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250 ACI Structural Journal/March-April 2003

ACI Structural Journal, V. 100, No. 2, March-April 2003.MS No. 02-100 received March 27, 2002, and reviewed under Institute publication

policies. Copyright © 2003, American Concrete Institute. All rights reserved, includ-ing the making of copies unless permission is obtained from the copyright proprietors.Pertinent discussion will be published in the January-February 2004 ACI StructuralJournal if received by September 1, 2003.

ACI STRUCTURAL JOURNAL TECHNICAL PAPER

This study focuses on the use of explicit finite element analysistools to predict the behavior of fiber-reinforced polymer (FRP)composite grid reinforced concrete beams subjected to four-pointbending. Predictions were obtained using LS-DYNA, an explicitfinite element program widely used for the nonlinear transientanalysis of structures. The composite grid was modeled in a discretemanner using beam and shell elements, connected to a concretesolid mesh. The load-deflection characteristics obtained from thesimulations show good correlation with the experimental data.Also, a detailed finite element substructure model was developed tofurther analyze the stress state of the main longitudinal reinforce-ment at ultimate conditions. Based on this analysis, a procedurewas proposed for the analysis of composite grid reinforced concretebeams that accounts for different failure modes. A comparison ofthe proposed approach with the experimental data indicated thatthe procedure provides a good lower bound for conservativepredictions of load-carrying capacity.

Keywords: beam; composite; concrete; fiber-reinforced polymer; reinforce-ment; shear; stress.

INTRODUCTIONIn recent years, research on fiber-reinforced polymer (FRP)

composite grids has demonstrated that these products may be aspractical and cost-effective as reinforcements for concretestructures.1-5 FRP grid reinforcement offers several advantagesin comparison with conventional steel reinforcement and FRPreinforcing bars. FRP grids are prefabricated, noncorrosive, andlightweight systems suitable for assembly automation and idealfor reducing field installation and maintenance costs. Researchon constructability issues and economics of FRP reinforcementcages for concrete members has shown the potential ofthese reinforcements to reduce life-cycle costs and significantlyincrease construction site productivity.6

Three-dimensional FRP composite grids provide a mechanicalanchorage within the concrete due to intersecting elements, andthus no bond is necessary for proper load transfer. This type ofreinforcement provides integrated axial, flexural, and shearreinforcement, and can also provide a concrete member withthe ability to fail in a pseudoductile manner. Continuingresearch is being conducted to fully understand the behavior ofcomposite grid reinforced concrete to commercialize its useand gain confidence in its design for widespread structuralapplications. For instance, there is a need to predict the correctfailure mode of composite grid reinforced concrete beamswhere there is significant flexural-shear cracking.7 This typeof information is critical for the development of designguidelines for FRP grid reinforced concrete members.

Current flexural design methods for FRP reinforced concretebeams are analogous to the design of concrete beams usingconventional reinforcement.8 The geometrical shape, ductility,modulus of elasticity, and force transfer characteristics of FRPcomposite grids, however, are likely to be different than

conventional steel or FRP bars. Therefore, the behavior ofconcrete beams with this type of reinforcement needs to bethoroughly investigated.

OBJECTIVESThe objectives of the present study were: 1) to investigate

the ability of explicit finite element analysis tools to predictthe behavior of composite grid reinforced concrete beams,including load-deflection characteristics and failure modes;2) to evaluate the effect of the shear span-depth ratio in thefailure mode of the beams and the stress state of the mainflexural reinforcement at ultimate conditions; and 3) todevelop an alternate procedure for the analysis of composite gridreinforced concrete beams considering multiple failure modes.

RESEARCH SIGNIFICANCEThe research work presented describes the use of advanced

numerical simulation for the analysis of FRP reinforcedconcrete. These numerical simulations can be used effectivelyto understand the complex behavior and phenomena observedin the response of composite grid reinforced concrete beams. Inparticular, this effort provides a basis for the understanding ofthe interaction between the composite grid and the concretewhen large flexural-shear cracks are present. As such, alternateanalysis and design techniques can be developed based on theunderstanding obtained from numerical simulations to ensurethe required capacity in FRP reinforced concrete structures.

BackgroundSeveral researchers have studied the viability of three-

dimensional FRP grids to reinforce concrete members.3,5,9,10

One specific type of three-dimensional FRP reinforcement isconstructed from commercially manufactured pultruded FRPprofiles (also referred to as FRP grating cages). Figure 1 showsa schematic of the structural members present in a concretebeam reinforced with the three-dimensional FRP reinforcementinvestigated in this study.

A pilot experimental and analytical study was conductedby Bank, Frostig, and Shapira3 to investigate the feasibilityof developing three-dimensional pultruded FRP grating cagesto reinforce concrete beams. Failure of all beams tested occurreddue to rupture of the FRP main longitudinal reinforcement inthe shear span of the beam. Experimental results also revealedthat most of the deflection at high loads appeared to occurdue to localized rotations at large flexural crack widths

Title no. 100-S27

Analysis of Fiber-Reinforced Polymer Composite Grid Reinforced Concrete Beamsby Federico A. Tavarez, Lawrence C. Bank, and Michael E. Plesha

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251ACI Structural Journal/March-April 2003

developed in the shear span near the load points. The studyconcluded that further research was needed to obtain a betterunderstanding of the stress state in the longitudinal rein-forcement at failure to predict the correct capacity and failuremodes of the beams.

Further experimental tests on concrete beams reinforcedwith three-dimensional FRP composite grids were conducted toinvestigate the behavior and performance of the grids whenused to reinforce beams that develop significant flexural-shearcracking.7 Different composite grid configurations weredesigned to study the influence of the FRP grid components(longitudinal bars, vertical bars, and transverse bars) on theload-deflection behavior and failure modes. Even though failuremodes of the beams were different depending upon thecharacteristics of the composite grid, all beams failed in theirshear spans. Failure modes included splitting and rupture ofthe main longitudinal bars and shear-out failure of thevertical bars. Research results concluded that the designof concrete beams with composite grid reinforcements mustaccount for failure of the main bars in the shear span.

A second phase of this experimental research was performedby Ozel and Bank5 to investigate the capacity and failure modesof composite grid reinforced concrete beams with different shearspan-to-effective depth ratios. Three different shear span-depth ratios (a/d) were investigated, with values of 3, 4.5, and 6,respectively.11 The data obtained from this recently completedexperimental study was compared with the finite element resultsobtained in the present study.

Experimental studies have shown that due to the develop-ment of large cracks in the FRP-reinforced concrete beams,most of the deformation takes place at a relatively smallnumber of cracks between rigid bodies.12 A schematic of thisbehavior is shown in Fig. 2. As a result, beams with relativelysmall shear span-depth ratios typically fail due to rupture of themain FRP longitudinal reinforcement at large flexural-shearcracks, even though they are over-reinforced according toconventional flexural design procedures.5,7,13,14 Due to theaforementioned behavior for beams reinforced with compositegrids, especially those that exhibit significant flexural-shearcracking, it is postulated that the longitudinal bars in themember are subjected to a uniform tensile stress distribution,plus a nonuniform stress distribution due to localized rota-tions at large cracks, which can be of great importance indetermining the ultimate flexural strength of the beam. Thepresent study investigates the stress-state at the flexural-shear cracks in the main longitudinal bars, using explicitfinite element tools to simulate this behavior and determinethe conditions that will cause failure in the beam.

Numerical analysis of FRP composite grid reinforced beams

Implicit finite element methods are usually desirable forthe analysis of quasistatic problems. Their efficiency andaccuracy, however, depend on mesh topology and severityof nonlinearities. In the problem at hand, it would be verydifficult to model the nonlinearities and progressive damage/failure using an implicit method, and thus an explicit methodwas chosen to perform the analysis.15

Using an explicit finite element method, especially tomodel a quasistatic experiment as the one presented herein,can result in long run times due to the large number of timesteps that are required. Because the time step depends on thesmallest element size, efficiency is compromised by meshrefinement. The three-dimensional finite element mesh forthis study was developed in HyperMesh16 and consisted ofbrick elements to represent the concrete, shell elements torepresent the bottom longitudinal reinforcement, and beamelements to represent the top reinforcement, stirrups, andcross rods. Figure 3 shows a schematic of the mesh used forthe models developed. Beams with span lengths of 2300,3050, and 3800 mm were modeled corresponding to shearspan-depth ratios of 3, 4.5, and 6, respectively. These modelsare referred to herein as short beam, medium beam, and longbeam, respectively. The cross-sectional properties wereidentical for the three models. As will be seen later, the longi-tudinal bars play an important role in the overall behavior of thesystem, and therefore they were modeled with greater detailthan the rest of the reinforcement. The concrete representationconsisted of 8-node solid elements with dimensions 25 x 25 x12.5 mm (shortest dimension parallel to the width of the beam),with one-point integration. The mesh discretization was estab-lished so that the reinforcement nodes coincided with theconcrete nodes. The reinforcement mesh was connected to theconcrete mesh by shared nodes between the concrete and the

Federico A. Tavarez is a graduate student in the Department of Engineering Physicsat the University of Wisconsin-Madison. He received his BS in civil engineering fromthe University of Puerto Rico-Mayagüez and his MSCE from the University ofWisconsin. His research interests include finite element analysis, the use of compositematerials for structural applications, and the use of discrete element methods formodeling concrete damage and fragmentation under impact.

ACI member Lawrence C. Bank is a professor in the Department of Civil andEnvironmental Engineering at the University of Wisconsin-Madison. He received hisPhD in civil engineering and engineering mechanics from Columbia University in1985. He is a member of ACI Committee 440, Fiber Reinforced Polymer Reinforcement.His research interests include FRP reinforcement systems for structures, progressivefailure of materials and structural systems, and durability of FRP materials.

Michael E. Plesha is a professor in the Engineering Mechanics and AstronauticsProgram in the Department of Engineering Physics at the University of Wisconsin-Madison. He received his PhD from Northwestern University in 1983. His researchinterests include finite element analysis, discrete element analysis, dynamics ofgeologic media, constitutive modeling of geologic discontinuity behavior, soil structureinteraction modeling, and continuum modeling of jointed saturated rock masses.

Fig. 1—Structural members in composite grid reinforcedconcrete beam.

Fig. 2—Deformation due to rotation of rigid bodies.

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252 ACI Structural Journal/March-April 2003

reinforcement. As such, a perfect bond is assumed between theconcrete and the composite grid.

The two-node Hughes-Liu beam element formulation with2 x 2 Gauss integration was used for modeling the top longi-tudinal bars, stirrups, and cross rods in the finite elementmodels. In this study, each model contains two top longitudinalbars with heights of 25 mm and thicknesses of 4 mm. Themodels also have four cross rods and three vertical membersat each stirrup location, as shown in Fig. 3. The verticalmembers have a width of 38 mm and a thickness of 6.4 mm.The cross rod elements have a circular cross-sectionalarea with a diameter of 12.7 mm. To model the bottomlongitudinal reinforcement, the four-node Belytschko-Lin-Tsay shell element formulation was used, as shownin Fig. 3, with two through-the-thickness integration points.

Boundary conditions and event simulation timeTo simulate simply supported conditions, the beam was

supported on two rigid plates made of solid elements. Thefinite element simulations were displacement controlled,which is usually the control method for plastic and nonlinearbehavior. That is, a displacement was prescribed on the rigidloading plates located on top of the beam. The prescribeddisplacement was linear, going from zero displacement at t =0.0 s to 60, 75, and 90 mm at t = 1.0 s for the short, medium,and long beams, respectively. The corresponding appliedload due to the prescribed displacement was then determinedby monitoring the vertical reaction forces at the concretenodes in contact with the support elements.

The algorithm CONTACT_AUTOMATIC_SINGLE_SURFACE in LS-DYNA was used to model the contact

Fig. 3—Finite element model for composite grid reinforced concrete beam.

Fig. 4—Short beam model at several stages in simulation.

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between the supports, load bars, and the concrete beam.This algorithm automatically generates slave and mastersurfaces and uses a penalty method where normal interfacesprings are used to resist interpenetration between elementsurfaces. The interface stiffness is computed as a functionof the bulk modulus, volume, and face area of the elementson the contact surface.

The finite element analysis was performed to representquasistatic experimental testing. As the time over which theload is applied approaches the period of the lowest naturalfrequency of vibration of the structural system, inertial forcesbecome more important in the response. Therefore, the loadapplication time was chosen to be long enough so that inertialeffects would be negligible. The flexural frequency of vibrationwas computed analytically for the three beams using conven-tional formulas for vibration theory.17 Accordingly, it wasdetermined that having a load application time of 1.0 swas sufficiently long so that inertial effects are negligibleand the analysis can be used to represent a quasistatic experi-ment. For the finite element simulations presented in thisstudy, the CPU run time varied approximately from 22 to65 h (depending on the length of the beam) for 1.0 s of loadapplication time on a 600 MHz PC with 512 MB RAM.

Material modelsMaterial Type 72 (MAT_CONCRETE_DAMAGE) in

LS-DYNA was chosen for the concrete representation in thepresent study. This material model has been used successfullyfor predicting the response of standard uniaxial, biaxial, andtriaxial concrete tests in both tension and compression. Theformulation has also been used successfully to model thebehavior of standard reinforced concrete dividing wallssubjected to blast loads.18 This concrete model is a plasticity-based formulation with three independent failure surfaces(yield, maximum, and residual) that change shape dependingon the hydrostatic pressure of the element. Tensile andcompressive meridians are defined for each surface, describingthe deviatoric part of the stress state, which governs failure inthe element. Detailed information about this concrete materialmodel can be found in Malvar et al.18 The values used inthe input file corresponded to a 34.5 MPa concrete compressivestrength with a 0.19 Poisson’s ratio and a tensile strength of3.4 MPa. The softening parameters in the model were chosen tobe 15, –50, and 0.01 for uniaxial tension, triaxial tension, andcompression, respectively.19

The longitudinal bars were modeled using an orthotropicmaterial model (MAT_ENHANCED_COMPOSITE_DAMAGE),which is material Type 54 in LS-DYNA. Properties used forthis model are shown in Table 1. Because the longitudinalbars were drilled with holes for cross rod connections, thetensile strength in the longitudinal direction of the FRP barswas taken from experimental tensile tests conducted onnotched bar specimens with a 12.7 mm hole to accountfor stress concentration effects at the cross rod locations.The tensile properties in the transverse direction weretaken from tests on unnotched specimens.11 Values forshear and compressive properties were chosen based ondata in the literature. The composite material model usesthe Chang/Chang failure criteria.20

The remaining reinforcement (top longitudinal bars, stirrups,and cross rods) was modeled using two-noded beam elementsusing a linear elastic material model (MAT_ELASTIC) withthe same properties used for the longitudinal direction in thebottom FRP longitudinal bars. A rigid material model

(MAT_RIGID) was used to model the supports and theloading plates.

FINITE ELEMENT RESULTS AND DISCUSSIONGraphical representations of the finite element model for

the short beam at several stages in the simulation are shownin Fig. 4. The lighter areas in the model represent damage(high effective plastic strain) in the concrete material model.As expected, there is considerable damage in the shear spanof the concrete beam. Figure 4 also shows the behavior of thecomposite grid inside the concrete beam. All displacementsin the simulation graphics were amplified using a factor of 5to enable viewing. Actual deflection values are given inFig. 5, which shows the applied load versus midspan deflec-tion behavior for the short, medium, and long beams for theexperimental and LS-DYNA results, respectively. Thejumps in the LS-DYNA curves in the figure represent theprogressive tensile and shear failure in the concrete elements. Asshown in this figure, the ultimate load value from the finiteelement model agrees well with the experimental result. Themodel slightly over-predicts the stiffness of the beam, however,and under-predicts the ultimate deflection.

The significant drop in load seen in the load-deflectioncurves produced in LS-DYNA is caused by failure in the

Fig. 5—Experimental and finite element load-deflectionresults for short, medium, and long beams.

Fig. 6—Typical failure of composite grid reinforced concretebeam (Ozel and Bank5).

Table 1—Material properties of FRP bottom barsEx 26.7 GPa Xt 266.8 MPa

Ey 14.6 GPa Yt 151.0 MPa

Gxy 3.6 GPa Sc 6.9 MPa

νxy 0.26 Xc 177.9 MPa

β 0.5 Yc 302.0 MPa

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longitudinal bars, as seen in Fig. 4. The deformed shapeseen in this figure indicates a peculiar behavior through-out the length of the beam. It appears to indicate that aftera certain level of damage in the shear span of the model,localized rotations occur in the beam near the load points.These rotations create a stress concentration that causesthe longitudinal bars to fail at those locations. This deflectionbehavior was also observed in the experimental tests.Figure 6 shows a typical failure in the longitudinal barsfrom the experiments conducted on these beams.11 Asshown in this figure, there is considerable damage in theshear span of the member. Large shear cracks develop inthe beam, causing the member to deform in the samefashion as the one seen in the finite element model.

Figure 7 shows the medium beam model at several stagesin the simulation. The figure also shows the behavior of themain longitudinal bars. Comparing this simulation with theone obtained for the short beam, it can be seen that the sheardamage is not as significant as in the previous simulation.The deflected shape seen in the longitudinal bars shows thatthis model does not have the abrupt changes in rotation that

were observed in the short beam, which would imply that thismodel does not exhibit significant flexural-shear damage. Forthis model, the finite element analysis slightly over-predictedboth the stiffness and the ultimate load value obtained fromthe experiment. On the other hand, the ultimate deflectionwas under-predicted. Failure in this model was also causedby rupture of the longitudinal bars at a location near the loadpoints. In the experimental test, failure was caused by acombination of rupture in the longitudinal bars as well asconcrete crushing in the compression zone. This compressivefailure was located near the load points, however, andcould have been initiated by cracks formed due to stressconcentrations produced by the rigid loading plates.11

Figure 8 shows the results for the long beam model.Comparing this simulation with the two previous ones, itcan be seen that this model exhibits the least shear damage,as expected. As a result, the longitudinal bars exhibit aparabolic shape, which would be the behavior predictedusing conventional moment-curvature methods based on thecurvature of the member. Once again, the stiffness of thebeam was slightly over-predicted. However, the ultimate load

Fig. 7—Medium beam model at several stages in simulation.

Fig. 8—Long beam model at several stages in simulation.

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value compares well with the experimental result. Failure inthe model was caused by rupture of the longitudinal bars.Failure in the experimental test was caused by a compressionfailure at a location near one of the load application bars,followed by rupture of the main longitudinal bars. Figure 5also shows the time at total failure for each beam, which canbe related to the simulation stages given in Fig. 4, 7, and 8for the short, medium, and long beam, respectively.

To investigate the stress state of a single longitudinal barat ultimate conditions, the tensile force and the internalmoment of the longitudinal bars at the failed location for thethree finite element models was determined, as shown inFig. 9(a) and (b). It is interesting to note that for the shortbeam model, the tensile force at failure was approximately51.6 kN, while for the medium beam model and the longbeam model the tensile force at failure was approximately76.5 kN. On the other hand, the internal moment in the shortbeam model was approximately 734 N-m, while the internalmoment was approximately 339 N-m for both the short beammodel and the long beam model. It is clear that the sheardamage in the short beam model causes a considerablelocalized effect in the stress state of the longitudinal bars,which is important to consider for design purposes.

According to Fig. 9(a), the total axial load in the longitudinalbars for the short beam model produces a uniform stress of130 MPa, which is not enough to fail the element in tensionat this location. However, the ultimate internal momentproduces a tensile stress at the bottom of the longitudinalbars of 141 MPa. The sum of these two components producesa tensile stress of 271 MPa. When this value is entered in theChang/Chang failure criterion for the tensile longitudinaldirection, the strength is exceeded and the elements fail.

Using conventional over-reinforced beam analysis formulas,the tensile force in the longitudinal bars at midspan wouldbe obtained by dividing the ultimate moment obtained fromthe experimental test by the internal moment arm. Thiswould imply that there is a uniform tensile force in eachlongitudinal bar of 88.1 kN. This tensile force is neverachieved in the finite element simulation due to considerableshear damage in the concrete elements. As a result of thisshear damage in the concrete, the curvature at the center ofthe beam is not large enough to produce a tensile force in thebars of this magnitude (88.1 kN). The internal moment in thelongitudinal bars shown in Fig. 9(b), however, continuesto develop, resulting in a total failure load comparable tothe experimental result. As mentioned before, the force inthe bars according to the simulation was approximately51.6 kN, which is approximately half the load predictedusing conventional methods. Therefore, the use of conventionalbeam analysis formulas to analyze this composite grid reinforcedbeam would not only erroneously predict the force in thelongitudinal bars, but it would also predict a concrete

compression failure mode, which was not the failuremode observed from the experimental tests.

The curves for the medium beam model and the long beammodel, shown in Fig. 9, show that for both cases, the beamshear span-depth ratio was sufficiently large so that the stressstate in the longitudinal bars would not be greatly affected by theshear damage produced in the beam. As such, the ultimate axialforce obtained in the longitudinal bars for both models wasclose to the ultimate axial load that would be predicted by usingconventional methods.

In summary, Table 2 presents the ultimate load capacityfor the three models, including experimental results, conven-tional flexural analysis results, and finite element results. Asshown in this table, conventional flexural analysis under-predictsthe actual ultimate load carried by the beams and a betterultimate load prediction was obtained using finite elementanalysis. The tensile load in the bars was computed (analytically)by dividing the experimental moment capacity by the internalmoment arm computed by using strain compatibility. Althoughthe finite element results over-predicted the ultimate load for themedium and long beams, the simulations provided a betterunderstanding of the complex phenomena involved in thebehavior of the beams, depending on their shear span-depthratio. The results for tensile load in the bars reported in this tablesuggest that composite grid reinforced concrete beamswith values of shear span-depth ratio greater than 4.5 can beanalyzed by using the current flexural theory.

It is important to mention that the concrete material modelparameters that govern the post-failure behavior of the materialplayed a key role in the finite element results for the three finiteelement models. In the concrete material formulation, theelements fail in an isotropic fashion and, therefore, once anelement fails in tension, it cannot transfer further shear.Because the concrete elements are connected to the reinforce-ment mesh, this behavior causes the beam to fail prematurelyas a result of tensile failure in the concrete. Therefore, theparameters that govern the post-failure behavior in theconcrete material model were chosen so that when an elementfails in tension, the element still has the capability to transfershear forces and the stresses will gradually decrease to zero.Because the failed elements can still transfer tensile stresses,however, the modifications caused an increase in the stiffnessof the beam. In real concrete behavior, when a crack opens,there is no tension transfer between the concrete at thatlocation, causing the member to lose stiffness as crackingprogresses. Regarding shear transfer, factors such as aggre-gate interlock and dowel action would contribute to transfershear forces in a concrete beam, and tensile failure in theconcrete would not affect the response as directly as in thefinite element model.

Table 2—Summary of experimental and finite element results

Beam

Total load capacity, kNTensile force in each

main bar, kN

ExperimentalFlexuralanalysis

Finiteelement analysis

Flexural analysis

Finiteelement analysis

Short 215.7 196.2 215.3 90.7 51.6

Medium 143.2 130.8 161.9 90.7 76.5

Long 108.1 97.9 113.0 90.7 76.5Fig. 9—(a) Tensile force in longitudinal bars; and (b) internalmoment in longitudinal bars.

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Stress analysis of FRP barsAs discussed previously, failure modes observed in experi-

mental tests performed on composite grid reinforced concretebeams suggest that the longitudinal bars are subjected to auniform tensile stress plus a nonuniform bending stress dueto localized rotations at locations of large cracks. This sectionpresents a simple analysis procedure to determine the stressconditions at which the longitudinal bars fail. As a result of thisanalysis, a procedure is presented to analyze/design a compositegrid reinforced concrete beam, considering a nonuniform stressstate in the longitudinal bars.

A more detailed finite element model of a section of thelongitudinal bars was developed in HyperMesh16 using shellelements, as shown in Fig. 10. A height of 50.8 mm wasspecified for the bar model, with a thickness of 4.1 mm. Thelength of the bar and the diameter of the hole were 152 and12.7 mm, respectively. The material formulation and propertieswere the same as the ones used for the longitudinal bars in theconcrete beam models, with the exception that now theunnotched tensile strength of the material (Xt = 521 MPa) wasused as an input parameter because the hole was incorporated inthe model.

The finite element model was first loaded in tension toestablish the tensile strength of the notched bar. The loadwas applied by prescribing a displacement at the end of thebar. Figure 10 shows the simulation results for the model atthree stages, including elastic deformation and ultimatefailure. As expected, a stress concentration developed on theboundary of the hole causing failure in the web of the model,followed by ultimate failure of the cross section. A tensilestrength of 274 MPa was obtained for the model. A valueof 267 MPa was obtained from experimental tests con-ducted on notched bars (tensile strength used in Table 2),demonstrating good agreement between experimentaland finite element results.

A similar procedure was performed to establish thestrength of the bar in pure bending. That is, displacementswere prescribed at the end nodes to induce bending in themodel. Figure 11 shows the simulation results for the modelat three stages, showing elastic bending and ultimate failurecaused by flexural failure at the tension flange. As shown inthis figure, the width of the top flange was modified toprevent buckling in the flange (which was present in theoriginal model). Because buckling would not be present in alongitudinal bar due to concrete confinement, it was decidedto modify the finite element model to avoid this behavior. Tomaintain an equivalent cross-sectional area, the thickness ofthe flange was increased. A maximum pure bending momentof 2.92 kN-m was obtained for the model.

Knowing the maximum force that the bar can withstand inpure tension and pure bending, the model was then loaded atdifferent values of tension and moment to cause failure. Thisprocedure was performed several times to develop a tension-moment interaction diagram for the bar, as shown in Fig. 12.The discrete points shown in the figure are combinations oftensile force and moment values that caused failure in thefinite element model. This interaction diagram can be usedto predict what combination of tensile force and momentwould cause failure in the FRP longitudinal bar.

Considerations for designThe strength design philosophy states that the flexural

capacity of a reinforced concrete member must exceed theflexural demand. The design capacity of a member refers

to the nominal strength of the member multiplied by a strength-reduction factor φ, as shown in the following equation

(1)

For FRP reinforced concrete beams, a compression failureis the preferred mode of failure, and, therefore, the beamshould be over-reinforced. As such, conventional formulasare used to ensure that the selected cross-sectional area of thelongitudinal bars is sufficiently large to have concretecompression failure before FRP rupture. Considering a concretecompression failure, the capacity of the beam is computed usingthe following8

(2)

(3)

(4)

Experimental tests have shown, however, that there isa critical value of shear Vs

crit in a beam where localized rotationsdue to large flexural-shear cracks begin to occur. Theultimate moment in the beam is assumed to be related tothis shear-critical value and it is determined according tothe following equation

(5)

where n is the number of longitudinal bars. Once the beam hasreached the shear-critical value, it is assumed (conservatively)that the tensile force t, which is the force in each bar at theshear-critical stage, remains constant and any additional load iscarried by localized internal moment m in the longitudinalbars. Furthermore, it is assumed that at this stage the concreteis still in its elastic range, and, therefore, the internal momentarm ie can be determined by equilibrium and elastic straincompatibility. The tensile force t in Eq. (5) is computed

φMn Mu≥

Mn Af ff d a2---–

=

aAf ff

β1 fc′ b--------------=

ff Ef εcuβ1d a–

a-----------------=

Mn n t ie m+⋅( )⋅=

Fig. 10—Failure on FRP bar subjected to pure tension.

Fig. 11—Failure on FRP bar subjected to pure bending.

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according to the following equation for a simply supportedbeam in four-point bending

(6)

where as is the shear span of the member. The obtained valuefor the tension t in each bar is then entered in Eq. (7), whichis the equation for the interaction diagram, to determine theultimate internal moment m in Eq. (5) that causes the bar tofail. In this equation, tmax and mmax are known properties ofthe notched composite bar.

(7)

The aforementioned procedure is a very simplified analysis todetermine the capacity of a composite grid reinforced concretebeam, and, as can be seen, it depends considerably on the shear-critical value Vs

crit established for the beam. This value issomewhat difficult to determine. Based on experimental data, avalue given by Eq. (8) (analogous to Eq. (9-1) of ACI440.1R-01) can be considered to be a lower bound forFRP reinforced beams with shear reinforcement.

(8)

where fc′ is the specified compressive strength of the concretein MPa. In summary, the ultimate moment capacity in the beamis determined according to one of the following equations

(9)

(10)

According to Eq. (9), if the ultimate shear force computedanalytically based on conventional theory does not exceedthe shear-critical value Vs

crit, the moment capacity can becomputed from flexural analysis. On the other hand, if thecomputed ultimate shear force is greater than Vs

crit, Eq. (10)is used. Table 3 presents a summary showing the load capacityfor the three beams obtained experimentally and analyticallyusing the present approach. As shown in this table, the equationused to determine the flexural capacity depends on the ultimateshear obtained for each beam.

As seen in this procedure, the only difficulty in applyingthese formulas is the fact that an equation needs to be determined

tV crit

s as⋅nie

---------------------=

m mmax 1 ttmax

--------- 2

– for t 0 m 0>;>=

V crits

7ρf Ef

90β1 f c′----------------- 1

6--- f c′ bd=

Mn Af ff d a2---–

for Vult Vscrit<=

Mn n t ie m+⋅( ) for Vult Vscrit>⋅=

to compute the maximum moment that the bar can carry as afunction of the tensile force acting in the bar. If a specific baris always used, however, this difficulty is eliminated, and ifthe flexural demand is not exceeded, a higher capacity can beobtained by increasing the number of longitudinal bars in thesection. According to the results obtained for the three beamsanalyzed herein, the proposed procedure will under-predictthe capacity of the composite grid reinforced concrete beam,but it will provide a good lower bound for a conservativedesign. Furthermore, it will ensure that the longitudinal barswill not fail prematurely as a result of the development oflarge flexural-shear cracks in the member, and thus themember will be able to meet and exceed the flexural demandfor which it was designed.

CONCLUSIONSBased on the explicit finite element results and comparison

with experimental data, the following conclusions can be made:1. Failure in the FRP longitudinal bars occurs due to a

combination of a uniform tensile stress plus a nonuniformstress caused by localized rotations at large flexural-shearcracks. Therefore, this failure mode has to be accounted forin the analysis and design of composite grid reinforced concretebeams, especially those that exhibit significant flexural-shear cracking;

2. The shear span for the medium beam and the long beamstudied was sufficiently large so that the stress state in thelongitudinal bars was not considerably affected by sheardamage in the beam. Therefore, the particular failure modeobserved by the short beam model is only characteristic of

Table 3—Summary of results for three beams using proposed approach

Beam

Experimentalultimate

shear, kNTheoretical shear

critical, kNEquation for

moment capacity

Total load capacity, kN

Tension in eachmain bar, kNExperimental

AnalyticalPn = Mn /as

Short 108.1 88.1 Mn = t · ie + m 216 199 70.7

Medium 71.6 88.1 Mn = Af f f (d – a/ 2) 143 131 90.7

Long 54.7 88.1 Mn = Af f f (d – a/2) 109 99 90.7

Fig. 12—Tension-moment interaction diagram for longi-tudinal bar.

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258 ACI Structural Journal/March-April 2003

beams with a low shear span-depth ratio. Moreover, accordingto the proposed analysis for such systems, both the mediumbeam and the long beam could be designed using conventionalflexural theory because the shear-critical value was neverreached for these beam lengths;

3. Numerical simulations can be used effectively to under-stand the complex behavior and phenomena observed in theresponse of composite grid reinforced concrete beams and,therefore, can be used as a complement to experimentaltesting to account for multiple failure modes in the designof composite grid reinforced concrete beams; and

4. The proposed method of analysis for composite gridreinforced concrete beams considering multiple failuremodes will under-predict the capacity of the reinforcedconcrete beam, but it will provide a good lower bound fora conservative design. These design considerations willensure that the longitudinal bars will not fail prematurely(or catastrophically) as a result of the development of largeflexural-shear cracks in the member, and thus the membercan develop a pseudoductile failure by concrete crushing,which is more desirable than a sudden FRP rupture.

ACKNOWLEDGMENTSThis work was supported by the National Science Foundation under

Grant. No. CMS 9896074. Javier Malvar and Karagozian & Case arethanked for providing information regarding the concrete material formulationused in LS-DYNA. Jim Day, Todd Slavik, and Khanh Bui of LivermoreSoftware Technology Corporation (LSTC) are also acknowledged for theirassistance in using the finite element software, as well as StrongwellChatfield, MN, for producing the custom composite grids.

NOTATIONa = depth of equivalent rectangular stress blockas = length of shear span in reinforced concrete beamb = width of rectangular cross sectiond = distance from extreme compression fiber to centroid of tension

reinforcementEf = modulus of elasticity for FRP barEx = modulus of elasticity in longitudinal direction of FRP grid materialEy = modulus of elasticity in transverse direction of FRP grid materialGxy = shear modulus of FRP grid membersf ′c = specified compressive strength of concreteff = stress in FRP reinforcement in tensionie = internal moment arm in the elastic rangeMn = nominal moment capacitym = internal moment in longitudinal FRP grid barsn = number of longitudinal FRP grid barsSc = shear strength of FRP grid materialt = tensile force in a longitudinal bar at the shear critical stageVs

crit = critical shear resistance provided by concrete in FRP grid rein-forced concrete

Vult = ultimate shear force in reinforced concrete beamXc = longitudinal compressive strength of FRP grid materialXt = longitudinal tensile strength of FRP grid materialYc = transverse compressive strength of FRP grid materialYt = transverse tensile strength of FRP grid materialβ = weighting factor for shear term in Chang/Chang failure criterionβ1 = ratio of the depth of Whitney’s stress block to depth to neu-

tral axisεcu = concrete ultimate strainρf = FRP reinforcement ratioνxy = Poisson’s ratio of FRP grid material

REFERENCES1. Sugita, M., “NEFMAC—Grid Type Reinforcement,” Fiber-Reinforced-

Plastic (FRP) Reinforcement for Concrete Structures: Properties andApplications, Developments in Civil Engineering, A. Nanni, ed., Elsevier,Amsterdam, V. 42, 1993, pp. 355-385.

2. Schmeckpeper, E. R., and Goodspeed, C. H., “Fiber-ReinforcedPlastic Grid for Reinforced Concrete Construction,” Journal of CompositeMaterials, V. 28, No. 14, 1994, pp. 1288-1304.

3. Bank, L. C.; Frostig, Y.; and Shapira, A., “Three-Dimensional Fiber-Reinforced Plastic Grating Cages for Concrete Beams: A Pilot Study,” ACIStructural Journal, V. 94, No. 6, Nov.-Dec. 1997, pp. 643-652.

4. Smart, C. W., and Jensen, D. W., “Flexure of Concrete BeamsReinforced with Advanced Composite Orthogrids,” Journal of AerospaceEngineering, V. 10, No. 1, Jan. 1997, pp. 7-15.

5. Ozel, M., and Bank, L. C., “Behavior of Concrete Beams Reinforcedwith 3-D Composite Grids,” CD-ROM Paper No. 069. Proceedings of the16th Annual Technical Conference, American Society for Composites,Virginia Tech, Va., Sept. 9-12, 2001.

6. Shapira, A., and Bank, L. C., “Constructability and Economics of FRPReinforcement Cages for Concrete Beams,” Journal of Composites forConstruction, V. 1, No. 3, Aug. 1997, pp. 82-89.

7. Bank, L. C., and Ozel, M., “Shear Failure of Concrete Beams Reinforcedwith 3-D Fiber Reinforced Plastic Grids,” Fourth International Symposium onFiber Reinforced Polymer Reinforcement for Reinforced Concrete Structures,SP-188, C. Dolan, S. Rizkalla, and A. Nanni, eds., American ConcreteInstitute, Farmington Hills, Mich., 1999, pp. 145-156.

8. ACI Committee 440, “Guide for the Design and Construction ofConcrete Reinforced with FRP Bars (ACI 440.1R-01),” American ConcreteInstitute, Farmington Hills, Mich., 2001, 41 pp.

9. Nakagawa, H.; Kobayashi. M.; Suenaga, T.; Ouchi, T.; Watanabe, S.;and Satoyama, K., “Three-Dimensional Fabric Reinforcement,” Fiber-Reinforced-Plastic (FRP) Reinforcement for Concrete Structures: Propertiesand Applications, Developments in Civil Engineering, V. 42, A. Nanni, ed.,Elsevier, Amsterdam, 1993, pp. 387-404.

10. Yonezawa, T.; Ohno, S.; Kakizawa, T.; Inoue, K.; Fukata, T.; andOkamoto, R., “A New Three-Dimensional FRP Reinforcement,” Fiber-Reinforced-Plastic (FRP) Reinforcement for Concrete Structures: Propertiesand Applications, Developments in Civil Engineering, V. 42, A. Nanni, ed.,Elsevier, Amsterdam, 1993, pp. 405-419.

11. Ozel, M., “Behavior of Concrete Beams Reinforced with 3-DFiber Reinforced Plastic Grids,” PhD thesis, University of Wisconsin-Madison, 2002.

12. Lees, J. M., and Burgoyne, C. J., “Analysis of Concrete Beams withPartially Bonded Composite Reinforcement,” ACI Structural Journal,V. 97, No. 2, Mar.-Apr. 2000, pp. 252-258.

13. Shehata, E.; Murphy, R.; and Rizkalla, S., “Fiber ReinforcedPolymer Reinforcement for Concrete Structures,” Fourth InternationalSymposium on Fiber Reinforced Polymer Reinforcement for ReinforcedConcrete Structures, SP-188, C. Dolan, S. Rizkalla, and A. Nanni, eds.,American Concrete Institute, Farmington Hills, Mich., 1999, pp. 157-167.

14. Guadagnini, M.; Pilakoutas, K.; and Waldron, P., “Investigation onShear Carrying Mechanisms in FRP RC Beams,” FRPRCS-5, Fibre-Reinforced Plastics for Reinforced Concrete Structures, Proceedings of theFifth International Conference, C. J. Burgoyne, ed., V. 2, Cambridge, July16-18, 2001, pp. 949-958.

15. Cook, R. D.; Malkus, D. S.; and Plesha, M. E., Concepts andApplications of Finite Element Analysis, 3rd Edition, John Wiley &Sons, N.Y., 1989, 832 pp.

16. Altair Computing, HyperMesh Version 2.0 User’s Manual, AltairComputing Inc., Troy, Mich., 1995.

17. Thompson, W. T., and Dahleh, M. D., Theory of Vibration withApplications, 5th Edition, Prentice Hall, N.J., 1998, 524 pp.

18. Malvar, L. J.; Crawford, J. E.; Wesevich, J. W.; and Simons, D., “APlasticity Concrete Material Model for DYNA3D,” International Journalof Impact Engineering, V. 19, No. 9/10, 1997, pp. 847-873.

19. Tavarez, F. A., “Simulation of Behavior of Composite Grid ReinforcedConcrete Beams Using Explicit Finite Element Methods,” MS thesis, Universityof Wisconsin-Madison, 2001.

20. Hallquist, J. O., LS-DYNA Keyword User’s Manual, LivermoreSoftware Technology Corporation, Livermore, Calif., Apr. 2000.

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616 ACI Structural Journal/September-October 2003

ACI Structural Journal, V. 100, No. 5, September-October 2003.MS No. 02-234 received July 2, 2002, and reviewed under Institute publication

policies. Copyright © 2003, American Concrete Institute. All rights reserved, includingthe making of copies unless permission is obtained from the copyright proprietors.Pertinent discussion including author’s closure, if any, will be published in the July-August 2004 ACI Structural Journal if the discussion is received by March 1, 2004.

ACI STRUCTURAL JOURNAL TECHNICAL PAPER

Constitutive formulations are presented for concrete subjected toreversed cyclic loading consistent with a compression fieldapproach. The proposed models are intended to provide substan-tial compatibility to nonlinear finite element analysis in the contextof smeared rotating cracks in both the compression and tensionstress regimes. The formulations are also easily adaptable to afixed crack approach or an algorithm based on fixed principalstress directions. Features of the modeling include: nonlinearunloading using a Ramberg-Osgood formulation; linear reloadingthat incorporates degradation in the reloading stiffness based onthe amount of strain recovered during the unloading phase; andimproved plastic offset formulations. Backbone curves from whichunloading paths originate and on which reloading paths terminateare represented by the monotonic response curves and account forcompression softening and tension stiffening in the compressionand tension regions, respectively. Also presented are formulationsfor partial unloading and partial reloading.

Keywords: cracks; load; reinforced concrete.

RESEARCH SIGNIFICANCEThe need for improved methods of analysis and modeling

of concrete subjected to reversed loading has been brought tothe fore by the seismic shear wall competition conducted bythe Nuclear Power Engineering Corporation of Japan.1 Theresults indicate that a method for predicting the peak strengthof structural walls is not well established. More important, inthe case of seismic analysis, was the apparent inability toaccurately predict structure ductility. Therefore, the state ofthe art in analytical modeling of concrete subjected to generalloading conditions requires improvement if the seismic responseand ultimate strength of structures are to be evaluated withsufficient confidence.

This paper presents a unified approach to constitutivemodeling of reinforced concrete that can be implementedinto finite element analysis procedures to provide accuratesimulations of concrete structures subjected to reversedloading. Improved analysis and design can be achieved bymodeling the main features of the hysteresis behavior ofconcrete and by addressing concrete in tension.

INTRODUCTIONThe analysis of reinforced concrete structures subjected to

general loading conditions requires realistic constitutive modelsand analytical procedures to produce reasonably accuratesimulations of behavior. However, models reported that havedemonstrated successful results under reversed cyclic loadingare less common than models applicable to monotonic loading.The smeared crack approach tends to be the most favored asdocumented by, among others, Okamura and Maekawa2 andSittipunt and Wood.3 Their approach, assuming fixed cracks,has demonstrated good correlation to experimental results;

however, the fixed crack assumption requires separateformulations to model the normal stress and shear stresshysteretic behavior. This is at odds with test observations. Analternative method of analysis, used herein, for reversedcyclic loading assumes smeared rotating cracks consistentwith a compression field approach. In the finite elementmethod of analysis, this approach is coupled with a secantstiffness formulation, which is marked by excellentconvergence and numerical stability. Furthermore, therotating crack model eliminates the need to model normalstresses and shear stresses separately. The procedure hasdemonstrated excellent correlation to experimental datafor structures subjected to monotonic loading.4 Morerecently, the secant stiffness method has successfullymodeled the response of structures subjected to reversedcyclic loading,5 addressing the criticism that it cannot beeffectively used to model general loading conditions.

While several cyclic models for concrete, includingOkamura and Maekawa;2 Mander, Priestley, and Park;6

and Mansour, Lee, and Hsu,7 among others, have beendocumented in the literature, most are not applicable to thealternative method of analysis used by the authors.

Documented herein are models, formulated in the context ofsmeared rotating cracks, for reinforced concrete subjected toreversed cyclic loading. To reproduce accurate simulations ofstructural behavior, the modeling considers the shape of theunloading and reloading curves of concrete to capture theenergy dissipation and the damage of the material due to loadcycling. Partial unloading/reloading is also considered, as struc-tural components may partially unload and then partially reloadduring a seismic event. The modeling is not limited to thecompressive regime alone, as the tensile behavior also plays akey role in the overall response of reinforced concrete struc-tures. A comprehensive review of cyclic models available in theliterature and those reported herein can be found elsewhere.8

It is important to note that the models presented are notintended for fatigue analysis and are best suited for a limitednumber of excursions to a displacement level. Further, themodels are derived from tests under quasistatic loading.

CONCRETE STRESS-STRAIN MODELSFor demonstrative purposes, Vecchio5 initially adopted

simple linear unloading/reloading rules for concrete. Theformulations were implemented into a secant stiffness-basedfinite element algorithm, using a smeared rotating crack

Title no. 100-S64

Compression Field Modeling of Reinforced Concrete Subjected to Reversed Loading: Formulationby Daniel Palermo and Frank J. Vecchio

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617ACI Structural Journal/September-October 2003

approach, to illustrate the analysis capability for arbitraryloading conditions, including reversed cyclic loading. Themodels presented herein have also been formulated in thecontext of smeared rotating cracks, and are intended to buildupon the preliminary constitutive formulations presented byVecchio.5 A companion paper9 documenting the resultsof nonlinear finite element analyses, incorporating theproposed models, will demonstrate accurate simulationsof structural behavior.

Compression responseFirst consider the compression response, illustrated in

Fig. 1, occurring in either of the principal strain directions.Figure 1(a) and (b) illustrate the compressive unloading andcompressive reloading responses, respectively. The backbonecurve typically follows the monotonic response, that is,Hognestad parabola10 or Popovics formulation,11 andincludes the compression softening effects according tothe Modified Compression Field Theory.12

The shape and slope of the unloading and reloading responsesare dependent on the plastic offset strain εc

p , which is essentiallythe amount of nonrecoverable damage resulting fromcrushing of the concrete, internal cracking, and compressing ofinternal voids. The plastic offset is used as a parameter indefining the unloading path and in determining the degree ofdamage in the concrete due to cycling. Further, the backbonecurve for the tension response is shifted such that its origincoincides with the compressive plastic offset strain.

Various plastic offset models for concrete in compressionhave been documented in the literature. Karsan and Jirsa13

were the first to report a plastic offset formulation for concretesubjected to cyclic compressive loading. The model illustratedthe dependence of the plastic offset strain on the strain at theonset of unloading from the backbone curve. A review ofvarious formulations in the literature reveals that, for themost part, the models best suit the data from which they werederived, and no one model seems to be most appropriate. Aunified model (refer to Fig. 2) has been derived herein consid-ering data from unconfined tests from Bahn and Hsu14 andKarsan and Jirsa,13 and confined tests from Buyukozturk andTseng.15 From the latter tests, the results indicated that theplastic offset was not affected by confining stresses or strains.The proposed plastic offset formulation is described as

(1)

where εcp is the plastic offset strain; εp is the strain at peak

stress; and ε2c is the strain at the onset of unloading from thebackbone curve. Figure 2 also illustrates the response of otherplastic offset models available in the literature.

The plot indicates that models proposed by Buyukozturkand Tseng15 and Karsan and Jirsa13 represent upper- and

ε pc εp 0.166

ε2c

εp

------- 2

0.132ε2c

εp

------- +=

lower-bound solutions, respectively. The proposed model(Palermo) predicts slightly larger residual strains than thelower limit, and the Bahn and Hsu14 model calculatesprogressively larger plastic offsets. Approximately 50% ofthe datum points were obtained from the experimental resultsof Karsan and Jirsa;13 therefore, it is not unexpected that thePalermo model is skewed towards the lower-bound Karsanand Jirsa13 model. The models reported in the literature werederived from their own set of experimental data and, thus,may be affected by the testing conditions. The proposedformulation alleviates dependence on one set of experimentaldata and test conditions. The Palermo model, by predicting

Daniel Palermo is a visiting assistant professor in the Department of Civil Engineering,University of Toronto, Toronto, Ontario, Canada. He received his PhD from the Universityof Toronto in 2002. His research interests include nonlinear analysis and design ofconcrete structures, constitutive modeling of reinforced concrete subjected to cyclicloading, and large-scale testing and analysis of structural walls.

ACI member Frank J. Vecchio is Professor and Associate Chair in the Department ofCivil Engineering, University of Toronto. He is a member of Joint ACI-ASCECommittee 441, Reinforced Concrete Columns, and 447, Finite Element Analysisof Reinforced Concrete Structures. His interests include nonlinear analysis anddesign of concrete structures.

