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1 INTRODUCTION

Brittle fractures of reinforced concrete members in-clude bond fracture and anchorage fracture. The fullunderstanding of bond fractures is, however, verydifficult owing to the interactions of microscopicstress transmission, the constitutive law of each ma-terial, linked fractures at the macroscopic level, etc.In recent years, the Finite Element Analysis (FEA)has become a useful tool to determine the behaviorof reinforced concrete members. The FEA is, how-ever, significantly affected by the model of bond be-havior, specifically the preconditions in a bond-softening region considerably affect the accuracy ofthe solution. The Japan Concrete Institute TechnicalCommittee on Bond in Concrete Structures carriedout research activities from 2009 to 2011 aiming tograsp the bond behavior on a microscopic scale, thusto review the information relating to bond behaviorand to make a proposal for using the informationspecifically through determination of the bond con-stitutive law in FEA aimed at its practical applica-tion.

In this paper, the results of the following twomain research programs are reported. The one is themeso-scale analysis using FEA and Rigid BodySpring Model (RBSM) by the modeling of bearing

between lugs of reinforcements and concrete, andthe other is the actual applications of bond modelsinto FEA of RC members. The objectives of theseprograms are to understand the bond mechanisms ofreinforcements and bond stress - slip relationships,and to abstract the problems of bond-link model inFEA.

2 MESO-SCALE ANALYSIS OF BONDBEHAVIOR

Studies began as early as the '80s to determine thebond behavior through direct modeling analysis ofmechanical mating between nodes of a reinforcingbar and concrete, which is the essence of the bondbetween a deformed reinforcing bar and concrete.(Ingarffea et at. 1984, Kokusho et al. 1981,Reinhardt et al. 1984, etc.) According to the analy-sis, the τ-s relation is not regarded as a constitutivelaw, and the bond phenomenon is reproduced as aresult of the propagation of fracture of concrete pe-ripheral to the reinforcing bar, such as the internalcracks occurring from the front end of a node andthe plasticizing of concrete at the front side of thenode. Consequently, it is expected that the analyticalresults give a macroscopic bond model.

Japan Concrete Institute TC Activities on Bond Behavior andConstitutive Laws in RC

(Part 3: Application of Constitutive Laws for FEA)

T. KanakuboUniversity of Tsukuba, Tsukuba, Japan

Y. SatoKyoto University, Kyoto, Japan

Y. UchidaGifu University, Gifu, Japan

K. WatanabeRailway Technical Research Institute, Kokubunji, Japan

H. ShimaKochi University of Technology, Kochi, Japan

ABSTRACT: Finite Element Analysis (FEA) is the one of the effective simulating methods to represent thebehavior of Reinforced Concrete (RC) members and structures. The Japan Concrete Institute TechnicalCommittee on Bond in Concrete Structures has conducted the following two main research programs in orderto discuss the effective usages of bond models in FEA. The one is the meso-scale analysis using FEA andRigid Body Spring Model (RBSM) by the modeling of bearing between lugs of reinforcements and concrete,and the other is the actual applications of bond models into FEA of RC members. The objectives of these pro-grams are to understand the bond mechanisms of reinforcements and bond stress - slip relationships, and toabstract the problems of bond-link model in FEA.

Bond in Concrete 2012 – General Aspects of Bond J. W. Cairns, G. Metelli and G. A. Plizzari (eds)

2012 Publisher creations, ISBN: 978 - 88 - 907078 - 1 - 0

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for the boundary elements are shown in Figure 6.The enough-lower stiffness for axial direction sur-face is set for the boundary element A.

