements of loads and displacements of end sections,they were also registered the relative beam to columnrotations through displacement transducers, placed atthe elements’ intersections.

The first cycle of the loading process consisted infour loading-unloading steps, ending with the forma-tion of plastic hinges in the beams, both for positiveand negative bending moments.The subsequent cycleswere obtained iterating the first one and increasingprogressively the displacements of the beams’ endsections.

2.2 The analytical model

Slip-bond analytical model, currently adopted inliterature, is derived considering equilibrium and

Figure 2. Equilibrium of the r.c. tie.

compatibility conditions of a r.c. infinitesimalcylindrical tie of length dx, limited on one side bya crack and formed by a rebar and the surroundingportion of concrete (Ac) (see fig. 2).

The equation governing the global equilibrium ofthe tie, at the crack level is

being F the total tensile force applied to the tie and Fsand Fc the quotas ofF , acting on steel and on concrete,respectively.

Differentiating, and assuming Ac to be constant,equation (1) gives

where ρ is the reinforcement ratio, referred to theconcrete area Ac.

Said τ the bond stress, function of the slip s, localequilibrium of the rebar over the length dx gives

and the compatibility condition

from which

Combining (2), (4) and (6), the following non lineardifferential equation is obtained

which can be solved numerically, once input the σ − εrelationships for both steel and tensile branch ofconcrete, as well as the τ − s law.

534

Figure 3. Slip bond correlation proposed by CEB (poorbond).

2.3 Constitutive relationships

The constitutive laws for concrete under tensilestresses and for steel were modelled by the Ramberg-Osgood formulations

setting a and n to fit the experimental σ − ε diagrams.Concrete ultimate strain was set as εcu = 0.115‰.As said, in the first elaboration, it was adopted the

slip-bond law proposed by CEB for concrete in poorbond conditions, as illustrated in figure 3. More detailscan be found in Beconcini et al. 2007.

2.4 Numerical algorithm

Assuming that effective area of concrete increasesfrom zero to Ac, according to an exponential law,within a distance from the crack equal to the distancebetween two adjacent bars, solution of equation (7) canbe implemented via finite differences method.

The process starts assigning the initial slip s0 andits first order derivate s′

0 at the first crack, localised atthe beam-column intersection. In this section, whereεc0 = 0, the value of s′

0 is linked to the steel strainεs0 through equation (6). The value of s0can be deter-mined as a function of the strain εs0. A valid solutionis achieved by a trail and error iterative procedure,when at the distance La from the crack, where s = 0,the condition εc = εs is satisfied. The distance La is theanchorage length, and its value increases, together withthe concrete tensile stress, accordingly to the strains inthe rebar εs0.

The limit condition is achieved as soon as the ten-sile stress in the concrete reaches its ultimate value,when a second crack opens at a distance Lf fromthe first one (see fig. 4). Once the second crack isopened it is possible to determine the steel-concrete

Figure 4. Slips and strains at opening of the 2nd crack andafter.

slip at each crack, applying backward the procedureillustrated above.

In this way it is possible to calculate, for eachvalue of the bending moment, the relevant param-eters, needed for deriving bending moment-rotationdiagrams, to be compared with experimental ones.

2.5 Bending moment-rotation diagrams

In figure 5 the experimental bending moment–rotationdiagram for one specimen is compared with the oneobtained by the numerical algorithm described above,adopting the CEB slip-bond law.

In the figure, the curve labelled CEB1 presents anevident discontinuity, with a sudden rotation reduc-tion occurring at the opening of the second crack inthe beam, located at a distance of 125 mm from thecolumn. This discontinuity is due to the concrete con-traction at cracking: in fact, for rotation calculation,deformations are integrated over 90 mm, according tothe transducers experimental set-up. Obviously, if thebase length in the model would have been taken greaterthan 125 mm, in order to include the second crack, thediagram would have anyway presented a discontinu-ity (opposite side), as illustrated by the curve CEB2.There is no experimental evidence of such disconti-nuities; furthermore diagrams thus obtained (CEB1and CEB2), closer to the experimental ones than thatobtained under full bond hypothesis, still overestimatethe element stiffness, in particular for higher values ofbending moment.