Fig. 1—Hysteresis models for concrete in compression: (a)unloading; and (b) reloading.

Fig. 2—Plastic offset models for concrete in compression.

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618 ACI Structural Journal/September-October 2003

relatively small plastic offsets, predicts more pinching inthe hysteresis behavior of the concrete. This pinchingphenomenon has been observed by Palermo and Vecchio8 andPilakoutas and Elnashai16 in the load-deformation response ofstructural walls dominated by shear-related mechanisms.

In analysis, the plastic offset strain remains unchangedunless the previous maximum strain in the history of loadingis exceeded.

The unloading response of concrete, in its simplest form,can be represented by a linear expression extending from theunloading strain to the plastic offset strain. This type ofrepresentation, however, is deficient in capturing the energydissipated during an unloading/reloading cycle in compression.Test data of concrete under cyclic loading confirm that theunloading branch is nonlinear. To derive an expression todescribe the unloading branch of concrete, a Ramberg-Osgood formulation similar to that used by Seckin17 wasadopted. The formulation is strongly influenced by theunloading and plastic offset strains. The general form ofthe unloading branch of the proposed model is expressed as

(2)

where fc is the stress in the concrete on the unloading curve,and ∆ε is the strain increment, measured from the instantaneousstrain on the unloading path to the unloading strain, A, B,and C are parameters used to define the general shape of thecurve, and N is the Ramberg-Osgood power term. Applyingboundary conditions from Fig. 1(a) and simplifying yields

(3)

where

(4)

and

(5)

ε is the instantaneous strain in the concrete. The initialunloading stiffness Ec2 is assigned a value equal to theinitial tangent stiffness of the concrete Ec, and is routinelycalculated as 2fc′ /ε′c. The unloading stiffness Ec3, which definesthe stiffness at the end of the unloading phase, is defined as0.071 Ec, and was adopted from Seckin.17 f2c is the stresscalculated from the backbone curve at the peak unloadingstrain ε2c.

Reloading can sufficiently be modeled by a linear responseand is done so by most researchers. An important characteristic,however, which is commonly ignored, is the degradation inthe reloading stiffness resulting from load cycling. Essentially,the reloading curve does not return to the backbone curve atthe previous maximum unloading strain (refer to Fig. 1 (b)).Further straining is required for the reloading response tointersect the backbone curve. Mander, Priestley, and Park6

attempted to incorporate this phenomenon by defining a new

fc ∆ε( ) A B∆ε C∆εN+ +=

fc ∆ε( ) f2c Ec2 ∆ε( )Ec3 Ec2–( )∆εN

N εcp ε2c–( )

N 1–--------------------------------------+ +=

∆ε ε ε2c–=

NEc2 Ec3–( ) εc

p ε2c–( )

fc2 Ec2+ εcp ε2c–( )

----------------------------------------------------=

stress point on the reloading path that corresponded to themaximum unloading strain. The new stress point was assumedto be a function of the previous unloading stress and thestress at reloading reversal. Their approach, however, wasstress-based and dependent on the backbone curve. Theapproach used herein is to define the reloading stiffnessas a degrading function to account for the damage induced in theconcrete due to load cycling. The degradation was observed tobe a function of the strain recovery during unloading. Thereloading response is then determined from

(6)

where fc and εc are the stress and strain on the reloading path;fro is the stress in the concrete at reloading reversal andcorresponds to a strain of εro ; and Ec1 is the reloadingstiffness, calculated as follows

(7)

where

(8)

and

(9)

and

(10)

βd is a damage indicator, fmax is the maximum stress in theconcrete for the current unloading loop, and εrec is theamount of strain recovered in the unloading process and isthe difference between the maximum strain εmax and theminimum strain εmin for the current hysteresis loop. Theminimum strain is limited by the compressive plastic offsetstrain. The damage indicator was derived from test data onplain concrete from four series of tests: Buyukozturk andTseng,15 Bahn and Hsu,14 Karsan and Jirsa,13 andYankelevsky and Reinhardt.18 A total of 31 datum pointswere collected for the prepeak range (Fig. 3(a)) and 33 datumpoints for the postpeak regime (Fig. 3(b)). Because there was anegligible amount of scatter among the test series, the datumpoints were combined to formulate the model. Figure 3(a) and(b) illustrate good correlation with experimental data, indi-cating the link between the strain recovery and the damage dueto load cycling. βd is calculated for the first unloading/reloadingcycle and retained until the previous maximum unloading strainis attained or exceeded. Therefore, no additional damage isinduced in the concrete for hysteresis loops occurring at strainsless than the maximum unloading strain. This phenomenon isfurther illustrated through the partial unloading and partialreloading formulations.

fc fro Ec1 εc εro–( )+=

Ec1βd fmax⋅( ) fro–

ε2c εro–------------------------------------=

βd1

1 0.10 εrec εp⁄( )0.5+------------------------------------------------ for εc εp<=

βd1

1 0.175 εrec εp⁄( )0.6+--------------------------------------------------- for εc εp>=

εrec εmax εmin–=

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ACI Structural Journal/September-October 2003 619

It is common for cyclic models in the literature to ignorethe behavior of concrete for the case of partial unloading/reloading. Some models establish rules for partial loadingsfrom the full unloading/reloading curves. Other modelsexplicitly consider the case of partial unloading followedby reloading to either the backbone curve or strains in excessof the previous maximum unloading strain. There exists,however, a lack of information considering the case wherepartial unloading is followed by partial reloading to strainsless than the previous maximum unloading strain. This moregeneral case was modeled using the experimental results ofBahn and Hsu.14 The proposed rule for the partial unloadingresponse is identical to that assumed for full unloading;however, the previous maximum unloading strain andcorresponding stress are replaced by a variable unloadingstrain and stress, respectively. The unloading path is definedby the unloading stress and strain and the plastic offset strain,which remains unchanged unless the previous maximumstrain is exceeded. For the case of partial unloading followedby reloading to a strain in excess of the previous maximumunloading strain, the reloading path is defined by the expressionsgoverning full reloading. The case where concrete is partiallyunloaded and partially reloaded to a strain less than theprevious maximum unloading strain is illustrated in Fig 4.

Five loading branches are required to construct the responseof Fig. 4. Unloading Curve 1 represents full unloading fromthe maximum unloading strain to the plastic offset and iscalculated from Eq. (3) to (5) for full unloading. Curve 2defines reloading from the plastic offset strain and isdefined by Eq. (6) to (10). Curve 3 represents the case ofpartial unloading from a reloading path at a strain less than theprevious maximum unloading strain. The expressions usedfor full unloading are applied, with the exception of substi-tuting the unloading stress and strain for the current hysteresisloop for the unloading stress and strain at the previousmaximum unloading point. Curve 4 describes partialreloading from a partial unloading branch. The responsefollows a linear path from the load reversal point to theprevious unloading point and assumes that damage is notaccumulated in loops forming at strains less than theprevious maximum unloading strain. This implies that thereloading stiffness of Curve 4 is greater than the reloadingstiffness of Curve 2 and is consistent with test data reportedby Bahn and Hsu.14 The reloading stiffness for Curve 4 isrepresented by the following expression

(11)

The reloading stress is then calculated using Eq. (6) forfull reloading.

In further straining beyond the intersection with Curve 2,the response of Curve 4 follows the reloading path of Curve 5.The latter retains the damage induced in the concrete fromthe first unloading phase, and the stiffness is calculated as

(12)

The reloading stresses are then determined from thefollowing

Ec1fmax fro–

εmax εro–-----------------------=

Ec1βd f2c⋅ fmax–

ε2c εmax–-------------------------------=

(13)

The proposed constitutive relations for concrete subjectedto compressive cyclic loading are tested in Fig. 5 against theexperimental results of Karsan and Jirsa.13 The Palermomodel generally captures the behavior of concrete under cycliccompressive loading. The nonlinear unloading and linearloading formulations agree well with the data, and the plasticoffset strains are well predicted. It is apparent, though, thatthe reloading curves become nonlinear beyond the point ofintersection with the unloading curves, often referred to as the

fc fmax Ec1 εc εmax–( )+=

Fig. 3—Damage indicator for concrete in compression:(a) prepeak regime; and (b) postpeak regime.

Fig. 4—Partial unloading/reloading for concrete in compression.

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620 ACI Structural Journal/September-October 2003

common point. The Palermo model can be easily modified toaccount for this phenomenon; however, unusually small loadsteps would be required in a finite element analysis to capturethis behavior, and it was thus ignored in the model. Further-more, the results tend to underestimate the intersection of thereloading path with the backbone curve. This is a direct resultof the postpeak response of the concrete and demonstrates theimportance of proper modeling of the postpeak behavior.

Tension responseMuch less attention has been directed towards the modeling

of concrete under cyclic tensile loading. Some researchersconsider little or no excursions into the tension stress regimeand those who have proposed models assume, for the most

part, linear unloading/reloading responses with no plasticoffsets. The latter was the approach used by Vecchio5 informulating a preliminary tension model. Stevens, Uzumeri,and Collins19 reported a nonlinear response based on definingthe stiffness along the unloading path; however, the modelswere verified with limited success. Okumura and Maekawa2

proposed a hysteretic model for cyclic tension, in which anonlinear unloading curve considered stresses through bondaction and through closing of cracks. A linear reloading pathwas also assumed. Hordijk20 used a fracture mechanicsapproach to formulate nonlinear unloading/reloading rulesin terms of applied stress and crack opening displacements.

The proposed tension model follows the philosophy used tomodel concrete under cyclic compression loadings. Figure 6 (a)and (b) illustrate the unloading and reloading responses,respectively. The backbone curve, which assumes themonotonic behavior, consists of two parts adopted from theModified Compression Field Theory12: that describing theprecracked response and that representing postcrackingtension-stiffened response.

A shortcoming of the current body of data is the lack oftheoretical models defining a plastic offset for concrete intension. The offsets occur when cracked surfaces come intocontact during unloading and do not realign due to shear slipalong the cracked surfaces. Test results from Yankelevskyand Reinhardt21 and Gopalaratnam and Shah22 provide datathat can be used to formulate a plastic offset model (refer toFig. 7). The researchers were able to capture the softeningbehavior of concrete beyond cracking in displacement-controlled testing machines. The plastic offset strain, in theproposed tension model, is used to define the shape of theunloading curve, the slope and damage of the reloading path,and the point at which cracked surfaces come into contact.Similar to concrete in compression, the offsets in tensionseem to be dependent on the unloading strain from the back-bone curve. The proposed offset model is expressed as

(14)

where εcp is the tensile plastic offset, and ε1c is the unloading

strain from the backbone curve. Figure 7 illustrates verygood correlation to experimental data.

Observations of test data suggest that the unloading responseof concrete subjected to tensile loading is nonlinear. Theaccepted approach has been to model the unloading branchas linear and to ignore the hysteretic behavior in the concrete

ε pc 146ε 2

1c 0.523ε1c+=

Fig. 5—Predicted response for cycles in compression.

Fig. 6—Hysteresis models for concrete in tension: (a)unloading; and (b) reloading. Fig. 7—Plastic offset model for concrete in tension.

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ACI Structural Journal/September-October 2003 621

due to cycles in tension. The approach used herein was toformulate a nonlinear expression for the concrete that wouldgenerate realistic hysteresis loops. To derive a model consistentwith the compression field approach, a Ramberg-Osgoodformulation, similar to that used for concrete in compression,was adopted and is expressed as

fc = D + F∆ε + G∆εN (15)

where fc is the tensile stress in the concrete; ∆ε is the strainincrement measured from the instantaneous strain on theunloading path to the unloading strain; D, F, and G areparameters that define the shape of the unloading curve; andN is a power term that describes the degree of nonlinearity.

Applying the boundary conditions from Fig. 6(a) andsimplifying yields

(16)

where

(17)

and

(18)

f1c is the unloading stress from the backbone curve, and Ec5is the initial unloading stiffness, assigned a value equal to theinitial tangent stiffness Ec. The unloading stiffness Ec6, whichdefines the stiffness at the end of the unloading phase, wasdetermined from unloading data reported by Yankelevsky andReinhardt.21 By varying the unloading stiffness Ec6, thefollowing models were found to agree well with test data

(19)

(20)

The Okamura and Maekawa2 model tends to overestimatethe unloading stresses for plain concrete, owing partly to thefact that the formulation is independent of a tensile plasticoffset strain. The formulations are a function of the unloadingpoint and a residual stress at the end of the unloading phase.The residual stress is dependent on the initial tangent stiffnessand the strain at the onset of unloading. The linear unloadingresponse suggested by Vecchio5 is a simple representation ofthe behavior but does not capture the nonlinear nature of theconcrete and underestimates the energy dissipation. Theproposed model captures the nonlinear behavior and energydissipation of the concrete.

The state of the art in modeling reloading of concrete intension is based on a linear representation, as described by,among others, Vecchio5 and Okamura and Maekawa.2 Theresponse is assumed to return to the backbone curve at theprevious unloading strain and ignores damage induced to the

fc ∆ε( ) f1c Ec5 ∆ε( )Ec5 Ec6–( )∆εN

N ε1c εcp–( )

N 1–--------------------------------------+–=

∆ε ε1c ε–=

NEc5 Ec6–( ) ε1c εc

p–( )

Ec5 ε1c εcp–( ) f1c–

----------------------------------------------------=

Ec6 0.071 Ec 0.001 ε1c⁄( ) ε1c 0.001≤⋅=

Ec6 0.053 Ec 0.001 ε1c⁄( ) ε1c 0.001>⋅=

concrete due to load cycling. Limited test data confirm thatlinear reloading sufficiently captures the general response ofthe concrete; however, it is evident that the reloading stiffnessaccumulates damage as the unloading strain increases. Theapproach suggested herein is to model the reloading behavioras linear and to account for a degrading reloading stiffness.The latter is assumed to be a function of the strain recoveredduring the unloading phase and is illustrated in Fig. 8 againstdata reported by Yankelevsky and Reinhardt.21 The reloadingstress is calculated from the following expression

(21)

where

(22)

fc is the tensile stress on the reloading curve and correspondsto a strain of εc. Ec4 is the reloading stiffness, βt is a tensiledamage indicator, tf max is the unloading stress for the currenthysteresis loop, and tfro is the stress in the concrete at reloadingreversal corresponding to a strain of tro. The damage parameterβt is calculated from the following relation

(23)

where

(24)

εrec is the strain recovered during an unloading phase. It isthe difference between the unloading strain εmax and theminimum strain at the onset of reloading εmin, which islimited by the plastic offset strain. Figure 8 depicts goodcorrelation between the proposed formulation and thelimited experimental data.

Following the philosophy for concrete in compression, βtis calculated for the first unloading/reloading phase and retaineduntil the previous maximum strain is at least attained.

The literature is further deficient in the matter of partialunloading followed by partial reloading in the tension stressregime. Proposed herein is a partial unloading/reloading

fc βt tfmax Ec4–⋅ ε1c εc–( )=

Ec4βt tfmax⋅( ) tfro–

ε1c tro–---------------------------------------=

βt1

1 1.15 εrec( )0.25+-----------------------------------------=

εrec εmax εmin–=

Fig. 8—Damage model for concrete in tension.

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622 ACI Structural Journal/September-October 2003

model that directly follows the rules established for concretein compression. No data exist, however, to corroborate themodel. Figure 9 depicts the proposed rules for a concreteelement, lightly reinforced to allow for a post-cracking response.

Curve 1 corresponds to a full unloading response and isidentical to that assumed by Eq. (16) to (18). Reloading froma full unloading curve is represented by Curve 2 and is computedfrom Eq. (21) to (24). Curve 3 represents the case of partialunloading from a reloading path at a strain less than theprevious maximum unloading strain. The expressions forfull unloading are used; however, the strain and stress atunloading, now variables, replace the strain and stress atthe previous peak unloading point on the backbone curve.Reloading from a partial unloading segment is describedby Curve 4. The response follows a linear path from thereloading strain to the previous unloading strain. The modelexplicitly assumes that damage does not accumulate forloops that form at strains less than the previous maximumunloading strain in the history of loading. Therefore, thereloading stiffness of Curve 4 is larger than the reloadingstiffness for the first unloading/reloading response ofCurve 2. The partial reloading stiffness, defining Curve 4,is calculated by the following expression

(25)

and the reloading stress is then determined from

Ec4tfmax tfro–

εmax tro–-------------------------=

(26)

As loading continues along the reloading path of Curve 4,a change in the reloading path occurs at the intersection withCurve 2. Beyond the intersection, the reloading responsefollows the response of Curve 5 and retains the damage inducedto the concrete from the first unloading/reloading phase. Thestiffness is then calculated as

(27)

The reloading stresses can then be calculated according to

(28)

The previous formulations for concrete in tension arepreliminary and require experimental data to corroborate. Themodels are, however, based on realistic assumptions derivedfrom the models suggested for concrete in compression.

CRACK-CLOSING MODELIn an excursion returning from the tensile domain,

compressive stresses do not remain at zero until thecracks completely close. Compressive stresses will ariseonce cracked surfaces come into contact. The recontactstrain is a function of factors such as crack-shear slip.There exists limited data to form an accurate model forcrack closing, and the preliminary model suggestedherein is based on the formulations and assumptionssuggested by Okamura and Maekawa.2 Figure 10 is aschematic of the proposed model.

The recontact strain is assumed equal to the plastic offsetstrain for concrete in tension. The stiffness of the concrete duringclosing of cracks, after the two cracked surfaces have come intocontact and before the cracks completely close, is smaller thanthat of crack-free concrete. Once the cracks completely close,the stiffness assumes the initial tangent stiffness value. Thecrack-closing stiffness Eclose is calculated from

(29)

where

fclose = –Ec(0.0016 ⋅ ε1c + 50 × 10–6) (30)

fclose, the stress imposed on the concrete as cracked surfacescome into contact, consists of two terms taken from theOkamura and Maekawa2 model for concrete in tension. Thefirst term represents a residual stress at the completion ofunloading due to stress transferred due to bond action.The second term represents the stress directly related toclosing of cracks. The stress on the closing-of-cracks path isthen determined from the following expression

(31)

fc tfro Ec4 εc tro–( )+=

Ec4βt f1c tfmax–⋅

ε1c εmax–--------------------------------=

fc tfmax Ec4 εc εmax–( )+=

Eclosefclose

εcp

-----------=

fc Eclose εc εcp–( )=

Fig. 9—Partial unloading/reloading for concrete in tension.

Fig. 10—Crack-closing model.

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ACI Structural Journal/September-October 2003 623

After the cracks have completely closed and loadingcontinues into the compression strain region, the reloadingrules for concrete in compression are applicable, with thestress in the concrete at the reloading reversal point assuminga value of fclose.

For reloading from the closing-of-cracks curve into thetensile strain region, the stress in the concrete is assumed tobe linear, following the reloading path previously establishedfor tensile reloading of concrete.

In lieu of implementing a crack-closing model, plastic off-sets in tension can be omitted, and the unloading stiffness atthe completion of unloading Ec6 can be modified to ensurethat the energy dissipation during unloading is properlycaptured. Using data reported by Yankelevsky and Reinhardt,21

a formulation was derived for the unloading stiffness at zeroloads and is proposed as a function of the unloading strain onthe backbone curve as follows

(32)

Implicit in the latter model is the assumption that, in anunloading excursion in the tensile strain region, the compressivestresses remain zero until the cracks completely close.

REINFORCEMENT MODELThe suggested reinforcement model is that reported by

Vecchio,5 and is illustrated in Fig. 11. The monotonic responseof the reinforcement is assumed to be trilinear. The initialresponse is linear elastic, followed by a yield plateau, and endingwith a strain-hardening portion. The hysteretic response of thereinforcement has been modeled after Seckin,17 and the Bausch-inger effect is represented by a Ramberg-Osgood formulation.

The monotonic response curve is assumed to represent thebackbone curve. The unloading portion of the responsefollows a linear path and is given by

(33)

where fs(εi) is the stress at the current strain of εi , fs – 1 and εs – 1are the stress and strain from the previous load step, and Eris the unloading modulus and is calculated as

(34)

if (35)

Er = 0.85Es if (εm – εo) > 4εy (36)

where Es is the initial tangent stiffness; εm is the maximumstrain attained during previous cycles; εo is the plastic offsetstrain; and εy is the yield strain.

The stresses experienced during the reloading phase aredetermined from

(37)

where

Ec6 1.1364 ε1c0.991–( )–=

fs εi( ) fs 1– Er εi εs 1––( )+=

Er Es if εm εo–( ) εy<=

Er Es 1.05 0.05εm εo–

εy

----------------– = εy εm εo–( ) 4εy< <

fs εi( ) Er εi εo–( )Em Er–

N εm εo–( )N 1–⋅--------------------------------------- εi εo–( )N⋅+=

(38)

fm is the stress corresponding to the maximum strain recordedduring previous loading; and Em is the tangent stiffness at εm.

The same formulations apply for reinforcement in tensionor compression. For the first reverse cycle, εm is taken aszero and fm = fy, the yield stress.

IMPLEMENTATION AND VERIFICATIONThe proposed formulations for concrete subjected to

reversed cyclic loading have been implemented into atwo-dimensional nonlinear finite element program, whichwas developed at the University of Toronto.23

The program is applicable to concrete membrane structuresand is based on a secant stiffness formulation using a total-load,iterative procedure, assuming smeared rotating cracks.The package employs the compatibility, equilibrium, andconstitutive relations of the Modified Compression FieldTheory.12 The reinforcement is typically modeled assmeared within the element but can also be discretelyrepresented by truss-bar elements.

The program was initially restricted to conditions ofmonotonic loading, and later developed to account formaterial prestrains, thermal loads, and expansion andconfinement effects. The ability to account for materialprestrains provided the framework for the analysis capability ofreversed cyclic loading conditions.5

For cyclic loading, the secant stiffness procedure separatesthe total concrete strain into two components: an elasticstrain and a plastic offset strain. The elastic strain is used tocompute an effective secant stiffness for the concrete, and,therefore, the plastic offset strain must be treated as a strainoffset, similar to an elastic offset as reported by Vecchio.4

The plastic offsets in the principal directions are resolvedinto components relative to the reference axes. From theprestrains, free joint displacements are determined as functionsof the element geometry. Then, plastic prestrain nodal forcescan be evaluated using the effective element stiffness matrixdue to the concrete component. The plastic offsets developed in

NEm Er–( ) εm εo–( )fm Er– εm εo–( )

---------------------------------------------=

Fig. 11—Hysteresis model for reinforcement, adapted fromSeckin (1981).

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624 ACI Structural Journal/September-October 2003

each of the reinforcement components are also handled in asimilar manner.

The total nodal forces for the element, arising from plasticoffsets, are calculated as the sum of the concrete and reinforce-ment contributions. These are added to prestrain forces arisingfrom elastic prestrain effects and nonlinear expansion effects.The finite element solution then proceeds.

The proposed hysteresis rules for concrete in this procedurerequire knowledge of the previous strains attained in the historyof loading, including, amongst others: the plastic offset strain,the previous unloading strain, and the strain at reloading reversal.In the rotating crack assumption, the principal strain directionsmay be rotating presenting a complication. A simple andeffective method of tracking and defining the strains isthe construction of Mohr’s circle. Further details of theprocedure used for reversed cyclic loading can be foundfrom Vecchio.5

A comprehensive study, aimed at verifying the proposedcyclic models using nonlinear finite element analyses, willbe presented in a companion paper.9 Structures consideredwill include shear panels and structural walls available in theliterature, demonstrating the applicability of the proposedformulations and the effectiveness of a secant stiffness-based algorithm employing the smeared crack approach. Thestructural walls will consist of slender walls, with height-width ratios greater than 2.0, which are heavily influenced byflexural mechanisms, and squat walls where the response isdominated by shear-related mechanisms. The former isgenerally not adequate to corroborate constitutive formulationsfor concrete.

CONCLUSIONSA unified approach to constitutive modeling of reversed

cyclic loading of reinforced concrete has been presented.The constitutive relations for concrete have been formulated

in the context of a smeared rotating crack model, consistentwith a compression field approach. The models are intendedfor a secant stiffness-based algorithm but are also easilyadaptable in programs assuming either fixed cracks or fixedprincipal stress directions.

The concrete cyclic models consider concrete in compressionand concrete in tension. The unloading and reloading rulesare linked to backbone curves, which are represented by themonotonic response curves. The backbone curves are adjustedfor compressive softening and confinement in the compressionregime, and for tension stiffening and tension softening inthe tensile region.

Unloading is assumed nonlinear and is modeled using aRamberg-Osgood formulation, which considers boundaryconditions at the onset of unloading and at zero stress.Unloading, in the case of full loading, terminates at the plasticoffset strain. Models for the compressive and tensile plasticoffset strains have been formulated as a function of themaximum unloading strain in the history of loading.

Reloading is modeled as linear with a degrading reloadingstiffness. The reloading response does not return to the backbonecurve at the previous unloading strain, and further straining isrequired to intersect the backbone curve. The degradingreloading stiffness is a function of the strain recoveredduring unloading and is bounded by the maximum unloadingstrain and the plastic offset strain.

The models also consider the general case of partial unloadingand partial reloading in the region below the previous maximumunloading strain.

NOTATIONEc = initial modulus of concreteEclose = crack-closing stiffness modulus of concrete in tensionEc1 = compressive reloading stiffness of concreteEc2 = initial unloading stiffness of concrete in compressionEc3 = compressive unloading stiffness at zero stress in concreteEc4 = reloading stiffness modulus of concrete in tensionEc5 = initial unloading stiffness modulus of concrete in tensionEc6 = unloading stiffness modulus at zero stress for concrete in tensionEm = tangent stiffness of reinforcement at previous maximum strainEr = unloading stiffness of reinforcementEs = initial modulus of reinforcementEsh = strain-hardening modulus of reinforcementf1c = unloading stress from backbone curve for concrete in tensionf2c = unloading stress on backbone curve for concrete in compressionfc = normal stress of concretef ′c = peak compressive strength of concrete cylinderfclose = crack-closing stress for concrete in tensionfcr = cracking stress of concrete in tensionfm = reinforcement stress corresponding to maximum strain in historyfmax = maximum compressive stress of concrete for current unloading

cyclefp = peak principal compressive stress of concretefro = compressive stress at onset of reloading in concretefs = average stress for reinforcementfs – 1 = stress in reinforcement from previous load stepfy = yield stress for reinforcementtfmax = maximum tensile stress of concrete for current unloading cycletfro = tensile stress of concrete at onset of reloadingtro = tensile strain of concrete at onset of reloadingβd = damage indicator for concrete in compressionβt = damage indicator for concrete in tension∆ε = strain increment on unloading curve in concreteε = instantaneous strain in concreteε0 = plastic offset strain of reinforcementε1c = unloading strain on backbone curve for concrete in tensionε2c = compressive unloading strain on backbone curve of concreteεc = compressive strain of concreteε′c = strain at peak compressive stress in concrete cylinderεc

p = residual (plastic offset) strain of concrete εcr = cracking strain for concrete in tensionεi , εs = current stress of reinforcementεm = maximum strain of reinforcement from previous cyclesεmax = maximum strain for current cycleεmin = minimum strain for current cycleεp = strain corresponding to maximum concrete compressive stressεrec = strain recovered during unloading in concreteεro = compressive strain at onset of reloading in concreteεsh = strain of reinforcement at which strain hardening beginsεs – 1 = strain of reinforcement from previous load stepεy = yield strain of reinforcement

REFERENCES1. Nuclear Power Engineering Corporation of Japan (NUPEC),

“Comparison Report: Seismic Shear Wall ISP, NUPEC’s Seismic UltimateDynamic Response Test,” Report No. NU-SSWISP-D014, Organization forEconomic Co-Operation and Development, Paris, France, 1996, 407 pp.

2. Okamura, H., and Maekawa, K., Nonlinear Analysis and ConstitutiveModels of Reinforced Concrete, Giho-do Press, University of Tokyo, Japan,1991, 182 pp.

3. Sittipunt, C., and Wood, S. L., “Influence of Web Reinforcement onthe Cyclic Response of Structural Walls,” ACI Structural Journal, V. 92,No. 6, Nov.-Dec. 1995, pp. 745-756.

4. Vecchio, F. J., “Finite Element Modeling of Concrete Expansion andConfinement,” Journal of Structural Engineering, ASCE, V. 118, No. 9,1992, pp. 2390-2406.

5. Vecchio, F. J., “Towards Cyclic Load Modeling of Reinforced Concrete,”ACI Structural Journal, V. 96, No. 2, Mar.-Apr. 1999, pp. 132-202.

6. Mander, J. B.; Priestley, M. J. N.; and Park, R., “Theoretical Stress-Strain Model for Confined Concrete,” Journal of Structural Engineering,ASCE, V. 114, No. 8, 1988, pp. 1804-1826.

7. Mansour, M.; Lee, J. Y.; and Hsu, T. T. C., “Cyclic Stress-StrainCurves of Concrete and Steel Bars in Membrane Elements,” Journal ofStructural Engineering, ASCE, V. 127, No. 12, 2001, pp. 1402-1411.

8. Palermo, D., and Vecchio, F. J., “Behaviour and Analysis of ReinforcedConcrete Walls Subjected to Reversed Cyclic Loading,” Publication No.2002-01, Department of Civil Engineering, University of Toronto, Canada,2002, 351 pp.

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625ACI Structural Journal/September-October 2003

9. Palermo, D., and Vecchio, F. J., “Compression Field Modeling ofReinforced Concrete Subjected to Reversed Loading: Verification,” ACIStructural Journal. (accepted for publication)

10. Hognestad, E.; Hansen, N. W.; and McHenry, D., “Concrete StressDistribution in Ultimate Strength Design,” ACI JOURNAL, Proceedings V. 52,No. 12, Dec. 1955, pp. 455-479.

11. Popovics, S., “A Numerical Approach to the Complete Stress-StrainCurve of Concrete,” Cement and Concrete Research, V. 3, No. 5, 1973, pp.583-599.

12. Vecchio, F. J., and Collins, M. P., “The Modified Compression-FieldTheory for Reinforced Concrete Elements Subjected to Shear,” ACIJOURNAL, Proceedings V. 83, No. 2, Mar.-Apr. 1986, pp. 219-231.

13. Karsan, I. K., and Jirsa, J. O., “Behaviour of Concrete UnderCompressive Loadings,” Journal of the Structural Division, ASCE, V. 95,No. 12, 1969, pp. 2543-2563.

14. Bahn, B. Y., and Hsu, C. T., “Stress-Strain Behaviour of ConcreteUnder Cyclic Loading,” ACI Materials Journal, V. 95, No. 2, Mar.-Apr.1998, pp. 178-193.

15. Buyukozturk, O., and Tseng, T. M., “Concrete in Biaxial CyclicCompression,” Journal of Structural Engineering, ASCE, V. 110, No. 3,Mar. 1984, pp. 461-476.

16. Pilakoutas, K., and Elnashai, A., “Cyclic Behavior of RC CantileverWalls, Part I: Experimental Results,” ACI Structural Journal, V. 92, No. 3,May-June 1995, pp. 271-281.

17. Seckin, M., “Hysteretic Behaviour of Cast-in-Place Exterior Beam-Column Sub-Assemblies,” PhD thesis, University of Toronto, Toronto,Canada, 1981, 266 pp.

18. Yankelevsky, D. Z., and Reinhardt, H. W., “Model for CyclicCompressive Behaviour of Concrete,” Journal of Structural Engineering,ASCE, V. 113, No. 2, Feb. 1987, pp. 228-240.

19. Stevens, N. J.; Uzumeri, S. M.; and Collins, M. P., “Analytical Modellingof Reinforced Concrete Subjected to Monotonic and Reversed Loadings,”Publication No. 87-1, Department of Civil Engineering, University ofToronto, Toronto, Canada, 1987, 201 pp.

20. Hordijk, D. A., “Local Approach to Fatigue of Concrete,” DelftUniversity of Technology, The Netherlands, 1991, pp. 210.

21. Yankelevsky, D. Z., and Reinhardt, H. W., “Uniaxial Behaviour ofConcrete in Cyclic Tension,” Journal of Structural Engineering, ASCE,V. 115, No. 1, 1989, pp. 166-182.

22. Gopalaratnam, V. S., and Shah, S. P., “Softening Response of PlainConcrete in Direct Tension,” ACI JOURNAL, Proceedings V. 82, No. 3, May-June 1985, pp. 310-323.

23. Vecchio, F. J., “Nonlinear Finite Element Analysis of ReinforcedConcrete Membranes,” ACI Structural Journal, V. 86, No. 1, Jan.-Feb.1989, pp. 26-35.

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ACI Structural Journal, V. 104, No. 4, July-August 2007.MS No. S-2006-230.R1 received June 21, 2006, and reviewed under Institute publi-

cation policies. Copyright © 2007, American Concrete Institute. All rights reserved, includ-ing the making of copies unless permission is obtained from the copyright proprietors.Pertinent discussion including author’s closure, if any, will be published in the May-June 2008 ACI Structural Journal if the discussion is received by January 1, 2008.

ACI STRUCTURAL JOURNAL TECHNICAL PAPER

The seismic performance of four one-half scale exterior beam-columnsubassemblages is examined. All subassemblages were typical of newstructures and incorporated full seismic details in current buildingcodes, such as a weak girder-strong column design philosophy.The subassemblages were subjected to a large number of inelasticcycles. The tests indicated that current design procedures couldsometimes result in excessive damage to the joint regions.

Keywords: beam-column frames; connections; cyclic loads; reinforcedconcrete; structural analysis.

INTRODUCTIONThe key to the design of ductile moment-resisting frames

is that the beam-to-column connections and columns mustremain essentially elastic throughout the load history toensure the lateral stability of the structure. If the connectionsor columns exhibit stiffness and/or strength deteriorationwith cycling, collapse due to P-Δ effects or to the formationof a story mechanism may be unavoidable.1,2

Four one-half scale beam-column subassemblages weredesigned and constructed in turn, according to Eurocode 23

and Eurocode 8,4 according to ACI 318-055 and ACI 352R-02,6

and according to the new Greek Earthquake ResistantCode7 and the new Greek Code for the Design of ReinforcedConcrete Structures.8

The subassemblages were subjected to cyclic lateral loadhistories so as to provide the equivalent of severe earthquakedamage. The results indicate that current design procedurescould sometimes result in severe damage to the joint, despitethe use of a weak girder-strong column design philosophy.

RESEARCH SIGNIFICANCEExperimental data and experience from earthquakes indicate

that loss of capacity might occur in joints that are part ofolder reinforced concrete (RC) frame structures.9-12 There isscarce experimental evidence and insufficient data, however,about the performance of joints designed according tocurrent codes during strong earthquakes. This researchprovides structural engineers with useful information aboutthe safety of new RC frame structures that incorporateseismic details from current building codes. In some cases,safety could be jeopardized during strong earthquakes bypremature joint shear failures. The joints could at timesremain the weak link even for structures designed in accordancewith current model building codes.

DESCRIPTION OF TEST SPECIMENS—MATERIAL PROPERTIES

Four one-half scale exterior beam-column subassemblageswere designed and constructed for this experimental andanalytical investigation. Reinforcement details of thesubassemblages are shown in Fig. 1(a) and (b). All the

subassemblages (A1, E1, E2, and G1) had the same generaland cross-sectional dimensions, as shown in Fig. 1.Subassemblages E1, E2, and G1 had the same longitudinalcolumn reinforcement, eight bars with a diameter of 14 mm,while the longitudinal column reinforcement of A1 consisted ofeight bars with a diameter of 10 mm (0.4 in.). The longitudinalcolumn reinforcement of A1 was lower than that of the otherthree subassemblages (E1, E2, and G1) due to the restrictionsof ACI 352R-026 for the column bars passing through thejoint. Subassemblages E1 and G1 had the same percentageof longitudinal beam reinforcement (ρE1 = ρG1 = 7.7 × 10–3)and Subassemblages A1 and E2 also had the same percentageof longitudinal beam reinforcement (ρA1 = 5.23 × 10–3 andρE2 = 5.2 × 10–3), but different from the percentage of E1 andG1. The longitudinal beam reinforcement of A1 consisted offour bars with a diameter of 10 mm, while the beam reinforce-ment of E2 consisted of two bars with a diameter of 14mm. Subassemblage A1 had smaller beam reinforcing barsthan Subassemblage E2 due to the restrictions of ACI 352R-026

for the beam bars passing through the joint. The joint shearreinforcements of the subassemblages used in the experiments,are as follows: Ø6 multiple hoop at 5 cm for Subassemblage A1(Fig. 1(a)), Ø6 multiple hoop at 5 cm for Subassemblage E1,(Fig. 1(b)), Ø6 multiple hoop at 4.8 cm for Subassemblage E2(Fig. 1(a)) and Ø8 multiple hoop at 10 cm for Subassemblage G1(Fig. 1(b)). All subassemblages incorporated seismic details.The purpose of Subassemblages A1, E1, E2, and G1 was torepresent details of new structures. As is clearly demon-strated in Fig. 1(a) and (b), all the subassemblages had highflexural strength ratios MR. The purpose of using an MR ratio(sum of the flexural capacity of columns to that of beam(s))significantly greater than 1.00 in earthquake-resistantconstructions is to push the formation of the plastic hinge inthe beams, so that the safety (that is, collapse prevention) ofthe structure is not jeopardized.1,2,4-7,9,10,13 Thus, in all thesesubassemblages, the beam is expected to fail in a flexural modeduring cyclic loading.

The concrete 28-day compressive strength of bothSubassemblages A1 and E2 was 35 MPa (5075 psi), while theconcrete 28-day compressive strength of both SubassemblagesE1 and G1 was 22 MPa (3190 psi). Reinforcement yield strengthsare as follows: Ø6 = 540 MPa (78 ksi), Ø10 = 500 MPa (73 ksi),and Ø14 = 495 MPa (72 ksi) (note: Ø6 [No. 2]), Ø10 [No. 3],and Ø14 [No. 4]) are bars with a diameter of 6, 10, and 14 mm).

Title no. 104-S45

Cyclic Load Behavior of Reinforced ConcreteBeam-Column Subassemblages of Modern Structuresby Alexandros G. Tsonos

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469ACI Structural Journal/July-August 2007

Approximately 10 electrical-resistance strain gauges werebonded on the reinforcing bars of each subassemblage ofthe program.

EXPERIMENTAL SETUP ANDLOADING SEQUENCE

The general arrangement of the experimental setup isshown in Fig. 2(a). All subassemblages were subjected to11 cycles applied by slowly displacing the beam’s free endaccording to the load history shown in Fig. 2(b) withoutreaching the actuator stroke limit. The amplitudes of thepeaks in the displacement history were 15, 20, 25, 30, 35, 40,45, 50, 55, 60, and 65 mm. One loading cycle was performedat each displacement amplitude. An axial load equal to 200 kNwas applied to the columns of the subassemblages and keptconstant throughout the test. The experimental loadingsequence used is a typical one, commonly used in previousstudies.1,11,13 It was not the objective of this study to investigatethe effect of other, nonstandard loading histories on theresponse of the subassemblages.

As previously mentioned, all the subassemblages were loadedslowly. The strain rate of the load applied corresponded tostatic conditions. In the case of seismic loading, the strainrate is higher than the rate corresponding to static conditions.Soroushian and Sim14 showed that an increase in with

ε·ε·

respect to static conditions leads to a moderate increase inthe strength of concrete

(1)

Scott et al.15 tested column subassemblages with variousamounts of hoop reinforcement under strain rates rangingfrom 0.33 × 10–5 sec–1 (static loading), to 0.0167 sec–1

(seismic loading). Their test results conformed with theresults obtained from Eq. (1).

Using the aforementioned expression, it is estimated that fora strain rate of = 0.0167 sec–1, concrete strengths increaseby approximately 20% (compared with the static one). Anexpression similar to Eq. (1) can be found in the CEB code.16

Thus, the strengths exhibited by Subassemblages A1, E1,E2, and G1 during the tests are somewhat lower than thestrengths they would exhibit if subjected to load historiessimilar to actual seismic events.

EXPERIMENTAL RESULTSFailure mode of Subassemblages A1, E1, E2, and G1

The failure mode of Subassemblages A1 and E2, asexpected, involved the formation of a plastic hinge in thebeam at the column face. The formation of plastic hingescaused severe cracking of the concrete near the fixed beam endof each subassemblage (Fig. 3). The behavior of Subassem-blages A1 and E2 was as expected and as documented in theseismic design philosophy of the modern codes as will beexplain in the following.4-7

fc dyn, 1.48 0.160 ε· 0.0127 ε·log( )2+log×+[ ] fc stat,×=

ε·

ACI member Alexandros G. Tsonos is a Professor of reinforced concrete structures,Department of Structural Engineering, the Aristotle University of Thessaloniki,Thessaloniki, Greece. He received his PhD from the Aristotle University of Thessalonikiin 1990. His research interests include the inelastic behavior of reinforced concretestructures, structural design, fiber-reinforced concrete, seismic repair and rehabilitationof reinforced concrete structures, and the seismic repair and restoration of monuments.

Fig. 1—Dimensions and cross-sectional details of: (a) Subassemblages A1 and E2; and (b) Subassemblages E1 and G1. (Note:dimensions are in cm. 1 cm = 0.0394 in.)

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Significant inelastic deformations occurred in the beams’longitudinal reinforcement in both Subassemblages A1 and E2(strains of over 40.000με were obtained in the beams’longitudinal bars), while joint shear reinforcement remainedelastic. Figure 4(a) shows strain gauge data of joint hoopreinforcement for both Subassemblages A1 and E2. As isclearly shown in Fig. 4(a), the maximum strain recorded in

the joint hoop reinforcement for both subassemblages wasbelow the yield strain of 2.500με, which was in agreement withthe observed failure modes of Subassemblages A1 and E2.17

One difference between the failure modes of SubassemblagesA1 and E2 was that hairline cracks appeared in the joint regionof E2, and partial loss of the concrete cover in the rear face ofthe joint of E2 took place during the three last cycles of loading(ninth, tenth, and eleventh) when drift Angle R ratiosexceeded 4.5 while the joint region of Subassemblage A1remained intact at the conclusion of the test (refer to Fig. 3).

The connections of both Subassemblages E1 and G1,contrary to expectations, exhibited shear failure during the

Fig. 2—(a) General arrangement of experimental setup andphotograph of test setup (dimensions are in m; 1 m = 3.28 ft);and (b) lateral displacement history. (Note: 1 mm = 0.039 in.)

Fig. 3—Views of collapsed Subassemblages A1, E1, E2, and G1.

Fig. 4—Applied shear versus strain in beam-column jointhoop reinforcement of: (a) Subassemblages A1 and E2;and (b) Subassemblages E1 and G1. (Note: 1 kN = 0.225 kip.)

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ACI Structural Journal/July-August 2007 471

early stages of cyclic loading. Damage occurred both in thejoint area and in the columns’ critical regions. Figure 4(b)shows strain gauge data for the joint hoop reinforcement forSubassemblages E1 and G1. As shown in Fig. 4(b), themaximum strain recorded in the joint hoop reinforcement ofboth Subassemblages E1 and G1 was significantly higherthan the yield strain 2.500 με. Joint shear damage has beenshown to occur after yielding of the joint hoop reinforcement,which is in agreement with the damage observed in the jointsof these subassemblages.18 The maximum strain recorded inthe longitudinal bars of the beams of both SubassemblagesE1 and G1 was below 2.500με (refer to Fig. 5). In Fig. 6, theprogression of cracking of Subassemblage E1 during the testis demonstrated.

Load-drift angle curvesPlots of applied shear force versus drift angles for all the

Subassemblages (A1, E1, E2, and G1) are shown in Fig. 7.The beam calculated flexural capacities of the subassemblagesare shown as dashed lines in Fig. 7.

A major concern in the seismic design of RC structures isthe ability of members to develop their flexural strength beforefailing in shear. This is especially true for members framingat a beam column joint (beams and columns), where it isimportant to develop their flexural strengths before jointshear failure. Moreover, by designing the flexural strengthsof columns in RC frame structures to meet the strong-columnweak-beam rule, all members against premature shear failure,and by detailing plastic hinge (critical) regions for ductility,RC frame structures have been shown to exhibit a controlledand very ductile inelastic response.2,4,9

Fig. 5—Maximum strain during each cycle of loading inbeam longitudinal reinforcement of Subassemblages A1, E1,E2, and G1.

Fig. 6—Gradual cracking configuration of Subassemblage E1during test.

Fig. 7—Hysteresis loops of Subassemblages A1, E1, E2, andG1. (Note: 1 kN = 0.225 kip.)

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472 ACI Structural Journal/July-August 2007

As can be seen in Fig. 7, the beam of Subassemblage A1developed maximum shear forces higher than those corre-sponding to its ultimate flexural strength until the sixth cycleof loading. This is an indication of the flexural response ofthis beam because it developed its flexural strength until adrift Angle R ratio of 4.0 was reached and exceeded. Also, aflexural failure was observed for this beam, caused bycrushing of the concrete cover of the longitudinal reinforcement,and subsequent inelastic buckling of the longitudinal bars.The beam of Subassemblage E2 also developed maximumshear forces higher than those corresponding to its ultimateflexural strength until the eleventh upper half cycle of loadingand until the seventh lower half cycle of loading. In particular,during the final cycles of loading beyond drift Angle R ratiosof 4.5 when large displacements were imposed, crushing ofthe concrete cover of the reinforcement took place and thebeam’s hoops could not provide adequate support to the longi-tudinal reinforcement. As a result, buckling of the beam longi-tudinal reinforcement in Subassemblages A1 and E2 occurredafter the sixth and seventh cycles of loading, respectively.

The beam of Subassemblage E1 developed maximumshear forces very close to those corresponding to its ultimateflexural strength only during the second and third cycle ofloading. For the remaining cycles (four through 11), thepremature joint shear failure did not allow the beam in thissubassemblage to develop its flexural capacity (Fig. 6 and 7).

The premature joint shear failure of Subassemblage G1also did not allow the beam in this subassemblage to developits flexural capacity. As can be seen in Fig. 7, the beam ofSubassemblage G1 developed maximum shear forcessignificantly lower than those corresponding to its ultimateflexural strength.

One of the basic provisions of all modern structural codesis to provide the structures with sufficient strength and sufficientductility to undergo post-elastic deformations without losinga large percentage of their strength.2,4,7,9 As can be seen inFig. 7, this criterion is fulfilled for Subassemblies A1 and E2.By contrast, it is not fulfilled for Subassemblies E1 and G1because they exhibited significant loss of strength duringcyclic loading.

The beam-column Subassemblages A1, E1, E2, and G1 aresimilar to real modern frame structures. If the sequence in thebreakdown of the chain of resistance of these real framestructures follows the desirable hierarchy during a catastrophicearthquake, the formation of plastic hinges in the beams ofthese structures would be expected, because the use of aweak girder-strong column design philosophy is adopted bythe modern codes.2,4,5,7,9 The aforementioned desirablefailure mode (with formation of a plastic hinge in the beam)was developed by Subassemblages A1 and E2. Thus, themagnitude of loads resisted by Subassemblages A1 and E2are consistent with the expected values from actual events.Story drifts allowed by modern codes are on the order of 2%of the story height.4,7,8 While it was reassuring that storydrifts of as much as 4% of the story height were achieved inmost reported tests referring to the seismic response of beam-column specimens, it should be remembered that drifts inexcess of 2% are not likely to be readily accommodated inhigh rise frames. This is due to significant and detrimentalinfluence of P-Δ phenomena on both lateral load resistanceand dynamic response.19

Subassemblages A1 and E2, which developed plastichinges in their beams (Fig. 3 and 7), showed stable hystereticbehavior up to drift Angle R ratios of 4.0. They showed a

considerable loss of strength, stiffness, and unstable hystereticbehavior, but beyond drift Angle R ratios of 4.5 (Fig. 7).