Figure 7 shows the relationship between stress ofreinforcing bar and whole elongation comparing ex-perimental results and FEA. Crack stress differs be-tween experiment and FEA. The reason is consid-ered that the shrinkage stress occurs in the specimen.However outline of FEA curve agrees with the ex-

perimental result. Figure 8 shows bond stress versusrelative slip curves between reinforcing bar elementsand concrete elements. The curves are plotted by thepositions from the loaded end of the specimen. Ex-cept for the curves of 0.5D, 1.5D (D=bar diameter)curves, curves behave similarly. After the crack oc-curs, the bond stresses decrease and become show-ing negative bond stresses (opposite direction). Thedifferences of 0.5D and 1.5D curves from other

Figure 4. Tensile bond specimen for FEA to perform meso-scale analysis. The cracks with the interval of approximately 500mmwere observed in the loading test. (unit: mm)

Figure 5. FE model of specimen and reinforcing bar, boundary elements

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Figure 6. Constitutive laws for boundary elements. Figure 7. Stress – elongation relationships.

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Figure 8. Bond stress – relative slip relationships plotted by the positions from the loaded end of specimen.

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curves are due to cone shaped crack which takesplace at the end of specimen.

3 ACTUAL APPLICATIONS OF BONDMODELS INTO FEA

The results of two series of FEA are reported in thischapter. Both FEAs are conducted to study the effectof bond models to the behavior of RC memberscomparing the experimental results. At first, an uni-axial tensile test is investigated. This is a basic ex-perimental method to investigate the bond character-istics between a reinforcing bar and concrete..

3.1 FEA of uniaxial tensile test for prism specimen

Engineers and researchers often use the bond linkelements or other similar elements in FEA to expressthe bond slip between reinforcing bar and concrete.On the other hand, the bond behavior is indirectlyexpressed by the tension stiffening model of con-crete in case of smeared crack model. The tension

stiffening model is boldly modeled of tensile stressof concrete transmitted by the bond. If the bond linkelements and tension stiffening model are simulta-neously used, it is anxious that the effect of bond isdouble counted as shown in Figure 9. In this section,“double counted effect” is investigated by the FEAusing both bond link elements for discrete reinforc-ing bar and tension stiffening model for concretecomparing with the experimental results.

3.1.1 Experiments and target specimensThe target specimens for FEA are uniaxial tensileprism specimens tested by Kanakubo et al. (2009)with square section of 80 mm, 100 mm and 120 mm.A deformed bar of 10 mm (D10) in diameter is ar-ranged at the center of the section, and the rein-forcement ratios () are 1.23 %, 0.79 % and 0.55 %,respectively. The length of specimen is 1600 mm.The four-node plane elements, two-node truss ele-ments and four-node bond link elements are used.

Before the investigation of double counted stress,the tension stiffening curve is modeled by an inte-gration calculation based on the assumption of bondstress – slip relationship as shown in Figure 10. The

Figure 9. Effect of tension stiffening model for uniaxial tensile specimen. Double counted stress is included by tension stiffeningand bond link model.

Figure 10. Assumption of bond stress – Figure 11. Comparison of tensile load – elongation curve between test resultsslip curve. and analytical results.

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calculation method is the integration of the one-dimensional elements under the force equilibriumcondition and compatibility condition including slipbetween reinforcing bar elements and concrete. Thecalculated region is between cracks. If the tensilestress of concrete elements becomes bigger than thetensile strength of concrete, new crack takes place atthe center of the region, and the length of calculatedregion is set to half length. The calculation is con-tinued until the new crack does not occur. The re-sults of calculation are shown in Figure 11 compar-ing with the test results. The shape of tensile load –elongation curves from the calculation almost agrees

with the test results. Figure 12 shows the analyticalresults of tensile concrete stress which is obtained bythe subtraction of the force of reinforcing bar fromthe whole tensile load, and its modeling by multi-linear model.

3.1.2 FEA modeling and resultsTwo series of FEAs by smeared model and discretemodel are performed. Figure 13 shows the modelingfor both cases. In the smeared-based analysis, thetension stiffening model shown in Figure 12 isadopted for all concrete elements. The bond linkmodel is also used. The sizes of elements range from

Figure 12. Tension stiffening model from integration calculation results. Tensile stress of concrete is modeled by multi linear.

Figure 13. Elements for FEA, (a) smeared concrete elements with bond link, (b) discrete elements in which only central elementbehaves under tension stiffening model.