Calculated slip values are much lower than thoseassociated with softening branch of CEB model,so that it seems practically impossible to reach theultimate limit state for bond, even at yielding of rebars.

These results ask for an improved bond model, capa-ble to fit more accurately the experimental curves.To this aim a parametrical study has been performed,obtaining the τ− s law shown in figure 6, which leadsto a theoretical bending moment–rotations diagram,

535

Figure 5. Bending moment-rotation, comparison of theoretical and experimental diagrams.

Figure 6. Modified bond-slip law.

very close to the experimental one, as illustrated in thesame figure 5.

3 EXPERIMENTAL EVALUATION OF THEBOND-SLIP RELATION

The relevant differences between the proposed bond-slip law and the CEB model put in evidence the needof further experimental investigation of the problem.

In pull-out tests, even if modified to limit thedisturb due to local compressive stresses in con-crete (Tastani & Pantazopoulou, 2002), the rebar isbonded for a limited length (usually 5 φ). This kindof test, in Authors’ opinion, does not reproduce theactual condition of a rebar embedded in a r.c. beam,which is usually fully anchored at its ends. Starting

from this remark, it has been studied an original testarrangement, suitable for the investigation of bond-slipbehaviour of fully anchored rebars.

The specimen, to be tested in tension, consists of aribbed steel bar, whose central part is embedded in aconcrete cylinder, leaving the bar’s ends free for theclamping in the testing machine.

Before concreting, the bar is instrumented with aconsiderable number of strain gauges, appropriatelyplaced in order to register steel strain all along theembedded segment and out of it. To avoid the mea-sure’s alterations due to local effects, the strain gaugesare placed into two opposite longitudinal grooves,obtained by milling the bar.

The total elongation of the concreted part and thelocal slip at its ends are measured through displace-ments transducers. The local strains of concrete arealso measured by means of appropriate strain gauges,applied to the surface of the cylinder.

An example of such specimen is shown in figure 7.In this case the concrete 13,2 cm diameter cylinder is100 cm in length, and the embedded φ16 mm bar is140 cm in length.

Global deformations are measured by four LVDTsA-type (fig. 7), while the relative slip is measured bya couple of LVDTs B-type (fig. 7).

The milled grooves are 2.5 mm deep, 3.3 mm wideand 1000 mm long. The strain gauges, whose measur-ing grid is 3 mm long, are spaced every 25 mm, andcover all the grooved segment.

Two couples of additional strain gauges are alsoplaced on the naked ends of the bar immediatelyoutside the concrete.

536

5

2,5

concrete

100

140

steel bar

1010

LVDT-A

LVDT-A

LVDT-B

13,2

strain gauges

LVDT-B

LVDT-B

LVDT-B

strain gauges2,5

3,3

Figure 7. Bond-slip test specimen.

Figure 8. Specimen under preparation.

The tensile load is increased monotonically up to astabilised cracking pattern, and further till the inelasticstrain of the steel bar reaches a suitable value.

The knowledge of steel strains along the rebarallows to calculate, through equation (1) and con-stitutive relations (8), the stresses and strains in theconcrete. From the equilibrium equation (4), appliedto the generic 25 mm tie segment, it is possible to cal-culate the bond stresses τ via numerical integration.Finally the slip s in each section can be calculatedthrough the numerical integration of equation (6) andthe τ − s relationship consequently derived.

Before the occurrence of the first crack, the bound-ary condition on s is trivial, as s = 0 where τ = 0.

Calculated εc strains can be checked with thosemeasured by the strain gauges applied on the concretesurface.

After cracking the boundary condition is derived bythe measurements of the B-type LVDT transducers,which give the relative slip at both the end sections ofthe cylinder.The crack opening can be checked againstthe total elongation of the concrete cylinder, measuredby the A-type LVDT transducers.

The feasibility of the above mentioned specimenand the effectiveness of test procedure have beenproved by a preliminary test carried on a simplifiedspecimen, as described in the following.

The full specimen is currently under preparation andin figure 8 it is illustrated a detail of the milled groove,with strain gauges.