Subassemblages E1 and G1, which exhibited premature jointshear failure (refer to Fig. 3 and 7) showed a considerable loss ofstrength, stiffness, and unstable degrading hysteresis beyonddrift Angle R ratios of 2.5 and 2.0%, respectively (Fig. 7).

CODE REQUIREMENTSDespite the fact that all the subassemblages were designed

according to their corresponding modern codes, two developedfailure modes dominated by joint shear failure (Fig. 3). Forthis reason, it is discussed how requirements of these codesused for the design of the joints of Subassemblages A1,5,6 E1,E2

3,4 (for DC”M” structures), and G17,8 were satisfied.

Table 1 clearly indicates that the joint of A1 satisfied thedesign requirements of ACI 318-055 and ACI 352R-026 forexterior beam-column joints for seismic loading.

Table 2 indicates that the joints of both E1 and E2 satisfied thedesign provisions for exterior beam-column joints ofEurocode 23 and Eurocode 84 for DC”M” structures.

In both subassemblages, two 8 mm diameter short barswere placed and were tightly connected on the top bends ofthe beam reinforcing bars and two on the bottom, running inthe transverse direction of the joint, as shown in Fig. 5. Thisis the setup recommended by Eurocode 8 when the requirementof limitation of beam bar diameter (dbl) to ensure appropriateanchorage through the joint is not satisfied (refer to Table 2).It was considered worthwhile, however, to determine thebeam bar pull-out. Strain gauge measurements were used todetermine beam bar pull-out. If the maximum strains in abeam’s longitudinal bar during each two consecutive cyclesof loading remained the same or decreased, as long as buckling

Table 1—Comparison of joint of Subassemblage A1 design parameters with ACI 318-055 and ACI 352R-026

Subassemblage γ ldh, cm Ash, cm2 sh, cm

hbeam/column bar

diameter MR

A1 0.67 < (1.0)*†

17 (15.65)*

(17)†0.95

(0.66)*†5.0

(5.0)*†30

(23.80)†1.72

(1.20)*†

*Numbers inside parentheses are required values of ACI 318-05.5†Numbers inside parentheses are required values of ACI 352R-02.6 Note: γ is shear strength factor reflecting confinement of joint by lateral members, ldhis development length of hooked beam bars, Ash is total cross-sectional area of transversesteel in joint, and sh is spacing of transverse reinforcement in joint. Numbers outsideparentheses are provided values. 1 cm = 0.394 in.

Table 2—Comparison of joints of Subassemblages E1 and E2 design parameters with Eurocode 84 and Eurocode 23

Subassemblage Vjh, kNAsh,

cm2Asv ,

cm2 dbl , mm MR

lb,net, cm

sw , cm

E1 126 < (168)*

6.85 (2.85)*

3.08 (1.06)*

14 (9.15)*

2.60 (1.20)* 45 (43)† 5 (5)*

E275.6 < (222)*

6.85 (2.85)*

3.08 (1.06)*

14 (11.20)*

3.30 (1.20)* 45 (32)† 5 (5)*

*Numbers inside parentheses are required values of Eurocode 8.4†Numbers inside parentheses are required values of Eurocode 2.3 Note: Vjh is horizontal joint shear force, Ash is total cross-sectional area of transversesteel of joint, Asv is vertical joint shear reinforcement, dbl is diameter of hooked beambars (in both E1 and E2 setup recommended by EC8 and shown in Fig. 5 was applied), lb,netis development length of hooked beam bars, and sw is spacing of transverse reinforcementof joint. Numbers outside parentheses are provided values. 1 m = 0.394in.; 1 mm =0.039 in.; 1 kN = 0.225 kip.

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ACI Structural Journal/July-August 2007 473

of this bar had not taken place, it was concluded that a pull-out of this bar had occurred.13,18 As shown in Fig. 5, thebeam’s longitudinal reinforcement in Subassemblages E1and E2 maintained adequate anchorage throughout the testsdue to the short bars placed and tightly connected under thebends of a group of reinforcing bars (refer to Fig. 5).

Table 3 also clearly indicates that the joint of G1 satisfiedthe design provisions for exterior beam-column joints ofboth the new Greek codes.7,8

The codes prescribe minimum MR values. So, as can beseen from Tables 1 through 3, the minimum value for the MRratio according to ACI 318-05 and ACI 352R-02, as well asaccording to Eurocode 8 (DC”M”), is 1.20.4-6 The minimumvalue for the MR ratio according to the new Greek EarthquakeResistant Code is 1.40.7 Thus, a good target MR for moststructures is between 1.20 and 1.40.

Neither the New Greek Code for the Design of RCStructures8 nor the new Greek Earthquake Resistant Code7

require limitations for the joint shear stress. Of course both ofthese codes need to add requirements to limit joint shear stress.

THEORETICAL CONSIDERATIONSA new formulation published in recent studies20-26 predicts

the beam-column joint ultimate shear strength and was used inthe present study to predict the failure modes of SubassemblagesA1, E1, E2, and G1. A summary of this formulation is presented.

Figure 8(a) shows an RC exterior beam-column joint for amoment resisting frame and Fig. 8(b) shows the internalforces around this joint.10,12 The shear forces acting in thejoint core are resisted partly by a diagonal compression strutthat acts between diagonally opposite corners of the jointcore (refer to Fig. 8(c)) and partly by a truss mechanismformed by horizontal and vertical reinforcement and concretecompression struts.10,12,19 The horizontal and verticalreinforcement is normally provided by horizontal hoops inthe joint core around the longitudinal column bars and bylongitudinal column bars between the corner bars in the sidefaces of the column.10,12,27 Both mechanisms depend on thecore concrete strength. Thus, the ultimate concrete strengthof the joint core under compression/tension controls theultimate strength of the connection. After failure of theconcrete, strength in the joint is limited by gradual crushingalong the cross-diagonal cracks and especially along thepotential failure planes (Fig. 8(a)).

For instance, consider Section I-I in the middle of the jointheight (Fig. 8(a)). In this section, the flexural moment isalmost zero. The forces acting in the concrete are shown inFig. 8(d).27,28 Each force acting in the joint core is analyzedinto two components along the x and y axes (Fig. 8(d)). Thevalues of Ti are the tension forces acting on the longitudinalcolumn bars between the corner bars in the side faces of thecolumn. Their resultant is ΣTi. An equal and opposing

Fig. 8—(a) Exterior beam-column joint; (b) internal forces around exterior beam-columnjoint as result of seismic actions;10,12 (c) two mechanisms of shear transfer (diagonal concretestrut and truss mechanism);10,12,19 and (d) forces acting in joint core concrete throughSection I-I from two mechanisms.27,28

Table 3—Comparison of joint of Subassemblage G1 design parameters with ERC-19957 and CDCS-19958

Subassemblage Ash, cm2 lb,net, cm MR

G1 2.01 (2.01)* 45 (43)* 2.60 (1.40)†

*Numbers inside parentheses are required values of CDCS-1995.8†Numbers inside parentheses are required values of ERC-1995.7 Note: Ash is total cross-sectional area of transverse steel of joint and lb,net isdevelopment length of hooked beam bars. Numbers outside parentheses are providedvalues. 1 cm = 0.394 in.

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474 ACI Structural Journal/July-August 2007

compression force (–ΣTi) must act in the joint core to balancethe vertical tensile forces generated in the reinforcement.This compression force was generated by the resultant of thevertical components of the truss mechanism’s diagonalcompression forces D1, D2 …Dv.

27 Thus, D1y + D2y + … +Dvy = ΣTi = T1 + T2 + T3 + T4.27 The column axial load isresisted by the compression strut mechanism.12 The summationof vertical forces equals the vertical joint shear force Vjv

↓ ↓compression strut truss model (2a)

The summation of horizontal forces equals the horizontaljoint shear force Vjh

(2b)

The vertical normal compressive stress σ and the shearstress τ uniformly distributed over Section I-I are given byEq. (3) and (4)

(3)

(4)

where h ′c and b ′c are the length and the width of the jointcore, respectively.

It is now necessary to establish a relationship between theaverage normal compressive stress σ and the average shearstress τ. From Eq. (3) and (4)

(5)

It has been shown that

(6)

where α is the joint aspect ratio.4,10,12 The principle (σI = maximum, σII = minimum) stresses are

calculated

(7)

Equation (8)29 was adopted for the representation of theconcrete biaxial strength curve30 by a fifth-degree parabola

(8)

where fc is the increased joint concrete compressive strengthdue to confinement by joint hoop reinforcement, which isgiven by the model of Scott et al.15 according to the equation

Dcy T1 ... T4+ +( )+ Dcy Dsy+ Vjv= =

Dcx D1x D2x… Dvx+ +( )+ Dcx Dsx+ Vjh= =

σDcy Dsy+

hc′ bc′×-----------------------

Vjv

hc′ bc′×--------------------= =

τVjh

hc′ bc′×--------------------=

σVjv

Vjh

------- τ×=

Vjv

Vjh

-------hb

hc

----- α= =

σI II,σ2--- σ

2--- 1 4τ2

σ2--------+±=

10σI

fc

-----–σII

fc

------5

+ 1=

(9a)

Also, f ′c is the concrete compressive strength and K is aparameter of the model15 expressed as

(9b)

where ρs is the volume ratio of transverse reinforcement andfyh is its yield strength.

Substituting Eq. (5) through (7) into Eq. (8) and using τ =γ gives the following expression

(10)

Assume herein that

(11)

and

(12)

Then Eq. (10) is transformed into

(13)

The solution of the system of Eq. (11) to (13) gives thebeam-column joint ultimate strength τult = γult (MPa).

This system is solved each time for a given value of the jointaspect ratio using standard mathematical analysis. The jointultimate strength τult depends on the increased joint concretecompressive strength due to confining fc and on the jointaspect ratio α. Thus, typical values of τult for comparison withthe values of ACI 318-05,5 ACI 352R-02,6 and Eurocode 84

are not possible to derive. A particular value, however, foreach joint would be calculated as in the following example.

Example for Subassemblage A1The value α = 1.5 and the solution of the system of Eq. (11)

to (13) gives x = 0.1485 and y = 0.248; f ′c(A1) = 35 MPa,K(A1) = 1.558 according to the Scott et al.15 model andfc(A1) = K(A1) × f ′c(A1) = 54.53 MPa.

Equation (11) gives

and finally τult(A1) = 1.46 MPa = 10.78 MPa (refer toTable 4).

COMPARISON OF PREDICTIONS AND EXPERIMENTAL RESULTS

The proposed shear strength formulation can be used topredict the failure mode of the subassemblages and thus the

fc K fc× ′=

K 1ρs fyh×

fc′-----------------+=

fc

αγ

2 fc

----------- 1 1 4

α2------++⎝ ⎠

⎛ ⎞5

5αγ

fc

--------- 1 4

α2------+ 1–⎝ ⎠

⎛ ⎞+ 1=

x αγ

2 fc

-----------=

ψ αγ

2 fc

----------- 1 4

α2------+=

x ψ+( )5 10ψ 10x–+ 1=

fc

γult A1( )2 0.1458( ) 54.53

1.5------------------------------------------ 1.46= =

54.53

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ACI Structural Journal/July-August 2007 475

actual values of connection shear stress. Therefore, when thecalculated joint shear stress τcal is greater or equal to the jointultimate strength (τcal = γcal ≥ τult = γult ), then thepredicted actual value of connection shear stress will be nearτult(τult = γult ). This is because the connection failsearlier than the adjacent beam(s). When the calculated jointshear stress τcal is lower than the connection ultimatestrength (τcal = γcal < τult = γult ), then the predictedactual value of connection shear stress will be near τcalbecause the connection permits its adjacent beam(s) to yield.τult = γult is calculated from the solution of the system ofEq. (11) to (13). The value of τcal is calculated from thehorizontal joint shear force assuming that the top reinforcementof the beam yields (Fig. 8(a)). In this case, the horizontaljoint shear force is expressed as

(14)

where As1 is the top longitudinal beam reinforcement (Fig. 8(a)),fy is the yield stress of this reinforcement, and Vcol is thecolumn shear force (Fig. 8(a)). For Type 2 joints, the designforces in the beam according to ACI 352R-026 should bedetermined using a stress value of α × fy for beam longitudinalreinforcement, where α = 1.25.

The improved retention of strength in the beam-columnsubassemblages, as the values of the ratio τcal/τult = γcal/γultdecrease was also demonstrated. For τcal/τult = γcal/γult ≤ 0.50,the beam-column joints of the subassemblages performedexcellently during the tests and remained intact at theconclusion of the tests.20-26

The validity of the formulation was checked using testdata from more than 120 exterior and interior beam-columnsubassemblages that were tested in the Structural EngineeringLaboratory at the Aristotle University of Thessaloniki,20-26 aswell as data from similar experiments carried out in the U.S.,Japan, and New Zealand.1,12,13,31-36 A part of this verificationis presented in Table 5 where the comparison is shownbetween experimental and predicted results by the precedingmethodology for 39 exterior and interior beam-column jointsubassemblages from the literature. A very good correlationis observed (Table 5). In Table 5, the limiting values of jointshear stress according to ACI 318-055 and ACI 352R-026

(1.0 MPa for exterior beam-column joints and 1.25MPa for interior beam-column joints) are included for eachreference subassemblage. In Table 5, the limiting values ofjoint shear stress according to Eurocode 84 (15τR MPa forexterior beam-column joints and 20τR MPa for interiorbeam-column joints) are also included.

The shear capacities of the connections of SubassemblagesA1, E1, E2, and G1 were also computed using the aforemen-tioned methodology. One of the motivations behind this

fc fc

fc

fc fc

fc

Vjhcal 1.25As1 fy Vcol–×=

fc′ fc′

study was the verification of the shear strength formulationpresented herein for beam-column joints designed accordingto modern codes.

The horizontal joint shear stresses are mainly producedby the longitudinal beam reinforcement as clearlydescribed by Eq. (14). The longitudinal beam reinforcement ofSubassemblages A1 and E2 was purposely chosen toproduce low joint shear stresses during the tests, that is, a ratioτcal/τult = γcal/γult less than 0.5.

Table 6 shows that γcal/γult is equal to 0.47 in SubassemblageA1 (that is, lower than 0.5) and γcal/γult is equal to 0.46 inSubassemblage E2 (that is, lower than 0.5). Thus, the formationof a plastic hinge in the beams near the columns is expectedwithout any serious damage in the joint regions and, as aresult, there will be satisfactory performance for bothSubassemblages A1 and E2. As predicted, both subassemblagesfailed in flexure, exhibiting remarkable seismic performance(Fig. 3 and 7). Values τpred of A1 and E2, which are shownin Table 6, are equal to their τcal values (because γcal < γult)and are significantly different from their τult values, whichare shown in Table 4.

The percentage of longitudinal beam reinforcement ofSubassemblages E1 and G1 was purposely chosen to behigher than that of Subassemblages A1 and E2 to producehigher joint shear stresses than those corresponding to theirultimate capacities. The joint region of E1, however, satisfiedall the design requirements of Eurocode 23 and Eurocode 84

and the joint regions of G1 satisfied all the design requirementsof the two Greek codes.7,8

Table 6 also shows that for both Subassemblages E1 andG1, the calculated joint shear stress τcal = γcal when thebeams reach their ultimate strength is higher than the jointultimate capacity τult = γult . Therefore, the joints of boththese subassemblages will fail earlier than their beamsaccording to the aforementioned methodology, because thejoints of both E1 and G1 reach their ultimate shear strengthduring the tests before the beams reach their ultimate strength.Thus, according to the aforementioned methodology, a jointshear failure is expected for both Subassemblages E1 and G1without any serious damage in their beams and, as a result, theperformance of both subassemblages will not be satisfactory.As expected, both Subassemblages E1 and G1 demonstratedpremature joint shear failure starting from the early stages ofseismic loading and damage concentrated mostly in thisregion (Fig. 3). As also predicted, both Subassemblages E1and G1 exhibited poor seismic performance, which wascharacterized by significant loss of strength, stiffness, andenergy dissipation capacity during the tests. Furthermore, thevolume ratios of joint transverse reinforcement for Subassem-blages E1 and G1 were 0.025 and 0.017, respectively. Thus,the joint of Subassemblage E1 was more confined than thejoint of Subassemblage G1, which explains why the hystereticresponse of the former was better than that of the latter (Fig. 7).The concrete compressive strength significantly increasesthe joint ultimate strength τult. Thus, if the SubassemblagesE1 and G1 had higher values with concrete compressivestrengths, they would have behaved as well as SubassemblagesA1 and E2. This would have happened for values withconcrete compressive strength of approximately 50 MPa,which would have resulted in values of ratio γcal/γult lowerthan 0.5. The value of concrete 28-day compressivestrengths of 22 MPa for both Subassemblages E1 and G1,however, is acceptable for Eurocode 2,3 Eurocode 8,4 and forboth Greek codes.7,8

fc

fc

Table 4—Joint ultimate strength and ratios τpred /τexp and γcal /γult for Subassemblages A1, E1, E2, and G1

Subassemblage

According to Park and Paulay10

According to proposed shear strength formulation

τult, MPa τpred /τexp γcal /γult τult, MPa τpred /τexp γcal /γult

A1 6.05 1.19 1.0 10.78 1.17 0.47

E1 8.94 1.31 1.0 6.92 1.19 1.08

E2 5.96 1.24 1.0 10.78 1.20 0.46

G1 8.34 1.28 1.0 6.60 1.19 1.04

Note: 1 MPa = 144.93 psi.

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476 ACI Structural Journal/July-August 2007

Table 5—Experimental verifications

ReferenceSub-

assemblage

Type of subassem-

blage*

Joint aspect ratio

α = hb/hc

Concrete compressive strength f ′c,

MPaτACI, MPa

τEC8, MPa

Longitudinal beam bar fy ,

MPa

Joint hoop fy ,

MPa γcal γexp γult

Predicted shear strength τpred,

MPa

Observed shear strength

τexp, MPaμ = τpred /

τexp

34

No. 1 E 1.00 31.10 5.58 7.80 391 250 0.78 0.88 0.92 4.46 5.03 0.89

No. 2 E 1.00 41.70 6.46 9.45 391 250 0.68 0.74 1.06 4.50 4.90 0.92

No. 3 E 1.00 41.70 6.46 9.45 391 250 0.68 0.67 1.06 4.50 4.43 1.01

No. 4 E 1.00 44.70 6.69 9.90 391 281 0.66 0.67 1.08 4.43 4.50 0.99

No. 5 E 1.00 36.70 6.06 8.63 391 281 0.74 0.69 0.99 4.50 4.20 1.07

No. 6 E 1.00 40.40 6.35 9.30 391 281 0.70 0.69 1.03 4.47 4.41 1.01

No. 7 E 1.00 32.20 5.67 7.95 391 250 0.77 0.82 0.93 4.47 4.76 0.94

No. 8 E 1.00 41.20 6.42 9.40 391 250 0.68 0.72 1.06 4.47 4.74 0.94

No. 9 E 1.00 40.60 6.37 9.30 391 250 0.69 0.67 1.05 4.51 4.40 1.03

No. 10 E 1.00 44.40 6.65 9.83 391 281 0.67 0.69 1.08 4.49 4.62 0.97

No. 11 E 1.00 41.90 6.47 9.48 391 281 0.69 0.70 1.05 4.49 4.55 0.99

No. 12 E 1.00 35.10 5.92 8.34 391 281 0.75 0.74 0.96 4.47 4.40 1.01

No. 13 E 1.00 46.40 6.81 10.16 391 250 0.64 0.64 1.12 4.47 4.47 1.00

No. 14 E 1.00 41.00 6.40 9.36 391 281 0.70 0.69 1.03 4.50 4.44 1.01

No. 15 E 1.00 30.70 5.54 7.74 391 281 0.71 0.74 1.02 3.95 4.12 0.96

No. 16 E 1.00 37.40 6.11 8.76 391 250 0.72 0.76 1.01 4.51 4.76 0.95

33

A1 I 1.14 40,20 7.93 12.33 1070 291 4.62 1.34 1.11 7.21 8.70 0.83

A2 I 1.14 40.20 7.93 12.33 409 291 1.76 1.23 1.11 7.21 7.99 0.90

A3 I 1.14 40.20 7.93 12.33 1070 291 4.62 1.34 1.11 7.21 8.70 0.83

A4 I 1.14 40.20 7.93 12.33 1070 291 4.48 1.33 1.14 7.62 8.88 0.86

B1 E 1.14 30.00 5.48 7.65 1070 291 2.68 0.93 0.96 5.39 5.22 1.03

B2 E 1.14 30.00 5.48 7.65 409 291 1.02 0,83 0.96 5.39 4.66 1.16

B3 E 1.14 30.00 5.48 7.65 1070 291 2.68 1.03 0.96 5.39 5.78 0.93

B4 E 1.14 30.00 5.48 7.65 1070 291 2.60 1.05 0.99 5.71 6.06 0.94

12

UNIT1 I 1.126 41.30 8.03 12.54 315 320 1.20 1.13 1.26 8.96 8.44 1.06

UNIT2 I 1.126 46.90 8.56 13.65 307 320 1.31 1.08 1.33 10.23 8.43 1.20

UNIT3 E† 1.126 38.20 6.18 8.70 473 321 1.17 0.90 1.09 7.06 5.85 1.21

UNIT4 E 1.126 38.90 6.23 8.55 473 321 2.32 0.90 1.11 7.29 5.91 1.23

36

SHC1 I 1.14 56.50 9.39 15.9 413 551 1.00 0.91 1.31 7.81 7.11 1.10

SHC2 I 1.14 59.50 9.64 16.5 413 551 0.97 0.91 1.36 7.90 7.41 1.07

SOC3 I 1.14 47.10 8.58 13.71 413 551 1.06 1.00 1.22 7.70 7.26 1.06

35

SP1 Eठ1.33 30.70 5.54 7.74 347 0 0.90 0.78 1.03 4.99 4.32 1.15

SP2 E‡ 1.33 31.10 5.58 7.80 349 0 0.90 0.77 1.04 5.02 4.30 1.17

SP3 E§ 1.33 27.00 5.20 7.11 350 427 0.94 0.83 1.00 5.17 4.56 1.13

SP4 E§ 1.33 31.00 5.57 7.79 349 379 0.86 0.87 1.09 5.13 5.19 0.90

SP5 Eठ1.33 32.00 5.66 7.92 347 0 0.88 0.75 1.05 4.97 4.24 1.17

SP6 E 1.33 36.20 6.02 8.55 352 357 0.78 0.78 1.20 5.16 5.16 1.00

SP7 E 1.33 30.70 5.54 7.74 352 365 0.87 0.83 1.08 5.16 4.93 1.05

SP8 E 1.33 26.30 5.13 7.00 352 365 1.19 1.02 1.11 6.44 5.92 1.09

Total 39 Average 1.02

COV 0.10

*I equals interior beam-column subassemblage; E equals exterior beam-column subassemblage.†Beam bars of UNIT3 were anchored in beam stub at far face of column.‡Unreinforced joints.§Subassemblages with one transverse beam for γcal < γult, γpred = γcal, τpred = τcal and for γcal ≥ γult, γpred = γult, τpred = τult.

Notes: τACI is the limiting values of joint stress according to ACI 318-055 and ACI 352R-02;6 τEC8 is the limiting values of joint shear stress according to Eurocode 8.4 Neither relevant Greek

codes7,8 provide information regarding limiting values for joint shear stress. All subassemblages have flexural strength ratios MR higher than 1.0. Overstrength factor a = 1.25 for

beam steel is included in computations of joint shear stress τcal = γcal MPa. 1 MPa = 144.93 psi; 1.0 MPa = 12.05 psi.fc fc fc

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ACI Structural Journal/July-August 2007 477

A question arises regarding how concrete slabs, which aretypical in buildings, affect the performance of the joints ofsubassemblages such as A1, E1, E2, and G1. Ehsani andWight31 found that “the flexural strength ratio MR at theconnections is reduced significantly due to the contributionof the slab longitudinal reinforcement.” They recommendedthat, to ensure flexural hinging in the beam, flexural strengthratios should be no less than 1.20.31 The flexural strengthratios of all the Subassemblages A1, E1, E2, and G1 tested inthis study were significantly higher than 1.20 (refer to Fig. 1(a)and (b)); thus, the presence of a concrete slab would not havehad any influence on the response of these subassemblages.

It would be of interest to learn whether simpler proceduresfor arriving to the beam-column joint ultimate strength suchas that proposed by Park and Paulay,10 would lead to similarfindings as those derived from the solution of the system ofEq. (11) to (13). To this end, Table 4 presents the joint ultimatestrength and ratios, τpred/τexp and γcal/γult for SubassemblagesA1, E1, E2, and G1 according to the aforementioned procedures.The ultimate joint shear strengths of Subassemblages A1, E1,E2, and G1 derived from the solution of the system of Eq. (11)to (13) depend on the increased joint concrete compressivestrength due to confining fc, as well as on the joint aspectratio α. These values differ significantly from those of Parkand Paulay,10 which mainly depend on the percentage of toplongitudinal beam reinforcement. Thus, Table 4 shows thatthe values of ultimate joint shear strengths of SubassemblagesA1 and E2 derived from the solution of the system of Eq. (11)to (13) are higher than those of Subassemblages E1 and G1derived by the same methodology. This clearly explains whythe Park and Paulay10 values of ultimate joint shear strengthin Table 4 are larger than the values from Eq. (11) to (13) forE1 and G1 and less than the values from Eq. (11) to (13) forA1 and E2. Finally, as can be seen from Table 4, the proposedshear strength formulation predicted the failure mode forSubassemblages A1, E1, E2, and G1 with significant accuracy,while the Park and Paulay10 procedure predicted only thefailure mode of Subassemblages A1 and E2.

CONCLUSIONSBased on the test results described in this paper, the

following conclusions can be drawn.1. The behavior of Subassemblages A1 and E2 was as

expected and as documented in the seismic design philosophyof ACI 318-05,5 ACI 352R-02,6 and Eurocode 8.4 The beam-column joints of both Subassemblages A1 and E2 performedsatisfactorily during the cyclic loading sequence to failure,allowing the formation of plastic hinges in their adjacent beams.Both subassemblages showed high strength without any appre-ciable deterioration after reaching their maximum capacity;

2. Despite the fact that Subassemblages E1 and G1 representedbeam-column subassemblages of contemporary structures, theyperformed poorly under reversed cyclic lateral deformations.The joints of both Subassemblages E1 and G1, contrary toexpectations based on Eurocode 2,3 Eurocode 8,4 and the twoGreek codes7,8 exhibited shear failure during the early stagesof cyclic loading. This happened because, for both Subas-semblages E1 and G1, the calculated joint shear stress τcalwas higher than the joint ultimate strength τult (Table 6).Damage occurred both in the joint area and in the columns’critical regions. This effect cannot be underestimated as it maylead to premature lateral instability in ductile moment-resisting frames of modern structures; and

3. It was demonstrated that the design assumptions of Euro-code 2,3 Eurocode 8,4 and those in the Greek codes7,8 did notavoid premature joint shear failures because the resultingdesign can not ensure that the joint shear stress will be signif-icantly lower than the joint ultimate strength τult and did notensure the development of the optimal failure mechanism withplastic hinges occurring in the beams while columns remainedelastic, according to the requisite strong column-weak beam.Thus, provisions in Eurocode 23 and Eurocode 84 and those inthe two Greek codes7,8 related to the design of beam-columnjoints need improvement.

NOTATION∅ = bar diametera = overstrength factorb ′c = width of joint coref ′c = compressive strength of concretehb = total depth of beamh ′c = length of joint corehc = total depth or width of square columnMR = sum of flexural capacity of columns to that of beamN = applied column axial load during testVjh = horizontal joint shear forceVjv = vertical joint shear forceα = hb/hcγcal = design values of parameter [γcal = (τcal/ )]γexp = actual values of parameter [γexp = (τexp/ )]γult = values of parameter γ at ultimate capacity of connection [γult =

(τult/ )]τ = joint shear stress

REFERENCES1. Leon, R. T., “Shear Strength and Hysteretic Behavior of Interior Beam-

Column Joints,” ACI Structural Journal, V. 87, No. 1, Jan.-Feb. 1990, pp. 3-11.2. Penelis, G. G., and Kappos, A. J., Earthquake-Resistant Concrete

Structures, E&FN Spon, London, 1997, 572 pp.3. CEN Technical Committee 250/SC2, “Eurocode 2: Design of Concrete

Structures—Part 1: General Rules and Rules for Buildings (ENV 1992-1-1),”CEN, Berlin, Germany, 1991, 61 pp.

4. CEN Technical Committee 250/SC8, “Eurocode 8: Earthquake ResistantDesign of Structures—Part 1: General Rules and Rules for Buildings (ENV1998-1-1/2/3),” CEN, Berlin, Germany, 1995, 192 pp.

5. ACI Committee 318, “Building Code Requirements for StructuralConcrete (ACI 318-05) and Commentary (318R-05),” American ConcreteInstitute, Farmington Hills, Mich., 2005, 430 pp.

6. Joint ACI-ASCE Committee 352, “Recommendations for Design ofBeam-Column Connections in Monolithic Reinforced Concrete Structures(ACI 352R-02),” American Concrete Institute, Farmington Hills, Mich.,2002, 37 pp.

7. “New Greek Earthquake Resistant Code (ERC-1995),” Athens,Greece, 1995, 145 pp. (in Greek)

8. “New Greek Code for the Design of Reinforced Concrete Structures(CDCS-1995),” Athens, Greece, 1995, 167 pp. (in Greek)

9. Hakuto, S.; Park, R.; and Tanaka, H., “Seismic Load Tests on Interiorand Exterior Beam-Column Joints with Substandard Reinforcing Details,”ACI Structural Journal, V. 97, No. 1, Jan.-Feb. 2000, pp. 11-25.

fc

fc

fc

Table 6—Experimental and predicted values of strength of Subassemblages A1, E1, E2, and G1

Sub-assem-blage

Joint aspect ratioα =

hb/hc K γcal γexp γult

Predicted shear

strength τpred, MPa

Observed shear

strength τexp, MPa

μ = τpred/τexp

γcal/γult

A1 1.50 1.558 0.685 0.584 1.46 5.05 4.31 1.17 0.47

E1 1.50 1.593 1.26 0.98 1.17 6.92 5.80 1.19 1.08

E2 1.50 1.558 0.675 0.554 1.46 5.00 4.10 1.20 0.46

G1 1.50 1.50 1.20 0.96 1.15 6.60 5.56 1.19 1.04

Notes: For γcal < γult, γpred = γcal, τpred = τcal and for γcal ≥ γult, γpred = γult, τpred = τult.

1 MPa = 144.93 psi; 1.0 MPa = 12.05 psi. Overstrength factor a = 1.25 for beam

steel is included incomputations of joint shear stress τcal = γcal MPa.

fc fc

fc

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478 ACI Structural Journal/July-August 2007

10. Park, R., and Paulay, T., Reinforced Concrete Structures, John WileyPublications, New York, 1975, 769 pp.

11. Park, R., “A Summary of Results of Simulated Seismic Load Testson Reinforced Concrete Beam-Column Joints, Beams and Columns withSubstandard Reinforcing Details,” Journal of Earthquake Engineering, V. 6,No. 2, 2000, pp. 147-174.

12. Paulay, T., and Park, R., “Joints of Reinforced Concrete FramesDesigned for Earthquake Resistance,” Research Report 84-9, Departmentof Civil Engineering, University of Canterbury, Christchurch, NewZealand, 1984, 71 pp.

13. Ehsani, M. R., and Wight, J. K., “Exterior Reinforced ConcreteBeam-to-Column Connections Subjected to Earthquake-Type Loading,”ACI JOURNAL, Proceedings V. 82, No. 4, July-Aug. 1985, pp. 492-499.

14. Soroushian, P., and Sim., J., “Axial Behavior of Reinforced ConcreteColumns under Dynamic Loads,” ACI JOURNAL, Proceedings V. 83, No. 6,Nov.-Dec. 1986, pp. 1018-1025.

15. Scott, B. D.; Park, R.; and Priestley, M. J. N., “Stress-Strain Behaviorof Concrete Confined by Overlapping Hoops at Low and High StrainRates,” ACI JOURNAL, Proceedings V. 79, No. 1, Jan.-Feb. 1982, pp. 13-27.

16. CEB-FIP, “Model Code 1990,” Bulletin d’ Information, CEB, Lausanne,Switzerland, 1993, 490 pp.

17. Mitchel, D., “Controversial Issues in the Seismic Design of Connectionsin Reinforced Concrete Frames,” Recent Developments in Lateral ForceTransfer in Buildings, SP-157, N. Priestley, M. P. Collins, and F. Seible,eds., American Concrete Institute, Farmington Hills, Mich., 1995, pp. 75-96.

18. Ehsani, M. R.; Moussa, A. E.; and Vallenilla, C. R., “Comparison ofInelastic Behavior of Reinforced Ordinary- and High-Strength ConcreteFrames,” ACI Structural Journal, V. 84, No. 2, Mar.-Apr. 1987, pp. 161-169.

19. Paulay, T., “Seismic Behavior of Beam-Column Joints in ReinforcedConcrete Space Frames, State-of-the Art Report,” Proceeding of the NinthWorld Conference on Earthquake Engineering, V. VIII, Tokyo, Japan,1988, pp. 557-568.

20. Tsonos, A. G., “Towards a New Approach in the Design of R/CBeam-Column Joints,” Technika Chronika, Scientific Journal of the TechnicalChamber of Greece, V. 16, No. 1-2, 1996, pp. 69-82.

21. Tsonos, A. G., “Shear Strength of Ductile Reinforced ConcreteBeam-to-Column Connections for Seismic Resistant Structures,” Journalof European Association for Earthquake Engineering, No. 2, 1997, pp. 54-64.

22. Tsonos, A. G., “Lateral Load Response of Strengthened ReinforcedConcrete Beam-to-Column Joints,” ACI Structural Journal, V. 96, No. 1,Jan.-Feb. 1999, pp. 46-56.

23. Tsonos, A. G., “Seismic Retrofit of R/C Beam-to-Column Jointsusing Local Three-Sided Jackets,” Journal of European Earthquake

Engineering, No. 1, 2001, pp. 48-64.24. Tsonos, A. G., “Seismic Rehabilitation of Reinforced Concrete

Joints by the Removal and Replacement Technique,” Journal of EuropeanEarthquake Engineering, No. 3, 2001, pp. 29-43.

25. Tsonos, A. G., “Seismic Repair of Exterior R/C Beam-to-ColumnJoints using Two-Sided and Three-Sided Jackets,” Structural Engineeringand Mechanics, V. 13, No. 1, 2002, pp. 17-34.

26. Tsonos, A. G., “Effectiveness of CFRP-Jackets and RC-Jackets in Post-Earthquake and Pre-Earthquake Retrofitting of Beam-Column Subassem-blages,” Final Report, Grant No. 100/11-10-2000, Earthquake Planning andProtection Organization (E.P.P.O.), Sept. 2003, 167 pp. (in Greek).

27. Paulay, T., “Equilibrium Criteria for Reinforced Concrete Beam-ColumnJoints,” ACI Structural Journal, V. 86, No. 6, Nov.-Dec. 1989, pp. 635-643.

28. Park, R., “The Paulay Years,” Recent Developments in Lateral ForceTransfer in Buildings, SP-157, N. Priestley, M. P. Collins, and F. Seible,eds., American Concrete Institute, Farmington Hills, Mich., 1995, pp. 1-30.

29. Tegos, I. A., “Contribution to the Study and Improvement of Earth-quake-Resistant Mechanical Properties of Low Slenderness StructuralElements,” PhD thesis, Appendix 13, V. 8, Aristotle University of Thessaloniki,1984, pp. 185. (in Greek)

30. Kupfer, H.; Hilsdorf, H. K.; and Rusch, H., “Behavior of Concreteunder Biaxial Stresses,” ACI JOURNAL, Proceedings V. 66, No. 8, Aug.1969, pp. 656-667.

31. Ehsani, M. R., and Wight, J. K., “Effect of Transverse Beams andSlab on Behavior of Reinforced Concrete Beam-to-Column Connections,”ACI JOURNAL, Proceedings V. 82, No. 2, Mar.-Apr. 1985, pp. 188-195.

32. Durrani, A. J., and Wight, J. K., “Behavior of Interior Beam-to-ColumnConnections under Earthquake-Type Loading,” ACI JOURNAL, ProceedingsV. 82, No. 3, May-June 1985, pp. 343-349.

33. Fujii, S., and Morita, S., “Comparison Between Interior and ExteriorRC Beam-Column Joint Behavior,” Design of Beam-Column Joints forSeismic Resistance, SP-123, J. O. Jirsa, ed., American Concrete Institute,Farmington Hills, Mich., 1991, pp. 145-166.

34. Kaku, T., and Asakusa, H., “Ductility Estimation of Exterior Beam-Column Subassemblages in Reinforced Concrete Frames,” Design ofBeam-Column Joints for Seismic Resistance, SP-123, J. O. Jirsa, ed.,American Concrete Institute, Farmington Hills, Mich., 1991, pp. 167-185.

35. Uzumeri, S. M., “Strength and Ductility of Cast-in-Place Beam-ColumnJoints,” Reinforced Concrete Structures in Seismic Zones, SP-53, AmericanConcrete Institute, Farmington Hills, Mich., 1977, pp. 293-350.

36. Attaalla, S. A., and Agbabian, M. S., “Performance of Interior Beam-Column Joints Cast from High Strength Concrete Under Seismic Loads,”Journal of Advances in Structural Engineering, V. 7, No. 2, 2004, pp. 147-157.

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ACI Structural Journal, V. 104, No. 4, July-August 2007.MS No. S-2006-226 received June 2, 2006, and reviewed under Institute publication

policies. Copyright © 2007, American Concrete Institute. All rights reserved, includingthe making of copies unless permission is obtained from the copyright proprietors. Pertinentdiscussion including author’s closure, if any, will be published in the May-June 2008ACI Structural Journal if the discussion is received by January 1, 2008.

ACI STRUCTURAL JOURNAL TECHNICAL PAPER

Cyclic loading responses of five reinforced concrete corner beam-column connections with one concentric or eccentric beam framinginto a rectangular column in the strong or weak direction arereported. The specimen variables are the shear direction and theeccentricity between the beam and column centerlines. Experimentalresults showed that two joints connecting a beam in the strongdirection were capable of supporting adjacent beam plasticmechanisms. The other three joints connecting a beam in the weakdirection, however, exhibited significant damage and loss ofstrength after beam flexural yielding. Eccentricity between beamand column centerlines had detrimental effects on the strength,energy dissipation capacity, and displacement ductility of thespecimens. Experimental verification shows that the current ACIdesign procedures are acceptable for seismic design purposes;however, it could not prevent the failure of corner connections atlarge drift levels.

Keywords: beam-column connections; joints; shear strength.

INTRODUCTIONShear failure in beam-column connections, leading to the

collapse of reinforced concrete (RC) buildings, has beenobserved in the post-earthquake reconnaissance.1-3 Thecause of collapse has been attributed to the lack of jointconfinement, especially for the exterior and corner beam-column connections without beams framing into all four sides.Since the late 1960s, amounts of experimental investigationson the seismic performance of RC beam-column connectionshave been extensively studied. The majority of the exper-imental programs have concentric beam-column connectionsisolated from a lateral-force-resisting frame at the nearestinflection points in the beams and columns framing into thejoint. Since 1976, Joint ACI-ASCE Committee 352 hasissued design recommendations for RC beam-columnjoints.4,5 Throughout the years, these guidelines evolved intostate-of-the-art reports6,7 by integrating results of newexperimental programs. Finally, a number of these designrecommendations for beam-column connections have beenadopted in Chapter 21 of the ACI 318 Building Code8 forseismic design. Current ACI design provisions are primarilydeveloped from test results of concentric beam-columnconnections, whereas eccentric beam-column connectionsare rather common in practice. Relatively few tests of eccentricRC beam-column connections have been reported in theliterature to date.9-19 To clarify the effect of eccentric beamson the behavior of connections, Joint ACI-ASCE Committee352 has called for additional research on this topic over thepast two decades,5-7 and appointed a task group to reviewand summarize previous research on eccentric RC beam-column connections.20

In the early 1990s, Joh et al.,9 Lawrance et al.,10 andRaffaelle and Wight11 tested six cruciform eccentric beam-

column connections with square columns. Early deteriorationof strength and ductility was observed in these eccentricconnections. The measured strains in joint hoop reinforcementand the joint shear deformations on the side near the beamcenterline were larger than those on the side away from thebeam centerline. Raffaelle and Wight11 suggested a formulafor reducing the effective joint width for shear resistance ofeccentric connections, and indicated that further study ofeccentric beam-column connections with rectangularcolumns is needed.

Chen and Chen12 first tested five T-shaped eccentriccorner beam-column connections in the late 1990s, whileVollum and Newman13 also tested 10 corner connectionswith two beams (one concentric and one eccentric)framing in from two perpendicular directions. Chen andChen12 concluded that the performance of eccentriccorner connections was inferior to that of concentric cornerconnections, and tapered width beams could eliminatethe detrimental effect of eccentric beams. On the otherhand, Vollum and Newman13 tested specimens withcombined loading in various load paths to investigate thebehavior of eccentric beam-column connections and toverify a previously proposed design method. The researchersconcluded that the performance of corner connections improvedsignificantly when reducing joint eccentricity. Notably, theaforementioned corner connections had square columns.

In the early 2000s, Teng and Zhou14 also tested fourcruciform eccentric beam-column connections with rectangularcolumns in aspect ratios of 2 and 1.33, and concluded thatjoint eccentricity slightly reduced the strength and stiffnessof the connections. Based on their analysis, Teng and Zhou14

also proposed an empirical equation for calculating thenominal shear strength of eccentric joints by reducing theeffective joint width.

Because floor slabs were typically not included inprevious tests of eccentric connections, Burak and Wight15

as well as Shin and LaFave16 tested five eccentric beam-column-slab connections. Each subassembly consisted ofeccentric edge beams, one concentric transverse beam, floorslabs, and rectangular columns with aspect ratios variedfrom 1.0 to 1.5. Burak and Wight’s15 three specimens weretested under sequential loading in two principal directions inwhich lateral loading was first applied in the edge beamdirection and then in the transverse beam direction. Shin andLaFave’s16 two specimens were tested under lateral loadingin the edge beam direction to simulate the behavior of an

Title no. 104-S44

Eccentric Reinforced Concrete Beam-Column Connections Subjected to Cyclic Loading in Principal Directionsby Hung-Jen Lee and Jen-Wen Ko

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ACI Structural Journal/July-August 2007460

edge connection in an exterior moment-resisting frame. Theresearchers15,16 reported that the damage in the joint regionof these eccentric beam-column-slab connections was notas severe as that of previous tests without floor slabs.9,11

Including floor slabs significantly improves the overallperformance of eccentric connections and delays thedeterioration of joint stiffness and strength. LaFave et al.20

pointed out that including floor slabs in cruciform eccentricconnections would not only raise the joint shear demand butwould also reduce the effect of joint eccentricity andenhance the joint shear-resisting mechanisms.

Test and analytical results of another nine cruciformeccentric beam-column connections were presented in the13th World Conference on Earthquake Engineering.17-19

Based on experimental results and finite element analysisof three cruciform eccentric connections, Goto and Joh17

concluded that the joint shear strength decreases as thejoint eccentricity increases due to the stress concentrationon the eccentric side. Similar observations were alsoconcluded by Kusuhara et al.18 who tested two cruciformeccentric connections (one with additional U-shaped reinforce-ment in the eccentric side). Finally, Kamimura et al.19 testedfour cruciform eccentric connections (three deep beam-widecolumn connections) and proposed an equation combiningshear and torsion to evaluate the joint shear strength.

Beam-column joints in RC buildings are probablysubjected to lateral loading in two principal directions duringan earthquake. Nevertheless, current ACI design procedures7,8

require that the joint shear strength be evaluated in eachdirection independently and implicitly assume an ellipticalinteraction relationship for biaxial loading. Notably, onlyone value of permissible shear stress is selected for a jointaccording to the effective confinement on the vertical facesof the joint, even though the column cross section is rectangular.Current ACI design procedures consider the effects of thecolumn’s aspect ratio and eccentric beam on joint shear

strength by limiting and reducing the effective joint width.More experimental results are needed to verify the effectivejoint width in eccentric connections.5-7 Thus, this experimentalprogram focuses on the behavior of eccentric corner connectionswith rectangular columns because they have not beenexperimentally verified.

RESEARCH SIGNIFICANCECurrent ACI design provisions for estimating joint shear

strength of eccentric beam-column connections are establishedbased on few experimental investigations. The effects of acolumn’s aspect ratio and eccentric beam on joint shearstrength are evaluated by the effective joint width. Additionalexperimental verification of the design provisions for eccentricconnections is needed, especially for eccentric cornerconnections with rectangular columns. This paper presentsexperimental results for five corner connections with oneconcentric or eccentric beam framing into a rectangular jointin the strong or weak direction. Experimental verificationson the ACI approach provided contribution to the under-standing of beam-column connections.

EXPERIMENTAL PROGRAMFive RC corner beam-column connections were

designed, constructed, and tested under reversed cyclicloading. A T-shaped assembly was used to represent theessential components of a corner beam-column connectionin a two-way building frame subjected to lateral loading ineach principal direction. The primary test variables werethe lateral loading directions and the eccentricity betweenthe beam and column centerlines. Neither transverse beamsnor floor slabs were constructed to ease testing. As a result,each subassembly had only one beam framing into onecorner column in each principal direction. For a corner,interstory connections, floor slabs, and transverse beamscould not only introduce additional demand on joint shearforce but also reduce the effect of joint eccentricity. Theenhancement on the joint shear capacity from confinementof floor slabs and transverse beams is questionable becausea corner joint is only confined on two adjacent faces and itis likely to sustain biaxial loading. Further study on thebehavior of corner beam-column-slab connectionssubjected to biaxial loading is needed.

Specimen geometry and reinforcementThe experimental program was designed using a concrete

compressive strength f ′c of 30 MPa (4.35 ksi) and a reinforce-ment yield stress fy of 420 MPa (60.9 ksi). Cross sections andreinforcement details of the five specimens, designated asS0, S50 (Series S), W0, W75, and W150 (Series W), areshown in Fig. 1. The first character (S or W) of the designationrepresents one south or west beam framing into the rectangularcolumn in the strong or weak direction. The subsequentnumerals denote the eccentricity between the beam andcolumn centerlines in mm. Thus, two concentric (S0 andW0) and three eccentric (S50, W75, W150) connectionswere tested in total.