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12.5 mm to 1600 mm (8 cases). In the discrete-basedanalysis, only the central element behaves under thetension stiffening model. The other elements are setto show elastic manner without bond link. After thecracking at the central element, the half lengthmodel is newly adopted. The same progress is con-tinued until no crack occurs.

The analysis results are shown in Figures 14 and15 for the smeared-based model and the discrete-based model, respectively. From the results of thesmeared-based model, it is observed that FEA re-sults show the almost similar curves with tension

stiffening model shown in Figure 12. This meansthat “double count” of tension stiffening effect doesnot occur. In case of smeared model, all elements forconcrete show the same strain, so that bond linkdoes not work. The use of combination of smearedconcrete element, discrete bar element and tensionstiffening model is effective only for expressing thelarge scale of slip of reinforcing bar such as slip outof bar from concrete. From the results of the discretebased model, the obtained curves show almost thesame tendency with the tension stiffening model.However, the crack spacing by the analysis (25 mm

Figure 14. Analysis results by smeared-based model shown by concrete stress.

Figure 15. Analysis results by discrete-based model shown by concrete stress.

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in case of =1.23%) differs from the experimentalresult (100 mm). This difference is due to the “dou-ble count” of tensile stress. Thus, the effect of dou-ble count appears as the difference of crack spacing.

3.2 FEA of RC beams failed by bond splitting

FEAs for RC beam specimens having a small shearspan ratio (a/d) under an anti-symmetrical bendingmoment are conducted to investigate the adaptabilityof bond models. It is known that small span ratiobeams under anti-symmetrical bending momentshow the cracks along the longitudinal bars withsplitting crack of concrete. This failure is called asbond splitting failure. The objective is to solve thebehavior of bond splitting beams considering bondstress transmission mechanism between reinforcingbar and concrete. The sectional size of beams is 400mm in effective height and 300 mm in width. Theshear span ratios are 1 (DB408), 1.5 (DB608) and 2(DB808), respectively. Four deformed longitudinalbars of 29 mm in diameter are arranged while de-formed stirrups of 13 mm in diameter are used with100 mm spacing. The concrete compressive strengthranges from 28.9 MPa to 29.3 MPa.

3.2.1 Analysis by DIANAThe analysis uses the commercially available FEAsoftware DIANA, and applies the model of smearedcracks - discrete reinforcing bars. The model appliedto the concrete is the fixed crack model in which theshear stress - shear strain relation is used consideringcompatibility with the loading experiment. Thenonlinear constitutive law for the concrete adoptedthe Thorenfeldt model and the Hordijk model, which

considered the fracture energy. To express the inter-action between the concrete and the longitudinal re-inforcing bar, a trial analysis was performed underthe condition that the bond characteristic is assumedas perfect bond, using the τ-s curve proposed byShima et al. (1987) and Suga et al. (2001).

Figure 16 shows examples of analytical results. Inthe analysis for DB408, many slant cracks occur atthe center of beams. Since the yield of stirrup is ob-served at the peak load, the analysis presumably canpursue the failure mode similar to experiment. Onthe other hand, for DB608 and DB808, bendingshear cracks and cracks along the longitudinal direc-tion of the reinforcing bar occur. However, manyslant cracks are observed at the center of the beam,showing a crack pattern different from that observedin the experiment.

3.2.2 Analysis by FINALFEA is conducted applying the bond model of Goto,Morita and Fujii (1981) to four-node bond link ele-ments, and coercive bond splitting cracks. The bondmodel is a multi-linear curve expressing the bondbehavior which accompanies the bond splittingstrength and slip based on many experimental re-sults. In FEA, the simplest method for expressingthe deterioration of bond is to insert the bond linkbetween the reinforcing bar elements and the con-crete elements. However, the deteriorating of bondwhich actually propagates around the reinforcing barin three-dimension is not modeled by bond link. Inthis FEA, a user subroutine to cause coercive bondsplitting cracks linking to the Goto, Morita and Fujiimodel is applied to the bond link.