4 PRELIMINARY TEST RESULTS

A preliminary test has been carried out on a simpli-fied specimen, having the overall geometry illustrated

Figure 9. Specimen after the preliminary test.

in figure 7, equipped with a lower number of straingauges (spacing 50 mm) directly placed on the smoothsurface of the ribbed steel bar, without any milling. Afurther simplification consists in the adoption of onlyone B-type LVDT.

The load was increased, in a first step, up to 83 kN;at that time the crack pattern was stabilised and fivecracks opened; afterwards, the load was decreaseddown to 16 kN, increased again well beyond theyielding point of the bar. In figure 9 it is illustratedthe specimen at the end of test.

537

Figure 10. Load-elongation diagrams.

Figure 11. Total slip curves.

Figure 12. εc, εs strain along the specimen.

In figure 10, the experimental load-elongation dia-gram of the r.c. cylinder, given by the A-type LVDTs(curve A), is compared with the diagrams correspond-ing to the linear elastic behaviour of the r.c. tie, underthe hypothesis of perfect bond (curve B), and to theelasto-plastic behaviour of the naked bar (curve C).Obviously the sudden increases of the elongation, reg-istered in the curve A, correspond to the opening ofthe cracks.

In figure 11 the total slip, obtained integrating thefunction s starting from the measures of the straingauges (curve D), is compared with the correspondingslip, deduced as difference between curves B and A of

Figure 13. Bond stress τ along the specimen.

Figure 14. Slip s along the specimen.

figure 10 (curve E); in figure it is also reported thelocal slip measured by B-type LVDT (curve F).

Comparing the curves in figure 11, some consider-ations can be done:

– curves (D) and (E) are practically coincident, con-firming that the test apparatus is suitable for thescope;

– when the first crack opens, the A-type LVDTs mea-sure an increase, while the B-type LVDT meaures adecrease in the slip, according to the expectations.

Some relevant results, for different load levels (20 to33.4 kN), are illustrated in figures 12, 13 and 14, whereon the abscissa it is indicated the distance x, in cm,from the end of the cylinder. In figure 12 they areshown the strains in steel and concrete, in figure 13the bond stress τ in N/mm2, and in figure 14 the slips, in mm, along the specimen.

Finally, in figure 15, it is illustrated the τ − scorrelation diagram deduced from values diagram-matically reported in figures 13 and 14, taken at theinterval between adjacent strain gauges near the endof the r.c. tie.

5 CONCLUSIONS

Starting from the analytical modelling of r.c. beamto column joints, it emerged the need to further

538

Figure 15. τ − s correlation.

investigate the bond-slip correlation. Consequently animproved bond-slip model is being studied on thebasis of a new proposed testing arrangement, able tobetter simulate the actual conditions of rebars in r.c.elements.

The proposed approach and test apparatus for theevaluation of the relevant mechanical parameters gov-erning the phenomenon, appear to be suitable for thescope.

Even if the illustrated results are to be consideredvery preliminary and affected by some approximations,due to the reduced amount of employed instruments,

they confirm that the maximum bond stress isassociated with very small slip values, if comparedwith the CEB model, as well as the softening branchof the curve.

As said, in the near future it is planned to refine thetest arrangement, using fully instrumented specimens,widening also the field of investigation, consideringdifferent concrete classes and bars’ diameters.

REFERENCES

Bartelletti, R. Beconcini, M.L. Favilli, A. Formichi, P. Froli,M. Nicotera, R. 2004. Ricerca sperimentale su nodi di telaiin cemento armato sotto azioni cicliche. 11◦ Congr. Naz.“L’Ingegneria Sismica in Italia”, Genova, Italy.

Beconcini, M.L. Croce, P. Formichi, P. Spadoni, I. 2007.Influenza del legame di aderenza sul comportamentoelasto-plastico di nodi trave-pilastro in cemento armato.Conv. AICAP, Salerno, Italy.

CEB-FIP-Bullettin 10, 2000. “Bond of Reinforcement inconcrete”.

Eligehausen, R. & Bigaj-van Vliet, A. 1999. Bond behaviourand models. Structural Concrete, the Textbook onBehaviour, Design and Performance (fib Bulletins 1, 2, 3).

Tastani, S.P. & Pantazopoulou, S.J. 2002. Experimental eval-uation of the direct tension – pullout bond test. Bond inConcrete – from research to standards, Budapest.

539