The corner column had a cross section of 400 x 600 mm(16 x 24 in.) and used 12 D22 (No. 7) longitudinal bars(gross reinforcement ratio of 1.9%) and D10 (No. 3) hoopswith crossties at a spacing of 100 mm (4 in.) throughout thecolumn. The total cross-sectional area of the lateral reinforce-ment for each direction of the column was approximately equalto the minimum amount required by ACI 318-05, Section

ACI member Hung-Jen Lee is an Assistant Professor in the Department of ConstructionEngineering and a Research Engineer of the Service Center for Construction Technologyand Materials in the National Yunlin University of Science and Technology, Yunlin,Taiwan. He received his PhD from the National Taiwan University of Science andTechnology, Taipei, Taiwan, in 2000. His research interests include seismic design ofreinforced concrete structures, behavior of beam-column connections, reinforcementdetailing, and strut-and-tie models.

Jen-Wen Ko is a PhD Student in the Department of Construction Engineering at theNational Taiwan University of Science and Technology. He received his MS from theNational Yunlin University of Science and Technology in 2005.

Fig. 1—Illustration of test specimens.

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ACI Structural Journal/July-August 2007 461

21.4.4.1.8 To control the demand of shear force acting on thejoint, the loading beam had a cross section of 300 x 450 mm(12 x 18 in.) and used four D22 (No. 7) longitudinal bars (steelratio of 1.29%) at both top and bottom. To avoid beam shearfailure and ensure adequate confinement in the beam plastichinge region, closed overlapping hoops were providedthrough the length of the beam. Figure 2 illustrates theoverall geometry of the specimens. The lengths of the beamand column that were chosen to simulate the nearest inflectionpoints in the beam and column framing into the joint. Ingeneral, the five specimens were nominally identical exceptfor the joint shear direction, the embedment lengths of thehooked beam bars, and the eccentricity between the beamand column centerlines.

Connection design parametersTable 1 shows the main design parameters for the specimens.

Due to column bending in the strong or weak direction, theratios of column-to-beam flexural strength Mr at the connectionsof Series S and W were equal to 5.10 and 3.46, respectively.Because both Mr values were much greater than the specifiedvalue of 1.2, flexural hinging in the beam was anticipated.

To ensure the anchorage of beam longitudinal bars and topromote the development of a diagonal compression strutwithin the joint, the beam longitudinal bars were anchoredusing a 90-degree standard hook bent into the joint andembedded as close as possible to the back of the column(Fig. 2). Leaving a 70 mm (2.8 in.) back cover behind thehook, the provided embedment lengths within the joint were24db for Series S and 15db for Series W. The requireddevelopment lengths of hooked beam bars, measured fromthe critical section, are given in ACI 318-05, Section21.5.4.1, and ACI 352R-02, Section 4.5.2.4, for Type 2connections. Per ACI 318-05,8 the critical section is taken atthe beam-column interface. Per ACI 352R-02,7 for Type 2connections, it is taken at the outside edge of the column core.For fy of 420 MPa (60.9 ksi) and f ′c of 30 MPa (4.35 ksi), theembedment lengths required by ACI 318-05 and ACI 352R-02are 14.2db and 16.8db, respectively. As shown in Table 1, theprovided embedment length within the joints in Series W is

105% of that required by ACI 318-05 but only 89% of thatrequired by ACI 352R-02.

Based on the capacity design concept, the demand of thejoint shear force Vu is dominated by the flexural capacity ofthe beam. When computing Vu values, a probable strengthof 1.25fy for the beam longitudinal reinforcement wasincluded. Due to small differences in beam lengths, thevalue of Vu is equal to 699 kN (157.1 kips) for the specimens inSeries S and 706 kN (158.7 kips) for the specimens inSeries W, respectively.

The current ACI design procedures for joint shear strengthare based on Eq. (1)

(1)

where φ is the strength reduction factor of 0.85; Vn is thenominal joint shear strength; γ is the nominal joint shearstress of 1.0 MPa (12 psi) for corner, interstoryconnections; hc is the column depth (mm or in.) in the directionof joint shear to be considered; and bj is the effective jointwidth (mm or in.) calculated using the following equations

ACI 318-058: (2)

φVn φγ fc′bjhc Vu≥=

fc′fc′ fc′

bj318 the smaller of

bb 2x+

bb hc+

bc⎩⎪⎨⎪⎧

=

Fig. 2—Overall geometry of test specimens.

Table 1—Connection design parametersSpecimen S0 S50 W0 W75 W150

Column width bc, mm (in.) 400 (16) 600 (24)

Column depth hc, mm (in.) 600 (24) 400 (16)

Moment strength ratio Mr* 5.10 3.46

Provided embedment length db† 24 15

Joint shear demand Vu, kN (kips) 699 (157.1) 706 (158.7)

Joint eccentricity e, mm (in.) 0 (0) 50 (2) 0 (0) 75 (3) 150 (6)

Effective joint width bj318, mm (in.) 400

(16)300 (12)

600 (24)

450 (18)

300 (12)

0.53 0.71 0.54 0.72 1.07

Effective joint width bj352, mm (in.) 350 (14) 450 (18) 360 (14.4)

0.61 0.72 0.90

*Mr = ΣMn(columns)/ΣMn(beams).†Embedment lengths required by ACI 318-05 and ACI 352R-02 are 14.2db and16.8db, respectively.Note: All values are computed with fc′ = 30 MPa (4.35 ksi) and fy = 420 MPa (60.9 ksi).

Vu

γ fc ′bj318hc

---------------------------

Vu

γ fc ′bj352hc

---------------------------

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462 ACI Structural Journal/July-August 2007

ACI 352R-027: (3)

where bb is the beam width (mm or in.); x is the smallerdistance between the beam and column edges (mm or in.); bcis the column width (mm or in.); and m is 0.3 when e isgreater than bc/8, otherwise m is 0.5. The summation term isapplied on each side of the joint where the column edgeextends beyond the beam edge. The joint eccentricity e wasdesigned to be bc/8 for Specimen S50 and W75, and to be bc/4for Specimen W150.

As shown in Table 1, only Specimen W150 had a targetjoint shear stress exceeding the nominal value of 1.0 MPa(12 psi) for the effective joint width per ACI 318-05.8

The other four specimens satisfied the requirement on thejoint shear stress when following ACI design procedureswith a strength reduction factor of 0.85.

Construction and material propertiesTwo sizes of standard reinforcement meeting ASTM A 706

were used for longitudinal and transverse reinforcement in allspecimens. The D22 (No. 7) longitudinal reinforcement hadan average yield stress of 455 MPa (66 ksi) and an averageultimate strength of 682 MPa (99 ksi). The average yield andultimate strengths were 471 and 715 MPa (68 and 104 ksi) forD10 (No. 3) transverse reinforcement, respectively.

Each specimen was cast in a wood form with the beam andcolumn lying on the ground and the exterior column side(east side for Series S and north side for Series W) facing up.Concrete was supplied by a local ready mix plant usingnormal concrete aggregate and delivered by pump using a125 mm (5 in.) diameter hose. Series S was cast at one timeusing a single batch of concrete, and then Series W was castusing another batch of concrete with the same mixtureproportions. The fresh concrete was covered with plastic

bj352 the smaller of

bb bc+( ) 2⁄

bb Σmhc

2---------+

bc⎩⎪⎪⎨⎪⎪⎧

=

fc′fc′

sheets and wet-cured for 1 week. For each batch of concrete,12 150 x 300 mm (6 x 12 in.) concrete cylinders were castand cured together with the beam-column assemblies. Threecylinders were tested at 28 days and the rest were tested atthe testing date of each beam-column assembly. Table 2summarizes the concrete compressive strengths at 28 daysand the testing date. The average of concrete compressivestrengths at the testing date are used for analytical f ′c in thispaper, because the variation of concrete compressive strengthswithin each batch of concrete is small.

Test setup and loading sequenceFigure 3 shows the elevation views of the test setup. To

restrain the column for twisting about the column axis, eachbeam-column assembly was rotated 90 degrees and tieddown to a strong floor with reaction steel beams, coverplates, and rods. In addition, four one-dimensional rollerswere seated beside the column to allow in-plane rotation atboth ends of the column. This arrangement was chosen toprovide stability against torsional action. The actuator loadwas applied at the beam centerline while the column axialload was applied along the column longitudinal axis. Thus, atwist of the column about its longitudinal axis was appliedfor the eccentric connections.

To simulate the displacement reversal of beam-columnconnections during earthquake events, the specimens weresubjected to reversed cyclic lateral displacements. Axial loadwas applied at the beginning of a test and held at a level of0.10Ag fc′ during testing. A typical lateral displacementhistory consisting of three cycles at monotonicallyincreasing drift levels (0.25, 0.50, 0.75, 1.0, 1.5, 2, 3, 4, 5, 6,and 7%) was used for all specimens. The actuator appliedeach target displacement in a quasi-static manner at a speedranging from 0.05 to 1.40 mm/s (0.002 to 0.056 in./s). Targetdisplacement amplitudes at the beam tip Δ were computedusing the following equation

Drift ratio (4)

where drift ratio θ is the angular rotation of the beam chordwith respect to the column chord; Lb + 0.5hc is the verticaldistance between the actuator and column centerlines, and itis equal to 2.15 m (86 in.) for Series W and 2.075 m (83 in.)for Series S (Fig. 2).

EXPERIMENTAL RESULTSExperimental results showed that two joints of Series S

were capable of supporting the complete formation of abeam plastic hinge. In contrast, three joints of Series Wexhibited significant damage and strength degradation afterthe beam flexural yielding. Measured responses are summarizedand discussed in the following subsections. Resultspresented include: 1) beam flexural failure for Series S; 2)joint failure after beam yielding for Series W; 3) discussionof joint shear capacity; and 4) effect of joint eccentricity. Theresults are used to evaluate the influence of joint eccentricityand loading directions on the seismic performance of cornerbeam-column connections.

Beam flexural failure for Series SFigure 4 depicts the normalized load-displacement hysteretic

curves for the test specimens. The actuator load P was

θ ΔLb 0.5hc+-------------------------=

Fig. 3—Test setup for Series W (similar setup for Series S).

Table 2—Concrete compressive strengthsSpecimen S0 S50 W0 W75 W150

Concrete batch 1 2

28-day f ′c, MPa (psi) 28.5 (4133) 25.2 (3655)

Test days 49 67 53 57 60

Test day f ′c, MPa (psi) 32.6 (4728)

34.2 (4960)

28.9 (4191)

30.4 (4409)

29.1 (4220)

Analytical f ′c, MPa (psi) 33.2 (4815) 29.5 (4278)

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normalized to the nominal yield load Pn that was calculatedat a given strain of 0.004 for extreme compression fiber ofthe critical beam section. When analyzing the beam section,the measured material properties were used to model theconcrete and reinforcing bars. In addition, the beam-tipdisplacement Δ was also normalized to the drift ratio anddisplacement ductility ratio. As shown in Fig. 4, the nominalyield displacement Δy was determined by extrapolation frommeasured displacement at 0.75Pn in the 1% drift cycle.Table 3 reports the nominal yield load and displacementfor each specimen.

The load-displacement responses for Specimens S0 andS50, as shown in Fig. 4, are very similar in stiffness, strength,and ductility. Beam bars initiated yielding in the 1.0% driftcycle and maximum load was recorded at 5% drift level. Thehysteretic curves show relatively little pinching, which istypical for a flexure-dominated system. The failure mech-anisms for specimens of Series S were core concretecrushing and subsequent buckling of longitudinal bars in thebeam plastic hinge region. The buckling of the beam bars ineccentric Specimen S50 appeared earlier than that of concentricSpecimen S0.

The failure mode for the specimens in Series S was classifiedas beam flexure failure (Mode B) due to buckling of the beambars. Figure 5 shows the final damage states for test specimens.For Specimens S0 and S50, only hairline shear (diagonal)cracks were observed on the east and west face of the jointduring testing. Concrete crushing in the beam plastic regionwas evident, but only minor cover concrete spalling appearedon the east face of the joint adjacent to the beam-columninterfaces. Further, the readings of shear deformations

measured on the east face of the joints remained in elasticrange during testing. Accordingly, it was concluded that bothjoints of Series S were capable of maintaining joint integrityand remaining elastic during the formation of adjacent beamplastic hinges.

Joint failure after beam yielding for Series WAs shown in Fig. 4, the load-displacement responses for

the specimens in Series W were similar up to 4% drift cyclesafter yielding of the beam bars (1% drift cycle) and jointtransverse reinforcement (2 to 3% drift cycle). All three

Fig. 4—Normalized load versus displacement response.Table 3—Test results

Specimen S0 S50 W0 W75 W150

Nominal yield load Pn, kN (kips) 158 (35.5)

158 (35.5)

147 (33.0)

147 (33.0)

147 (33.0)

Nominal yield displacement Δy, mm (in.) 18.9 (0.74)

20.1 (0.79)

23.5 (0.93)

23.5 (0.93)

24.8 (0.98)

Over strength factor Pmax /Pn 1.22 1.20 1.11 1.11 1.05

Ductility ratio Δmax/Δy 5.41 5.12 4.58 4.60 3.41

Maximum joint shear Vj,max, kN (kips) 827 (186)

814 (183)

778 (175)

781 (176)

739 (166)

0.60 0.78 0.60 0.80 1.13

0.68 0.67 0.80 0.80 0.94

Failure mode* B B BJ BJ BJ

*Failure Mode B means beam flexural failure and BJ means joint shear failure afterbeam yielding.Note: All values are computed with analytical f ′c (refer to Table 2) of concrete andmeasured strengths of reinforcement.

Vj max,

γ fc ′bj318hc

---------------------------

Vj max,

γ fc ′bj352hc

---------------------------

Fig. 5—Final damage states for test specimens.

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464 ACI Structural Journal/July-August 2007

joints were capable of supporting beam flexural yielding upto 4% drift; however, a considerable strength degradationwas observed after the maximum loads recorded at the 4%drift level (Specimen W150) or 5% drift levels (Specimens W0and W75). Eventually, specimens in Series W exhibitedsignificant pinching curves in Fig. 4, which were typicalresponses of the shear or bond-slip mechanism.

The beam bar strains were measured using electricalresistance strain gauges attached to reinforcing bars atselected locations. Figure 6 shows the strain distributionsalong the beam bars at peak drift values for Specimens W0and W150. The hooked beam bars initiated yielding at thecritical section (Gauge 10) during the 1% drift cycle, andthen spread plasticity into the plastic hinge region (Gauges 11and 12) during the 2 and 3% drift cycle. Meanwhile, thestrain readings of Gauge 9 within the joint remained elasticup to the 3% drift level. This denoted that some bond stillexisted along the straight part of the bar embedded within thejoint. Figure 7 depicts the available strain histories of Gauge 9for Specimens W0 and W150 during testing. Both gauge

readings remained elastic in the 3% drift cycles and thenwent into yielding plateau in the first or second cycle of the4% drift level. It is evident that the beam bar was adequatelydeveloped up to 4% drift. The bond along the straight portionof the bar was lost at this stage, and therefore the bearinginside the bent portion of the hook resisted most of thetension force. The stress of the bar would begin to drop aftercrushing of the diagonal strut within the joint. In this paper,this type of failure is classified as diagonal shear compressionfailure of the joint rather than premature anchorage failure ofthe beam bars.

Figure 8 shows the cracking pattern and measurement ofjoint shear deformation on the north (flush) face of the jointfor Specimen W150. The initial joint shear cracks appeareddiagonally during the 0.5% drift cycle, followed by propagationof diagonal cracks up to a 4% drift level. After strengthdegradation commenced at 4% drift, however, no new jointshear cracks appeared while crushing and spalling ofconcrete started on the north face of the joint. The measuredjoint shear deformation rapidly increased after the maximumload recorded at 4% drift, followed by significant degradationon strength and stiffness. The joint shear failure after beamyielding (Mode BJ) was evident due to the nonlinear sheardeformation, wide-opened diagonal shear cracks, and visibleexpansion from crushing of concrete in the joint region.

Specimens W0 and W75, which had similar behavior withSpecimen W150, also failed in Mode BJ. Visible cracking,crushing, and spalling of concrete in Specimens W0 andW75 were less than those in Specimen W150 (Fig. 5). Dueto the distance between beam and column edges (Fig. 1), theappearance of initial joint shear cracks on the north face ofthe joint was delayed to the 1.0 and 1.5% drift cycle forSpecimens W75 and W0, respectively. Strength degradationafter the 5% drift cycle was attributed to the crushing ofconcrete within the joints, followed by extensive pushoutcracks distributed on the east face of the joint behind thehooked beam bars (Fig. 5). Due to crushing of the concretewithin the joint, the hooked beam bars might gradually loseits bond and anchorage within the joint. As a result, thepushout movement of the beam compression bars inducedthe pushout cracks on the east face of the joints in Series W.The joint failure and subsequent pushout cracks wereobserved at a drift level of 5% or more, which is large for a well-designed building system. Therefore, the observed behaviorappears to be acceptable for the seismic design purpose.

Discussion of joint shear capacityPaulay et al.12 first discussed that there are two shear-resisting

mechanisms exiting in joints, the truss mechanism and thediagonal strut mechanism. The truss mechanism transfers the

Fig. 6—Strain profiles of hooked beam bars for Specimens W0and W150.

Fig. 7—Strain histories of Gauge 9 on hooked beam bars forSpecimens W0 and W150.

Fig. 8—Measurement of joint shear deformation on northface of Specimen W150: (a) cracking patterns on north faceat 5% drift; and (b) load versus joint shear deformation.

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ACI Structural Journal/July-August 2007 465

forces uniformly from the beam and column bars through thebond mechanism. Adequate bond must exist between thereinforcement and concrete to necessitate a truss mechanism,which also requires considerable amounts of horizontal andvertical tie forces in the truss panel to be in equilibrium.

Figure 9 illustrates a conceptual model for the degradationof joint shear capacity under increasing drift or ductilityratio. Joints subjected to inelastic displacement reversalsoften undergo significant bond deterioration along thereinforcing bars from the adjacent beam plastic hinge. Atthis stage, a part of the joint shear is transferred through thehorizontal hoops with fan-shaped struts, while the remainderis carried by the diagonal strut. As the drift or ductility ratioincreases, the horizontal hoops would yield progressively,the joint concrete may crack excessively, and the bond of thereinforcing bars within the joint might be lost. Eventually,the joint shear force is directly transferred by the diagonalstrut mechanism.

Real shear-transferring mechanisms in joints may be acombination of the diagonal strut and the truss mechanism,with the bond deterioration being at a certain degree of longitu-dinal reinforcement during cyclic loading (Fig. 9). Hence,the joint shear capacity decreases as the cyclic inelasticloading increases, which is referred to as the degradation ofthe joint shear capacity. When the joint shear capacity fallsbelow the shear demand from beam hinging, the joint willfail in the shear after beam yielding (Mode BJ). If the jointshear capacity is greater than the demand, the maximumstrength is limited by the beam flexure capacity (Mode B).

Three levels of strength and ductility ratios for the testspecimens are shown in Table 3. Because the maximumstrengths of Specimens S0 and S50 were dominated by thebeam flexure capacity rather than the joint shear capacity,Specimens S0 and S50 had over-strength factors ofapproximately 1.2 and ductility ratios greater than 5. In contrast,Specimens W0 and W75 had over-strength factors of approxi-mately 1.1 and ductility factors of approximately 4.6 due tothe joint shear failure at 5% drift level. Further, the large-joint-eccentricity Specimen W150 barely reached the nominalyield load and deteriorated at a ductility ratio of only 3.4.

Corresponding to the maximum actuator load, themaximum shear force acting on the horizontal cross sectionwithin the joint can be estimated by

(5)

where Tmax is the maximum force in the tension reinforcementof the beam (N or lb); Vcol is the column shear in equilibriumwith the applied loading (N or lb); and jd is the internal levelarm of the beam section (mm or in.). From standard moment-curvature analysis for each specimen, jd is approximately 7/8 ofthe effective depth of the beam section. Thus, jd is simplyassumed to be 350 mm (13.8 in.) for the following evaluationof maximum joint shear forces.

Table 3 compares the maximum joint shear force withthe nominal joint shear strength following the methods inACI 318-058 or ACI 352R-02.7 When following ACI 318-05,8

Specimens S0 and W0 had equal effective joint area.Thus, the maximum joint shear forces were only 60% ofthe nominal strength for concentric Specimens S0 and W0(Table 3), but different failure modes occurred duringtesting (Fig. 4). For the flexure-dominated Specimen S0,

Vj max, Tmax Vcol– PmaxLb

jd-----

Lb 0.5hc+( )Lc

-----------------------------–⎝ ⎠⎛ ⎞= =

the maximum shear force acting on the joint was less thanthe joint shear capacity (Fig. 9). In contrast, Specimen W0failed in Mode BJ when the joint shear force reached thejoint shear capacity at 5% drift. Clearly, the joint shearcapacity in the strong direction of the rectangular joint(Specimen S0) was greater than that in the weak direction(Specimen W0). Comparing eccentric Specimens S50 andW75 can also find similar observation. This point cannot berationally reflected on the calculation of a cross-sectionalapproach within the joint, especially for the effective jointwidth given by Eq. (2).

When following ACI 352R-02,7 the maximum joint shearforces were approximately 70% of the nominal strengths forSeries S, 80% of those for Specimens W0 and W75, and 94%of that for Specimen W150. Three levels of demand-to-capacityratios reasonably reflected three levels of performance onstrength and ductility ratios shown in Table 3. This shows thatthe effective joint width bj

352 is more rational than bj318 for test

specimens. Although following the ACI 352R-027 procedurescould not avoid joint shear failure at a large drift level of 4 or5%, it is considered acceptable in a real structural system.

In this experimental program, each specimen was able tocarry the applied column axial load of 0.10Ag f ′c over theentire displacement history. Strain readings of gauges confirmedthat all column longitudinal bars remained elastic duringtesting. For a building frame during earthquake events,however, the axial load in a corner column may be higherthan 0.10Agf ′c , or even in tension, due to overturning momentfrom lateral loads. Therefore, more research on the behaviorof eccentric beam-column connections under high axialloads is still needed.

Effect of joint eccentricityThe relative energy dissipation ratio β and the equivalent

viscous damping ratio ξeq, as shown in Fig. 10(a), were usedto evaluate the energy dissipation capacities of the testspecimens. The first index β represents a fatter or narrowerhysteretic curve (pinching) with respect to an elastic perfectlyplastic model. Another quantitative index ξeq describes thehysteretic damping (or energy dissipation per cycle) withrespect to an equivalent linear elastic system.

Average β and ξeq of three cycles at each drift level forthe test specimens are compared in Fig. 10(b) and (c).Three performance levels of energy dissipation capacities

Fig. 9—Conceptual model for degradation of joint shearcapacity.

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466 ACI Structural Journal/July-August 2007

are evident. The flexure-dominated Specimens S0 and S50had a highest performance while Specimen W150 hadlowest performance. A small joint eccentricity of bc /8(Specimens S0 and W75) had a slight influence on thisexperimental program. Obviously, the large joint eccentricity ofbc/4 had significant detrimental effects on the seismicperformance of Specimen W150.

Strain histories for the joint hoops and crossties were usedto plot the strain distribution along the joint width at peak

drift values. There were three layers of transverse reinforcementat a spacing of 100 mm (4 in.) in each joint. Only the strainprofiles of the hoop legs and crossties in the direction ofshear and at the central layer of the transverse reinforcementwere compared in Fig. 11 and 12.

For the corresponding drift ratios shown in Fig. 11, thestrain readings of Gauge 24 in Specimens W75 and W150were larger than those in Specimen W0. These profilesconfirm the observations of more extensive shear or torsioncracks on the north side of the eccentric joints. On the southside, the strain readings of Gauge 20 in Specimens W75 andW150 were less than those in Specimen W0 because theshear and torsional stresses counteract each other.11

The effective joint width bj352 is also displayed in Fig. 11.

For the joints of Series W, strain gauges on hoop legs andcrossties within bj

352 yielded during the 2 or 3% drift cycleswhile the outside strain gauges remained elastic at the samedrift level. During testing of Series W, crushing of jointconcrete was observed within bj

352 on the west side of thejoint. These observations agreed well with the strain profilesshown in Fig. 11.

For the specimens in Series S, all joint hoops andcrossties remained elastic over the entire displacementhistory. Figure 12 shows the strain distributions of hooplegs and crossties along the joint width. Due to torsionalstresses from joint eccentricity, Specimen S50 had asymmetricstrain distribution with respect to concentric Specimen S0. Itshould be noted that the total cross-sectional area of jointtransverse reinforcement in two principal directions wasdifferent. Although the maximum joint shear forces in

Fig. 10—Normalized energy dissipation at each drift levelfor test specimens.

Fig. 11—Strain profiles at central layer of joint shearreinforcement in Series W.

Fig. 12—Strain profiles at central layer of joint shearreinforcement in Series S.

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Series S and W were similar (Table 3), the joint shearforces transferring by the lateral joint reinforcement inSeries S were obviously less than those in Series W. Theseprofiles agree well with Hwang and Lee’s model,22 whichproposed that the fraction of shear carried by the jointtransverse reinforcement depends on the inclination of thediagonal strut. Due to a deeper joint depth, the joints inSeries S had a flatter diagonal strut that can resist horizontaljoint shear more efficiently.23 As a result, the shear forcestransferring by the lateral joint reinforcement was reducedand then the lateral joint reinforcement remained elasticduring testing.

CONCLUSIONSBased on the evaluation of the cyclic loading responses of

five reinforced concrete beam-column corner connections inthis experimental program, the conclusions are as follows:

1. The joint shear capacity in the strong direction of arectangular joint is greater than that in the weak direction. Inthis experimental program, two joints subjected to lateralloading in the strong direction were capable of supportingthe complete formation of a beam plastic hinge. The otherthree joints exhibited significant damage at the joints withthe joint shear acting along the weak direction of the column;

2. Joint eccentricity between the beam and column center-lines had detrimental effects on the seismic performance ofbeam-column connections. Slight influence on the connectionperformance was found when the joint eccentricity was equal tohalf-quarter width of the column. As the joint eccentricityincreasing to one-quarter of the column width, significantreductions in the strength, ductility, and energy dissipationcapacity was observed; and

3. Compared with seismic performance levels, straindistributions, joint damage of the test specimens, the effectivejoint width recommend by ACI 352R-02 is a better choicethan that given in the ACI 318 code. Experimental verificationsshow that the current ACI design procedures are acceptable forseismic design purposes but could not prevent the failure ofcorner connections at a large drift level of 4 or 5%.

ACKNOWLEDGMENTSThe authors are grateful to the funding support (NSC 93-2211-E-224-010) of

the National Science Council in Taiwan. The assistance of graduate studentsfor the construction and testing of the beam-column connections in thestructural laboratory of the National Yunlin University of Science andTechnology is also acknowledged.

REFERENCES1. Moehle, J. P., and Mahin, S. A., “Observations on the Behavior of

Reinforced Concrete Buildings during Earthquakes,” Earthquake-ResistantConcrete Structures—Inelastic Response and Design, SP-127, S. K.Ghosh, ed., American Concrete Institute, Farmington Hills, Mich., 1991,pp. 67-89.

2. Sezen, H.; Whittaker, A. S.; Elwood K. J.; and Mosalam, K. M.,“Performance of Reinforced Concrete Buildings during the August 17, 1999,Kocaeli, Turkey, Earthquake, and Seismic Design and Construction Practicein Turkey,” Engineering Structures, V. 25, No. 1, Jan. 2003, pp. 103-114.

3. Earthquake Engineering Research Institute (EERI), “Chi-Chi, Taiwan,Earthquake of September 21, 1999,” Reconnaissance Report No. 2001-02,Earthquake Engineering Research Institute (EERI), Oakland, Calif.

4. Joint ACI-ASCE Committee 352, “Recommendations for Design of

Beam-Column Joints in Monolithic Reinforced Concrete Structures,” ACIJOURNAL, Proceedings V. 73, No. 7, July 1976, pp. 375-393.

5. Joint ACI-ASCE Committee 352, “Recommendations for Design ofBeam-Column Joints in Monolithic Reinforced Concrete Structures,” ACIJOURNAL, Proceedings V. 82, No. 3, May-June 1985, pp. 266-283.

6. Joint ACI-ASCE Committee 352, “Recommendations for Design ofBeam-Column Joints in Monolithic Reinforced Concrete Structures (ACI352R-91),” American Concrete Institute, Farmington Hills, Mich., 1991,18 pp.

7. Joint ACI-ASCE Committee 352, “Recommendations for Design ofBeam-Column Connections in Monolithic Reinforced Concrete Structures(ACI 352R-02),” American Concrete Institute, Farmington Hills, Mich.,2002, 40 pp.

8. ACI Committee 318, “Building Code Requirements for StructuralConcrete (ACI 318-05) and Commentary (318R-05),” American ConcreteInstitute, Farmington Hills, Mich., 2005, 430 pp.

9. Joh, O.; Goto, Y.; and Shibata, T., “Behavior of Reinforced ConcreteBeam-Column Joints with Eccentricity,” Design of Beam-Column Jointsfor Seismic Resistance, SP-123, J. O. Jirsa, ed., American Concrete Institute,Farmington Hills, Mich., 1991, pp. 317-357.

10. Lawrance, G. M.; Beattie, G. J.; and Jacks, D. H., “The Cyclic LoadPerformance of an Eccentric Beam-Column Joint,” Central LaboratoriesReport 91-25126, Central Laboratories, Lower Hutt, New Zealand, Aug.1991, 81 pp.

11. Raffaelle, G. S., and Wight, J. K., “Reinforced Concrete EccentricBeam-Column Connections Subjected to Earthquake-Type Loading,” ACIStructural Journal, V. 92, No. 1, Jan.-Feb. 1995, pp. 45-55.

12. Chen, C. C., and Chen, G. K., “Cyclic Behavior of Reinforced ConcreteEccentric Beam-Column Corner Joints Connecting Spread-Ended Beams,”ACI Structural Journal, V. 96, No. 3, May-June 1999, pp. 443-449.

13. Vollum, R. L., and Newman, J. B., “Towards the Design of ReinforcedConcrete Eccentric Beam-Column Joints,” Magazine of ConcreteResearch, V. 51, No. 6, Dec. 1999, pp. 397-407.

14. Teng, S., and Zhou, H., “Eccentric Reinforced Concrete Beam-ColumnJoints Subjected to Cyclic Loading,” ACI Structural Journal, V. 100, No. 2,Mar.-Apr. 2003, pp. 139-148.

15. Burak, B., and Wight, J. K., “Seismic Behavior of Eccentric R/CBeam-Column-Slab Connections under Sequential Loading in Two PrincipalDirections,” ACI Fifth International Conference on Innovations in Designwith Emphasis on Seismic, Wind and Environmental Loading, Quality Control,and Innovation in Materials/Hot Weather Concreting, SP-209, V. M. Malhotra,ed., American Concrete Institute, Farmington Hills, Mich., 2002, pp. 863-880.

16. Shin, M., and LaFave, J. M., “Seismic Performance of ReinforcedConcrete Eccentric Beam-Column Connections with Floor Slabs,” ACIStructural Journal, V. 101, No. 3, May-June 2004, pp. 403-412.

17. Goto, Y., and Joh, O., “Shear Resistance of RC Interior EccentricBeam-Column Joints,” Proceedings of the 13th World Conference onEarthquake Engineering, Paper No. 649, Vancouver, British Columbia,Canada, 2004, 13 pp.

18. Kusuhara, F.; Azukawa, K.; Shiohara, H.; and Otani, S., “Tests ofReinforced Concrete Interior Beam-Column Joint Subassemblage withEccentric Beams,” Proceedings of 13th World Conference on EarthquakeEngineering, Paper No. 185, Vancouver, British Columbia, Canada, 2004,14 pp.

19. Kamimura, T.; Takimoto, H.; and Tanaka, S., “Mechanical Behaviorof Reinforced Concrete Beam-Column Assemblages with Eccentricity,”Proceedings of the 13th World Conference on Earthquake Engineering,Paper No. 4, Vancouver, British Columbia, Canada, 2004, 10 pp.

20. LaFave, J. M.; Bonacci, J. F.; Burak, B.; and Shin, M., “EccentricBeam-Column Connections,” Concrete International, V. 27, No. 9, Sept.2005, pp. 58-62.

21. Paulay, T.; Park, R.; and Priestley, M. J. N., “Reinforced ConcreteBeam-Column Joints under Seismic Actions,” ACI JOURNAL, ProceedingsV. 75, No. 11, Nov. 1978, pp. 585-593.

22. Hwang, S. J., and Lee, H. J., “Strength Prediction for DiscontinuityRegions by Softened Strut-and-Tie Model,” Journal of Structural Engineering,ASCE, V. 128, No. 12, Dec. 2002, pp. 1519-1526.

23. Hwang, S. J.; Lee, H. J.; Liao, T. F.; Wang, K. C.; and Tsai, H. H.,“Role of Hoops on Shear Strength of Reinforced Concrete Beam-ColumnJoints,” ACI Structural Journal, V. 102, No. 3, May-June 2005, pp. 445-453.

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ACI Structural Journal, V. 104, No. 4, July-August 2007.MS No. S-2006-212 received May 25, 2006, and reviewed under Institute publication

policies. Copyright © 2007, American Concrete Institute. All rights reserved, includingthe making of copies unless permission is obtained from the copyright proprietors.Pertinent discussion including author’s closure, if any, will be published in the May-June 2008 ACI Structural Journal if the discussion is received by January 1, 2008.

ACI STRUCTURAL JOURNAL TECHNICAL PAPER

Flexural strengthening using near-surface mounted (NSM) fiber-reinforced polymer (FRP) materials is a promising technology. AsNSM reinforcement, the FRP is surrounded by concrete on threesides so the bond and damage problems associated with externallybonded FRP strengthening systems are reduced or eliminated. Thispaper presents experimental results from 12 full-scale concretebeams strengthened with NSM carbon FRP (CFRP) strips. Threecompanion unstrengthened specimens were also tested to serve as acontrol. Experimental variables include three different ratios of steelreinforcement ρs and two different ratios of CFRP reinforcementρfrp. Yield and ultimate strengths, flexural failure modes, and ductilityare discussed based on measured load, deflection, and strain data.Test results show measurable increases in yield and ultimatestrengths; predictable nominal strengths and failure modes; andeffective force transfer between the CFRP, epoxy grout, andsurrounding concrete. Also, strengthening with CFRP resulted in adecrease in both energy ductility and deflection ductility.

Keywords: beam; polymer; reinforcement; strength.

INTRODUCTIONIn-service steel-reinforced concrete flexural members may

require strengthening due to material decay of the internalreinforcement and surrounding concrete, errant design andconstruction practice, increased service loads, and unforeseensettlement and structural damage. These conditions requirestructural retrofit to increase the flexural strength of thesection. A popular method of increasing the flexural strengthof beams, walls, and slabs is through external bonding offiber-reinforced polymer (FRP) plates and sheets. FRP materialsare characterized by high tensile strength and low unitweight, and they are noncorrosive when exposed to chlorideenvironments. An excellent summary of research in this areais available by Teng et al. (2002) and ACI has published adesign guide for strengthening concrete structures withexternally-bonded FRP materials (ACI Committee 440 2002).

Premature failure of externally-bonded FRP plates andsheets can occur before the ultimate flexural capacity of thestrengthened section is achieved. This is typically due tobond failure between the FRP and concrete or tensile peelingof the cover concrete. Available research documenting thisbehavior is abundant. Brena et al. (2003) reported debondingof longitudinal carbon FRP (CFRP) sheets at deformationlevels less than half the deformation capacity of controlspecimens. Nguyen et al. (2001) observed only a limitedincrease in flexural capacity for beams strengthened withpartial length longitudinal CFRP sheets due to prematuredelamination, or ripping, of the concrete cover surroundingthe steel reinforcement. For beams strengthened with CFRPplate and fabric systems, Grace et al. (2002) identified brittlefailure by shear tension and debonding, respectively. Shinand Lee (2003) reported failure of beams held undersustained load and strengthened with CFRP laminates due to

rip-off type failure of the CFRP at loads well below the ultimateflexural capacity of the sections. Similar results have beenreported by Rahimi and Hutchinson (2001), Bencardino et al.(2002), Arduini and Nanni (1997), Sharif et al. (1994),Saadatmanesh (1994), and Mukhopadhyaya and Swamy(1999). In addition to problems associated with bond failure,external FRP plates are vulnerable to mechanical, thermal,and environmental damage. It should be noted, however, thatmechanical anchors can be used to improve the peel resistanceof externally bonded FRP.

In response to the detrimental conditions associated withexternally bonded FRP, engineers have proposed relocatingthe strengthening FRP material from the unprotected exterior ofthe concrete to the protected interior. This technology isreferred to as near-surface mounted (NSM) strengtheningand is shown in Fig. 1. The surrounding concrete now protectsthe FRP so that mechanical and thermal damage is unlikely.Other advantages of using NSM FRP technology includeimproved bond and force transfer with the surrounding concreteand the ability to increase the negative bending strength ofbridge decks, pavements, and other structural riding surfaces.

RESEARCH SIGNIFICANCEThis paper documents behavior of full-scale test beams

strengthened in flexure with NSM CFRP strips and tested tofailure in four-point bending. The parameters of steel andFRP reinforcement ratios are investigated. Concrete strength,shear span-to-depth ratio, and steel reinforcement ratios wereselected as typical for concrete flexural components in thecivil infrastructure. Theory related to failure modes and strengthmodels are evaluated based on comparison with the test data.It is expected that the conclusions reported will ultimatelycontribute to the development of a design guide for using NSMFRP for flexural strengthening of concrete beams and slabs.

Title no. 104-S41

Flexural Behavior of Concrete Beams Strengthened with Near-Surface-Mounted CFRP Stripsby Joseph Robert Yost, Shawn P. Gross, David W. Dinehart, and Jason J. Mildenberg

Fig. 1—Concrete member strengthened in flexure withNSM FRP.

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431ACI Structural Journal/July-August 2007

BACKGROUND AND LITERATURE REVIEWNanni (2000) compared the behavior of full-scale simply

supported highway bridge deck panels strengthened in flexurewith either externally bonded CFRP laminates or internallyplaced NSM CFRP bars. Failure of the CFRP laminatereinforced deck spans was through a combination of ruptureand peeling of the CFRP laminates. The NSM CFRP-reinforcedspan failed by tensile rupture of the CFRP bars. Relative to thecapacity of an unstrengthened control deck, moment strengthincreases of 17 and 29% were reported for decks retrofittedwith externally bonded CFRP laminates and internally placedNSM CFRP bars, respectively.

DeLorenzis et al. (2000) tested three steel-reinforcedconcrete T-beams strengthened in flexure with NSM glassFRP (GFRP) and NSM CFRP bars. The CFRP retrofittedbeams experienced increases in strength of 30% (two No. 3CFRP bars) and 44% (two No. 4 CFRP bars) over anunstrengthened control specimen. Both CFRP strengthenedbeams failed due to debonding of the NSM rods. The specimenstrengthened with two No. 4 GFRP bars also failed due todebonding of the NSM GFRP bars at a load 26% higher thanthe control specimen. The authors reported that bond is criticalto using this technology effectively. Bond failure of theNSM FRP bars was also identified by DeLorenzis and Nanni(2001) as in need of further investigation. Debonding of theNSM FRP bars due to splitting of the epoxy used for holdingthe rod in place was reported. It was suggested that thisfailure limit-state could possibly be avoided by increasingbond lengths or anchoring the NSM rods in the flange.Significantly, the authors reported that, where debonding ofthe NSM FRP bars is prevented, splitting of the concretecover surrounding the longitudinal steel bars might becomethe controlling ultimate limit-state. Loss of anchorage wasobserved in several of their test specimens. In a relatedexperimental bond study, DeLorenzis et al. (2004) state thatepoxy is superior to cement paste as the groove filler material,a groove size-to-bar diameter of 2.0 is optimal, and a smoothgrove surface yields slightly lower local bond strengths, but ispreferable because it yields a more ductile bond-slip behavior.

Taljsten and Carolin (2001) evaluated four rectangularconcrete beams subjected to four-point bending andmonotonically loaded in deformation control. Three of thetest beams were strengthened with NSM CFRP strips and thefourth served as a control specimen. Two of the threestrengthened beams used an epoxy for bonding the FRP andthe third used a cement grout. Test results showed that two ofthe three retrofitted beams failed due to anchorage loss

between the NSM FRP strips and concrete. The thirdstrengthened beam failed due to tensile rupture of the FRPstrip. Predicted failure loads overestimated measured strengths.

El-Hacha and Rizkalla (2004) compared the behavior ofbeams strengthened on an equal axial stiffness basis usingNSM FRP bars and strips and externally bonded FRP laminates.Their research showed that higher ultimate strengths andincreased ductility were achieved by the NSM strengthenedspecimens. They also noted that bond integrity of NSM FRPbars was less effective than for NSM FRP strips.

Together, these research findings demonstrate that bondintegrity can not be taken for granted and that bond relatedlimit states must also be considered for NSM FRP. DeLorenzisand Nanni (2002) suggest that bond performance will beinfluenced by multiple factors including bond length, NSMFRP bar diameter and surface characteristic, material charac-teristics of the FRP, groove geometry, and properties of theepoxy grout. Their experimental bond tests showed threebond related failure modes, namely, splitting of the epoxycover, cracking of the concrete surrounding the grove, andpullout of the NSM FRP rod.

EXPERIMENTAL PROGRAMThis experimental investigation consisted of testing 15 simply

supported full-scale concrete beams in flexure and materialcharacterization of the CFRP, steel reinforcement, and concrete.All test beams had a shear-span-to-steel-reinforcement-depthratio av /ds of 8.4. This ratio was intentionally selected so thatultimate strength would be controlled by flexural failure andnot shear failure. The test setup and associated specimendetails are shown in Fig. 2.

The 15 test beams were separated into three groups of fivebeams, with all beams in a given group having the samecross section and steel reinforcement ratio ρs. Within eachgroup of five beams, two beams had one CFRP strip(designated 6-1Fa&b, 9-1Fa&b, and 12-1Fa&b), two beamshad two CFRP strips (designated 6-2Fa&b, 9-2Fa&b, and12-2Fa&b), and one beam acted as a control with no CFRP(designated 6-C, 9-C, and 12-C). Note that beams identified asa and b are replicate specimens. Thus, the two parametersinvestigated in the study are the amount of steel and CFRPreinforcements. Table 1 presents the unstrengthened steelreinforcement ratio ρs relative to a balanced design ρs/ρsb.

Joseph Robert Yost is an Associate Professor of Civil and Environmental Engineering atVillanova University, Villanova, Pa. His research interests include the use of innovativematerials in transportation infrastructure, nondestructive methods for health monitoringof structures, and seismic design and analysis of bridges.

ACI member Shawn P. Gross is an Associate Professor in the Department of Civiland Environmental Engineering at Villanova University. He is Secretary of JointACI-ASCE Committee 423, Prestressed Concrete, and a member of ACI Committees 213,Lightweight Aggregate and Concrete; 363, High-Strength Concrete; 435, Deflection ofConcrete Building Structures; 440, Fiber Reinforced Polymer Reinforcement; andE803, Faculty Network Coordinating Committee. His research interests include thedesign and behavior of reinforced and prestressed concrete structures, including theuse of high-strength concrete and fiber-reinforced polymer reinforcement.

David W. Dinehart is an Associate Professor of Civil and Environmental Engineering atVillanova University. His research interests include seismic evaluation of wood structures,passive damping systems, and the design and behavior of concrete and steel structures.

Jason J. Mildenberg is a Structural Engineer with Schoor De Palma of Brick, Manalapan,N.J. He received an MS in civil engineering from Villanova University.

Fig. 2—Test setup.

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432 ACI Structural Journal/July-August 2007

The ratios of 0.353, 0.470, and 0.684 were selected as typicalfor existing structures.

All specimens were instrumented with a concrete straingauge located on the top compression fiber at the centerspan. Strengthened Specimens 6-1Fb, 6-2Fb, 9-1Fb, 9-2Fb,12-1Fb, and 12-2Fb had an additional strain gauge bonded tothe CFRP at the center span. Linear variable displacementtransducers (LVDTs) were used to measure displacement atthe center span.

Concrete for the test specimens was delivered to thelaboratory by a concrete supplier. The concrete was inaccordance with Pennsylvania Department of Transportation(PennDOT) Class AAA, Concrete for Bridge Decks, withdesign specifications and properties given in BD-601M

(PennDOT 2001). The mixture design was selected as typicalfor bridge decks and is given as follows: water 1530 N/m3

(263 lb/yd3), cement 3967 N/m3 (682 lb/yd3), coarse aggregate1784 lb/yd3, fine aggregate 7242 N/m3 (1245 lb/yd3), air entrain-ment 30 N/m3 (3 oz/yd3), and retarder 196 N/m3 (20 oz/yd3). Theslump at specimen casting was 101.6 mm (4 in.), and the33-day compressive strength as determined by ASTM C 684-99(ASTM 1999) using 100 mm (4 in.) diameter by 200 mm (8 in.)high cylinders was 37.2 MPa (5.4 ksi) for all beams. Yieldstrength of the steel reinforcement was determined fromuniaxial coupon testing to be 510 MPa (74 ksi) for No. 4 barsand 490 MPa (71 ksi) for the No. 5 bars. Elastic modulus Esis taken as 200 GPa (29,000 ksi).

The CFRP strips have a thin rectangular cross section thatmeasures approximately 15 x 2.5 mm (0.60 x 0.10 in.), and thesurface of the wide face is roughened to enhance forcetransfer with the concrete epoxy grout. A photo of the CFRPreinforcement with associated instrumentation detail can beseen in Fig. 3(a). The material composition is 60% 4137 MPa(600 ksi) carbon fiber by volume in a bisphenol epoxyvinylester resin matrix. The CFRP elastic modulus Ef andultimate tensile strength ffu were determined from testinguniaxial coupon specimens according to ACI Committee440 (2004). Test results are shown in Fig. 3(b) from whichEf and ffu were determined to be 136 GPa and 1648 MPa(19,765 and 239 ksi), respectively.

Installation of the NSM CFRP strips is shown in Fig. 4 anddescribed as follows. First, the beams were rotated 180 degreesabout the long axis so that the steel reinforcement was at thetop of the beam. Next a rectangular groove approximately6.4 mm (1/4 in.) wide by 19 mm (3/4 in.) deep was cutlongitudinally in the concrete where the CFRP was to beinstalled. The groove was cut using a hand-held circular withan 18 cm (7 in.) diameter diamond-tooth, abrasive cuttingblade. The saw was fitted with a rip guide, so that the distancefrom the edge of the beam to the blade could be set andmaintained during cutting. The depth of the blade was set to19 mm (3/4 in.) by adjusting the saw. The saw blade was justover 3.2 mm (1/8 in.) wide so that two passes were made toachieve the required width. For test specimens having oneCFRP strip, the longitudinal groove was located at the centerof the cross section; and for specimens having two CFRPstrips, the grooves were located at the 1/3 points in the crosssection. Next, the groove was thoroughly cleaned of debriswith compressed air and then partially filled with a structuralepoxy material that bonds with the concrete and FRP to

Table 1—Specimen design and predicted strength parameters

Specimen ρs/ρsb* Afb, mm2 (in.2) Af /Afb Failure type†‡ ff-ult, MPa (ksi) Mn, kN-mm (kip-in.) Pn , kN (kip)§ Pn/PnC

6-C

0.684 –38.86 (–0.060)

NA SY/CC NA 23,068 (204.2) 18.92 (4.25) 1

6-1Fa&b –0.85 CC 810 (117.4) 26,606 (235.5) 21.82 (4.91) 1.15

6-2Fa&b –1.69 CC 709 (102.8) 29,168 (258.2) 23.92 (5.38) 1.26

9-C

0.470 –1.04 (–0.0016)

NA SY/CC NA 25,104 (222.2) 20.59 (4.63) 1

9-1Fa&b –31.66 CC 1276 (185) 31,221 (276.3) 25.61 (5.76) 1.24

9-2Fa&b –63.31 CC 1091 (158) 35,415 (313.5) 29.05 (6.53) 1.41

12-C

0.353 38.94 (0.060)

NA SY/CC NA 25,790 (228.3) 21.15 (4.76) 1

12-1Fa&b 0.84 TR 1648 (239) 34,071 (301.6) 27.95 (6.28) 1.32

12-2Fa&b 1.69 CC 1436 (208.2) 40,023 (354.2) 32.83 (7.38) 1.55*ρs = As/bds and ρsb = 0.85(fc′ /fy)β1(εcu)/(εcu + εsy) is unstrengthened balanced reinforcement ratio.†SY = steel yield, CC = concrete compression failure, TR = tensile rupture of FRP.‡For all samples with CC failure, steel has yielded at ultimate as per analysis of Eq. (4).§Pn = Mn/1219 mm (Mn/48 in.).