Figure 17 shows the relationships between theshear force and the displacement and the crack pat-

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Figure 16. Analysis results of bond splitting failed beams by DIANA.

Shear force (kN) Shear force (kN) Shear force (kN)

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ExperimentAnalysis

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terns. In the shear force-displacement relationships,the thin lines represent the test results, the thick linesthe analyses without the splitting cracks, and the dot-ted lines the analyses with the splitting cracks. Sincethe splitting cracks appear after the peak load, thepre-peak behaviors are the same whereas the post-peak behaviors differ. The differences, however, arenot significant in the shear force-displacement rela-tionships. Regarding the cracks at the peak loads, theexperiments showed regions without the cracks nearthe center of the beam. In the analysis, however,cracks are generated over the entire span.

In the experiment, before growing the shearcracks over the entire region of the span, splittingcracks were presumably generated along the longi-tudinal bars, thus a region of no cracks remained.The objective of introducing splitting cracks in theanalysis is to reproduce the above phenomenon. Asillustrated in Figure 17, however, many shear crackshave already formed at the point of reaching maxi-mum load, and an effect of suppressing shear cracksby introducing splitting cracks is not observed.

4 CONCLUSIONS

In the meso-scale analyses of the reinforcing bar andsurrounding concrete, a possibility is found to un-derstand the bond behavior and the bond models byRBSM and FEA. By the FEA of uniaxial tensile testof RC prism, the "double count" tensile stress byadopting both the tension stiffening model and thebond link element is discussed. In the FEA for thebond splitting beams, it is difficult to simulate thecrack patterns caused by the bond splitting althoughthe shear force-displacement behaviors are wellsimulated.

5 ACKNOWLEDGEMENT

The authors fully acknowledge the members of Ja-pan Concrete Institute Technical Committee onBond in Concrete Structures for their dedicated ac-tivities.

6 PREFERENCES

Goto, S., Morita, S., Fujii, S., Tokuno, M., and Fukui, K.,1981: Influence of Lateral Reinforcement on Bond SplittingFailure Modes. Annual Reports of Architectural Institute ofJapan, Kinki Branch, Vol. 21, 197-200

Ingarffea, A.R. et.al., 1984: Fracture Mechanic of Bond in Re-inforced Concrete, ASCE Structural Division, Vol.110,No.4, pp.871-890

Kanakubo, T., Yamato, N., 2009: Prediction Me-thod of CrackWidth in Reinforced Concrete Members, Journal of Struc-tural and Construction Engineering, Transactions of Archi-tectural Insti-tute of Japan, Vol.74, No.640, pp.1137-1143

Kokusho, S., Hayashi, S. and Yoshida, H., 1981: Study onBond between Deformed Bar and Con-crete, Summaries ofTechnical Papers of Annual Meeting, Architectural Instituteof Japan, pp.1513-1514

Muto, S., Nakamura, H., Tanabe, T., Srisoros, W. and Lee,S.H., 2004: Analysis of Bond Character-istics BetweenConcrete and Deformed Bar by Meso-Scale Analysis, Jour-nal of Applied Me-chanic, JSCE, Vol. 7, No. 2, pp. 767-774

Reinhardt, H.W. et.al., 1984: Prediction of Bond between Steeland Concrete by Numerical Analy-sis, Material and Struc-tures, Vol.17, No.100, pp.311-320

Shima, H., Chou, L., and Okamura, H., 1987: Bond-Slip-StrainRelationship of Deformed Bars Embedded in Massive Con-crete, Journal of Japan Society of Civil Engineers, No. 378,Vol. 6，pp.165-174

Suga, M., Nakamura, H., Higai, T, and Saito, S., 2001: Effectof Bond Properties on The Mechanical Behavior of RCBeam, Proceedings of the Japan Concrete Institute,Vol.23，No.3，pp.295-300

DB408 DB608 DB808

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Figure 17. Analysis results of bond splitting failed beams by FINAL.

Experiment

Analysis

- - - - Analysis with split crack