Fig. 3—CFRP and tensile test results.

Fig. 4—Specimen preparation.

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ACI Structural Journal/July-August 2007 433

provide a mechanism for force transfer. The epoxy groutused was a two-part epoxy. Finally, the FRP was depressedinto the groove, where care was taken to ensure that no airvoids were trapped within the epoxy gel. Excess epoxy gelwas then cleaned from the concrete surface and curing wasdone for a minimum of 2 weeks.

All beams were tested monotonically from an uncrackedcondition. Two 90 kN (10 ton) hydraulic cylinders, located152 mm (6 in.) on either side of center span and controlledby a manually-operated pump, were used to apply load at anapproximate rate of 4.5 kN/minute (1 kip/minute). A loadcell was located under each hydraulic cylinder to measureapplied load. Electronic signals from the strain gauges(concrete and CFRP), LVDTs, and load cell were recordedby a 16-bit data acquisition system at a frequency of 1 Hz.

ANALYTICAL STRENGTHFigure 5 illustrates the assumed basic analytical conditions

of internal strain, stress, and resultant force for a crackedsection at ultimate that is under-reinforced with steel (ρs < ρsb)and strengthened with FRP. From Fig. 5, the followingassumptions are implicit: strain varies linearly through thecross section, the section is initially uncracked, perfect bondexists between the steel and FRP reinforcements andconcrete, the concrete strain at compression failure is 0.003,the Whitney rectangular stress block in the compressionzone is a valid substitution for a nonlinear stress distributionat ultimate, and the steel stress-strain behavior is assumed tobe elastic-plastic. Also noted in Fig. 5, because the section isinitially uncracked and df > ds, the FRP strain εf will slightlyexceed the steel strain εs.

The theoretical nominal flexural strength Mn of an initiallyuncracked beam that is under-reinforced with steel (ρs < ρsb)and strengthened with FRP is dependent on the amount ofFRP provided (Af) relative to the FRP area corresponding toa balanced-strengthened strain condition (Afb). In this context,balanced-strengthened represents simultaneous tensile ruptureof the FRP and compression failure of the concrete. Again,for an initially uncracked section with df > ds and εf = εfu inFig. 5, by default the steel for a balanced-strengtheneddesign will have yielded (εs > εsy). Using these assumptionsand strain limits, and considering compatibility and equilibrium,the theoretical balanced-strengthened area of FRP is

(1)

Using Eq. (1) as a theoretical FRP reinforcement limit, failurewill be tensile rupture of the FRP when Af > Afb, or compressionfailure of the concrete, when Af < Afb. It is noted that Afb can beeither positive or negative, depending on the existing amount ofsteel reinforcement present (As). For a negative result fromEq. (1), Af provided will always be greater than Afb, indicating acompression failure of the concrete. Strain distributions for FRPfailure, balanced-strengthened, and compression failure areshown in Fig. 5(b). For sections controlled by FRP failure, thecompression block depth a and nominal moment strength atultimate Mn are calculated from equilibrium as follows

(2a)

Afb

0.85f′cbβ1dfεcu

εcu εfu+-------------------

⎩ ⎭⎨ ⎬⎧ ⎫

As fy–

ffu

-------------------------------------------------------------------------=

aAf ffu As fy+

0.85fc′b-------------------------- for Af Afb<=

(2b)

For sections controlled by concrete crushing, the stress levelin the steel is initially unknown, as is shown in Fig. 5(b). Itcan be determined by fixing the steel and concrete strains atyield εsy and crushing εcu, respectively, calculating the steelarea corresponding to yield Asy, and comparing this with thearea of steel present As. From Fig. 5(b), this is as follows

(3)

Accordingly, for As ≤ Asy, the steel stress is equal to fy. Likewise,for As > Asy, the steel stress is less than fy and must be determinedfrom compatibility and equilibrium. Using this procedure,the steel stress at ultimate for all specimens controlled byconcrete failure in this study was equal to yield. With thesteel stress at yield, the compression block a, stress in theFRP reinforcement ff, and nominal moment capacity Mn forsections controlled by concrete failure are found fromcompatibility and equilibrium as follows

(4a)

(4b)

(4c)

The preceding analysis is offered as an alternative to the trialand error procedure set forth by ACI Committee 440 (2002)and yields identical results as would be obtained using theACI 440.2R procedure. Table 1 summarizes relevant designand strength parameters. Moment strength Mn was calculatedusing the measured material strengths for the steel, CFRP,and concrete. It is evident from Table 1 that, for a given areaof FRP Af , the relative increase in strength Pn/PnC isinversely proportional to the amount of steel reinforcement.

TEST RESULTSLoad-deflection and load-strain results are shown in Fig. 6 and

summarized in Table 2. Typical photos at failure are shown inFig. 7. The applied cylinder loads plotted in Fig. 6 and recordedin Table 2 have been corrected to include the self-weightbending effects of the beam. Moment equivalence at center span

Mn Af ffu dfa2---–⎝ ⎠

⎛ ⎞ As fy dsa2---–⎝ ⎠

⎛ ⎞ for Af Afb<+=

Asy

0.85fc′bβ1dsεcu

εcu εsy+--------------------⎝ ⎠⎛ ⎞ AfEfεsy

df

ds

----⎝ ⎠⎛ ⎞–

fy

-------------------------------------------------------------------------------------------=

aAfEfεcu Asfy–( )

2 4 0.85( )fc ′bβ1AfEfεcudf+ AfEfεcu Asfy–( )–2( )0.85fc ′b

------------------------------------------------------------------------------------------------------------------------------------------------------=

ff Efεcudf α β1⁄–( )α β1⁄

---------------------------- ffu≤=

Mu Af ff dfa2---–⎝ ⎠

⎛ ⎞ As fy dsa2---–⎝ ⎠

⎛ ⎞+=

Fig. 5—Analytical model at ultimate.

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434 ACI Structural Journal/July-August 2007

was used to calculate an equivalent concentrated force Peq thatwas added to all laboratory measured load data. Momentequivalence at center span is expressed as {1/8wbeamL2} ={Peqav}. From Fig. 3, Peq for the 152, 230, and 305 mm (6,9, and 12 in.) wide specimens is calculated to be 0.50, 0.77,and 1.0 kN (0.115, 0.172, and 0.230 kips), respectively.

From Fig. 6, the physical effects of supplemental strengtheningwith CFRP are clearly evident when strengthened specimensare compared with companion control (unstrengthened)specimens. All specimens strengthened with CFRP showed

a significant increase in ultimate strength when comparedwith the companion control specimens. To a lesser degree,strengthening with CFRP increased stiffness and yield load.Detailed discussions of the test results for control andstrengthened specimens are presented in the following sections.

Control specimens: 6-C, 9-C, and 12-CReferring to the load-deflection behavior of control

Specimens 6-C, 9-C, and 12-C, the ductile behavior charac-teristic of under-reinforced steel flexural (ρs < ρsb) members

Fig. 6—Load-deflection and load-strain results.

Table 2—Summary of test results

Theory

Measured

ComparisonYield Ultimate

Sample ID Pn, kN (kip) Py, kN (kip) Mechanism type* Pmax, kN (k) Py /PyC Average Pmax/PmaxC Average Pmax/Py Average Pmax/Pn

6-C (control) 18.9 (4.25) 19 (4.28) SY/CC 21.12 (4.75) 1 1 1 — 1.11 — 1.12

6-1Fa21.8 (4.91)

20.9 (4.69) CC 24.83 (5.58) 1.101.11

1.181.14

1.191.14

1.14

6-1Fb 21.3 (4.78) CC 23.24 (5.23) 1.12 1.10 1.09 1.06

6-2Fa23.9 (5.38)

24.4 (5.48) CC 24.99 (5.62) 1.281.29

1.181.23

1.021.06

1.04

6-2Fb 24.7 (5.56) CC 26.94 (6.06) 1.30 1.28 1.09 1.13

9-C (control) 20.6 (4.63) 22.4 (5.03) SY/CC 25.29 (5.69) 1 1 1 — 1.13 — 1.23

9-1Fa25.6 (5.76)

25.3 (5.70) CC 28.22 (6.34) 1.131.11

1.121.11

1.111.13

1.10

9-1Fb 24.5 (5.50) CC 27.93 (6.28) 1.09 1.10 1.14 1.09

9-2Fa29.0 (6.53)

27.7 (6.22) CC 37.05 (8.33) 1.241.18

1.471.44

1.341.38

1.28

9-2Fb 25.0 (5.63) CC 35.82 (8.05) 1.12 1.42 1.43 1.23

12-C (control) 21.2 (4.76) 21.5 (4.84) SY/CC 23.52 (5.29) 1 1 1 — 1.09 — 1.11

12-1Fa27.9 (6.28)

24.7 (5.56) TR 29.59 (6.65) 1.151.18

1.261.29

1.201.20

1.06

12-1Fb 25.9 (5.81) TR 31.01 (6.97) 1.20 1.32 1.20 1.11

12-2Fa32.8 (7.38)

26.5 (5.97) CC 33.80 (7.60) 1.231.27

1.441.61

1.271.38

1.03

12-2Fb 28.0 (6.30) CC 41.77 (9.39) 1.30 1.78 1.49 1.27*SY = steel yield, CC = concrete crushing, TR = CFRP tensile rupture.

Fig. 7—Test specimens at failure.

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435ACI Structural Journal/July-August 2007

is apparent. Initially, all sections are uncracked and grosssection properties apply (Ig). At the cracking load Pcr,behavior changes from uncracked to cracked-elastic. As loadis increased further, the section responds elastically until theyield strength of the steel reinforcement fy is reached. At theyield load Py, behavior changes from cracked-elastic toinelastic. For Specimens 6-C, 9-C, and 12-C, steel yieldoccurred at 19, 22.4, and 21.5 kN (4.28, 5.03, and 4.84 kips),respectively. The yield load corresponds to a flattening of theload-deflection trace and simultaneous inflection in theconcrete load-strain response. Yield is followed by a load plateauwhere the moment capacity of the section remains roughlyconstant. The load plateau is clearly visible for Specimens 9-Cand 12-C, and to a lesser degree for Specimen 6-C.

At the ultimate load Pmax, failure occurred by concretecrushing. Ultimate load for Specimens 6-C, 9-C, and 12-Cwas 21.1, 25.3, and 23.5 kN (4.75, 5.69, and 5.29 kips),respectively. For all control specimens, the ultimate loadPmax was approximately 12% greater than the yield load Py.The measured failure loads for Specimens 6-C, 9-C, and 12-Cwere 12, 23, and 11%, respectively, greater than the theoreticalnominal capacity Pn.

Specimens strengthened with one CFRP strip:6-1Fa&b, 9-1Fa&b, and 12-1Fa&b

For specimens strengthened with one CFRP strip, the changefrom cracked-elastic to inelastic behavior (yield point) is lessabrupt and the associated reduction in the slope of the load-deflection curve is less than for the control specimens. Thisis especially true for specimens with a large relative amount ofsteel reinforcement ρs/ρsb. Referring to Fig. 6, for Specimens6-1Fa&b, the change in stiffness at ensuing nonlinear load-deflection response associated with steel yielding is negligible.These specimens have the largest relative area of steelreinforcement equal to 0.68ρsb. For Specimens 9-1Fa&b and12-1Fa&b, however, the change in stiffness after steel yieldis more apparent. These specimens were reinforced with0.47ρsb and 0.34ρsb, respectively.

The mechanism of failure at ultimate for all specimens inthis group is consistent with that predicted using the theoryoutlined previously and summarized in Table 1. As can beseen in Table 2, all 152 and 230 mm (6 and 9 in.) wide specimensstrengthened with one CFRP strip failed by crushing of theconcrete. For these specimens, the CFRP did not ruptureprior to concrete crushing, indicating that the strain level wasless than the ultimate material strength. For the 305 mm (12 in.)wide specimens with one CFRP strip, however, the CFRPreinforcement did rupture at ultimate. This was followed bycompression failure in the concrete. Thus, the bond betweenthe CFRP and concrete for Specimens 12-1Fa&b was able todevelop the tensile strength of the CFRP strip. Also, for allsamples in this group, no debonding or slip between theCFRP strip and concrete was observed (refer to Fig. 7(b)).

When compared with control specimens, the average yieldand ultimate loads for Specimens 6-1Fa&b, 9-1Fa&b, and12-1Fa&b increased by 11%, 11, and 18%, and 14%, 11, and29%, respectively. Thus, the relative increase in yield Py andultimate Pmax loads for the 152 and 230 mm (6 and 9 in.)wide specimens strengthened with one CFRP strip relative tothe respective control specimens (PyC and PmaxC) was roughlythe same and taken approximately as 11%. For the 305 mm(12 in.) wide specimens, the yield load increased by 18% andthe ultimate load increased by 29%. Therefore, a greaterincrease in both yield and ultimate load capacities was

achieved for the 305 mm (12 in.) wide specimens than for the152 and 230 mm (6 and 9 in.) wide specimens. This isverification that the increase in strength is inverselyproportional to the relative area of steel reinforcement (ρs/ρbs).

For the strengthened specimens in this group, the averageultimate loads Pmax were between 13 and 20% greater thanthe average yield loads Py. Thus, the strength increasebetween yield and ultimate limit states is slightly greater forthese specimens than for the control specimens (which wasapproximately 12%). This is expected and represents theadditional tensile capacity provided by the CFRP after steelyield, which is not available for the control specimens.

All specimens failed at loads slightly in excess of theirrespective predicted nominal flexural strength Pn. Referringto Table 2, the measured failure loads Pmax were between 6%(6-1Fb) and 14% (6-1Fa) greater than the theoretical strengthPn. The magnitude and range of this comparison suggest thatthe analytical model and associated assumptions used in Eq. (2)and (4) are acceptable for predicting the flexural capacity ofthese four test specimens.

Specimens strengthened with two CFRP strips: 6-2Fa&b, 9-2Fa&b, and 12-2Fa&b

Referring to Fig. 6, the change from cracked-elastic toinelastic behavior for the 230 and 305 mm (9 and 12 in.)wide specimens reinforced with two CFRP strips can still beseen. For the 152 mm (6 in.) wide specimens strengthenedwith two CFRP strips, however, this change from elastic toinelastic behavior is much less obvious from the load-deflectiongraphs. The load-strain curve for Specimen 6-2Fb, however,shows a clear redistribution of tensile force to the CFRP as aresult of steel yield. It is therefore concluded that the steeldid yield for these specimens (6-2Fa&b).

Failure of all 152, 230, and 305 mm (6, 9, and 12 in.) widespecimens reinforced with two CFRP strips occurred byconcrete crushing. This is consistent with the failure modepredicted in Table 1. After concrete crushing, the 305 mm(12 in.) wide specimens were further deformed until ruptureof the CFRP occurred. This rupture is significant in that itagain confirmed that force transfer is sufficient to developthe full tensile capacity of the CFRP strip.

For all specimens, there was a significant increase in yieldload Py relative to the respective companion control specimensPyC. Referring to Table 2, the yield loads for 152, 230, and305 mm (6, 9, and 12 in.) wide specimens reinforced withtwo CFRP strips increased by 29, 18, and 27% over thecontrol, respectively. Comparing results, the yield loadincrease for specimens with two CFRP strips was significantlyhigher than for specimens with one CFRP strip. Relative to thecontrol specimens, the increase in ultimate load Pmax for the152, 230, and 305 mm (6, 9, and 12 in.) wide specimens was23, 44, and 61%, respectively. The trend in these values isconsistent with those listed in Table 1, where the gain inultimate strength increases with decreasing steel reinforcementratio. Thus, in design, the expected additional strength fromthe CFRP must consider the existing relative amount of steelin the unstrengthened condition.

For the 152 mm (6 in.) wide specimens with two CFRPstrips, the average ultimate load was only 6% greater than theyield load. This indicates that at steel yield, the concretestrain was near ultimate so that any increase in strength islimited by the threshold level corresponding to concretecompression failure. For the 230 and 305 mm (9 and 12 in.)wide specimens, the average ultimate loads increased by

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436 ACI Structural Journal/July-August 2007

38% over the yield loads. This is expected and represents theincreased available capacity in the concrete at steel yield. Thisbehavior is reflective of the relative amounts of both steel andCFRP reinforcement and how these reinforcement areascompare with that required for a balanced-strengthened design.

Predicted flexural strength of all specimens with twoCFRP strips was less than measured values, indicating theanalytical model is conservative. Referring to Table 2, themeasured loads were between 3 and 28% greater thanpredicted strengths. Thus, the model is an acceptable analyticaltool for strength prediction in design.

Ductility and energyThe reported effect of flexural strengthening with external

FRP reinforcement is a reduction in flexural ductility relativeto the unstrengthened condition (ACI Committee 440 2002,Bencardino et al. 2002). Typically, ductility is calculated in termsof dimensionless deflection or energy ratios. Using these param-eters ductility μ relative to the yield condition is defined as

Deflection ductility: μd = Δu /Δy (5a)

Energy ductility: μE = Eu /Ey (5b)

In Eq. (5) Δu and Δy are the ultimate and yield center-spandeflections, respectively, and Eu and Ey are the areas underthe load-deflection diagrams at ultimate and yield, respectively.Numerical integration of the measured load-deflectiondiagrams was used to determine Eu and Ey. Ductility resultsare summarized in Table 3 where it is observed that mostspecimens experience a decrease in both deflection ductility andenergy ductility relative to the control beams. The exceptionsare Specimens 6-1Fa and 12-2Fb, which experienced anincrease in both deflection and energy ductilities, andSpecimen 9-2Fb, which experienced a slight increase inenergy ductility. Under closer scrutiny, Specimen 12-2Fb,experienced a major crack at approximately 35 kN (7.84 kips).It could be argued that in a load controlled test this would havebeen the ultimate limit state for which Δu, Eu, μd , and μE are32.3 mm (1.27 in.), 724.2 kN-mm (6.41 k-in.), 1.62, and2.17, respectively. This reduces the deflection ductility and

energy ductility ratios to 0.64 and 0.60, respectively,resulting in a decrease in both ductility indexes.

The experimental ductility analysis presented previouslyis subjective for two reasons. First, for some specimens, theyield limit state is not an instantaneous condition that occursat a clearly defined load, deflection, or strain. Secondly, theultimate limit state is also subject to interpretation. Thus,depending on the selection for the yield and ultimate limitstates, a range of ductility results can be expected that maybe slightly different from those reported in Table 3. Thegeneral conclusion, however, must be that ductility isdecreased relative to the unstrengthened condition. Furtherparametric investigation of ductility using theoreticalmodeling to calculate deflection and strain is recommended.

CONCLUSIONSThe research presented in this study evaluated strength and

ductility of steel reinforced concrete beams strengthened withnear surface mounted CFRP strips. Experimental variableswere the amount of steel and CFRP reinforcements. Steelreinforcement ratios ρs and concrete strength were selectedas typical for existing concrete flexural members that wouldbe found in nonprestressed bridge and building flexuralmembers. The conclusions reported are restricted to thematerial properties (for concrete and CFRP), reinforcementratios (ρs and ρf), type of CFRP (thin rectangular strips), andtesting procedures that were used in this study. From the datapresented, the following conclusions are made.

1. The strengthened beams failed in flexure as predictedaccording to the amounts of steel and CFRP reinforcement.All 152 and 230 mm (6 and 9 in.) wide specimens, and 305 mm(12 in.) wide specimens with two CFRP strips failed by steelyield followed by concrete crushing. The CFRP remainedintact at concrete failure and no debonding was detected.These beams were predicted to fail in compression. The305 mm (12 in.) wide specimens strengthened with oneCFRP strip failed by steel yield followed by CFRP rupture.These beams were predicted to fail by CFRP rupture. In allcases, no debonding of the CFRP was detected;

2. All beams strengthened with CFRP failed at loadsgreater than their respective control beams. Relative tocontrol specimen capacity, CFRP strengthened specimens

Table 3—Ductility results

Sample ID

Yield Ultimate Deflection ductility Energy ductility

Δy , mm (in.) Ey*, kN-mm (kip-in.) Δu, mm (in.) Eu

*, kN-mm (kip-in.) μd = Δu/Δy Ratio† μE = Eu/Ey Ratio†

6-C 22.17 (0.87) 233 (2.07) 30.23 (1.19) 395 (3.50) 1.36 1.00 1.69 1.006-1Fa 19.51 (0.77) 235 (2.08) 28.98 (1.14) 455 (4.02) 1.49 1.09 1.93 1.146-1Fb 23.06 (0.91) 2823 (2.50) 29.30 (1.15) 423 (3.74) 1.27 0.93 1.50 0.886-2Fa 24.66 (0.97) 353 (3.12) 26.19 (1.03) 389 (3.45) 1.06 0.78 1.10 0.656-2Fb 25.26 (0.99) 354 (3.13) 31.04 (1.22) 503 (4.45) 1.23 0.90 1.42 0.849-C 21.05 (0.83) 280 (2.48) 47.03 (1.85) 909 (8.05) 2.23 1.00 3.24 1.00

9-1Fa 21.14 (0.83) 323 (2.86) 36.80 (1.45) 729 (6.46) 1.74 0.78 2.26 0.709-1Fb 24.16 (0.95) 331 (2.93) 44.45 (1.75) 863 (7.64) 1.84 0.82 2.61 0.809-2Fa 20.76 (0.82) 344 (3.05) 40.81 (1.61) 989 (8.75) 1.97 0.88 2.87 0.889-2Fb 22.15 (0.87) 323 (2.86) 47.87 (1.88) 1125 (9.96) 2.16 0.97 3.49 1.0812-C 17.55 (0.69) 228 (2.02) 44.68 (1.76) 845 (6.80) 2.55 1.00 3.70 1.00

12-1Fa 19.50 (0.77) 296 (2.62) 44.09 (1.74) 976 (8.64) 2.26 0.89 3.29 0.8912-1Fb 20.56 (0.81) 317 (2.80) 47.36 (1.86) 1081 (9.50) 2.30 0.90 3.42 0.9212-2Fa 20.23 (0.80) 334 (2.96) 46.10 (1.81) 1147 (10.15) 2.28 0.89 3.43 0.9312-2Fb 19.90 (0.78) 334 (2.95) 58.55 (2.31) 1732 (15.33) 2.94 1.16 5.19 1.40

* .

†Ratio = {strengthened sample}/{control sample}.

E P Δd∫=

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had measured increases in yield strength ranging from 9 to30%, and measured increases in ultimate strength rangingfrom 10 to 78%. In general, the increase in strength wasinversely proportional to the relative amount of steelreinforcement normalized to a balanced design ρs/ρsb;

3. The measured ultimate capacity of CFRP strengthenedbeams was between 6 and 28% greater than the respectivepredicted nominal strength. Nominal strength was calculatedusing a simplified closed-form analysis that yields identicalresults to the trial and error procedure given in ACI 440.2R-02.For unstrengthened beams, the measured ultimate strength wasbetween 11 and 23% greater than the section’s predicted nominalstrength. These ratios suggest that the CFRP strengthenedsection nominal flexural capacity is appropriately predicted usingthe simplified closed-form or ACI 440.2R-02 methodologies;

4. Force transfer between the CFRP, epoxy grout, andsurrounding concrete was able to develop the full tensile strengthof the CFRP strips. Tensile rupture of the single CFRP strip wasachieved in the 305 mm (12 in.) wide specimens with noapparent slip or damage to the concrete cover or epoxy grout.For all other specimens where the CFRP did not fail, therewas no apparent loss in force transfer between the CFRP,epoxy grout, and surrounding concrete. Thus, the CFRP strip’sthin rectangular cross section and roughened surface providean effective mechanism of force transfer with this epoxy; and

5. For the specimens tested, there was no discernable trendbetween the change in ductility (energy and deflection) andthe relative amount of steel reinforcement ρs/ρsb or CFRPstrengthening reinforcement Afrp. With the exception of twostrengthened beam, energy and deflection ductilities werereduced for CFRP strengthened beams.

The authors suggest that additional research is required tostudy the strength and ductility behavior of a beam strengthenedwith wider range of combinations of steel and FRP reinforce-ment ratios. Furthermore, NSM FRP splice and bond behavior,appropriate code mandated design limitations for strength,deflection, and ductility need to be investigated.

ACKNOWLEDGMENTSThe authors wish to thank Hughes Brothers, Inc., for donating the CFRP

reinforcement and the Office of Research and Sponsored Projects at VillanovaUniversity for providing financial support for this research.

NOTATIONAf , As = area of CFRP and steel reinforcement, respectivelyAfb = balanced-strengthened area of CFRPAsy = steel area corresponding to simultaneous concrete crushing

and steel yieldinga, av = depth of compression block at ultimate and shear span,

respectivelyb, c = beam width and depth on neutral axis, respectivelydf , ds = depth to CFRP and steel reinforcement, respectivelyEf , fc′ = FRP elastic modulus and concrete strength, respectivelyff , fs = stress in CFRP and steel, respectivelyffu, fy = ultimate strength of FRP (1648 MPa [239 ksi]) and steel

yield strength, respectivelyff-ult = calculated CFRP stress at sections theoretical moment strengthMn = theoretical nominal moment strengthPn = theoretical applied load corresponding to MnPnC = theoretical applied load for control specimens correspond-

ing to MnPy, Pmax = measured load at steel yield and ultimate, respectivelyPyC, PmaxC = measured load for control specimen at steel yield and ultimate,

respectivelyTf , Ts = tensile force in CFRP and steel, respectivelywbeam = self-weight of beamβ1 = ratio of a/cεf, εs = strain in CFRP and steel, respectivelyεcu, εfu = ultimate strain of concrete (0.003) and FRP (0.012), respectively

ρs, ρf = steel As/bds and CFRP Af/bdf reinforcement ratio, respectivelyρsb = balanced steel reinforcement ratio for unstrengthened section

REFERENCESACI Committee 440, 2002, “Design and Construction of Externally

Bonded FRP Systems for Strengthening Concrete Structures (ACI 440.2R-02),”American Concrete Institute, Farmington Hills, Mich., 45 pp.

ACI Committee 440, 2004, “Guide Test Methods of Fiber-ReinforcedPolymers (FRPs) for Reinforcing or Strengthening Concrete Structures (ACI440.3R-04),” American Concrete Institute, Farmington Hills, Mich., 40 pp.

ASTM C 684-99, 1999, “Standard Test Method for Making, AcceleratedCuring, and Testing Concrete Compression Test Specimens,” ASTMInternational, West Conshohocken, Pa., 10 pp.

Arduini, M., and Nanni, A., 1997, “Behavior of Precracked RC BeamsStrengthened with Carbon FRP Sheets,” Journal of Composites forConstruction, ASCE, V. 1, No. 2, pp. 63-70.

Bencardino, F.; Spadea, G.; and Swamy, R., 2002, “Strength and Ductility ofReinforced Concrete Beams Externally Reinforced with Carbon Fiber Fabric,”ACI Structural Journal, V. 99, No. 2, Mar.-Apr., pp. 163-171.

Brena, S. F.; Bramblett, R. M.; Wood, S. L.; and Kreger, M. E., 2003,“Increasing Flexural Capacity of Reinforced Concrete Beams Using CarbonFiber-Reinforced Polymer Composites,” ACI Structural Journal, V. 100,No. 1, Jan.-Feb., pp. 36-46.

DeLorenzis, L. A.; Nanni, A.; and Tegila, A. L., 2000, “Flexural andShear Strengthening of Reinforced Concrete Structures with Near SurfaceMounted FRP Bars,” Proceedings of the 3rd International Conference onAdvanced Composite Materials in Bridges and Structures, Ottawa, Canada,Aug. 15-18, pp. 521-528.

DeLorenzis, L., and Nanni, A., 2001, “Shear Strengthening of ReinforcedConcrete Beams with Near-Surface Mounted Fiber-Reinforced PolymerRods,” ACI Structural Journal, V. 98, No. 1, Jan.-Feb., pp. 60-68.

DeLorenzis, L., and Nanni, A., 2002, “Bond between Near-SurfaceMounted Fiber-Reinforced Polymer Rods and Concrete in StructuralStrengthening,” ACI Structural Journal, V. 99, No. 2, Mar.-Apr., pp. 123-132.

DeLorenzis, L.; Lundgren, K.; and Rizzo, A., 2004, “Anchorage Lengthof Near-Surface Mounted Fiber-Reinforced Polymer Bars for ConcreteStrengthening—Experimental Investigation and Numerical Modeling,”ACI Structural Journal, V. 101, No. 2, Mar.-Apr., pp. 269-278.

El-Hacha, R., and Rizkalla, S., 2004, “Near-Surface-Mounted Fiber-Reinforced Polymer Reinforcements for Flexural Strengthening of ConcreteStructures,” ACI Structural Journal, V. 101, V. 5, Sept.-Oct., pp. 717-726.

Grace, N.; Abdel-Sayed, G.; and Ragheb, W., 2002, “Strengthening ofConcrete Beams Using Innovative Ductile Fiber-Reinforced Polymer Fabric,”ACI Structural Journal, V. 99, No. 5, Sept.-Oct., pp. 692-700.

Mukhopadhyaya, P., and Swamy, R. N., 1999, “Critical Review of PlateAnchorage Stresses in Premature Debonding Failures of Plate BondedReinforced Concrete Beams,” Fourth International Symposium on FiberReinforced Polymer Reinforcement for Reinforced Concrete Structures, SP-188,C. W. Dolan, S. H. Rizkalla, and A. Nanni, eds., American Concrete Institute,Farmington Hills, Mich., pp. 359-368.

Nanni, A., 2000, “FRP Reinforcement for Bridge Structures,” Proceedings,Structural Engineering Conference, University of Kansas, Lawrence,Kans., Mar. 16, pp. 1-5.

Nguyen, D.; Chan, T.; and Cheong, H., 2001, “Brittle Failure and BondDevelopment Length of CFRP-Concrete Beams,” Journal of Compositesfor Construction, ASCE, V. 5, No. 1, pp. 12-17.

Pennsylvania Department of Transportation (PennDOT), 2001, “TheBridge Design Specification Sheet, BD-601M,” Specifications for theConcrete, Class AAA.

Rahimi, H., and Hutchinson, A., 2001, “Concrete Beams Strengthenedwith Externally Bonded FRP Plates,” Journal of Composites for Construction,ASCE, V. 5, No. 1, Jan., pp. 44-55.

Saadatmanesh, H., 1994, “Fiber Composites for New and ExistingStructures,” ACI Structural Journal, V. 91, No. 3, May-June, pp. 346-354.

Sharif, A.; Al-Sulaimani, G. J.; Basunbul, I. A.; Baluch, M. H.; andGhaleb, B. N., 1994, “Strengthening of Initially Loaded ReinforcedConcrete Beams Using FRP Plates,” ACI Structural Journal, V. 91, No. 2,Mar.-Apr., pp. 160-168.

Shin, Y. S.; and Lee, C., 2003, “Flexural Behavior of Reinforced ConcreteBeams Strengthened with Carbon Fiber-Reinforced Polymer Laminates atDifferent Levels of Sustaining Load,” ACI Structural Journal, V. 100, No. 2,Mar.-Apr., pp. 231-239.

Taljsten, B., and Carolin, A., 2001, “Concrete Beams Strengthened withNear Surface Mounted CFRP Laminates,” Proceedings of the Non-MetallicReinforcement for Concrete Structures, FRP RCS-5 Conference, July 16-18,Cambridge, UK, pp. 107-116.

Teng, J. G.; Chen, J. F.; Smith, S. T.; and Lam, L., 2002, FRP-StrengthenedRC Structures, John Wiley & Sons, West Sussex, UK, 266 pp.

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ACI Structural Journal, V. 104, No. 4, July-August 2007.MS No. S-2006-206.R1 received May 20, 2006, and reviewed under Institute publication

policies. Copyright © 2007, American Concrete Institute. All rights reserved, includingthe making of copies unless permission is obtained from the copyright proprietors. Pertinentdiscussion including author’s closure, if any, will be published in the May-June 2008ACI Structural Journal if the discussion is received by January 1, 2008.

ACI STRUCTURAL JOURNAL TECHNICAL PAPER

Test results of 24 reinforced concrete continuous deep beams arereported. The main variables studied were concrete strength, shearspan-to-overall depth ratio (a/h) and the amount and configuration ofshear reinforcement. The results of this study show that the loadtransfer capacity of shear reinforcement was much more prominentin continuous deep beams than in simply supported deep beams.For beams having an a/ h of 0.5, horizontal shear reinforcementwas always more effective than vertical shear reinforcement. Theratio of the load capacity measured and that predicted by the strut-and-tie model recommended by ACI 318-05 dropped against theincrease of a/h. This decrease rate was more remarkable in continuousdeep beams than that in simple deep beams. The strut-and-tiemodel recommended by ACI 318-05 overestimated the strength ofcontinuous deep beams having a/ h more than 1.0.

Keywords: beams; load; shear reinforcement; strut-and-tie model.

INTRODUCTIONReinforced concrete deep beams are used in structures as

load distribution elements such as transfer girders, pile caps,and foundation walls in tall buildings. Although these memberscommonly have several supports, extensive experimentalinvestigations have brought simple deep beams into focus.The behavior of continuous deep beams is significantlydifferent from that of simply supported deep beams. Thecoexistence of high shear and high moment within the interiorshear span in continuous deep beams has a considerableeffect on the development of cracks, leading to a significantreduction in the effective strength of the concrete strut,which is the main load transfer element in deep beams.1

Indeed, few experiments1-3 were carried out on continuousdeep beams of shear span-to-overall depth ratio (a/h) greaterthan 1.08. The results of simple deep beams tested by Tan et al.4

and Smith and Vantsiotis,5 however, showed that the relativeeffectiveness of horizontal and vertical shear reinforcementon controlling diagonal cracks and enhancing load capacityreversed for deep beams having an a/h less than 1.0, that is,horizontal shear reinforcement was more effective for an a/hbelow 1.0, whereas vertical shear reinforcement was moreeffective for an a/h lager than 1.0. Therefore, a reasonableevaluation of the influence of shear reinforcement oncontinuous deep beams having an a/h less than 1.0 requiresfurther investigation.

The current codes6-8 and several researchers9-12 haverecommended the design of deep beams using the strut-and-tie model. In these strut-and-tie models, the main function ofshear reinforcement is to restrain diagonal cracks near theends of bottle-shaped struts and to give some ductility tostruts. ACI 318-05, Section A.3.3, allows the use of aneffectiveness factor of 0.75 when computing the effectiveconcrete compressive strength of bottle-shaped struts withreinforcement satisfying ACI 318-05, Section A.3.3. Thevalue of the effectiveness factor drops to 0.6 if shear reinforce-

ment as recommended by ACI 318-05, Section A.3.3, is notprovided. This implies that shear reinforcement satisfyingACI 318-05, Section A.3.3, would increase the ultimatestrength of beams predicted by the strut-and-tie model by25%. Studies on the validity of the strut-and-tie modelrecommended by ACI 318-05, however, are very rare evenin simple deep beams.12-14

This paper presents test results of 24 two-span reinforcedconcrete deep beams. The main variables included concretestrength, a/h, and the amount and configuration of shearreinforcement. The influence of shear reinforcement on theultimate shear strength in continuous deep beams wascompared with that in the corresponding simple ones. Theload capacity predictions of reinforced concrete continuousdeep beams by the strut-and-tie model of ACI 318-05 wereevaluated by comparison with test results.

RESEARCH SIGNIFICANCEA great deal of research has focused on simply supported

deep beams. Even the few tests on continuous deep beamswere carried out on beams having an a/h exceeding 1.0 andconcrete strength less than 35 MPa (5.0 ksi). Test results inthis study clearly showed the influence of shear reinforcementon the structural behavior of continuous deep beams accordingto the variation of concrete strength and a/h. The ultimateshear strength of continuous deep beams and load transfercapacity of shear reinforcement were compared with thoseof the corresponding simple deep beams and the predictionsobtained from the strut-and-tie model recommended inACI 318-05.

EXPERIMENTAL INVESTIGATIONThe details of geometrical dimensions and reinforcement

of test specimens are shown in Table 1 and Fig. 1. The mainvariables studied were compressive strength of concrete fc′ ,a/h, and the amount and configuration of shear reinforcement.Beams tested were classified into two groups according tothe concrete compressive strength: L-series for designconcrete strength of 30 MPa (4350 psi) and H-series fordesign concrete strength of 60 MPa (8700 psi). The a/h wereinitially designed to be 0.5 and 1.0 to allow comparison withcurrent results with those reported by Yang13 for simpledeep beams. The value of a/h in H-series, however, wasincreased from 0.5 to 0.6, as the capacity of beams having fc′of 60 MPa (8700 psi) and an a/h of 0.5 had exceeded thecapacity of the loading machine in the pilot test. The

Title no. 104-S40

Influence of Shear Reinforcement on Reinforced Concrete Continuous Deep Beamsby Keun-Hyeok Yang, Heon-Soo Chung, and Ashraf F. Ashour

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configuration of shear reinforcement included four differentarrangements as shown in Fig. 1: none, only vertical, onlyhorizontal, and orthogonal reinforcement. The spacing of shearreinforcement was chosen to be 60 and 120 mm (2.36 and4.72 in.) and the corresponding shear reinforcement ratios, ρ(= Aw /bws, where Aw equals the area of shear reinforcementat spacing s, and bw equals the beam width), were 0.003 and0.006, respectively, to satisfy the maximum spacing speci-fied in ACI 318-05, Section 11.8, and the minimum amountrecommended in ACI 318-05, Section A.3.3.2. The beamnotation given in Table 1 includes four parts. The first partrefers to the concrete design strength: L for low compressivestrength and H for high compressive strength. The secondpart is used to identify the a/h. The third and fourth parts givethe amount of horizontal and vertical shear reinforcement,

respectively: N for no shear reinforcement, and S and T forshear reinforcement ratios of 0.003 and 0.006, respectively.For example, L5-SS is a continuous deep beam having designconcrete strength of 30 MPa (4350 psi), an a/h of 0.5, and bothhorizontal and vertical shear reinforcement ratios of 0.003.

All beams tested had the same section width bw of 160 mm(6.3 in.) and overall section depth h of 600 mm (23.6 in.). Bothlongitudinal top, ρs′ = (As′ /bwd), and bottom, ρs = (As/bwd),reinforcement ratios were kept constant in all beams as 1%,which were calculated from nonlinear FE analysis,15 to ensureno flexural yielding of longitudinal reinforcement prior tofailure of concrete struts. The length of each span L variedaccording to a/h, as given in Table 1. The clear covers tolongitudinal top and bottom reinforcement, and shear reinforce-ment were 35 and 29 mm (1.38 and 1.14 in.), respectively. Thelongitudinal bottom reinforcement was continuous over thefull length of the beam and welded to 160 x 100 x 10 mm (6.3x 3.9 x 0.39 in.) end plates, whereas longitudinal top rein-forcement was anchored in the outside of the exteriorsupports by 90-degree hooks according to ACI 318-05. Thevertical shear reinforcement was closed stirrups and thehorizontal shear reinforcement with 90 degree hooks wasarranged along the longitudinal axis in both sides of the beams.

Material propertiesThe mechanical properties of reinforcement are given in

Table 2. All longitudinal and shear reinforcing bars weredeformed bars of a 19 mm (0.75 in.) diameter, having anominal area of 287 mm2 (0.44 in.2) and yield strength of562 MPa (81.6 ksi) and a 6 mm (0.23 in.) diameter, having anominal area of 28.2 mm2 (0.04 in.2) and yield strength of

Keun-Hyeok Yang is a Visiting Research Fellow at the University of Bradford, UK,and an Assistant Professor at Mokpo National University, Korea. He received his MScand PhD from Chungang University, Korea. His research interests include ductility,strengthening, and shear of reinforced, high-strength concrete structures.

Heon-Soo Chung is a Professor at Chungang University, Korea. He received his MScand PhD from Tokyo Institute of Technology, Japan. His research interests includeflexure, shear, and bond behavior of reinforced, high-strength concrete members.

Ashraf F. Ashour is a Senior Lecturer at the University of Bradford, UK. He receivedhis BSc and MSc from Mansoura University, Egypt, and his PhD from CambridgeUniversity, UK. His research interests include shear, plasticity, and optimization ofreinforced concrete and masonry structures.

Fig. 1—Geometrical dimensions and reinforcement of testspecimens. (Note: all dimensions are in mm and • indicateslocations of strain gauges. 1 mm = 0.039 in.)

Table 1—Details of test specimens

Specimenfc′ ,

MPa a/h a/jd L, mm

Details of shear reinforcement

Horizontal Vertical

sh, mm ρh sv , mm ρv

L5NN

32.4 0.5 0.58 600

— — — —

L5NS — — 120 0.003

L5NT — — 60 0.006

L5SN 120 0.003 — —

L5SS 120 0.003 120 0.003

L5TN 60 0.006 — —

L10NN

32.1 1.0 1.17 1200

— — — —

L10NS — — 120 0.003

L10NT — — 60 0.006

L10SN 120 0.003 — —

L10SS 120 0.003 120 0.003

L10TN 60 0.006 — —

H6NN

65.1 0.6 0.7 720

— — — —

H6NS — — 120 0.003

H6NT — — 60 0.006

H6SN 120 0.003 — —

H6SS 120 0.003 120 0.003

H6TN 60 0.006 — —

H10NN

68.2 1.0 1.17 1200

— — — —

H10NS — — 120 0.003

H10NT — — 60 0.006

H10SN 120 0.003 — —

H10SS 120 0.003 120 0.003

H10TN 60 0.006 — —

Note: 1 MPa = 145 psi; 1 mm = 0.039 in.

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483 MPa (70 ksi), respectively. The yield strength of 6 mm(0.23 in.) diameter reinforcement was obtained by 0.2%offset method.

The ingredients of ready mixed concrete were ordinaryportland cement, fly ash, irregular gravel of a maximum sizeof 25 mm (0.98 in.), and sand. The water-binder ratios of theL-series added with fly ash of 12% and of the H-series addedwith fly ash of 20% were 0.41 and 0.27, respectively. Allspecimens were cast in a vertical position in the samewooden mold. Control specimens, which were 100 mm(3.94 in.) diameter by 200 mm (7.87 in.) high cylinders, werecast and cured simultaneously with beams to determine thecompressive strength. They were tested soon after the beamtest. The results of the cylinder compressive strength givenin Table 1 are the average value from testing nine cylinders.

Test setupLoading and instrumentation arrangements are shown in

Fig. 2. All beams having two spans were tested to failureunder a symmetrical two-point top loading system with aloading rate of 30 kN/minute (6.7 kip/minute) using a 3000 kN(675 kip) capacity universal testing machine (UTM). Eachspan was identified as E-span or W-span, as shown in Fig. 1.The two exterior end supports were designed to allowhorizontal and rotational movements, whereas the intermediatesupport prevented horizontal movement but allowed rotation.To evaluate the shear force and loading distribution, 1000 kN(225 kip) capacity load cells were installed in both exteriorend supports. At the location of loading or support point, asteel plate of 100, 150, or 200 mm (3.94, 5.9, or 7.88 in.)wide was provided to prevent premature crushing or bearingfailure, as shown in Fig. 2. All steel plates were 50 mm (1.97 in.)thick and 300 mm (11.8 in.) long to cover the full width oftest specimen. All beams were preloaded up to a total load of150 kN (33.7 kip) before testing, which wouldn’t produceany cracks, to assure a similar loading distribution to supportsaccording to the result of the linear two-dimensional finiteelement (2-D FE) analysis.

Vertical deflections at a distance of 0.45L to 0.47L fromthe exterior support, which is the location of the maximum

deflection predicted by the linear 2-D FE analysis, and at themidspan of each span were measured using linear variabledifferential transformers (LVDTs). Both surfaces of thebeams tested were whitewashed to aid in the observation ofcrack development during testing. The inclined crack widthof concrete struts joining the edges of load and support plateswas monitored by the π-shape displacement transducers (PIgauges) as shown in Fig. 2. The strains of shear reinforcementwere measured by 5 mm (0.2 in.) electrical resistance straingauges (ERS) at the region crossing the line joining the edgesof load and intermediate support plates as shown in Fig. 1. Ateach load increment, the test data were captured by a datalogger and automatically stored.

Support settlementsContinuous deep beams are sensitive to differential

support settlements causing additional moment and shear.To assess the effect of differential settlements on the beamstested, a linear 2-D FE analysis considering shear deformationeffect was performed on the beams shown in Fig. 1. For thebeams tested, sources of relative support settlements werethe elastic shortening of the load cell and plates and elasticdeformation of the bed of the testing machine. The secondmoment of area of the testing machine bed cross sectionabout the bending axis was 3.2 × 1010 mm4 (7.69 × 104 in.4),then the elastic deformation under a point load R (in kN) at adistance 1500 mm (59 in.) from the center of the testingmachine is 0.000176R mm. The amount of elastic shorteningdue to a load at the exterior and intermediate supportsinvolving the load cell and plates was considered indesigning the support size as follows. When a/h is 0.5, thereactions of the exterior and intermediate supports due to thetotal applied load P, from the linear 2-D FE analysis, are 0.2Pand 0.6P, respectively. As the height of the intermediatesupport was equal to that of the exterior load cell, the contactarea of the intermediate support with the bed of the testingmachine was designed to be three times wider than that of theload cell at the exterior support to produce the same elasticshortening. The pilot test results showed that the maximumsettlement of the exterior support relative to the intermediatesupport was in order of L/25,000. For a differential settlementbetween the exterior and intermediate supports of L/25,000,the maximum additional shear forces obtained from linear 2-DFE analysis are 25 and 7 kN (5.62 and 1.57 kip) for beams

Fig. 2—Test setup. (Note: all dimensions are in mm. 1 mm =0.039 in.)

Table 2—Mechanical properties of reinforcementDiameter, mm fy, MPa εy fsu, MPa Es, GPa

6* 483 0.0044 549 199

19 562 0.00284 741 198*Yield stress of 6 mm diameter reinforcement was obtained by 0.2% offset method.Note: 1 mm = 0.039 in.; 1 MPa = 145 psi.

Fig. 3—Crack patterns and failure of concrete strut. Numbersindicate total load in kN at which crack occurred. (Note:1 kN = 0.2248 kips.)

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having an a/h of 0.5 and 1.0, respectively. This indicates thatthe differential settlement had no significant effect on thetest arrangement.

EXPERIMENTAL RESULTS AND DISCUSSIONCrack propagation and failure mode

The crack propagation was significantly influenced by thea/h as shown in Fig. 3 and Table 3. The crack pattern in theL-series was similar to that in the H-series; therefore, it is notshown in Fig. 3. For beams with a/h = 0.5, the first cracksuddenly developed in the diagonal direction at approximately40% of the ultimate strength at the middepth of the concretestrut within the interior shear span, and then a flexural crackin the sagging region immediately followed. The first flexuralcrack over the intermediate support generally occurred atapproximately 80% of the ultimate strength, and was lessthan 0.2h deep at failure. As the load increased, more flexuraland diagonal cracks were formed and a major diagonal crackextended to join the edges of the load and intermediatesupport plates. A diagonal crack within the exterior shearspan occurred suddenly near the failure load. Cracks inbeams with a/h = 1.0 developed in a different order from thatdescribed previously for beams with a/h = 0.5. In thosebeams, the first crack occurred vertically in the hogging

zone, followed by a diagonal crack in the interior shear span,and then a vertical crack took place in the sagging zone, butdiagonal cracks within exterior shear spans were seldomdeveloped. The influence of shear reinforcement on the firstflexural and diagonal crack loads was not significant (referto Table 3) as also observed in simple deep beams given inAppendix A.

Just before failure, the two spans showed nearly the samecrack patterns. All beams developed the same mode offailure as observed in other experiments.3 The failure planesevolved along the diagonal crack formed at the concrete strutalong the edges of the load and intermediate support plates.Two rigid blocks separated from original beams at failuredue to the significant diagonal cracking. An end blockrotated about the exterior support leaving the other blockfixed over the other two supports as shown in Fig. 3.

Load versus midspan deflectionThe beam deflection at midspan was less than that

measured at 0.45L to 0.47L from the exterior support untilthe occurrence of the first diagonal crack as predicted by the2-D FE analysis. After the first diagonal crack, however, themidspan deflection was higher. Therefore, the midspan

Table 3—Details of test results and predictions obtained from ACI 318-05

Specimen

Load Pcr and shear force Vcr at first diagonal crack, kNFailure load Pn and ultimate shear

force (Vn)I at interior shear spans, kN ACI 318-05

(Pn)Exp./(Pn)ACI

(Vn)I-Exp./(Vn)I-ACI

W-span E-span

Pn

(Vn)I

Pn, kN (Vn)I, kN

Interior Exterior Interior Exterior

W-span E-span(Pcr)I (Vcr)I (Pcr)E (Vcr)E (Pcr)I (Vcr)I (Pcr)E (Vcr)E

L5NN 852 255 902 180 816 244 937 187 1635 473 456* 1298 342 1.260 1.334

L5NS 849 247 1028 210 857 262 1330 281 1710 486 475* 1298 342 1.317 1.389

L5NT 1017 278 1380 284 850 230 1260 262 1789 512* 494 1298 342 1.378 1.498

L5SN 864 255 1268 252 867 257 927 179 1887 537* 546 1298 342 1.454 1.571

L5SS 814 247 990 192 980 293 1020 202 2117 607* 583 1623 427 1.305 1.420

L5TN 912 266 1130 230 910 278 966 185 2317 655 640* 1298 342 1.785 1.872

L10NN 537 173 — — 537 171 — — 880 264* 262 1000 265 0.880 0.997

L10NS 477 156 — — 596 195 — — 1153 349 348* 1000 265 1.153 1.314

L10NT 635 206 1023 230 647 208 — — 1541 446* 439 1000 265 1.541 1.684

L10SN 498 153 — — 490 151 782 146 884 266 265* 1000 265 0.884 1.000

L10SS 521 166 — — 452 148 713 129 1177 357 352* 1250 331 0.942 1.063

L10TN 538 175 — — 621 193 775 143 935 287 288* 1000 265 0.935 1.087

H6NN 1046 305 1562 321 1236 303 1960 407 2248 633* 634 2520 668 0.892 0.950

H6NS 1261 379 1646 316 978 300 2280 457 2289 684 683* 2520 668 0.908 1.023

H6NT 1116 324 2550 550 915 264 2480 531 2625 757 757* 2520 668 1.042 1.134

H6SN 1322 393 2420 517 1022 297 2420 513 2427 703* 708 2520 668 0.963 1.053

H6SS 1207 367 2630 548 825 256 2630 542 2763 792 799* 3150 834 0.877 0.958

H6TN 1442 439 — — 980 297 2648 540 2966 854 852* 2520 668 1.177 1.276

H10NN 690 228 868 149 690 228 840 143 1276 373 372* 2124 563 0.601 0.661

H10NS 759 237 — — 751 234 — — 1443 413* 414 2124 563 0.679 0.734

H10NT 788 251 — — 717 224 — — 2116 638 637* 2124 563 0.996 1.132

H10SN 757 255 — — 757 252 — — 1309 387* 378 2124 563 0.616 0.688

H10SS 718 232 — — 768 244 — — 1575 492* 484 2655 703 0.593 0.699

H10TN 754 234 — — 704 220 — — 1287 393 388* 2124 563 0.606 0.689

*Failure occurred in this shear span.Note: 1 kN = 0.2248 kips.

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424 ACI Structural Journal/July-August 2007

deflection of the failed span for different beams tested areonly presented in Fig. 4 against the total applied load: Fig. 4(a)for beams in the L-series and Fig. 4(b) for beams in the H-series.The initial stiffness of beams tested increased in accordancewith the increase of concrete strength and the decrease of thea/h, but it seems to be independent of the amount andconfiguration of shear reinforcement. The development offlexural cracks in sagging and hogging zones has little influenceon the stiffness of beams tested. But the occurrence of diagonalcracks in the interior shear span caused a sharp decrease inthe beam stiffness and an increase of the beam deflection.

This stiffness reduction was prominent in case of lowerconcrete strength and higher a/h.

Support reactionFigure 5 shows the amount of the load transferred to the

end and intermediate supports against the total applied loadin the L-series beams having a/h = 0.5. On the same figure,the support reactions obtained from the linear 2-D FE analysisare also presented. The end and intermediate support reactionsof the L-series beams having a/h = 1.0 and the H-seriesbeams were similar to those of the L-series beams having ana/h = 0.5; therefore, not presented herein. Before the firstdiagonal crack, the relationship of the end and intermediatesupport reactions against the total applied load in all beamstested shows good agreement with the prediction of thelinear 2-D FE analysis. The amount of loads transferred tothe end support, however, was slightly higher than thatpredicted by the linear 2-D FE analysis after the occurrenceof the first diagonal crack within the interior shear span. Atfailure, the difference between the measured end supportreaction and prediction of the linear 2-D FE analysis was inorder of 7 and 12%, for beams with a/h = 0.5 and a/h = 1.0,respectively. The distribution of applied load to supports wasindependent of the amount and configuration of shearreinforcement. This means that, although after the occurrenceof diagonal cracks the beam stiffness has reduced, as shownin Fig. 4, the internal redistribution of forces is limited.

Width of diagonal crackFigure 6 shows the variation of the diagonal crack width in

the interior shear span according to the configuration ofshear reinforcement: Fig. 6(a) at the first diagonal crackingload and Fig. 6(b) at the same load as the ultimate failure loadof the corresponding deep beam without shear reinforcement.For the same concrete compressive strength, the larger the a/h,the wider the diagonal crack width. Shear reinforcement hadan important role in restraining the development of the diagonalcrack width, which significantly depended on the a/h. Amore prominent reduction of diagonal crack width appearedin beams with horizontal shear reinforcement only ororthogonal shear reinforcement than in beams with verticalshear reinforcement only when a/h was 0.5. On the otherhand, for beams with a/h = 1.0, a smaller diagonal crackwidth was observed in beams with vertical shear reinforcementonly than in beams with orthogonal shear reinforcement, eventhough the total shear reinforcement ratio in these beams wasthe same (ρv + ρh = 0.006). It seems possible to reduce thediagonal crack width by more than twice if shear reinforcementis suitably arranged according to the variation of a/h.

Fig. 4—Total load versus midspan deflection. (Note: 1 kN =0.2248 kips.)

Fig. 5—Total applied load versus support reactions for L-seriesbeams tested having a/ h of 0.5. (Note: 1 kN = 0.2248 kips.)

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ACI Structural Journal/July-August 2007 425

Figure 7 shows the strain in shear reinforcement againstthe total applied load in the H-series beams: Fig. 7(a) forvertical shear reinforcement in beams having either verticalor orthogonal shear reinforcement, and Fig. 7(b) for horizontalshear reinforcement in beams having either horizontal ororthogonal shear reinforcement. The relation between strainsin shear reinforcement and the total applied load in the L-seriesbeams was similar to that in the H-series beams; therefore,not presented herein. The strains of shear reinforcementwere recorded by ERS gauges at different locations, asshown in Fig. 1. Shear reinforcement was not generallystrained at initial stages of loading. However, strains suddenlyincreased with the occurrence of the first diagonal crack. Inbeams with a/h = 0.6, only horizontal reinforcing bars yielded,whereas in beams with a/h = 1.0, only vertical reinforcingbars yielded. This indicates that the reinforcement ability totransfer tension across cracks strongly depends on the anglebetween the reinforcement and the axis of the strut.

Ultimate shear stressThe normalized ultimate shear strength, λ = Vn/bwd ,

plotted against a/h, is given in Fig. 8: Fig. 8(a) for simplysupported deep beams given in Appendix A, and Fig. 8(b)for continuous deep beams including the test results ofRogowsky et al.1 and Ashour.2 It can be seen that the ultimateshear strength of all beams without or with shear reinforcementdropped due to the increase of a/h. The reduction of the ultimateshear strength was also dependent on the configuration of shearreinforcement. For deep beams without shear reinforcement,the normalized ultimate shear strength λ in continuous deepbeams was less than that in simply supported ones by anaverage of 26% due to higher transverse tensile strainsproduced by the tie action of longitudinal top and bottom

fc′

reinforcement. When shear reinforcement is provided, thenormalized ultimate shear strength λ in continuous deepbeams matched that of the corresponding simply supportedones. The influence of the horizontal and vertical shearreinforcement on the ultimate shear strength is influenced bythe a/h. The lower the a/h, the more effective the horizontalshear reinforcement and the less effective the vertical shearreinforcement. When a/h was below 0.6, the shear strength

Fig. 7—Total load versus strains in shear reinforcement forbeams in H-series. (Note: 1 kN = 0.2248 kips.)

Fig. 6—Configuration of shear reinforcement versus diagonalcrack width. (Note: 1 mm = 0.039 in.)

Fig. 8—Normalized ultimate shear strength versus shearspan-to-overall depth ratio.

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426 ACI Structural Journal/July-August 2007

of deep beams with minimum horizontal shear reinforcementhad an average value of 150% higher than the upper boundvalue, 0.83 bwd, specified in ACI 318-05, Section 11.8.3.

Load transfer capacity of shear reinforcementThe shear strength of deep beams Vn can be described

as follows

Vn = Vc + Vs (1)

where Vc and Vs equal the load capacity of concrete and loadtransfer capacity of shear reinforcement, respectively.

As the load capacity of concrete is usually regarded as thestrength of beams without shear reinforcement, (Vn)W/O, theratio of the load transfer capacity of shear reinforcement tothe shear strength of beams Vs/Vn is

fc′

(2)

The variations of Vs /Vn at the failed shear span against theincrease of a/h are given in Fig. 9: Fig. 9(a), (b), and (c) forbeams with vertical shear reinforcement only, with horizontalshear reinforcement only, and with orthogonal shear reinforce-ment, respectively. On the same figure, the test results ofsimply supported deep beams given in Appendix A, whichhad the same material and geometrical properties as continuousdeep beams tested in the current study, are also presented.The load transfer capacity of shear reinforcement is morepronounced in continuous deep beams than that in simple ones.

The load transfer capacity of shear reinforcement is dependenton a/h. The load transfer capacity of vertical shear reinforcementwas higher in beams having a/h = 1.0 than those havinga/h = 0.5 as shown in Fig. 9(a). On the other hand, the loadtransfer capacity of horizontal shear reinforcement washigher in beams having a/h = 0.5 than those having a/h = 1.0,as shown in Fig. 9(b). Existing test results of continuousdeep beams carried out by Rogowsky et al.1 and Ashour,2

and the comments of ACI 318-05, Section 11.8, havesuggested that horizontal shear reinforcement has littleinfluence on the shear strength improvement and crackcontrol. In the current tests, horizontal shear reinforcement ismore effective than vertical shear reinforcement for beamswith a/h of 0.5, as shown in Fig. 8 and 9.

Comparison with current codesIt has been shown by several researchers,1,2,4 that the

shear capacity prediction of reinforced concrete deep beamsobtained from ACI 318-9916 (unchanged since 1983) wasunconservative. For the design of deep beams, ACI 318-05requires the use of either nonlinear analysis or strut-and-tiemodel. Figure 10 shows a schematic strut-and-tie model ofcontinuous deep beams in accordance with ACI 318-05,Appendix A. The strut-and-tie model shown in Fig. 10 identifiestwo main load transfer systems: one of which is the strut-and-tieaction formed with the longitudinal bottom reinforcementacting as a tie and the other is the strut-and-tie action due tothe longitudinal top reinforcement. As the applied loads inthe two-span continuous deep beams are carried to supportsthrough concrete struts of exterior and interior shear spans(refer to Fig. 10), the total load capacity of two-span continuousdeep beams Pn due to failure of concrete struts is

Pn = 2(FE – FI)sinθ (3)

where FE and FI equal the load capacities of exterior andinterior concrete struts, respectively, and θ equals the anglebetween the concrete strut and the longitudinal axis of thedeep beam, which can be expressed as tan–1(jd/a). Thedistance between the center of top and bottom nodes jd couldbe approximately assumed as the distance between the centerof longitudinal top and bottom reinforcing bars as

jd = h – c – c′ (4)

where h equals the overall section depth and c and c′ equalthe cover of longitudinal bottom and top reinforcement,respectively, as shown in Fig. 10.

The nodes at the applied load point could be classified asa CCC type, which is a hydrostatic node connecting both

Vs

Vn

-----Vn Vn( )W O⁄–

Vn

--------------------------------=

Fig. 9—Shear reinforcement ratios versus Vs/ Vn.

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ACI Structural Journal/July-August 2007 427

exterior and interior compressive struts in sagging zone anda CCT type for longitudinal top reinforcement in the hoggingzone. It was proved by Marti10 that the width of the strut at aCCC node is in proportion to the principal stress normal tothe node face to make the state of stress in the whole noderegion constant. To accommodate both CCC type and CCTtype, the loading plate width can be assumed to be subdividedinto two parts in accordance with the ratio of the exteriorreaction to the applied load β, each to form the nodeconnecting the exterior and the interior struts, respectively.The β values of tested beams are 0.4 and 0.346 when a/hratios are 0.5 and 1.0, respectively, as estimated from thelinear 2-D FE analysis. If enough anchorage of longitudinalreinforcement is provided, average widths of concrete strutsin interior (ws)I and exterior shear spans (ws)E are

(5a)

(5b)

where (lp)P, (lp)E, and (lp)I equal the widths of loading, exteriorsupport, and interior support plates, respectively, and wt′equals the smaller of the height of the plate anchored tolongitudinal bottom reinforcement wt and twice of the coverof longitudinal bottom reinforcement 2c as shown in Fig. 10.

The load transfer capacity of the concrete strut depends onthe area of the strut and the effective concrete compressivestrength. Hence, the load capacities of the exterior and interiorconcrete struts are

FE = ve f ′cbw(ws)E (6a)

FI = ve f ′cbw(ws)I (6b)

where ve equals the effectiveness factor of concrete. Theshear capacity at the interior shear span (Vn)I, where thefailure is expected to occur in continuous deep beams, can becalculated from FI sinθ.

The minimum amount of shear reinforcement required inbottle-shaped struts, which is recommended to be placed intwo orthogonal directions in each face, is suggested byACI 318-05 as follows

(7)

where Asi and si equal the total area and spacing in the i-thlayer of reinforcement crossing a strut, respectively, and αiequals the angle between i-th layer of reinforcement andthe strut.

The effectiveness factors for concrete strength notexceeding 40 MPa (5.8 ksi) in ACI 318-05 are suggested as0.75 and 0.6 when shear reinforcement satisfying Eq. (7) isarranged and is not provided, respectively. The truss modelrepresenting the load transfer mechanism of horizontal andvertical shear reinforcement is not included in ACI 318-05.This implies that shear reinforcement satisfying Eq. (7)enables the strength of beams to be increased by 25%.

Comparisons between test results and predictions obtainedfrom the strut-and-tie model recommended by ACI 318-05

ws( )I

wt ′ 2c ′+( ) θ 0.5 lp( )I 1.0 β–( ) lp( )P+[ ] θsin+cos2

-----------------------------------------------------------------------------------------------------------------------=

ws( )E

wt ′ 2c ′+( ) θ lp( )E β lp( )P+[ ] θsin+cos2

----------------------------------------------------------------------------------------------=

Asi

bwsi

---------- αi 0.003≥sin∑

as developed previously are shown in Table 3 and Fig. 11:Fig. 11(a) for simple deep beams given in Appendix A andFig. 11(b) for continuous deep beams including Rogowskyet al.’s and Ashour’s test results. In simple deep beams, thewidth of the strut can be calculated from wt′cosθ + (lp)Esinθ,and the total load capacity is 2FEsinθ. Although Eq. (7)proposed by ACI 318-05 is recommended for deep beamshaving concrete strength of less than 40 MPa, the loadcapacity of the H-series beams were also predicted using thisequation to evaluate its conservatism in case of high-strengthconcrete deep beams. The mean and standard deviation ofthe ratio, (Pn)Exp./(Pn)ACI, between the experimental andpredicted load capacities are 1.229 and 0.326, respectively,for simply supported deep beams, and 0.969 and 0.306,

Fig. 10—Qualitative strut-and-tie model of continuous deepbeams according to ACI 318-05.

Fig. 11—Comparison of test results and predictions byACI 318-05.

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428 ACI Structural Journal/July-August 2007

respectively, for two-span continuous deep beams as shownin Fig. 11. The ratio of the test result to prediction generallydropped with the increase of a/h. This decrease rate wasmore remarkable in continuous deep beams than that insimple ones. In particular, the predictions for several continuousdeep beams having a/h exceeding 1.0 were unconservative,even though the effectiveness factor used in the beams witheither horizontal or vertical shear reinforcement was 0.6regardless of the amount of shear reinforcement. In addition,for high-strength concrete continuous deep beams having a/h =1.0, the ratio, (Vn)I-Exp./(Vn)I-ACI, between the experimental andpredicted shear capacities in the interior shear span wasgenerally below 1.0 as given in Table 3; namely, the strut-and-tie model recommended by ACI 318-05 overestimatedthe shear capacity of high-strength concrete continuous deepbeams having a/h = 1.0.

CONCLUSIONSTests were performed to study the influence of the amount

and configuration of shear reinforcement on the structuralbehavior of continuous deep beams according to the variationof concrete strength and a/h. The following conclusionsare drawn:

1. In beams having a/h of 0.6, only horizontal shearreinforcement reached its yield strength with a sharpincrease of stress after the first diagonal crack. On the otherhand, only vertical shear reinforcement yielded in beamswith a/h of 1.0;

2. For deep beams without shear reinforcement, thenormalized ultimate shear strength was 26% lower incontinuous beams than that in simple ones. When shearreinforcement was provided, however, the normalized ultimateshear strength in continuous deep beams matched that insimply supported deep beams;

3. The load transfer capacity of all shear reinforcementwas much more prominent in continuous deep beams thanthat in simple ones. Horizontal shear reinforcement wasalways more effective than vertical shear reinforcementwhen the a/h was 0.5. However, vertical shear reinforcementwas more effective for a/h higher than 1.0;

4. In deep beams with a/h not exceeding 0.6, the criticalupper bound on shear strength suggested in ACI 318-05,0.83 bwd, highly underestimated the actual measuredshear strength, as if it was a lower limit; and

5. The ratios of measured load capacity to that obtainedfrom the strut-and-tie model recommended by ACI 318-05dropped with the increase of the a/h. This decrease rate wasmore remarkable in continuous deep beams than that insimple ones. The strut-and-tie model recommended by ACI318-05 overestimated the shear capacity of high-strengthconcrete continuous deep beams having a/h more than 1.0.

ACKNOWLEDGMENTSThis work was supported by the Korea Research Foundation Grant

(KRF-2003-041-D00586) and the Regional Research Centers Program(Bio-housing Research Institute), granted by the Korean Ministry of Educationand Human Resources Development. The authors wish to express theirgratitude for financial support.

NOTATIONAh = area of horizontal shear reinforcementAs = area of longitudinal bottom reinforcementAs′ = area of longitudinal top reinforcementAw = area of shear reinforcementa = shear spanbw = width of beam section

c = cover of longitudinal bottom reinforcementc′ = cover of longitudinal top reinforcementd = effective depth of beam sectionh = overall depth of beam sectionEs = elastic modulus of steelFE = load capacity of concrete strut in exterior shear spanFI = load capacity of concrete strut in interior shear spanfc′ = concrete compressive strengthfsu = tensile strength of reinforcementfy = yield strength of reinforcementjd = distance between center of top and bottom nodesL = span lengthlp = width of loading platePcr = diagonal crack loadPn = ultimate load at failuresh = spacing of horizontal shear reinforcementsv = spacing of vertical shear reinforcementT = tensile force in longitudinal reinforcementVc = load capacity of concreteVcr = diagonal crack shear forceVn = ultimate shear force at failureVs = load transfer capacity of shear reinforcementve = effectiveness factor of concretews = width of concrete strutwt = height of plate anchored to longitudinal reinforcementα = angle between shear reinforcement and axis of concrete strutβ = ratio of exterior reaction to applied loadεy = yield strain of reinforcementλ = normalized ultimate shear strengthθ = angle between concrete strut and longitudinal axis of beamρh = horizontal shear reinforcement ratio (Ah/bwsh)ρst = longitudinal bottom reinforcement ratio (As/bwd)ρst′ = longitudinal top reinforcement ratio (As′ /bwd)ρv = vertical shear reinforcement ratio (Av /bwsv)

REFERENCES1. Rogowsky, D. M.; MacGregor, J. G.; and Ong, S. Y., “Tests of Reinforced

Concrete Deep Beams,” ACI JOURNAL, Proceedings V. 83, No. 4, July-Aug. 1986, pp. 614-623.

2. Ashour, A. F., “Tests of Reinforced Concrete Continuous DeepBeams,” ACI Structural Journal, V. 94, No. 1, Jan.-Feb. 1997, pp. 3-12.

3. Subedi, N. K., “Reinforced Concrete Two-Span Continuous DeepBeams,” Proceedings of the Institution of Civil Engineers, Structures &Buildings, V. 128, Feb. 1998, pp. 12-25.

4. Tan, K. H.; Kong, F. K.; Teng, S.; and Weng, L. W., “Effect of WebReinforcement on High-Strength Concrete Deep Beams,” ACI StructuralJournal, V. 94, No. 5, Sept.-Oct. 1997, pp. 572-582.

5. Smith, K. N., and Vantsiotis, A. S., “Shear Strength of Deep Beams,”ACI JOURNAL, Proceedings V. 79, No. 3, May-June 1982, pp. 201-213.

6. ACI Committee 318, “Building Code Requirements for StructuralConcrete (ACI 318-05) and Commentary (318R-05),” American ConcreteInstitute, Farmington Hills, Mich., 2005, 430 pp.

7. Canadian Standards Association (CSA), “Design of Concrete Structures,”A23.3-94, Canadian Standards Association, Rexdale, Ontario, Canada,Dec. 1994, 199 pp.

8. FIP Recommendations: Practical Design of Structural Concrete. 1999.9. MacGregor, J. G., Reinforced Concrete: Mechanics and Design, Prentice-

Hall International, Inc., 1997.10. Marti, P., “Basic Tools of Reinforced Concrete Beam Design.” ACI

JOURNAL, Proceedings V. 82, No. 1, Jan.-Feb. 1985, pp. 46-56.11. Schlaich, J.; Schafer, K.; and Jennewein, M., “Toward a Consistent

Design of Structural Concrete,” Journal of the Prestressed Concrete Institute,V. 32, No. 3, May-June 1987, pp. 74-150.

12. Tjhin, T. N., and Kuchma, D. A., “Example 1b: Alternative Designfor the Non-Slender Beam (Deep Beam),” Strut-and-Tie Models, SP-208,K.-H. Reineck, ed., American Concrete Institute, Farmington Hills, Mich.,2002, pp. 81-90.

13. Yang, K. H., “Evaluation on the Shear Strength of High-StrengthConcrete Deep Beams,” PhD Thesis, Chungang University, Korea, Feb.2002, 120 pp.

14. ACI Committee 445, “Shear and Torsion,” Strut-and-Tie Bibliography,ACI Bibliography No. 16, American Concrete Institute, Farmington Hills,Mich., Sept. 1997, 50 pp.

15. Cervenka, V.; Jendele, L.; and Cervenka, J., “ATENA ComputerProgram Documentation: Part 1,” Cervenka Consultant, 2003, 106 pp.

16. ACI Committee 318, “Building Code Requirements for StructuralConcrete (ACI 318-99) and Commentary (318R-99),” American ConcreteInstitute, Farmington Hills, Mich., 1999, 369 pp.

fc′

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429ACI Structural Journal/July-August 2007

APPENDIX A

Table A1—Details and test results of simple deep beams13

Simple f ′c , MPa a/h a/jd ρh ρv Vcr, kN

Pn , kN

(Pn)Exp./(Pn)ACIExp. ACI 318-05

No.1

31.4

0.5 0.59

0 0 254.0 958.0 684.1 1.400

No. 2 0 0.006 259.0 992.0 684.1 1.450

No. 3 0 0.012 260.0 1111.3 684.1 1.624

No. 4 0.006 0 249.9 1042.7 684.1 1.524

No. 5 0.006 0.006 262.6 1323.0 855.2 1.547

No. 6 0.012 0 270.5 1391.6 684.1 2.034

No. 7

0.7 0.82

0 0.006 188.0 876.1 624.7 1.402

No. 8 0.006 0 215.6 993.7 624.7 1.591

No. 9 0.006 0.006 205.8 1044.7 780.9 1.338

No. 10

1.0 1.18

0 0 173.5 750.7 520.0 1.444

No. 11 0 0.006 172.5 762.4 520.0 1.466

No. 12 0 0.012 195.0 1107.4 520.0 2.130

No. 13 0.006 0 178.4 601.7 520.0 1.157

No. 14 0.006 0.006 181.0 905.5 650.0 1.393

No. 15 0.012 0 185.0 707.6 520.0 1.361

No. 161.5 1.76

0 0 107.8 409.6 378.8 1.081

No. 17 0.006 0.006 142.1 721.3 473.5 1.523

No. 18

52.9

0.5 0.590 0 290.0 1540.6 1154.5 1.334

No. 19 0.006 0.006 318.5 1775.8 1443.1 1.230

No. 201.0 1.18

0 0 225.4 952.6 877.4 1.086

No. 21 0.006 0.006 245.0 1129.0 1096.8 1.029

No. 22

78.4

0.5 0.59

0 0 347.9 1646.4 1710.4 0.963

No. 23 0 0.006 357.7 1789.5 1710.4 1.046

No. 24 0 0.012 347.9 1934.5 1710.4 1.131

No. 25 0.006 0 392.0 1962.0 1710.4 1.147

No. 26 0.006 0.006 345.0 2061.9 2138.0 0.964

No. 27 0.012 0 401.8 2269.7 1710.4 1.327

No. 28

0.7 0.82

0 0.006 289.1 1622.9 1561.8 1.039

No. 29 0.006 0 303.8 1395.5 1561.8 0.894

No. 30 0.006 0.006 308.7 1701.3 1952.2 0.871

No. 31

1.0 1.18

0 0 254.8 1146.6 1299.9 0.882

No. 32 0 0.006 240.1 1356.3 1299.9 1.043

No. 33 0 0.012 294.0 1558.2 1299.9 1.199

No. 34 0.006 0 249.9 1213.2 1299.9 0.933

No. 35 0.006 0.006 281.3 1295.6 1624.9 0.797

No. 36 0.012 0 291.1 1215.2 1299.9 0.935

No. 371.5 1.76

0 0 173.5 656.6 947.0 0.693

No. 38 0.006 0.006 181.3 836.9 1183.7 0.707

Mean 1.229

Standard deviation 0.326

Note: 1 MPa = 145 psi; 1 kN = 0.2248 kips.

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ACI Structural Journal, V. 104, No. 4, July-August 2007.MS No. S-2006-250 received June 16, 2006, and reviewed under Institute publication

policies. Copyright © 2007, American Concrete Institute. All rights reserved, includingthe making of copies unless permission is obtained from the copyright proprietors. Pertinentdiscussion including author’s closure, if any, will be published in the May-June 2008ACI Structural Journal if the discussion is received by January 1, 2008.

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An experimental program on 12 series of specimens with differentembedment lengths to determine the transfer length was conducted.Transfer length test results of seven-wire strand on twelve differentconcrete mixtures were analyzed. A testing technique based on theanalysis of bond behavior by means of measuring the force supportedby the tendon has been used. The specimens had been instrumentedwith slip measurement devices at each end of the specimen. Asequence of slip values at each end of the specimen after release versusthe embedment length has been analyzed. The expressions relatingthe transfer length to the tendon end slip are presented. A value ofGuyon’s factor for tendon stress distribution shape has beenobtained. Two criteria to determine the transfer length from theslip sequences at both ends of the specimens have been analyzed.

Keywords: bond; precast concrete; prestressing; pretensioning; slip;strand; transfer length.

INTRODUCTIONThe force in a prestressing strand is transferred by bond to

the concrete in the release operation. At this stage, strandstress varies from zero at the free end of the member to amaximum value (effective stress). Transfer length is definedas the distance required to develop the effective stress in theprestressing strand.1 Variation in strand stress along thetransfer length involves slip between the strand and theconcrete. The measurement of the strand end slip is an indirectmethod to determine the transfer length.2 Most experimentalstandards3-6 are based on this method, and it has been proposedas a simple nondestructive assurance procedure by which thequality of bond can be monitored within precasting plants.7

Guyon8 proposed the following expression from a theoreticalanalysis

(1)

where Lt is the transfer length, δ is the strand end slip at thefree end of a prestressed concrete member, εpi is the initialstrand strain, and the α coefficient represents the shapefactor of the bond stress distribution along the transfer zone.Two hypotheses were considered8: α = 2 for uniform bondstress distribution (linear variation in strand stress); and α = 3for linear descending bond stress distribution (parabolicvariation in strand stress)

Equation (1) can be rewritten as follows

(2)

where Ep is the modulus of elasticity of the prestressingstrand and fpi is the strand stress immediately before release.

Several researchers have proposed different values of αfor the bond stress distribution along the transfer zone from

Lt αδ

εpi

------=

Lt αδEp

fpt

---------=

experimental results and theoretical studies. Table 1 indicatesthe different assigned values of α.

Table 2 shows other expressions that relate the transferlength to the strand end slip at the free end of a pretensionedconcrete member, where db is the diameter of prestressingstrand, and fci′ is the compressive strength of concrete at thetime of prestress transfer.

Title no. 104-S47

Reliability of Transfer Length Estimation from Strand End Slipby José R. Martí-Vargas, César A. Arbeláez, Pedro Serna-Ros, and Carmen Castro-Bugallo

Table 1—Proposed α coefficient values from Guyon’s formula

Reference Coefficient Origin of value

FIP4 4Indicated value when stress inprestressing strand is rapidly

increasing

Guyon8

3By hypothesis

FIP4* Adopted value

Olesniewicz9

2.86 ExperimentalFIP10

RILEM3

2.8 Adopted valueIRANOR5

LCPC6

Balázs11 2/(1 – b)† 2.67 By theoretical studies

den Uijl12 2.3 to 2.6 Experimental value and by theoretical studies

Jonsson13 2.5 Assumed value

Guyon8

2 By hypothesis

Brooks et al.14

Balogh15

Russell and Burns16

Logan17

Steinberg et al.18

Oh and Kim19

Wan et al.20

CEB-FIP21*

Rose and Russell22

den Uijl12

1.5 Indicated value for linear ascend-ing bond stress distributionfib23

Lopes and do Carmo24

*Substituting fpi by effective stress in strand immediately after release.†b is experimental constant value that must be fixed for each type of prestressingstrand according to its bond characteristics (for 12.7 mm [0.5 in.] seven-wirestrand, b = 0.25 and α = 2.67).

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Some researchers conducted experimental studies to obtainthe transfer length from the strand end slip at the free end inhollow-core slabs,7,13-15 in beams,16,18,19,22,27-29 in piles,20,30

in prisms,31 and in specimens to simulate bond behavior alongtransfer length.32

Several authors7,14,20,30 have established an allowablefree end slip as the strand end slip which results in a transferlength equal to that computed by the ACI provisions fortransfer length (Eq. (7)).1 By setting Eq. (2) to be equal to theEq. (7) and substituting α = 2 and α = 3 in Eq. (2), theimplied allowable value of end slip can be calculated byEq. (8) and (9), respectively.

(7)

(8)

Lt13---fsedb (U.S. units) Lt

120.7----------fsedb (SI units)= =

δall216---

fpi

Ep

-----fsedb (U.S. units) =

δall21

41.4----------

fpi

Ep

-----fsedb (SI units)=

(9)

where fse is the effective stress in the prestressing strand afterallowance for all prestress losses, db is the nominal diameter ofprestressing strand, δall2 is the implied allowable value of freeend slip when α = 2, and δall3 is the implied allowable value offree end slip when α = 3 (Lt , fpi, and Ep, as previously described).

To apply Guyon’s end slip theory to determine transferlength is easy, but the measurements of slips are affected bythe local bond loss at the ends. Equations (1) to (6) are notapplicable to elements of a poor bond quality.14 In this case,greater slips are measured resulting in incorrect transferlength estimation.

The other disadvantages of Guyon’s method are largerscatter of experimental results,15 difficulty to measureaccurately smaller slips,13 breakage of gauges to measurethe strand end slip when a flame cutting process is applied,27

and excessive free end slip in prestressed members with poorconcrete consolidation around the strand.7

RESEARCH SIGNIFICANCEThis research study provides information on the transfer

length of a seven-wire prestressing strand in twelveconcretes of different compositions and properties. A testmethod based on the measurement and analysis of the forcesupported by the strand has been used. This paper analyzesthe reliability of transfer length determination from free endslips according with proposed expressions in the literature.Findings of the research are presented in procedures for theexperimental determination of transfer length measuring forcesor slips. The information is valuable for all parties involvedin the precast/prestressed concrete industry: manufacturers,producers, designers, builders, and owners.

EXPERIMENTAL INVESTIGATIONAn experimental program has been conducted to determine

the transfer length of prestressing strands: the ECADA*

test method33-34 (*Ensayo para Caracterizar la Adherenciamediante Destesado y Arrancamiento [Test to Characterizethe Bond by Release and Pull-out]).

MaterialsTwelve different concretes with a range of water-cement

ratios (w/c) from 0.3 to 0.5, cement content from 590 to843 lb/yd3 (350 to 500 kg/m3) and a compressive strength at thetime of testing fci′ from 3.5 to 8 ksi (24 to 55 MPa) were tested.

Concrete components were a) cement CEM I 52.5 R;35 b)crushed limestone aggregate (0.275 to 0.472 in. [7 to 12 mm]);c) washed rolled limestone sand (0 to 0.157 in. [0 to 4 mm]);and d) policarboxilic ether high-range water-reducing additive.The mixtures of the tested concretes are shown in Table 3.

The prestressing strand was a low-relaxation seven-wirestrand specified as UNE 36094:97 Y 1860 S7 13.036 with aguaranteed ultimate strength of 270 ksi (1860 MPa). Themain characteristics were adopted from the manufacturer:diameter 0.5 in. (12.7 mm), cross-sectional area 0.154 in.2

(99.69 mm2), ultimate strength 43.3 kips (192.60 kN), yieldstress at 0.2% 40 kips (177.50 kN), and modulus of elasticity28,507 ksi (196,700 MPa). The prestressing strand was

δall319---

fpi

Ep

-----fsedb (U.S. units) =

δall31

62.1----------

fpi

Ep

-----fsedb (SI units)=

José R. Martí-Vargas is an Associate Professor of civil engineering in the Department ofConstruction Engineering and Civil Engineering Projects, Polytechnic University ofValencia (UPV), Valencia, Spain. He is member of the Institute of Science and ConcreteTechnology (ICITECH) at UPV. He received his degree in civil engineering and his PhDfrom UPV. His research interests include bond behavior of reinforced and prestressedconcrete structural elements, durability of concrete structures, and strut-and-ties models.

César A. Arbeláez is a PhD Assistant Researcher in the Department of ConstructionEngineering and Civil Engineering Projects at Polytechnic University of Valencia. Heis member of ICITECH at UPV. He received his civil engineering degree from QuindíoUniversity, Armenia, Quindío, Colombia, and his PhD from UPV. His research interestsinclude bond properties of prestressed concrete structures and the use of advancecement-based materials in structural applications.

Pedro Serna-Ros is a Professor of civil engineering in the Department of ConstructionEngineering and Civil Engineering Projects at Polytechnic University of Valencia. Heis a member of ICITECH at UPV. He received his degree in civil engineering fromUPV and his PhD from l’Ecole National des Ponts et Chaussées, Paris, France. Hisresearch interests include self-consolidating concrete, fiber-reinforced concrete, andbond behavior of reinforced and prestressed concrete.

Carmen Castro-Bugallo is a PhD candidate in the Department of ConstructionEngineering and Civil Engineering Projects at Polytechnic University of Valencia.She is member of ICITECH at UPV. She received her degree in civil engineering fromUPV. Her research interests include bond properties of reinforced concrete andprestressed concrete structures and strut-and-ties models.

Table 2—Proposed equations for transfer length from strand end slip

ReferenceEquation

no. Equation (U.S. units) Equation (SI units)

Marshall and Krishnamurthy25 (3)

K = 0.00009 in.–1 for 0.5 in. seven-wire

strand

K = 0.0000035 mm–1 for 12.7 mm seven-

wire strand

Balázs26 (4) Lt = 218db Lt = 105db

Balázs11 (5)

Rose and Russell22 (6)

Notes: For U.S. units: fpi, f ′ci , and Ep in ksi; db, δ, and Lt in inches; for SI units: fpi,f ′ci, and Ep in MPa; db, δ, and Lt in mm. 1 in. = 25 mm; 1 MPa = 0.145 ksi.

LtδK----= Lt

δK----=

δ3 2⁄

fci′---------4

δ3 2⁄

fci′---------4

Lt24.7δ0.625

fci′0.15 fpi

Ep

-----⎝ ⎠⎛ ⎞

0.4------------------------------= Lt

111δ0.625

fci′0.15 fpi

Ep

-----⎝ ⎠⎛ ⎞

0.4------------------------------=

Lt 2δEp

fpi

----- 5.4+= Lt 2δEp

fpi

----- 137.16+=

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ACI Structural Journal/July-August 2007 489

tested in the as-received condition (free of rust and free oflubricant). The strand was no treated in any special manner.The strand was stored indoors, and care was taken not to dragthe strand on the floor.

Testing techniqueThe ECADA test method is based on the measurement and

analysis of the force supported by the strand in a series ofpretensioned concrete specimens with different embedmentlengths. Figure 1 shows the test equipment layout.

An anchorage-measurement-access (AMA) system isplaced at one end (stressed end) of a pretensioning frame tosimulate the sectional stiffness of the specimen. The AMAsystem is made up of a sleeve in the final stretch of the specimento prevent the influence of the confinement caused by theend frame plate, the stressed end frame plate, and ananchorage plate supported on the frame by two separators.

The step-by-step test procedure was described in detail inMartí-Vargas et al.,34 and may be summarized as follows:

Preparation stage—1. The strand is placed in the frame;2. Strand tensioning;3. Strand anchorage by an adjustable strand anchorage;4. The concrete is mixed, placed into the formwork in the

frame, and consolidated; and5. After concrete placement, the specimen is cured to

achieve the desired concrete properties at the time of testing.Testing stage—1. The adjustable strand anchorage is relieved using the

hydraulic jack; and2. Strand release is produced at a controlled speed, and the

prestressing force transfer to the concrete is performed. Thestrand is completely released. The specimen is supported atthe stressed end frame plate.

Stabilization period—The level of force during this time iszero at the free end. The force in the strand at the stressed enddepends on the strain compatibility with the concrete specimen.This force requires a stabilization period to guarantee itsmeasurement. The strand force in the AMA system isrecorded continuously during the test.

Although it is not included in this study, the test cancontinue with the pull-out operation positioning the hydraulicjack at the stressed end to increase the force in the strand,separating the anchorage plate of the AMA system fromthe frame.

Test parametersThe specimens had a 4 x 4 in.2 (100 x 100 mm2) cross section

with a concentrically located single strand at a prestress levelbefore release of 75% of guaranteed ultimate strand strength.All specimens were subjected to the same consolidation andcuring conditions. Release was gradually performed 24 hoursafter concreting at a controlled speed of 0.18 kips/s (0.80 kN/s).A stabilization period of 2 hours from release was established.With these test parameters, visible splitting cracks have nothappened in any of the tested specimens.

InstrumentationThe instrumentation used was a hydraulic jack pressure

sensor to control tensioning and release operations; a hollowforce transducer included in the AMA system to measure theforce supported by the strand; and two linear variable differentialtransducers (LVDTs), one at the free end (Fig. 2) to measurethe draw-in (δ, free end slip), and another at the stressed end(Fig. 3) to measure the strand slip to the last embedmentconcrete cross-section of the specimen (δl, stressed end slip).No internal measuring devices were used in the test specimensso as to not distort the bond phenomenon.

Criterion to determine transfer lengthWith the ECADA test method, the transfer length is

obtained with a series of specimens with different embedment

Table 3—Concrete mixtures from test program

DesignationCement,

lb/yd3 (kg/m3) w/cGravel/sand

ratio

f ′ci (at time of testing, 24 hours),

ksi (MPa)

M-350-0.50

590 (350)

0.50

1.14

3.8 (26.1)

M-350-0.45 0.45 5.4 (37.3)

M-350-0.40 0.40 6.8 (46.7)

M-400-0.50

674 (400)

0.50 3.5 (24.2)

M-400-0.45 0.45 4.1 (28.3)

M-400-0.40 0.40 6.0 (41.4)

M-400-0.35 0.35 6.6 (45.3)

M-450-0.40758 (450)

0.40 5.3 (36.3)

M-450-0.35 0.35 6.7 (46.6)

M-500-0.40

843 (500)

0.40 4.5 (30.8)

M-500-0.35 0.35 6.8 (46.6)

M-500-0.30 0.30 7.9 (54.8)

Fig. 1—Test equipment layout.

Fig. 2—LVDT at free end of specimen.

Fig. 3—LVDT at stressed end of specimen.

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490 ACI Structural Journal/July-August 2007

lengths. For each specimen, the strand force loss in the AMAsystem directly after the stabilization period is measured.

The force loss values are arranged according to the specimenembedment length (Fig. 4). The obtained curve shows abilinear tendency. The transfer length corresponds to thesmallest specimen embedment length that marks the beginningof the horizontal branch.33,34

The resolution in the determination of the transfer lengthwill depend on the sequence of lengths of the specimenstested. For an embedment length sequence of 2 in. (50 mm),the transfer length obtained by the ECADA test method isrepeated when a same concrete mixture is tested.34

Transfer length over-estimationThe ideal AMA system must have the same sectional

rigidity as the specimen. This rigidity depends on theconcrete properties, the age of the concrete at the time oftesting, and the specimen cross section. It would not really befeasible to design a system for each specific test conditions.

For this reason, in this experimental work, the rigidity ofthe AMA system designed is slightly greater than thesectional rigidity of the specimens. A discontinuity section isgenerated in the border between the specimen and the AMAsystem. In these conditions, the strand force measured in theAMA system after release will be slightly higher than theeffective prestressing force of the strand in the specimen.This difference of forces gives rise to a small over-estimation ofthe real transfer length.34 Consequently, even if the specimenembedment length is greater than the transfer length, a smallslip of the strand at the stressed end is registered.

EXPERIMENTAL RESULTS AND DISCUSSIONDetermination of transfer length

Transfer length is determined for each concrete mixture inaccordance with the exposed criterion. As an example, Fig. 5shows the results of force loss versus the embedment lengthfor the concrete M-350-0.50 (designation according withTable 3). Two curves are shown, one with the force lossesregistered just after release (ΔP), and another with the forcelosses registered after the stabilization period (ΔP). Bothcurves present a bilinear tendency with a descendent initialbranch with a strong slope and a practically horizontalbranch starting from 21.7 in. (550 mm) embedment length.The transfer length determined by the ECADA test methodfor this concrete mixture is 21.7 in. (550 mm).

The difference between the two curves corresponds to theincrement of force loss registered during the stabilizationperiod. When specimens have an embedment length below21.7 in. (550 mm), the force loss after the stabilization periodis greater than the force loss registered just after release.When specimens have an embedment length equal to orgreater than 21.7 in. (550 mm), however, the force loss issimilar at both points of time.

As it can be observed in Fig. 5, for this concrete, thebeginning of the horizontal branches coincides at bothpoints of time. In some cases, however, increases of forceloss have taken place during the stabilization period in thefirst point of the resulting horizontal branch just after release.For this reason, the transfer length must be always determinedon the curve measured after the stabilization period.

Comparison of test results with Guyon’s formulaFigure 6 shows the transfer length results obtained by the

ECADA test method for each concrete mixture, as well asthe transfer length obtained from the free end slips byapplying Guyon’s formula (Eq. (2)). This formula has beenapplied to free end slips registered after the stabilizationperiod in specimens with an embedment length equal to orgreater than the transfer length. Between four and 18 specimensfor each concrete mixture, with a total of 121 specimens,have been considered.

Two intervals are drawn for each concrete mixture. Theinterior interval corresponds to the extreme transfer lengthvalues obtained by applying Guyon’s formula with α = 2.8(adopted by RILEM3) to the minimum and maximum freeend slips. The exterior interval corresponds to the extremetransfer length values according to the hypotheses by Guyonobtained as follows: the lower limit was calculated byapplying α = 2 to the minimum free end slip, and the upperlimit was calculated by applying α = 3 to the maximum freeend slip.

The amplitude of the transfer length intervals is very variablefor the different concrete mixtures, as shown in Fig. 6. Theresults obtained by the ECADA test method are locatedwithin both intervals in all cases except for the M-500-0.30concrete mixture for the interior interval.

Figure 7 shows the transfer length results obtained by theECADA test method in the corresponding series versus thefree end slip registered after the stabilization period in eachspecimen. Only the specimens with an embedment lengthequal to or greater than the transfer length have beenincluded. The predicted transfer lengths by Guyon’s formulaare also plotted in Fig. 7. It is shown that 38.8% of theexperimental results fall outside the limits (33.0% show atransfer length greater than the predicted maximum values,

Fig. 4—Determination of transfer length through ECADAtest method.

Fig. 5—Force loss versus embedment length for ConcreteM-350-0.50.

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and 5.8% show a transfer length smaller than the predictedminimum values). A value of α = 2.44 from the regressionanalysis of the test results has been obtained.

Figure 8 shows the experimental transfer lengths versusthe registered free end slips obtained in beams by severalauthors. The predicted transfer length according to ACI 318-051

(LtACI) and the allowable free end slips δall2 (Eq. (8)) and δall3(Eq. (9)) are also plotted in Fig. 8. The LtACI, δall2, and δall3values have been calculated by considering that fpi = 202 ksi,fse = 0.8fpi = 162 ksi, Ep = 28,528 ksi and db = 0.5 in. (fpi =1395 MPa, fse = 0.8fpi = 1116 MPa, Ep = 196,700 MPa anddb = 12.7 mm). The percentages of results included in eachsector delimited by LtACI, δall2, and δall3 are indicated in Fig. 8.

The range of free end slip registered is very ample for onesame transfer length, as observed in Fig. 8. Also the range oftransfer length values is very variable for one same free endslip. Figure 8 also shows that when a transfer length is smallerthan LtACI, the δall2 limit is exceeded in 2.8% of the cases, andthe δall3 is exceeded in 32.3% of the cases (2.8% + 29.5%).On the other hand, for registered free end slips smaller thanδall3 or δall2, transfer lengths greater than LtACI are measuredin some cases (2.3 and 4.6%, respectively). Consequently, theuse of an assurance procedure for bond quality based on a limitvalue for the allowable free end slip is not completely reliable.

Comparison of test results with other expressionsThe experimental results obtained with both the ECADA

test method and the theoretical predictions from Eq. (3) to (6)have been compared. As an example, Fig. 9 illustrates thecomparison with Eq. (6). Table 4 summarizes thesecomparisons. Besides, the comparison with Eq. (2) bysubstituting α = 2.44 (obtained value from the experimentalresults of this study) is included. It can be observed that theexpressions based on Guyon’s formula (Eq. (6) and Eq. (2)with α = 2.44) show a good prediction of the averagemeasured transfer length. The coefficient of correlationimproves when the expressions include, in addition to theslips, other parameters like the concrete compressive strength.

Use of end slips sequences to determine transfer length

The possibility of determining the transfer length from thesequences of end slip values at both ends versus the embedmentlength of specimens was considered.

Fig. 7—Transfer length versus free end slip for specimens withembedment length equal to or greater than transfer length.

Fig. 8—Comparison between results of present tests andthose of other researchers.

Fig. 9—Comparison of measured transfer lengths withcalculated values according to Eq. (6).

Table 4—Comparison between measured and calculated transfer lengths

Equation no.Average

Lt(calculated)/Lt(measured) Coefficient of correlation R2

(3) 1.18 0.07

(4) 1.17 0.54

(5) 1.11 0.35

(6) 1.01 0.21

(2) with α = 2.44 0.95 0.20

Fig. 6—Graphical comparison between experimental transferlength and predicted transfer lengths from Guyon’s formulaand RILEM provisions.

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492 ACI Structural Journal/July-August 2007

Figure 10 shows the free end slip results versus theembedment length for the concrete mixture M-350-0.50.Two curves are shown, one with the free end slip just afterrelease δ, and the other with the free end slip registered after thestabilization period δ. Both curves present a bilinear tendency,with a descendent initial branch and a practically horizontalbranch starting from 21.7 in. (550 mm) embedment length.This embedment length coincides with the result obtained bythe ECADA test method (refer to Fig. 5). The free end slipincreases during the stabilization period in all the specimens.

Similarly, Fig. 11 shows the stressed end slip just afterrelease δl, and the stressed end slip after the stabilizationperiod δl versus the embedment length for the same concretemixture. Both curves present a bilinear tendency. The beginningof the horizontal branch coincides with the result obtained bythe ECADA test method (21.7 in. [550 mm]). In regard to theforce losses, the stressed end slip only increases during thestabilization period in specimens whose embedment lengthis smaller than the transfer length.

Figure 12 summarizes the results of the three variables(force loss and slip at both ends) for the concrete M-350-0.50after the stabilization period. The shown ratios are thequotient between each specimen test result (ΔP, δ, and δl),and the average test results (ΔPAVE, δAVE, and δlAVE) ofspecimens with an embedment length equal to or greaterthan the transfer length. Again, a bilinear tendency is observedwith a descendent initial branch and a perceptibly horizontalbranch from 21.7 in. (550 mm) embedment length. Althoughthe slope of the descendent initial branch is very pronouncedin the cases of force loss and stressed end slip, it is very weak inthe case of free end slip. Consequently, the beginning of thehorizontal branch is more easily identifiable by analyzing theforce loss and stressed end slip than the free end slip.

This procedure of test results analysis for each concretemixture has been applied. The transfer lengths from the threesequences of results obtained from the test instrumentation(ΔP, δ, and δl) versus the embedment length have beendetermined. Table 5 summarizes the obtained results. Thetransfer lengths obtained from the stressed end slip and bythe ECADA test method coincide in 11 out of the 12 concretemixtures, and only a 2 in. (50 mm) difference is observed inthe concrete M-400-0.35. The transfer lengths obtained fromthe free end slip coincide in eight out of the 12 concretemixtures. Given the wide dispersion of the measured freeend slip, no bilinear behavior was detected in the remainingcases (see range of free end slip to one same transfer lengthin Fig. 7). It was not possible to determine the transfer lengthif the beginning of the horizontal branch was not clearlydefined. These cases correspond to concrete mixtures withgreater water content in their mixture.

CONCLUSIONSBased on the results of this experimental investigation, the

following conclusions are drawn:1. The feasibility of applying the ECADA test method to

determine the transfer length of prestressing strands has beenverified, even in concretes with a low compressive strength;

Fig. 10—Free end slip versus embedment length forConcrete M-350-0.50.

Fig. 11—Stressed end slip versus embedment length forConcrete M-350-0.50.

Fig. 12—Ratios ΔP/ΔPAVE , δ/δAVE , and δl/δlAVE versusembedment length for Concrete M-350-0.50.

Table 5—Transfer length obtained from three sequences of results (ΔP, δ, and δl)

Designation

Transfer length, in. (mm)

ECADA test method ΔP Free end slip δ Stressed end slip δl

M-350-0.50 21.7 (550) 21.7 (550) 21.7 (550)

M-350-0.45 21.7 (550) 21.7 (550) 21.7 (550)

M-350-0.40 21.7 (550) 21.7 (550) 21.7 (550)

M-400-0.50 25.6 (650) — 25.6 (650)

M-400-0.45 21.7 (550) — 21.7 (550)

M-400-0.40 21.7 (550) 21.7 (550) 21.7 (550)

M-400-0.35 19.7 (500) 19.7 (500) 17.7 (450)

M-450-0.40 21.7 (550) — 21.7 (550)

M-450-0.35 19.7 (500) 19.7 (500) 19.7 (500)

M-500-0.40 23.6 (600) — 23.6 (600)

M-500-0.35 17.7 (450) 17.7 (450) 17.7 (450)

M-500-0.30 15.7 (400) 15.7 (400) 15.7 (400)

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2. An average value of α = 2.44 for Guyon’s formula hasbeen obtained from the experimental results of this study. Anample range of free end slip values has been obtained for onesame transfer length. Furthermore, the range of transferlength values for one same free end slip is very variable;

3. Consequently, a great variability of results for one sameconcrete mixture has been observed in transfer length estimationfrom the experimental free end slips when Guyon’s formulawas applied;

4. The prediction range of transfer lengths from expressionsproposed by several authors relating the transfer length to thefree end slip is very ample;

5. Determining transfer length from the free end slip isrelatively easy, although it can lead to a false perception thatthe transfer length value is very variable;

6. Using a limit value for the allowable free end slip as anassurance procedure for bond quality may give rise touncertain situations;

7. In relation to the results from the ECADA test method,the sequence of stressed end slip values versus the embedmentlength is a reliable assurance procedure for the experimentaldetermination of transfer length; and

8. The sequence of free end slip values versus the embedmentlength is not a reliable assurance procedure for the experimentaldetermination of transfer length. The beginning of the horizontalbranch is not clearly defined when the dispersion of measuredfree end slip is wide. This particularly occurs when concretehas a low compressive strength.

ACKNOWLEDGMENTSThe contents of this paper are within the framework of a line of research

that is currently being carried out by the Concrete Technology and ScienceInstitute (ICITECH) of the Polytechnic University of Valencia, Valencia, Spain,in collaboration with the companies PREVALESA and ISOCRON. Financialsupport provided by the Ministry of Education and Science and FEDER funds(Project MAT2003-07157 and Project BIA2006-05521) made this researchpossible. The authors appreciate the collaboration of the aforementionedcompanies and organizations, as well as the participation of the technicalstaff of the Concrete Structures Laboratory at the Polytechnic University ofValencia for their assistance in preparing and testing specimens.

NOTATIONdb = nominal diameter of prestressing strandEp = modulus of elasticity of prestressing strandfci′ = compressive strength of concrete at time of prestress transfer

(cylinder)fpi = strand stress immediately before releasefse = effective stress in the prestressing strand after allowance for all

prestress lossesLt = transfer lengthLtACI = predicted transfer length according ACI 318-05α = coefficient to take into account assumed shape of bond stress

distributionδ = strand end slip at free endδall2 = allowable free end slip when α = 2δall3 = allowable free end slip when α = 3δ = free end slip just after releaseδAVE = average free end slip after stabilization period obtained in all

specimens of series with embedment length equal to or greaterthan transfer length

δl = stressed (loaded) end slipδl = stressed end slip just after releaseδlAVE = average stressed end slip after stabilization period obtained in all

specimens of series with embedment length equal to or greaterthan transfer length

ΔP = force loss after stabilization periodΔPAVE = average force loss after stabilization period obtained in all

specimens of series with embedment length equal to or greaterthan transfer length

ΔP = force loss just after releaseεpi = initial strand strain

REFERENCES1. ACI Committee 318, “Building Code Requirements for Structural

Concrete (ACI 318-05) and Commentary (318R-05),” American ConcreteInstitute, Farmington Hills, Mich., 2005, 430 pp.

2. Thorsen, N., “Use of Large Tendons in Pretensioned Concrete,” ACIJOURNAL, Proceedings V. 53, No. 6, Feb. 1956, pp. 649-659.

3. RILEM RPC6, “Specification for the Test to Determine the BondProperties of Prestressing Tendons,” Réunion Internationale des Laboratoires etExperts des Matériaux, Systèmes de Constructions et Ouvrages, Bagneux,France, 1979, 7 pp.

4. Fédération Internationale de la Précontrainte, “Report on PrestressingSteel: 7. Test for the Determination of Tendon Transfer Length under StaticConditions,” FIP, London, UK, 1982, 19 pp.

5. IRANOR UNE 7436, “Bond Test of Steel Wires for PrestressedConcrete,” Instituto Nacional de Racionalización y Normalización, Madrid,Spain, 1982, 13 pp.

6. Laboratoire Central des Ponts et Chaussées, “Transfer LengthDetermination. Test Method Applicable for Prestressed Reinforcement,”Techniques et Méthodes, No. 53, LCPC, Paris, France, 1999, pp. 45-55.

7. Anderson, A. R., and Anderson, R. G., “An Assurance Criterionfor Flexural Bond in Pretensioned Hollow Core Units,” ACI JOURNAL,Proceedings V. 73, No. 8, Aug. 1976, pp. 457-464.

8. Guyon, Y., Pretensioned Concrete: Theoretical and ExperimentalStudy, Paris, France, 1953, 711 pp.

9. Olesniewicz, A., “Statistical Evaluation of Transfer Length of Strand,”Research and Design Centre for Industrial Building (BISTYP), Warsaw,Poland, 1975.

10. Fédération Internationale de la Précontrainte, “Report on PrestressingSteel: 2. Anchorage and Application of Pretensioned 7-Wire Strands,” FIP,London, UK, 1978, 45 pp.

11. Balázs, G., “Transfer Length of Prestressing Strand as a Function ofDraw-In and Initial Prestress,” PCI Journal, V. 38, No. 2, Mar.-Apr. 1993,pp. 86-93.

12. den Uijl, J. A., “Bond Modelling of Prestressing Strand,” Bond andDevelopment of Reinforcement, SP-180, R. Leon, ed., American ConcreteInstitute, Farmington Hills, Mich., 1998, pp. 145-169.

13. Jonsson, E., “Anchorage of Strands in Prestressed Extruded Hollow-Core Slabs,” Proceedings of the International Symposium Bond in Concrete:From Research to Practice, Riga Technical University and CEB, eds., Riga,Latvia, 1992, pp. 2.20-2.28.

14. Brooks, M. D.; Gerstle, K. H.; and Logan, D. R., “Effect of InitialStrand Slip on the Strength of Hollow-Core Slabs,” PCI Journal, V. 33,No. 1, Jan.-Feb. 1988, pp. 90-111.

15. Balogh, T., “Statistical Distribution of Draw-in of Seven-WireStrands,” Proceedings of the International Symposium Bond in Concrete:From Research to Practice, Riga Technical University and CEB, eds., Riga,Latvia, 1992, pp. 2.10-2.19.

16. Russell, B. W., and Burns, N. H., “Measured Transfer Lengths of 0.5and 0.6 in. Strands in Pretensioned Concrete,” PCI Journal, V. 44, No. 5,Sept.-Oct. 1996, pp. 44-65.

17. Logan, D. R., “Acceptance Criteria for Bond Quality of Strand forPretensioned Prestressed Concrete Applications,” PCI Journal, V. 42, No. 2,Mar.-Apr. 1997, pp. 52-90.

18. Steinberg, E.; Beier, J. T.; and Sargand, S., “Effects of Sudden PrestressForce Transfer in Pretensioned Concrete Beams,” PCI Journal, V. 46, No. 1,Jan.-Feb. 2001, pp. 64-75.

19. Oh, B. H., and Kim, E. S., “Realistic Evaluation of Transfer Lengthsin Pretensioned, Prestressed Concrete Members,” ACI Structural Journal,V. 97, No. 6, Nov.-Dec. 2000, pp. 821-830.

20. Wan, B.; Harries, K. A.; and Petrou, M. F., “Transfer Length ofStrands in Prestressed Concrete Piles,” ACI Structural Journal, V. 99, No. 5,Sept.-Oct. 2002, pp. 577-585.

21. Comité Euro-International du Béton-Fédération Internationale de laPrécontrainte, “Model Code for Concrete Structures,” CEB-FIP, Lausanne,Switzerland, 1993, 437 pp.

22. Rose, D. R., and Russell, B. W., “Investigation of Standardized Teststo Measure the Bond Performance of Prestressing Strand,” PCI Journal,V. 42, No. 4, July-Aug. 1997, pp. 56-80.

23. Fédération Internationale du Béton, “Bond of Reinforcement inConcrete: State-of-Art Report,” Bulletin d’Information, No. 10, fib,Lausanne, Switzerland, 2000, 427 pp.

24. Lopes, S. M., and do Carmo, R. N., “Bond of Prestressed Strands toConcrete: Transfer Rate and Relationship between Transfer Length andTendon Draw-in,” Structural Concrete, V. 3, No. 3, 2002, pp. 117-126.

25. Marshall, W. T., and Krishnamurthy, D., “Transfer Length ofPrestressing Tendons from Concrete Cube Strength at Transfer,” TheIndian Concrete Journal, V. 43, No. 7, July 1969, pp. 244-275.

26. Balázs, G., “Transfer Control of Prestressing Strands,” PCI Journal,

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V. 37, No. 6, Nov.-Dec. 1992, pp. 60-71.27. Kahn, L. F.; Dill, J. C.; and Reutlinger, C. G., “Transfer and

Development Length of 15-mm Strand in High-Performance ConcreteGirders,” Journal of Structural Engineering, V. 128, No. 7, July 2002,pp. 913-921.

28. Cousins, T. E.; Johnston, D. W.; and Zia, P., “Transfer Length ofEpoxy-Coated Prestressing Strand,” ACI Materials Journal, V. 87, No. 3,May-June 1990, pp. 193-203.

29. Cousins, T. E.; Stallings, J. M.; and Simmons, M. B., “ReducedStrand Spacing in Pretensioned, Prestressed Members,” ACI StructuralJournal, V. 91, No. 3, May-June 1994, pp. 277-286.

30. Petrou, M. F.; Wan, B.; Joiner, W. S.; Trezos, C. G.; and Harries, K. A.,“Excessive Strand End Slip in Prestressed Piles,” ACI Structural Journal,V. 97, No. 5, Sept.-Oct. 2000, pp. 774-782.

31. Mahmoud, Z. I.; Rizkalla, S. H.; and Zaghloul, E. R., “Transferand Development Lengths of Carbon Fiber Reinforcement PolymersPrestressing Reinforcing,” ACI Structural Journal, V. 96, No. 4, July-Aug.1999, pp. 594-602.

32. Abrishami, H. H., and Mitchell, D., “Bond Characteristics of Preten-sioned Strand,” ACI Materials Journal, V. 90, No. 3, May-June 1993,pp. 228-235.

33. Martí, J. R., “Experimental Study on Bond of Prestressing Strand inHigh-Strength Concrete,” PhD thesis, Polytechnic University of Valencia,ProQuest Information and Learning Co., UMI number 3041710, Mich.,2003, 355 pp. (in Spanish)

34. Martí-Vargas, J. R.; Serna-Ros, P.; Fernández-Prada, M. A.; Miguel-Sosa, P. F.; and Arbeláez, C. A., “Test Method for Determination of theTransmission and Anchorage Lengths in Prestressed Reinforcement,”Magazine of Concrete Research, V. 58, No. 1, Feb. 2006, pp. 21-29.

35. Comité Européen de Normalisation, “Cement. Part 1: Compositions,Specifications and Conformity Criteria for Common Cements,” EuropeanStandard EN 197-1:2000, CEN, Brussels, Belgium, 2000, 30 pp.

36. Asociación Española de Normalización y Certificación, “UNE36094: Steel Wire and Strand for Prestressed Concrete,” AENOR, Madrid,Spain, 1997, 21 pp.

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ACI Structural Journal, V. 104, No. 4, July-August 2007.MS No. S-2006-222 received June 1, 2006, and reviewed under Institute publication

policies. Copyright © 2007, American Concrete Institute. All rights reserved, includingthe making of copies unless permission is obtained from the copyright proprietors. Pertinentdiscussion including author’s closure, if any, will be published in the May-June 2008ACI Structural Journal if the discussion is received by January 1, 2008.

ACI STRUCTURAL JOURNAL TECHNICAL PAPER

Two-way slabs without beams are popular floor systems because oftheir relatively simple formwork and the potential for shorter storyheights. Earthquakes, however, have demonstrated that slab-columnframes are vulnerable to brittle punching shear failures in the slab-column connection region and dropping of the slab, which arecostly to repair. This paper focuses on the behavior and design ofslab-column connections under combined gravity and lateral loadingand reviews current design procedures, performance-based designapproaches, and relevant experimental data. An equation relatingthe gravity shear ratio at a slab-column connection to drift capacity ispresented. Finally, practical recommendations are provided fordefining specific performance objectives.

Keywords: deformation capacity; effective slab width; performance-baseddesign; punching shear.

INTRODUCTIONTwo-way slabs without beams are popular floor systems

because of their relatively simple formwork and the potentialfor shorter story heights due to their shallow profile. Thisstructural system is common in regions of low to moderateseismic risk, where it is allowed as a lateral-force-resistingsystem (LFRS), as well as in regions of high seismic risk forgravity systems where moment frames or shear walls areprovided as the main LFRS. Earthquakes, however, havedemonstrated that slab-column frames are not suitable as a mainLFRS in regions of high seismic risk because they are relativelyflexible and because of the potential for brittle punching shearfailures in the slab-column connection region.

In the last 40 years, a significant number of experimentshave been conducted to evaluate the performance of slab-column connections under cyclic lateral loading. This infor-mation has formed the basis of current code provisions andguidelines for the design of slab-column connections undercombined gravity and lateral loading. As performance-basedseismic design (PBSD) becomes more common in structuralengineering practice, it is important to evaluate therecommended limits for various structural systems withrespect to the latest experimental data and post-earthquakeobservations. This paper focuses on the behavior anddesign of interior slab-column connections under combinedgravity and lateral loading and serves to review current designprocedures, PBSD approaches, and relevant experimental data.Equation (23), for drift capacity of these systems in terms ofthe gravity shear ratio, is derived using the collected experi-mental data. Finally, practical recommendations are providedfor the PBSD of slab-column connections.

RESEARCH SIGNIFICANCEThe objectives of this paper, developed by a task group

within ACI Committee 374, Performance-Based SeismicDesign of Concrete Buildings, are: 1) to review the current stateof practice and PBSD approaches for slab-column connections;2) to summarize experimental data for slab-column connectionstested under combined gravity and lateral loads; and 3) to

present a practical approach for PBSD of slab-columnconnections. The PBSD material is presented in a formatconsistent with the limit states suggested in FEMA 356(ASCE 2000) and is intended to provide guidance primarily fornew construction. The criteria, however, could also be appliedto existing structures that contain subpar seismic details wherea moderate seismic demand is expected. As a significant benefitfor design approaches outside the PBSD framework, a practicalequation that relates drift capacity to gravity shear ratio ispresented (Eq. (23)).

SLAB-COLUMN FRAMES AND CONNECTIONSSlab-column frame construction can deliver several

desirable architectural features, including larger open space,lower building heights for a given number of stories, andefficient construction. The FEMA 356, “Prestandard andCommentary for the Seismic Rehabilitation of Buildings”(ASCE 2000) classifies slab-column moment frames asframes that meet the following conditions:

1. Framing components shall be slabs (with or without beamsin the transverse direction), columns, and their connections;

2. Frames shall be of monolithic construction that providesfor moment transfer between slabs and columns; and

3. Primary reinforcement in slabs contributing to lateralload resistance shall include nonprestressed reinforcement,prestressed reinforcement, or both.

This classification includes both frames that are or are notintended to be part of the LFRS for new, existing, andrehabilitated structures.

The connections between the slab and a column can beaccomplished in several ways including direct connection(whether from solid or waffle slab construction), withcolumn drop panels, and with column or shear capitals.Shear capitals are provided to increase the shear capacity atthe slab-column connection and are defined by Joint ACI-ASCE Committee 352 (1989) as a thickened portion of theslab around a column that does not meet the ACI 318 plandimension requirements for drop panels. A column capital isdefined as a flared portion of the column below the slab thatis cast monolithically with the slab.

Slab-column connections in structures subjected toearthquake or wind loading must transfer forces due to bothgravity and lateral loads. This combination can create largeshear and unbalanced moment demands at the connection.Without proper detailing, the connection can be susceptibleto two-way (punching) shear failure during response tolateral loads. The flexibility of a slab-column frame can leadto large lateral deformations, which may increase the potential

Title no. 104-S43

Seismic Design Criteria for Slab-Column Connectionsby Mary Beth D. Hueste, JoAnn Browning, Andres Lepage, and John W. Wallace

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for punching failures; therefore, in regions of high seismicrisk, slab-column frames are used in conjunction with beam-column moment frames or shear walls. Compatibility oflateral deformations between the slab-column frame and theLFRS, however, must be considered to determine thedemands on the connections.

The seismic performance of reinforced concrete structureswith flat-slab construction has demonstrated the vulnerabilitiesof the system. For example, following the 1985 Mexico Cityearthquake, punching shear failures were noted in a 15-storybuilding with waffle flat-plate construction (Rodriguez andDiaz 1989). This failure was partly attributed to a highflexibility combined with low-ductility capacities of thewaffle slab-to-column connection. In a department storeduring the 1994 Northridge earthquake, discontinuousflexural reinforcement at slab-column connections led topunching failures at column drop panels (Holmes andSomers 1996). Punching failures around shear capitals werealso noted in the post-tensioned floor slabs of a four-storybuilding during the same event (Hueste and Wight 1997).

CURRENT DESIGN APPROACHGeneral

The shear strength of slabs in the vicinity of columns isgoverned by the more severe of two conditions, either beamaction or two-way action. In beam action, the slab acts as awide beam with the critical section for shear extendingacross the entire width of the slab. This critical section isassumed to be located at a distance d (effective slab depth)from the face of the column or shear capital. For this condition,conventional beam theory applies and will not be discussedin detail herein. For the condition of two-way action, the criticalsection is assumed to be located at a distance d/2 from theperimeter of the column or shear capital, with potential diagonaltension cracks occurring along a truncated cone or pyramidpassing through the critical section (refer to Fig. 1, where d1

is the effective slab depth within the thickened shear capitalregion and d2 is the effective slab depth).

Existing methods for calculating the shear strength of slab-column connections include applications of elastic platetheory, beam analogies, truss analogies, strip design methods,and others. The design method specified by ACI 318-05(ACI Committee 318 2005) provides acceptable estimates ofshear strength with reasonable computational effort. Theprocedure is based on the results of a significant number ofexperimental tests involving slab-column specimens.

The eccentric shear stress model is the basis of the generaldesign procedure embodied in ACI 318 for determining theshear strength of slab-column connections transferring shearand moment. The model was adopted by the 1971 version ofthe ACI 318 and only minor modifications have beenincluded in subsequent versions. Recently, ACI 318-05 hasincorporated special provisions related to the lateral-loadcapacity of slab-column connections in structures located inregions of high seismic risk or structures assigned to highseismic performance or design categories.

The design approach presented in this section of the paper isbased on the design procedures given in ACI 318-05 comple-mented by ACI 421.1R-99 (Joint ACI-ASCE Committee 4211999) and 352.1R-89 (Joint ACI-ASCE Committee 352 1989).

ACI 318 eccentric shear stress modelSlab-column connections experience very complex

behavior when subjected to lateral displacements or unbalancedgravity loads. This involves transfer of flexure, shear, andtorsion in the portion of the slab around the column.Combined flexural and diagonal cracking are coupled withsignificant in-plane compressive forces in the slab inducedby the restraint of the surrounding unyielding slab portions.

Relatively simple design equations have been derived byconsidering the critical section to be located at d/2 awayfrom the face of the column and by assuming that shear stresson the critical perimeter varies linearly with distance fromthe centroidal axis. This eccentric shear stress model is basedon the work by DiStasio and Van Buren (1960) and reviewedby Joint ACI-ASCE Committee 326 (1962).

For a slab-column connection transferring shear andmoment, the ACI 318-05 design equations for limiting theshear stresses vu are given by

vu ≤ φvn (1)

vu = ± (2)Vu

bod--------

γvMuc

J---------------

ACI member Mary Beth D. Hueste is an Associate Professor in the Department ofCivil Engineering at Texas A&M University, College Station, Tex. She is a member ofACI Committees 374; Performance-Based Seismic Design of Concrete Buildings; 375,Performance-Based Design of Concrete Buildings for Wind Loads; E803, FacultyNetwork Coordinating Committee; and Joint ACI-ASCE Committee 352, Joints andConnections in Monolithic Concrete Structures. Her research interests includeearthquake-resistant design of reinforced concrete structures, structural rehabilitationincluding seismic retrofitting, performance-based seismic design, and design andevaluation of prestressed concrete bridge structures.

ACI member JoAnn Browning is an Associate Professor in the Department of Civil,Environmental, and Architectural Engineering at the University of Kansas, Lawrence,Kans. She is a member of ACI Committees 314, Simplified Design of Concrete Buildings;318-D, Flexure and Axial Loads; Beams, Slabs, and Columns; 341, Earthquake-Resistant Concrete Bridges; 374, Performance-Based Seismic Design of ConcreteBuildings; and 408, Bond and Development of Reinforcement. Her research interestsinclude the performance of reinforced concrete structures under seismic loads,design and analysis of concrete structures, and durability of concrete structures.

Andres Lepage, FACI, is an Assistant Professor in the Department of ArchitecturalEngineering at Pennsylvania State University, University Park, Pa. He is a member ofACI Committees 318-H, Seismic Provisions; 335, Composite and Hybrid Structures;369, Seismic Repair and Rehabilitation; 374, Performance-Based Seismic Design ofConcrete Buildings; and 375, Performance-Based Design of Concrete Buildings forWind Loads. His research interests include the design of concrete, steel, and hybridstructural systems subjected to extreme events.

John W. Wallace, FACI, is a Professor of civil engineering at the University ofCalifornia-Los Angeles, Los Angeles, Calif. He is a member of ACI Committee 318-H,Seismic Provisions; 335, Composite and Hybrid Structures; 369, Seismic Repair andRehabilitation; 374, Performance-Based Seismic Design of Concrete Buildings;E803, Faculty Network Coordinating Committee; and Joint ACI-ASCE Committee352, Joints and Connections in Monolithic Concrete Structures. His research interestsinclude response and design of buildings and bridges to earthquake actions, laboratoryand field testing of structural components and systems, and structural health monitoringand use of sensor networks.

Fig. 1—Critical sections for two-way shear for interiorslab-column connection with shear capital.

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450 ACI Structural Journal/July-August 2007

where vu is the factored shear stress; φ is the strength reductionfactor for shear; vn is the nominal shear stress; Vu is thefactored shear force acting at the centroid of the criticalsection; Mu is the factored unbalanced bending momentacting about the centroid of the critical section; d is thedistance from the extreme compression fiber to the centroidof the longitudinal tension reinforcement; bo is the length ofthe perimeter of the critical section; c is the distance from thecentroidal axis of the critical section to the point where shearstress is being computed; J is a property of the criticalsection analogous to the polar moment of inertia; and γv isthe fraction of the unbalanced moment considered to be trans-ferred by eccentricity of shear, defined by

(3)

where b1 and b2 are the widths of the critical section measuredin the direction of the span for which Mu is determined(Direction 1) and in the perpendicular direction (Direction 2).

For an interior column and a critical section of rectangularshape, bo and J are determined by

bo = 2(b1 + b2) (4)

(5)

The first term of Eq. (2), the shear stresses due to directshear, is assumed uniformly distributed on the criticalsection, and the fraction γvMu is assumed to be resisted bylinear variation of shear stresses on the critical section. Theportion of the moment not carried by eccentric shear is to becarried by slab flexural reinforcement placed within lines 1.5h oneither side of the column (h is the slab thickness, including droppanel, if any). This flexural reinforcement is also used to resistslab design moments within the column strip.

The provisions of the ACI 318 specify that in absence of shearreinforcement, the nominal shear strength (in stress units)carried by the concrete vc in nonprestressed slabs is given by

(6)

γv 1 1

1 23---

b1

b2

-----+

----------------------–=

Jdb1

3

6--------

b1d3

6-----------

db2b12

2--------------+ +=

vc min

4 fc′ (psi)

2 4βc

-----+⎝ ⎠⎛ ⎞ fc′ (psi)

αsd

bo

--------- 2+⎝ ⎠⎛ ⎞ fc′ (psi)

⎩⎪⎪⎪⎨⎪⎪⎪⎧

=

or vc min =

0.33 fc′ (MPa)

0.17 1 2βc

-----+⎝ ⎠⎛ ⎞ fc′ (MPa)

0.083αsd

bo

--------- 2+⎝ ⎠⎛ ⎞ fc′ (MPa)

⎩⎪⎪⎪⎨⎪⎪⎪⎧

For prestressed slabs without shear reinforcement, Eq. (6)is replaced by

(7)

where αs equals 40, 30, and 20 for interior, edge, andcorner columns, respectively; bo and d are defined previously;βc is the ratio of long side to short side of column; fc′ is thespecified concrete compressive strength (psi units); fpc is theaverage compressive stress in two vertical slab sections inperpendicular directions, after allowance for all prestresslosses; and Vp is the vertical component of all effectiveprestress forces crossing the critical section.

The use of Eq. (7) is restricted to cases where fc′ is less than5000 psi (35 MPa); fpc ranges between 125 and 500 psi (0.9 and3.5 MPa) in each direction; and no portion of the columncross section is closer than four times the slab thickness to adiscontinuous edge. If these conditions are not satisfied, the slabshould be treated as nonprestressed and Eq. (6) applies.

When vu > φvn, the slab shear capacity can be increased by: (a)thickening the slab in the vicinity of the column with acolumn capital, shear capital, or drop panel; (b) addingshear reinforcement; (c) increasing the specified compressivestrength of concrete; or (d) increasing the column size. In a flatslab with shear capitals or drop panels, stresses must be checkedat all critical locations—both at the thickened portion of the slabnear the face of the column and at the section outside the shearcapital or drop panels (refer to Fig. 1).

Shear reinforcement, which can be in the form of bars orwires and single- or multiple-leg stirrups properly anchored,increases both the shear strength and the ductility of theconnection when transferring moment and shear. Shearreinforcement consisting of structural steel shapes (shearheads)is also effective in increasing the shear strength and ductilityof slab-column connections. Design procedures for shear-head reinforcement are presented in Corley and Hawkins(1968) and are not discussed in this paper. For members withshear reinforcement other than shearheads, the nominalshear strength (in stress units) is calculated using

(8)

(9)

(10)

vc min 3.5 fc′ (psi) 0.3fpc

Vp

bod--------+ +

αsd

bo--------- 1.5+⎝ ⎠⎛ ⎞ fc′ (psi) 0.3fpc

Vp

bod--------+ +

⎩⎪⎪⎨⎪⎪⎧

=

vc min 0.29 fc′ (MPa) 0.3fpc

Vp

bod--------+ +

0.083αsd

bo--------- 1.5+⎝ ⎠⎛ ⎞ fc′ (MPa) 0.3fpc

Vp

bod--------+ +

⎩⎪⎪⎨⎪⎪⎧

=

vc vc vs 6 fc′ (psi) or 0.5 fc′ (MPa)≤+=

vc 2 fc′ (psi) or 0.17 fc′ (MPa)=

vsAv fyv

bos------------=

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ACI Structural Journal/July-August 2007 451

where vs is the nominal shear stress provided by shearreinforcement; Av is the area of shear reinforcement; fyv isthe specified yield strength of shear reinforcement; s is thespacing of shear reinforcement; and vc, fc′ , and bo are definedpreviously.

When lightweight aggregate concrete is used, the value of in Eq. (6) through (9) is multiplied by 0.75 for all

lightweight concrete or by 0.85 for sand-lightweightconcrete. The extent of the shear-reinforced zone isdetermined to ensure that punching shear failure does not occurimmediately outside this region for the design actions.

The nominal ultimate concrete shear stress along the criticalsection acting with shear reinforcement is taken as

because at approximately thisstress, diagonal tension cracks begin to form and cracking isneeded to mobilize the shear reinforcement. The shearreinforcement or shear capital must be extended for a sufficientdistance until the critical section outside the reinforcedregion satisfies Eq. (9). In nonprestressed slabs, the maximumspacing of shear reinforcement is 0.5d. In prestressed slabs,the spacing of shear reinforcement is allowed to reach 0.75hbut not to exceed 24 in. (0.61 m).

For both prestressed and nonprestressed slabs, ACI 318mandates continuity reinforcement to give the slab someresidual capacity following a single punching shear failure ata single support. Thus, in nonprestressed slabs, all bottombars within the column strip shall be continuous and at leasttwo of the column strip bottom bars in each direction shallpass through the column core (ACI Committee 318 2005,Section 13.3.8.5). In prestressed slabs, a minimum of twotendons shall be provided in each direction through thecritical shear section over columns (ACI Committee 3182005, Section 18.12.4).

ACI 421.1R-99 refinementsACI 318 sets out the principles of design for slab shear

reinforcement but does not make specific reference tomechanically anchored shear reinforcement, also referred toas shear studs. ACI 421.1R-99 (Joint ACI-ASCE Committee421 1999) gives recommendations for the design of shearreinforcement using shear studs in slabs. This report alsoincludes equations for calculating shear stresses onnonrectangular critical sections.

Shear studs have proven to be effective in increasingthe strength and ductility of slab-column connections.ACI 421.1R-99 suggests treating a shear stud as the equivalentof a vertical branch of a stirrup and to use higher limits onsome of the design parameters used in ACI 318. In particular,ACI 421.1R-99 suggests higher allowable values for vn, vc,s, and fyv, as follows

(11)

(12)

(13)

fc′

2 fc′ (psi) 0.17 fc′ [MPa]( )

vc vc vs 8 fc′ (psi) or 0.66 fc′ (MPa)≤+=

vc 3 fc′ (psi) or 0.25 fc′ (MPa)=

s0.75d when

vu

φ---- 6≤ fc′ (psi) or 0.5 fc′ (MPa)

0.5d when vu

φ---- 6 fc′ (psi)> or 0.5 fc′ (MPa)

⎩⎪⎪⎨⎪⎪⎧

fyv ≤ 72,000 psi (500 MPa) (14)

The justification for these higher values is mainly due tothe almost slip-free anchorage of the studs and that themechanical anchorage at the top and bottom of the stud iscapable of developing forces in excess of the specified yieldstrength at all sections of the stud stem.

ACI 352.1R-89 recommendationsACI 352.1R-89 (Joint ACI-ASCE Committee 352 1989)

includes recommendations for the determination of connectionproportions and details to ensure adequate performance ofmonolithic, reinforced concrete slab-column connections. Therecommendations address connection strength, ductility, andstructural integrity for resisting gravity and lateral forces.

ACI 352.1R-89 only applies to nonprestressed slab-column connections with fc′ less than 6000 psi (42 MPa),with or without drop panels or shear capitals, and withoutslab shear reinforcement. The provisions are limited toconnections where severe inelastic load reversals are notexpected, and do not apply to slab-column connections thatare part of a primary LFRS in regions of high seismic riskbecause slab-column frames are generally considered to beinadequate for multi-story buildings in these areas.

ACI 352.1R-89 classifies slab-column connections as oneof two types: 1) Type 1—connections not expected toundergo deformations into the inelastic range; and 2) Type 2—connections requiring sustained strength under moderatedeformations into the inelastic range. In structures subjectedto high winds or seismic loads, a slab-column connectionshould be classified as Type 2 even though it is not designatedas part of the primary LFRS.

To ensure a minimal level of ductility, ACI 352.1R-89references the work by Pan and Moehle (1989) andrecommends that for all Type 2 connections—withoutshear reinforcement—the direct factored shear Vu acting onthe connection, for which inelastic moment transfer isanticipated, must satisfy

Vu ≤ 0.4Vc = 0.4vcbod (15)

where vc is determined by either Eq. (6) or (7). The limitation defined by Eq. (15) was based on a review

of test data that revealed that the deformation capacity ofinterior connections without shear reinforcement is inverselyrelated to the direct shear on the connection. Connections notcomplying with Eq. (15) exhibit virtually no post-yielddeformation capacity under lateral loading. Pan and Moehle(1989) found that when the stress due to direct shearapproaches 0.4vc, the connection experiences a brittle failurefor story drift ratios of approximately 1.5%. No additionalstatements are made in ACI 352.1R-89 regarding othercombinations of shear stress and story drift ratio. The reportstates that Eq. (15) may be waived if calculations demonstratethat the imposed displacement will not induce yield in theslab system. For example, the use of structural walls mayadequately limit the imposed drifts on slab-column framessuch that yield at the slab-column connection may not occur.

The approach by ACI 352.1R-89 suggests that the deformationcapacity of slab-column connections may be defined as afunction of the shear stress due to direct shear only. Thisapproach has been developed further by Moehle (1996) andMegally and Ghali (2000). ACI 318-05 has incorporated this

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452 ACI Structural Journal/July-August 2007

concept into a general approach for addressing the deformationcapacity of slab-column connections not designated as partof the LFRS.

Requirements of ACI 318-05, Section 21.11.5Model building codes (SEI/ASCE 2005) have deformation

compatibility requirements for members that are not designatedas part of the LFRS. These members should be able to resistthe gravity loads at lateral displacements corresponding tothe design level earthquake. ACI 318-05, Section 21.11.5,has incorporated a design provision to account for thedeformation compatibility of slab-column connections.

Instead of calculating the induced effects under the designdisplacement, ACI 318-05 describes a prescriptive approach.The connection is evaluated based on a simple relationshipbetween the design story drift ratio (DR) and the shear stressdue to factored gravity loads. The design DR (story driftdivided by story height) should be taken as the largest valuefor the adjacent stories above and below the connection. Themaximum DR (in percent) that a slab-column connection cantolerate, in the absence of shear reinforcement, is given bythe following relationship and illustrated in Fig. 2.

(16)

where VR is the shear ratio, defined as

DR 3.5 5.0VR– for VR 0.6<( )0.5 for VR 0.6≥( )⎩

⎨⎧

=

(17)

The term vc is calculated using Eq. (6) or (7). The factoredshear force Vu on the slab critical section for two-way actionis determined for the load combination 1.2D + 1.0L + 0.2S,where D, L, and S are the dead, live, and snow loads.

If the DR exceeds the limit given by Eq. (16), shearreinforcement must be provided (or the connection can beredesigned). When adding shear reinforcement, ACI 318-05prescribes that the term vs, defined by Eq. (10), must exceed

and the shear reinforcementmust extend at least four times the slab thickness from theface of the support. Given that this approach is relatively simple,and that the added cost of providing shear reinforcement atconnections is not significant for structures designed forhigh seismic performance categories, use of this prescriptiveapproach is likely to be common. The representative designsteps are shown in Fig. 3.

If shear capitals, column capitals, or drop panels are used,all potential critical sections must be investigated. ACI 318-05does not prescribe a minimum extension of shear capitals.Wey and Durrani (1992), however, recommend a minimumlength equal to two times the slab thickness from the face ofthe column.

ANALYTICAL MODELINGThe shear stresses due to the combined factored shear and

moment transferred between the slab and the column underthe design displacement can be determined by creating anappropriate analytical model of the slab-column frame anddirectly assessing the potential for punching. Recommendationsby Hwang and Moehle (2000) may be used to establish theeffective stiffness of the slab and to include the impact ofcracking. Hwang and Moehle (2000) recommend that theuncracked effective stiffness for a model with rigid joints,for ratios of c2/c1 from 1/2 to 2 and a slab aspect ratio l2/l1greater than 2/3, be determined using an effective beamwidth represented as

(18)

where bint is the effective width for interior frame connections(interior connections and edge connections with bendingperpendicular to the edge); c1 and l1 are the column dimensionand slab span parallel to the direction of load being considered;and c2 and l2 correspond to the orthogonal direction. Forexterior frame connections (corner connections and edgeconnections with bending parallel to the edge), half the widthdefined in Eq. (18) is used. Effects of cross section changes,such as slab openings, are to be considered. One way toaccomplish this is to vary the width of the effective beamalong the span (Hwang and Moehle 1990).

To account for cracking, a stiffness reduction factor β hasbeen proposed by Hwang and Moehle (2000) for nonpre-stressed slabs and is given by

(19)

VRVu

φvcbod-----------------=

3.5 fc′ (psi) 0.29 fc′ [MPa]( )

bint 2c1l1

3---+=

β 4cl-- 1

3--->=

Fig. 2—ACI 318-05 relationship for determining adequacyof slab-column connections in seismic regions.

Fig. 3—Design steps when adding shear reinforcement.

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ACI Structural Journal/July-August 2007 453

where c and l are the column dimension and slab spanparallel to the load direction. Kang and Wallace (2005)recommend β = 0.5 for post-tensioned floor systems withapproximate values for span-to-slab thickness ratios of 40,c1/l1 of 1/14, and precompression of 200 psi (1.4 MPa).

The analytical model of the slab-column frame shouldcapture the potential for both slab yielding and connectionfailure due to punching as recommended in FEMA 356.Figure 4 shows an approach where yielding within the slabcolumn strip is modeled using slab-beam elements (in thiscase, an elastic slab-beam with stiffness properties definedby the effective beam width model, and zero-length plastichinges on either side of the connection). Further details ofthis model are described by Kang et al. (2006). Punchingfailures can occur if the capacity of the connection elementis reached or if a limiting story drift ratio is reached for agiven gravity shear ratio. Hueste and Wight (1999)suggested an approach for incorporating this behavior into anonlinear analysis program, where, after prediction of apunching shear failure, the member behavior is modified toaccount for the significant reduction in stiffness andstrength. Kang and Wallace (2005) suggest a direct approachby employing a limit state model.

The FEMA 356 guidelines note that the analytical model fora slab-column frame should consider all potential failuresincluding flexure, shear, shear-moment transfer, and reinforce-ment development at any section. The modeling informationmentioned previously gives a convenient and relativelystraightforward approach to modeling the behavior of slab-column frames for nonlinear static and dynamic analysis.

PERFORMANCE-BASED DESIGN CRITERIAA review of current practice with respect to performance-based

design is needed to provide context to the material presentedsubsequently on performance objectives for slab-columnconnections. The FEMA 356 prestandard (ASCE 2000)provides analytical procedures and criteria for the performance-based evaluation of existing buildings and for designingseismic rehabilitation alternatives. This prestandard includesrecommended limits for deformation capacities based on thecalculated gravity shear ratio, as well as a general frameworkfor creating performance levels and objectives.

In FEMA 356, performance levels describe limitations onthe maximum damage sustained during a ground motion,while performance objectives define the target performancelevel to be achieved for a particular intensity of groundmotion. Structural performance levels in FEMA 356 includeimmediate occupancy, life safety, and collapse prevention.Structures at collapse prevention are expected to remainstanding, but with little margin against collapse. Structures atlife safety may have sustained significant damage, but stillprovide an appreciable margin against collapse. Structures atimmediate occupancy should have only minor damage. InFEMA 356, the Basic Safety Objective is defined as lifesafety-performance for the basic safety earthquake 1 (BSE-1)earthquake hazard level and collapse prevention performancefor the BSE-2 earthquake hazard level. BSE-1 is the smallerevent corresponding to 10% probability of exceedance in50 years (10% in 50 years) and 2/3 of the BSE-2 (2% in50 years) event.

For a given design event and a target performance level,FEMA 356 provides acceptance criteria when using eitherstatic or dynamic analysis based on linear and nonlinearprocedures. To evaluate acceptability using linear procedures,

an action is classified as either deformation-controlled orforce-controlled. Deformation-controlled actions are applicablefor components that have the capacity to undergo deformationsinto the inelastic range without failure. Based on the demandto capacity ratio (DCR), calculated using the linear static ordynamic analysis procedures, components are classified ashaving low (DCR < 2), moderate (2 ≤ DCR ≤ 4), or high(DCR > 4) ductility demands.

The acceptance criteria based on linear analysis proceduresare expressed in terms of m-factors. The factor m is intended toprovide an indirect measure of the total deformationcapacity of a structural element or component. As such,the factor m is only used to evaluate the acceptability ofdeformation-controlled actions

mκQCE ≥ QUD (20)

where κ is the knowledge factor used to reduce the strength ofexisting components based on quality of information, QCE is theexpected strength of a component or element at the deformationlevel considered, and QUD is the deformation-controlleddesign action. Equation (20) can be rearranged for directcomparison of the DCR to m to determine acceptability

DCR ≤ m = (21)

The FEMA 356 limiting values for m-factors for two-wayslabs and slab-column connections are provided in Table 1.The m-factors for slab-column connections range from 1 to4 and depend on several parameters: the gravity-shear ratio,the presence of continuity reinforcement through the columncage, the development of reinforcement, and the selectedperformance level. The connections must also be classifiedas primary or secondary elements to determine the limitsfor life safety and collapse prevention. Secondaryelements are those typically not considered to provideresistance to earthquake effects.

For nonlinear static and dynamic analysis procedures,FEMA 356 restricts inelastic response values determinedfrom the analytical model in terms of maximum plasticrotations. Generally, plastic rotation is computed as the

QUD

κQCE

-------------

Fig. 4—Modeling of slab-column connection (adapted fromKang et al. 2006).

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454 ACI Structural Journal/July-August 2007

difference between the maximum rotation during analysisand the yield rotation at the member end. Therefore, it iscritical for the nonlinear model to represent the maximumplastic rotation for a certain level of demand. The plasticrotation limits in FEMA 356 range from 0.0 to 0.02 radiansfor primary slab-column connections and from 0.0 to 0.05radians for secondary slab-column connections. These limits arebased on the gravity-shear ratio, the presence of continuityreinforcement through the column cage, the developmentof reinforcement, and the selected performance level (immediateoccupancy, life safety, or collapse prevention).

EXPERIMENTAL DATAOver the past 40 years, experimental studies have been

conducted by researchers at a number of universities. Muchof the earlier data has been summarized by Pan and Moehle(1989), Megally and Ghali (1994), and Luo and Durrani(1995). Tables 2 and 3 provide information on interior slab-column connection test specimens, with and without shearreinforcement. Limited tests have been conducted fornonductile slab-column connections where the bottom slabreinforcement is discontinuous at the interior slab-columnconnection (Durrani et al. 1995; Dovich and Wight 1996;Robertson and Johnson 2006) and available data is includedin Table 2. The failure mode for each specimen is provided,when available, as either: punching shear P, flexure F, or a

combination of flexure and punching shear (F-P) where apunching shear failure occurred at a higher drift levelfollowing yielding of the slab reinforcement. The gravityshear ratio and peak drift are also provided for each specimen.The peak drift is defined as the drift corresponding to thepeak lateral load. Therefore, the maximum drift attained for aparticular specimen may be larger than the reported peak drift.

The maximum drift at which an interior connection willfail can be estimated from the gravity shear ratio Vg/Vo (Panand Moehle 1989; Luo and Durrani 1995). The gravity shearratio represents the unfactored vertical gravity shear Vgdivided by the theoretical punching shear strength withoutmoment transfer Vo determined using

Vo = vcbo d (22)

The term vc is calculated using Eq. (6) or (7). A similarratio can be computed for slabs with shear reinforcement byreplacing vc with vn defined by Eq. (8) through (10).

Figure 5 provides a plot of peak drift as a function of Vg/Vo forinterior slab-column connection specimens with no shearreinforcement. The figure shows the direct influence of thegravity shear ratio on the lateral drift capacity of slab-column connections. It may be observed that punching shearoccurs for a large range of Vg/Vo values (approximately 0.1 to0.9), while flexural failures primarily occur for Vg/Vo valuesof 0.3 or less.

Figure 6 provides a similar plot for interior slab-columnconnection specimens with shear reinforcement. Theexperimental data indicates that larger drift ratios are possiblewhen shear reinforcement is used. In particular, a number ofslab-column specimens with stud-shear reinforcement (SSR)attained story drift ratios well over 3% before failure.

The data from slab-column connection tests, with andwithout shear reinforcement, are compared in Fig. 7, along

Table 1—Acceptance criteria for linear procedures—two-way slabs and slab-column connections (adapted from FEMA 356 [ASCE 2000])

Conditions

m-factors by performance level*

IO

Component type

Primary Secondary

LS CP LS CP

1. Slab controlled by flexure and slab-column connections†

Vg/Vo‡ Continuity

reinforcement§

≤ 0.2 Yes 2 2 3 3 4

≥ 0.4 Yes 1 1 1 2 3

≤ 0.2 No 2 2 3 2 3

≥ 0.4 No 1 1 1 1 1

2. Slabs controlled by inadequate development or splicing along span†

— — — 3 4

3. Slabs controlled by inadequate embedment into slab-column joint†

2 2 3 3 4*IO = immediate occupancy; LS = life safety; and CP = collapse prevention.†When more than one of Conditions 1, 2, and 3 occurs for given component, useminimum appropriate numerical value from table.‡Vg = gravity shear acting on slab critical section and Vo = direct punching shearstrength as defined by ACI 318.§Under heading “Continuity reinforcement,” use “Yes” where at least one of the mainbottom bars in each direction is effectively continuous through column cage. Wherethat slab is post-tensioned, use “Yes” where at least one of post-tensioning tendons ineach direction passes through column cage. Otherwise, use “No.”

Fig. 5—Test data for interior slab-column connectionspecimens with no shear reinforcement.

Fig. 6—Test data for interior slab-column connectionspecimens with shear reinforcement.

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455ACI Structural Journal/July-August 2007

Table 2—Test data for interior slab-column connection test specimens with no shear reinforcement

Source Label Vg/Vo Peak drift, % Mode Source Label Vg/Vo Peak drift, % Mode

Dilger and Cao (1991)

CD 1 0.85 0.90 NA

Luo and Durrani (1995)

I.I 0.08 5.00 F

CD 2 0.65 1.20 NA INT1 0.43 NA P

CD 8 0.52 1.40 NAINT2 0.50 NA P

Durrani et al. (1995)

DNY 1* 0.20 3.00 F

DNY 2* 0.30 2.00 P

Megally and Ghali (2000)

MG-2A 0.58 1.17 P

DNY 3* 0.24 2.00 F MG-7 0.29 3.10 F-P

DNY 4* 0.28 2.60 F-P MG-8 0.42 2.30 F-P

Elgabry and Ghali (1987) 1 0.46 NA P MG-9 0.36 2.17 F-P

Farhey et al. (1993)

1 0.00 4.81 F

Morrison and Sozen (1983)

S1 0.03 4.70 F

2 0.00 4.04 F S2 0.03 2.80 F

3 0.26 3.56 P S3 0.03 4.20 F

4 0.30 2.40 P S4 0.07 4.50 F

Ghali et al. (1976)

SM 0.5 0.31 6.00 F S5 0.15 4.80 F

SM 1.0 0.33 2.70 F-P

Pan and Moehle (1989)

AP 1 0.37 1.60 F-P

SM 1.5 0.30 2.70 F-P AP 2 0.36 1.50 F-P

Hanson and Hanson (1968)

A12 0.29 NA P AP 3 0.18 3.70 F-P

A13L 0.29 NA P AP 4 0.19 3.50 F-P

B16 0.29 NA P

Pan and Moehle (1992)

1 0.35 1.50 P

B7 0.04 3.80 F-P 2 0.35 1.50EW/0.79NS P

C17 0.24 NA F-P 3 0.22 3.10 F-P

C8 0.05 5.80 F 4 0.22 3.20EW/1.75NS P

Hawkins et al. (1974)

S1 0.33 3.75 P

Robertson and Durrani (1990)

1 0.21 2.75 F

S2 0.45 2.00 P 2C 0.22 3.50 F-P

S3 0.45 2.00 P 3SE 0.19 3.50 F

S4 0.40 2.60 P 5SO 0.21 3.50 F

Hwang and Moehle (1990) 4 Int. Joints 0.24 4.00 NA 6LL 0.54 0.85 P

Islam and Park (1976)

1 0.25 3.67 P 7L 0.40 1.45 P

2 0.23 3.33 P 8I 0.18 3.50 F-P

3C 0.23 4.00 F-P Robertson et al. (2002) 1C 0.17 3.50 P

Robertson and Johnson (2006)

ND1C* 0.23 3.00 to 5.00 F-PSymonds et al. (1976)

S6 0.86 1.10 P

ND4LL* 0.28 3.00 F-P S7 0.81 1.00 P

ND5XL* 0.47 1.50 P

Wey and Durrani (1992)

SC 0 0.25 3.50 P

ND6HR* 0.29 3.00 P SC 2 0.18 6.00 F

NC7LR* 0.26 3.00 F-P SC 4 0.15 6.00 F

ND8BU* 0.26 3.00 F-PSC 6 0.15 5.00 P

Zee and Moehle (1984) INT 0.21 3.30 F-P*Bottom slab reinforcement is discontinuous at interior connection.Note: EW = east-west lateral load for biaxial test; NS = north-south lateral load for biaxial test; F = flexural failure; P = punching shear failure; and F-P = flexural and punchingshear failure. NA: Not available.

with the ACI 318-05 limits for assessing the need for shearreinforcement. The line defined by ACI 318-05 is a reasonablelower-bound limit for the data corresponding to specimenswithout shear reinforcement. A strength reduction factorof φ = 1 is used when determining Vg/Vo for the test data.

PERFORMANCE-BASED SEISMICDESIGN RECOMMENDATIONS

Research studies and past structural performance haveshown that slab-column frames provide lateral stiffnesscontributions to the overall LFRS and, as such, they do resistlateral loads during a seismic event even if they weredesigned for gravity loads only. For this reason, compatibility ofdeformations must be considered to calculate the demands atthe slab-column connections. Likewise, the analytical modelshould include the strength and stiffness of the slab-column

frames to ensure an accurate representation of the overallbuilding stiffness and allow an evaluation of the magnitudeof the lateral load that must be resisted by the slab-columnframe members. The appropriate parameters that should beincluded in such a model were highlighted previously (effectiveslab width for equivalent beams, cracked section properties,and hysteretic behavior for nonlinear models).

Performance-based seismic design (PBSD) criteria aresuggested in the following. The criteria are based onexperimental data of interior slab-column connections undercombined gravity and lateral load. The suggested criteriareference FEMA 356 performance levels (immediateoccupancy, life safety, and collapse prevention) and seismicdesign requirements for slab-column connections that areadopted in ACI 318-05. As noted previously, in regions ofhigh seismic risk, the slab-column connections of two-way

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456 ACI Structural Journal/July-August 2007

slabs without beams must be checked for the induced effectscaused by the lateral displacement expected for the design-basis earthquake. It is important to note the direct influenceof the gravity shear ratio on the lateral drift capacity of slab-column connections without shear reinforcement illustrated bythe test data in Fig. 5. As suggested by the FEMA 356 limitsfor slab-column connections, this relationship is critical to thedevelopment of appropriate PBSD criteria for slab-columnconnections. The ACI 318-05 seismic design limits for slab-column connections given in Eq. (16) also underscore the directrelationship between these two parameters.

Linear regression analysis on the experimental data forslab-column connections without shear reinforcement andhaving a gravity shear ratio Vg/Vo less than 0.6, results in aline defined by a slope of –6.95 and a zero intercept of 4.97.

Thus, the mean for the data gives the following expressionfor the maximum story drift ratio (in percent)

(23)

The PBSD criteria suggested herein use Eq. (23) as areference for selecting the collapse prevention performancelevel limits. The life safety performance level was initiallydefined as 2/3 of the values used for collapse prevention; andfor immediate occupancy, 1/3 of the values for collapseprevention was used. The drift limits determined using theaforementioned parameters were the basis for finalizing thekey points of the graphed PBSD criteria. Table 4 summarizesthe key points for the recommended PBSD criteria and thevalues are shown graphically in relationship to the test datain Fig. 7.

For the suggested PBSD criteria, the drift limits for theimmediate occupancy performance level are relatively lowso that the slab-column frame members remain at or near theelastic range of behavior. The suggested line for life safetycorresponds to the ACI 318-05 design limits (refer to Fig. 2).The life safety performance level includes the combinationof Vg/Vo = 0.4 and a drift of 1.5%, which is consistent withthe recommendation in ACI 352.1R-89 that the gravity shearratio should be kept below 0.4 to ensure some minimalductility with the availability of approximately 1.5% driftcapacity. The collapse prevention limits correspond to approx-imately the mean of the experimental data for specimenswithout shear reinforcement. For all performance levels, aconstant story drift ratio capacity is assigned for gravityshear ratios in excess of 0.6.

As the approximate mean of the data for specimenswithout shear reinforcement (Fig. 7), the collapse preventionlimits correspond to a 50% probability of failure (withoutconsidering the load and resistance factors provided in the

DR 5 7Vg

Vo

-----–=

Table 3—Test data for interior slab-column connection test specimens with shear reinforcement

Source Label Vg /Vo Peak drift, %Shear

reinforcement Mode

Dilger and Brown (1995)

SJB-1 0.48 5.50 SSR S1

SJB-2 0.47 5.70 SSR S1

SJB-3 0.48 5.00 SSR S2

SJB-4 0.43 6.40 SSR S2

SJB-5 0.47 7.60 SSR S1

SJB-8 0.46 5.70 SSR S2

SJB-9 0.49 7.10 SSR S2

Dilger and Cao (1991)

CD 3 0.91 3.50 SSR NA

CD 4 0.62 4.80 SSR NA

CD 6 0.64 5.40 SSR NA

CD 7 0.51 5.60 SSR NA

Elgabry and Ghali (1987)

2 0.47 NA SSR P

3 0.87 NA SSR P

4 0.85 NA SSR P

5 1.20 NA SSR P

Hawkins et al. (1975)

SS1 0.49 3.50 Stirrups C3

SS2 0.47 3.43 Stirrups P

SS3 0.48 4.10 Stirrups F

SS4 0.47 5.50 Stirrups NA

SS5 0.42 4.90 Stirrups F

Islam and Park (1976)

4S 0.23 4.33 Bent up P

5S 0.23 4.17 Shear head P

6CS 0.24 4.00 Stirrups P

7CS 0.24 3.70 Stirrups P

8CS 0.27 5.00 Stirrups P

Robertson et al. (2002)

2CS 0.16 4.50 Closed hoop F

4HS 0.15 5.00 Headed stud F

3SL 0.10 4.50 Single leg F

Megally and Ghali (2000)

MG-10 0.60 5.20 SSR NA

MG-3 0.56 5.40 SSR NA

MG-4 0.86 4.60 SSR F-P

MG-5 0.31 6.50 SSR F-P

MG-6 0.59 6.00 SSR F-P

Robertson and Durrani (1990) 4S 0.19 3.50 Closed hoop F

Note: SSR = stud-shear reinforcement; S1 = shear failure outside shear reinforced zone; S2

= shear failure in shear in zones without shear reinforcement; C3 = crushing failure atcolumn face without apparent punching shear failure; F = flexural failure; P = punchingshear failure; and F-P = flexural and punching shear failure. NA: not available.

Table 4—Key points for recommended PBSD criteria for interior slab-column connections

Gravity shear ratio (Vg /Vo)

Drift ratio, %, by performance level

IO LS CP

0.0 1.75 3.5 5.0

0.6 0.25 0.5 0.75

1.0 0.25 0.5 0.75

Note: IO = immediate occupancy; LS = life safety; and CP = collapse prevention.

Fig. 7—Comparison of recommended performance-basedseismic design limits with slab-column connection test data.

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457ACI Structural Journal/July-August 2007

code). Assuming a normal distribution, the life safetylimits, defined as 2/3 of collapse prevention, correspondto approximately 5% probability of failure, and theimmediate occupancy limits, defined as 1/3 of collapseprevention, correspond to less than 1% probability of failure.

When the story drift limit corresponding to the actinggravity shear ratio is exceeded for the performance levelconsidered, various options exist, including: 1) reduce thegravity shear ratio by thickening the slab, adding shear capitals,or adding drop panels; 2) reduce the story drift ratio to bewithin the allowable limit by stiffening the lateral system; or3) add shear reinforcement as prescribed by ACI 318-05. ForOptions 1 and 2, consideration must be given to increased lateralforces resulting from the structural modification. For Option 3,the experimental data indicates that larger drift ratios arepossible when shear reinforcement is used (refer to Fig. 6).The data for the shear reinforced specimens are included inFig. 7 for comparison.

A direct comparison of the suggested PBSD criteria to theFEMA 356 acceptance criteria is not simply accomplishedbecause FEMA limits are in terms of plastic rotations ratherthan drift ratios. FEMA 356 is intended for assessingexisting structures and also addresses cases involvingseveral possible deficiencies, including: 1) inadequatedevelopment or splicing along the slab span; 2) inadequateembedment into the slab-column joint; and 3) lack of continuityreinforcement through the column cage. In addition, adistinction is made between primary and secondary components.In general, the proposed PBSD limits appear to be in the range ofthe corresponding FEMA 356 limits. One exception is thatFEMA 356 does not allow plastic rotation in primarycomponents when the gravity shear ratio is above 0.4.

The aforementioned PBSD criteria are intended primarilyfor new construction. The criteria, however, could also beapplied to existing structures that contain subpar seismicdetails where a moderate seismic demand is expected. Forassessing the expected performance of a structure, the valueof Vo should be computed with φ = 1.0, whereas for newbuilding design, Vo should include a strength reductionfactor for shear, currently φ = 0.75 in ACI 318-05.

SUMMARY AND CONCLUSIONSThis paper focuses on the behavior and design of interior

slab-column connections under combined gravity and lateralloading and serves to review current design procedures,performance-based seismic design (PBSD) approaches, andrelevant experimental data. Practical recommendations areprovided for PBSD of slab-column connections underseismic loading conditions that can be readily implementedinto design practice.

An assessment of the experimental data versus the ACI 318-05recommendations for slab-column connections indicatethat the limits for determining the necessity of slab shearreinforcement are a reasonable lower bound of the test data.Very few reports for slab-column connection specimensinclude plastic rotation data. FEMA 356, however, provideslimits in terms of plastic rotations for nonlinear analysisprocedures that are determined in part by the gravity shearratio at the slab-column connections. The recommendedPBSD criteria in this paper use two key parameters forassessing slab-column connections: the gravity shear ratio atthe connection and the maximum story drift ratio. The use ofstory drift ratio allows a direct comparison to the experimentaldata and is readily available when conducting a structural

analysis. A relationship between drift capacity and gravityshear ratio is provided in Eq. (23), representing an average ofthe collected experimental data. Three performance levelsare used to match those in FEMA 356: immediate occupancy,life safety, and collapse prevention. The proposed limitscorrelate well with the ACI 318-05 seismic design provisionsfor slab-column connections and provide a practical approachfor conducting PBSD for slab-column connections.

ACKNOWLEDGMENTSThe authors wish to thank the members of ACI Committee 374, Performance-

Based Seismic Design of Concrete Buildings, for their input. The contribution ofY.-H. Kim, a graduate student at Texas A&M University, College Station,Tex., is also appreciated.

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Hueste, M. D., and Wight, J. K., 1997, “Evaluation of a Four-StoryReinforced Concrete Building Damaged During the Northridge Earthquake,”Earthquake Spectra, V. 13, No. 3, pp. 387-414.

Hueste, M. D., and Wight, J. K., 1999, “Nonlinear Punching Shear FailureModel for Interior Slab-Column Connections,” Journal of StructuralEngineering, ASCE, V. 125, No. 9, pp. 997-1008.

Hwang, S. J., and Moehle, J. P., 1990, “An Experimental Study of Flat-Plate Structures Under Vertical and Lateral Loads,” Report No. UCB/SEMM-90/11, University of California-Berkeley, Berkeley, Calif., 271 pp.

Hwang, S. J., and Moehle, J. P., 2000, “Models for Laterally Loaded Slab-Column Frames,” ACI Structural Journal, V. 97, No. 2, Mar.-Apr., pp. 345-353.

Islam, S., and Park, R., 1976, “Tests on Slab-Column Connections withShear and Unbalanced Flexure,” Journal of the Structural Division, V. 102,No. ST3, ASCE, pp. 549-568.

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Joint ACI-ASCE Committee 326, 1962, “Shear and Diagonal Tension,Slabs,” ACI JOURNAL, Proceedings V. 59, No. 3, Mar., pp. 353-396.

Joint ACI-ASCE Committee 352, 1989, “Recommendations for Designof Slab-Column Connections in Monolithic Reinforced ConcreteStructures (ACI 352.1R-89),” American Concrete Institute, FarmingtonHills, Mich., 22 pp.

Joint ACI-ASCE Committee 421, 1999, “Shear Reinforcement for Slabs(ACI 421.1R-99),” American Concrete Institute, Farmington Hills, Mich.,15 pp.

Kang, T. H. K., and Wallace, J. W., 2005, “Dynamic Response of FlatPlate Systems with Shear Reinforcement,” ACI Structural Journal, V. 102,No. 5, Sept.-Oct., pp. 763-773.

Kang, T. H. K.; Elwood, K. J.; and Wallace, J. W., 2006, “Dynamic Testsand Modeling of RC and PT Slab-Column Connections,” Paper 0362,Proceedings of the 8th U.S. National Conference on Earthquake Engineering,San Francisco, Calif., 10 pp. (CD-ROM)

Luo, Y., and Durrani, A. J., 1995, “Equivalent Beam Model for Flat-SlabBuildings—Part 1: Interior Connections,” ACI Structural Journal, V. 92,No. 1, Jan.-Feb., pp. 115-124.

Megally, S., and Ghali, A., 1994, “Design Considerations for Slab-ColumnConnections in Seismic Zones,” ACI Structural Journal, V. 91, No. 3, May-June, pp. 303-314.

Megally, S., and Ghali, A., 2000, “Punching Shear Design of Earth-quake-Resistant Slab-Column Connections,” ACI Structural Journal, V. 97,No. 5, Sept.-Oct., pp. 720-730.

Moehle, J. P., 1996, “Seismic Design Considerations for Flat PlateConstruction,” Mete A. Sozen Symposium: A Tribute from his Students,SP-162, J. K. Wight and M. E. Kreger, eds., American Concrete Institute,Farmington Hills, Mich., pp. 1-35.

Morrison, D. G., and Sozen, M. A., 1983, “Lateral Load Tests of R/CSlab-Column Connections,” Journal of the Structural Division, ASCE,V. 109, No. 11, pp. 2699-2714.

Pan, A., and Moehle, J. P., 1989, “Lateral Displacement Ductility ofReinforced Concrete Flat Plates,” ACI Structural Journal, V. 86, No. 3,May-June, pp. 250-258.

Pan, A., and Moehle, J. P., 1992, “An Experimental Study of Slab-Column Connections,” ACI Structural Journal, V. 89, No. 6, Nov.-Dec.,pp. 626-638.

Robertson, I., and Durrani, A. J., 1990, “Seismic Response of Connections inIndeterminate Flat-Slab Subassemblies,” Report No. 41, Department ofCivil Engineering, Rice University, Houston, Tex., 266 pp.

Robertson, I.; Kawai, T.; Lee, J.; and Enomoto, B., 2002, “Cyclic Testing ofSlab-Column Connections with Shear Reinforcement,” ACI StructuralJournal, V. 99, No. 5, Sept.-Oct., pp. 605-613.

Robertson, I., and Johnson, G., 2006, “Cyclic Lateral Loading of NonductileSlab-Column Connections,” ACI Structural Journal, V. 103, No. 3, May-June, pp. 356-364.

Rodriguez, M., and Diaz, C., 1989, “Analysis of the Seismic Performance ofa Medium Rise, Waffle Flat Plate Building,” Earthquake Spectra, V. 5, No. 1,pp. 25-40.

SEI/ASCE, 2005, “Minimum Design Loads for Buildings and otherStructures (SEI/ASCE 7-05),” Structural Engineering Institute, ASCE,Reston, Va., 376 pp.

Symonds, D. W.; Mitchell, D.; and Hawkins, N. M., 1976, “Slab-ColumnConnections Subjected to High Intensity Shears and Transferring ReversedMoments,” Progress Report on NSF Project GI-38717, Department ofCivil Engineering, University of Washington, Seattle, Wash., 80 pp.

Wey, E. H., and Durrani, A. J., 1992, “Seismic Response of InteriorSlab-Column Connections with Shear Capitals,” ACI Structural Journal,V. 89, No. 6, Nov.-Dec., pp. 682-691.

Zee, H. L., and Moehle, J. P., 1984, “Behavior of Interior and ExteriorFlat Plate Connections Subjected to Inelastic Load Reversals,” ReportNo. UCB/EERC-84/07, Earthquake Engineering Research Center, Universityof California-Berkeley, Berkeley, Calif., 130 pp.

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ACI Structural Journal, V. 104, No. 4, July-August 2007.MS No. S-2005-200 received August 8, 2005, and reviewed under Institute publication

policies. Copyright © 2007, American Concrete Institute. All rights reserved, includingthe making of copies unless permission is obtained from the copyright proprietors.Pertinent discussion including author’s closure, if any, will be published in the May-June2008 ACI Structural Journal if the discussion is received by January 1, 2008.

ACI STRUCTURAL JOURNAL TECHNICAL PAPER

The California Department of Transportation (Caltrans) codeprovides the only guidelines in the U.S. for the design of columnswith interlocking spirals. Previous studies have shown thatcolumns with interlocking spirals have a satisfactory behavior, butnone of them have addressed the Caltrans upper limit onhorizontal spacing between centers of the spirals in detail andnone used dynamic testing. Six large-scale column models weredesigned and tested on a shake table at the University of Nevada-Reno to study the effects of the shear level, spiral distance, andcrossties. The observed damage progression, load-displacementresponses, reinforcement strains, and the apparent plastic hingelengths were examined to evaluate the response. The resultsrevealed that the Caltrans upper spiral distance limit of 1.5 timesthe spiral radius is satisfactory. However, supplementary crosstiesare needed to prevent premature vertical shear cracking andstrength degradation in columns with relatively high shear.

Keywords: bridge; columns; interlocking spirals; seismic behavior.

INTRODUCTIONThe current seismic design philosophy for reinforced

concrete structures relies on confinement of concrete toprovide the necessary ductility and energy dissipationcapacity of structural members. Confinement is mainlyprovided by the transverse reinforcement, which in columnsusually consists of spirals in members with circular or squareshape and ties in those with square or rectangular crosssections. Spirals confine concrete more effectively thanrectilinear ties because they counteract the dilation ofconcrete through hoop action instead of a combination ofbending and hoop action that takes place in rectilinear ties.As a result, to provide the same level of confinement, theamount of tie reinforcement is greater than that provided byspirals. Another advantage of spirals is that they are generallyeasier to construct. The circular shape of spirals makes themsuitable for circular and square columns. To use the benefitsof spirals in rectangular columns, two or more sets ofinterlocking spirals are used.

The Caltrans Bridge Design Specifications (BDS)1 andSeismic Design Criteria (SDC)2 are currently the only codesin the U.S. that include provisions for the design of columnswith interlocking spirals. Because the amount of research oninterlocking spirals has been limited, the design provisionsare driven mainly by research on single spirals. Studies3-5

were conducted on the effect of several design parameters,including a comparison between interlocking spirals andties, horizontal distance between centers of the spirals,quantity of transverse reinforcement, variation of the axialload ratios, appropriate size and spacing of longitudinal barsin the interlocking region, and cross section shape. Thesestudies generally concluded that flexural and shearcapacities of columns with interlocking spirals can beconservatively estimated using current procedures. Conflicting

recommendations exist, however, with respect to thedistance between spiral sets and uncertainties about the needfor supplemental crossties between adjacent spiral sets. Forexample, the BDS upper limit on the distance between thecenters of adjacent spirals is 1.5 times the radius of the spiralR, whereas the study in Reference 3 places an upper limit of1.2R. To address these issues, a study was undertaken usinglarge-scale testing of bridge column models on one of theshake tables of the University of Nevada-Reno. The studyincluded both experimental and analytical components toevaluate the seismic performance of bridge columns withdouble interlocking spirals with different parameters, includingthe spread between the spiral sets, the level of shear, andcrossties. The focus of this paper is on the experimentalphase of the investigation. Details of all aspects of the studyare presented in Reference 6.

RESEARCH SIGNIFICANCEInterlocking spirals are used in the columns of many

bridges. The spirals are designed based on provisions thathave yet to be verified and, in part, are in conflict with someof the recommendations that are based on the limited availablepast studies. The research presented in this paper was usedto: 1) evaluate the dynamic performance of bridge columnsthat are designed based on the current Caltrans provisions; 2)determine if the limits in the provisions are satisfactory; and3) identify cases and limit states in which supplementalcrossties are needed.

EXPERIMENTAL STUDIESTest specimens

Six large-scale specimens were designed, constructed, andtested. The limit of 1.2R on the horizontal distance of thecenters of the spirals, di, recommended in Reference 3 is toensure sufficient vertical shear transfer between adjacentspiral sets. Because vertical shear is a function of horizontalshear, the test parameters were selected to capture the effect ofa range of realistic horizontal shear stresses. The test variableswere: 1) the level of average shear stress; 2) the horizontaldistance between the centers of the spirals, di; and 3) supple-mentary horizontal crossties. The test variables are listed inTable 1. The effect of other parameters such as axial load andmaterial strength was not considered because the variation ofthese parameters in real bridges is relatively small.

The average horizontal shear stress was calculated as thelateral load divided by the effective shear area taken equal to

Title no. 104-S37

Shake Table Studies of Bridge Columns with Double Interlocking Spiralsby Juan F. Correal, M. Saiid Saiidi, David Sanders, and Saad El-Azazy

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80% of the gross area (SDC).2 A shear stress index wasdefined as the average shear stress divided by 0.083√f ′c(MPa) (√f ′c [psi]). This index represents the level of shear inthe column. In this study, two levels of shear were selected:low index equal to 3 and high index equal to 7. These indexesrepresent column shear stresses in real bridges. Actualbridge columns are designed to be ductile and the loadcapacity is controlled by flexure, although shear damage isexpected to increase as the shear index increases.

The Caltrans BDS1 states that when more than one cage isused to confine an oblong column core, the spirals must beinterlocked or the pier must be designed as though it consistsof multiple single columns. A maximum limit of 1.5 timesthe radius of the spirals, R, (where R is measured to theoutside edge of the spiral) for the horizontal distance of thespirals, measured center-to-center of the spirals, di, isspecified. A minimum distance of 1.0R is recommended toavoid overlaps of more than two spirals in multiple spiralcases. Of the six models used in this study, two weredesigned with a di of 1.0R, one with a di of 1.25R, and threewith a di of 1.5R.

Three alphabetical characters followed by a number wereused to identify the test specimens. The initials I and Srepresented interlocking and spirals, respectively. The thirdinitial L or H was for the shear index of low or high,respectively. A numeral indicated the fraction of R used fordi. In one specimen an initial T was added at the end of thespecimen, designation to indicate the presence ofsupplementary crossties (Fig. 1).

The experimental program was developed to use one ofthe shake tables at the Large-Scale Structures Laboratory atthe University of Nevada-Reno. Scale factors of 1/4 for thespecimens with low shear and 1/5 for the columns with highshear were selected. These were the largest scales that couldbe used without exceeding shake table capacity. Thedisplacement-based design procedure in the SDC2 was usedfor a target displacement ductility capacity of 5. In the SDC,2

the displacement ductility is defined as the displacementdivided by the effective yield displacement excluding bondslip and shear deformations. Typical steel ratios of 2.0% and

2.8% were selected for the longitudinal reinforcement. Thetransverse steel ratio was designed to provide sufficientconfinement for the columns to reach the target displacementductility capacity. Additional crossties with the same barsize as the spirals and spacing of two times the spacing of thespirals were used based on a design recommendationdescribed in Reference 6. An axial load index, defined as theaxial load divided by the product of the gross cross-sectionalarea and the specified concrete compressive strength of 10%,was used to represent the axial load level in real bridge columns.

The details of the cross section and the elevations of thespecimens are shown in Fig. 1 and 2, respectively. Thespirals were continuous with constant pitch throughout theheight of the specimens. The spirals were extended along theheight of the footing and the top loading head. Thelongitudinal reinforcement was continuous with 90-degreestandard hooks at the ends. In the specimens with low shear,the height was taken from the top of the footing to the centerof the lateral loading head because these columns weretested in single curvature cantilever mode. The height forothers was taken as the clear distance between the top of thefooting and the bottom of the loading head because thesecolumns were tested in double curvature.

The specified concrete compressive strength was 34.5 MPa(5000 psi) with 9.52 mm (3/8 in.) maximum aggregate size. Theaverage measured concrete strength of the standard cylindrical

Juan F. Correal is an Assistant Professor of civil and environmental engineering atthe University of Los Andes, Colombia, where he received his BS and MSCE. Hereceived his PhD in 2004 from the University of Nevada-Reno, Reno, Nev. Hisresearch interests include the seismic design of bridges and applications of innovativematerials for design, repair, and rehabilitation of structures.

M. Saiid Saiidi, FACI, is a Professor of civil and environmental engineering and is theDirector of the Office of Undergraduate Research at the University of Nevada-Reno. Heis a Past Chair and a member of ACI Committee 341, Earthquake-Resistant ConcreteBridges, and is member of ACI Committees 342, Bridge Evaluation; E803, FacultyNetwork Coordinating Committee; and Joint ACI-ASCE Committee 352, Joints andConnections in Monolithic Concrete Structures. His research interests include analysisand shake table studies of reinforced concrete bridges and application of innovativematerials.

David Sanders, FACI, is an Associate Professor of civil and environmentalengineering at the University of Nevada-Reno. He is Chair of ACI Committee 445,Shear and Torsion, is Past Chair of ACI Committee 341, Earthquake-ResistantConcrete Bridges, and is a member of the ACI Technical Activities Committee; ACICommittees 318, Structural Concrete Building Code; 369, Seismic Repair andRehabilitation; 544, Fiber Reinforced Concrete; E803, Faculty Network CoordinatingCommittee; E804, Educational Awards Nomination Committee; and Joint ACI-ASCECommittee 423, Prestressed Concrete. His research interests include shake tablestudies of reinforced concrete bridges.

Saad EI-Azazy is the Seismic Research Program Manager at the California Departmentof Transportation (Caltrans). He received his BS from Cairo University, Giza, Egypt,and his MS and PhD from Ohio State University, Columbus, Ohio. His researchinterests include bridge seismic retrofit and performance of new bridges.

Table 1—Test variables

SpecimenScalefactor

Shear index

Aspect ratio di (× R)

Steel reinforcement

ρl , % ρs ,* %

ISL1.00.25

3.0 3.3 1.0 2.0 1.1

ISL1.5 3.0 3.6 1.5 2.0 1.1

ISH1.0

0.2

7.0 2.0 1.0 2.9 0.6

ISH1.25 7.0 2.0 1.25 2.8 0.9

ISH1.5 7.0 2.1 1.5 2.9 0.9

ISH1.5T 7.0 2.1 1.5 2.9 0.9*

*Steel ratio from additional crossties is not included.Note: ρl = ratio of longitudinal reinforcement and ρs = ratio of transversal reinforcement.

Fig. 1—Test specimens cross sections.

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samples on the day of testing was 36.8 MPa (5339 psi) forSpecimens ISL1.0 and ISL1.5, 31.1 MPa (4514 psi) forSpecimens ISH1.0 and ISH1.5, and 45.1 MPa (6542 psi) forSpecimens ISH1.25 and ISH1.5T. The specified yield stress forall the reinforcement was 420 MPa (60 ksi). The averagemeasured yield stress of the steel samples was 462 MPa (67 ksi)for Specimens ISL1.0 and ISL1.5, 443 MPa (64 ksi) forSpecimens ISH1.0 and ISH1.5, and 431 MPa (63 ksi) forSpecimens ISH1.25 and ISH1.5T.

Test setup, instrumentation, and testing procedureFigure 3 shows the shake table setup for the high shear

specimens. The test setup for the low-shear specimens wassimilar but with only one link between the mass rig and thecolumn to achieve single-curvature testing. All specimenswere tested in the strong direction. The lateral load wasapplied through an inertial mass system off the table for betterstability. Two sets of swiveled links were used to connect theinertial mass system to the specimens. One set consisted ofone link connected at the column loading head to test thespecimens as a cantilever member with single curvature. Theother set consisted of two links connected at the top of thecolumn, allowing the specimens to be tested in doublecurvature. The double-link system was designed to preventrotation of the loading head. The specimens with low shear(ISL1.0 and ISL1.5) were tested in single curvature whereasthe specimens with high shear (ISH1.0, ISH1.25, ISH1.5, andISH1.5T) were tested in double curvature. The total equivalentweight of the inertia mass was 445 and 356 kN (100 and80 kips) for specimens tested in single and double curvature,respectively. The axial load was applied through a steelspreader beam by prestressed bars connected to hydraulic jacksand an accumulator to limit axial load fluctuation. Electricalstrain gauges were attached to the longitudinal and transversesteel to measure strain variation. A series of curvaturemeasurement instruments were installed in the plastichinge zones. Additional displacement transducers formingpanels were placed along the height of the columns with highshear. Load cells were used to measure both the axial andlateral forces. The acceleration at the top of the columns wasmeasured using an accelerometer placed on the linkconnecting the mass rig to the specimens. Wire potentiometerswere used to measure the lateral displacements of the columns.

Preliminary moment-curvature analysis was performed toestimate the lateral load and displacement capacities of thespecimens. Once the capacity was estimated, a series ofdynamic analyses were conducted to select the input motionto be simulated in the shake table tests. The 1994 Northridgeearthquake, recorded at the Sylmar Hospital (0.606g peakground acceleration [PGA]) was selected as the input motionbased on the maximum displacement ductility demandplaced on the columns without exceeding the shake tablecapacity. The earthquake record is referred to as “Sylmar”hereafter. The time axis of the input record was compressedto account for the scale effect and the minor differencesbetween the axial load and the effective mass.

Each column was subjected to multiple simulatedearthquakes, each referred to as a “run.” The amplitude ofthe motions was increased in subsequent runs. Smallincrements of the Sylmar record (10 to 20% of the fullSylmar amplitude) were initially applied to the specimens todetermine the initial stiffness, the elastic response, and theeffective yield point. Once the effective yield was reached,the amplitude of the input record was increased until failure.

Intermittent free vibration tests were conducted to measurethe change in frequency and damping ratio of the columns.

EXPERIMENTAL RESULTSImportant aspects of the seismic performance of the test

columns were evaluated. The observed damage progression,load-displacement response, and strains were used to judgethe behavior of the columns. Additional response parameters,the curvature and plastic hinge length, were computed basedon the measured data and used in performance evaluation.

Observed responseSpecimens with low shear—The observed performance was

correlated with the displacement ductility μd, which representsthe displacement divided by the effective measured yielddisplacement. Only flexural cracks were observed during thefirst three runs (displacement ductility of up to 0.8) inSpecimen ISL1.0 and during the first six runs (μd of up to 1.5)in Specimen ISL1.5. Most of these cracks were located in thelower 1/3 of the column height. First spalling and shear crackswere seen in Specimen ISL1.0 after 0.5 × Sylmar (μd = 1.5) andin Specimen ISL1.5 after 1.25 × Sylmar (μd = 2.4). The shearcracks were located in the interlocking region in the lower 1/3of the height of the column and were extensions of the flexuralcracks. Considerable spalling at the bottom of the column, aswell as propagation of flexural and shear cracks, was observedafter 1.25 × Sylmar (μd = 2.8) in Specimen ISL1.0 and 1.5 ×Sylmar (μd = 3.1) in Specimen ISL1.5. Spirals were visible

Fig. 2—Test specimens elevations.

Fig. 3—Double cantilever test setup.

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after 1.5 × Sylmar (μd = 4.1) and longitudinal bars wereexposed after 1.75 × Sylmar (μd = 5.6) in Specimen ISL1.0.The spirals were visible in Specimen ISL1.5 after 1.75 ×Sylmar (μd = 4.5) and became exposed over a large area after2.0 × Slymar (μd = 7.5). There was no visible core damage ineither specimen. Specimens ISL1.0 and ISL1.5 failed during2.0 × Sylmar (1.21g PGA and μd = 9.6) and 2.125 × Sylmar(1.29g PGA and μd = 10.4), respectively. Figure 4 shows thedamage after failure in Specimen ISL1.0. The failure in bothcolumns was similar and was due to rupture of the spirals andbuckling of the longitudinal bars at the bottom of the column inthe plastic hinge zone.

Specimens with high shear—Even though these columnshad a relatively high shear index, they were flexural membersand, hence, only flexural cracks were formed during the initialthree or four runs. The measured displacement ductilitiesassociated with initial flexural cracks were 0.4, 0.6, 0.7, and0.6 in Specimens ISH1.0, ISH1.25, ISH1.5, and ISH1.5T,respectively. The flexural cracks were located in the plastichinge zones near the top and bottom of the columns. Thesecracks were concentrated mainly at the top and bottom 1/3 ofthe column height. A vertical crack in the interlocking regionextending from the top of the column to the midheight wasobserved after 0.4 × Slymar (μd = 0.7) in Specimen ISH1.5(Fig. 5). Diagonal cracks were formed in the interlockingregion in the plastic hinge zones of all the specimens. Thesecracks began to form starting with 0.5 × Sylmar (μd = 0.6) andbecame noticeable under 0.75 × Sylmar (μd = 0.9) in SpecimenISH1.0, and 1.0 × Sylmar (μd = 1.4) in Specimen ISH1.25. InSpecimen ISH1.5, shear cracks were visible starting with0.75 × Sylmar (μd = 1.0) and in Specimen ISH1.5T under

1.0 × Sylmar (μd = 1.2). Localized small vertical cracks wereobserved in Specimen ISH1.5T under 1.0 × Sylmar. After1.0 × Sylmar (μd = 1.4), first spalling at the top and bottom ofthe column was observed in Specimens ISH1.0 and ISH1.5,whereas in Specimens ISH1.25 and ISH1.5T, the first spallingwas observed during 1.25 × Sylmar (μd = 1.6 in SpecimenISH1.25 and 1.7 in Specimen ISH1.5T). Flexural and shearcracks propagated and more concrete spalled during 1.5 ×Sylmar (μd = 2.5) in Specimen ISH1.0, 1.75 × Sylmar (μd = 2.2)in Specimen ISH1.25, 1.25 × Sylmar (μd = 1.7) in SpecimenISH1.5, and 1.75 × Sylmar (μd = 2.5) in Specimen ISH1.5T.The spirals were visible at the top and bottom of the columnafter 2.125 × Sylmar (μd = 2.9) in Specimen ISH1.25. Thelongitudinal bars were exposed after 1.75 × Sylmar (μd = 3.6)in Specimen ISH1.0, 2.25 × Sylmar (μd = 3.7) in SpecimenISH1.25, 1.5 × Sylmar (μd = 2.2) in Specimen ISH1.5, and2.0 × Sylmar (μd = 2.8) in Specimen ISH1.5T. SpecimensISH1.0 and ISH1.25 (Fig. 6) failed in flexure/shear during2.0 × Sylmar (μd = 4.7) near the bottom and 2.375 × Sylmar(μd = 4.7) near the top, respectively.

Damage in the core was observed in Specimen ISH1.5 after2.125 × Sylmar (μd = 4.7) and in Specimen ISH1.5T after2.25 × Sylmar (μd = 3.0). The longitudinal bars buckled at thebottom of the column during 2.25 × Sylmar (μd = 3.4) inSpecimen ISH1.5 and 2.5 × Sylmar (μd = 3.4) in SpecimenISH1.5T Specimens ISH1.5 and ISH.5T (Fig. 7) failedduring 2.375 × Sylmar (μd = 4.0) and 2.625 × Sylmar (μd =3.8), respectively. Failure in Specimen ISH1.5, was due tofracture of the spirals and buckling of the longitudinal bars,whereas in Specimen ISH1.5T, failure was due to fracture ofthe spirals and one of the longitudinal bars.

Fig. 4—Specimen ISL1.0 after failure.

Fig. 5—Vertical crack (µd = 0.7) Specimen ISH1.5.

Fig. 6—Specimen ISH1.25 after failure.

Fig. 7—Specimen ISH1.5T after failure.

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Force-displacement relationshipsThe accumulated measured hysteresis curves for the ISL

and ISH groups are plotted in Fig. 8 and 9, respectively. Foreach column, a backbone force-displacement envelope wasdeveloped based on the peak forces with correspondingdisplacements for all the motions before failure. The failurepoint in the backbone curve was assumed either at the pointof maximum displacement or at a point with 80% of the peakforce with the corresponding displacement. The latter wasused when the force at the maximum displacement droppedmore than 20% of the pick force (Fig. 8 and 9). Thebackbone curves for the predominant direction of the motionwere idealized by elasto-plastic curves to quantify theductility capacity. The force corresponding to the firstreinforcement yield and the corresponding displacement onthe measured envelope was used to define the elastic portionof the idealized curve. Once the elastic portion was defined,the yield level was established by equalizing the areabetween the measured backbone and the idealized curves.Figures 8 and 9 show the idealized curves for specimens with

low and high shear, respectively. Based on the elasto-plasticcurves, displacement ductility capacities of 9.5 and 10.4 wereobtained for Specimens ISL1.0 and ISL1.5, respectively. InSpecimens ISH1.0, ISH1.25, ISH1.5, and ISH1.5T, themeasured displacement ductility capacities were 4.7, 5.0,4.0, and 3.8, respectively.

The column section total depths were different within eachspecimen group due to different distances between the spiralsets. As a result, the lateral load capacity varied among thecolumns. To compare the performance of the specimens,forces were normalized with respect to the effective yieldforce of each specimen and the normalized force-displacement envelopes were compared (Fig. 10). The effectof a large distance between the spiral sets in low-shearcolumns can be seen in Fig. 10(a). The overall ductilitycapacity of the two low-shear specimens was comparable.The strength of the specimen with di of 1.5R (Specimen ISL1.5),however, degraded starting with displacement ductility of7.4, whereas the strength of the column with di of 1.0R

Fig. 8—Hysteretic curves and envelopes for low-shear specimens.

Fig. 9—Hysteretic curves and envelopes for high-shear specimens.

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(Specimen ISL1.0) did not drop until failure. At adisplacement ductility of 9, the strength degradation in thecolumn with di of 1.5R was 10% whereas it was 4% when diwas 1.0R. Nevertheless, degradation started at a relativelyhigh ductility and hence is not of concern. Note that thetarget design displacement ductility for the columns was 5.

In specimens with high shear, the displacement ductilitycapacity was comparable in the two columns with di of 1.0Ror 1.25R. The ductility capacity dropped by approximately20% when di was increased to 1.5R. The slightly lowerductility of Specimen ISH1.5T versus Specimen ISH1.5 (3.8versus 4) suggests that the addition of crossties had littleeffect on the ductility capacity. A comparison of Fig. 9(c)and (d) indicates that the response of Specimen ISH1.5contained limited excursions into the negative displacementrange, whereas the Specimen ISH1.5T response was some-what symmetric. Variations of concrete strength properties,column stiffness, and the shake table response are attributed

to the difference in the column responses. Symmetric cyclicdisplacements tend to place higher demands on reinforcedconcrete members. It is hence concluded that, had thedisplacements in the two columns been identical, SpecimenISH1.5T would have shown a higher ductility capacity.Nonetheless, the ductility capacity of approximately 4measured in Specimens ISH1.5 and ISH1.5T was consideredto be satisfactory. The displacement ductility at whichstrength degradation began in columns with di of 1.0R and1.25R was approximately 3.7, and in those with di of 1.5Rwas approximately 3. The larger spread of the spirals clearlyshows some effect on the overall load-displacementresponse. The addition of the crossties reduced the slope ofthe degradation part of the responses (Fig. 10(b)).

The displacement ductility capacity versus the averageshear stress index is shown in Fig. 11. The measuredconcrete compressive strengths were used in this graph. Ingeneral, the displacement ductility capacity decreased whenthe average shear stress index increased. This was becausecolumns subjected to high shear failed in shear/flexuralmode, whereas those with low shear failed in flexure with nosignificant shear damage.

Measured curvaturesDisplacement transducers were used to measure curvature

in the plastic hinge regions at the bottom of the ISL groupand at the top and bottom of the ISH group. The strain oneach side of the column was calculated from the verticaldisplacement measured by each external transducer dividedby the gauge length. The average curvature over the gaugelength was computed as the difference between the strains onthe opposite sides of the column divided by the horizontaldistance between the instruments. This procedure assumesthat sections remained plane.

The curvature profiles for the predominant direction ofmotion are shown in Fig. 12 and 13 for specimens with lowand high shear, respectively. High values of curvature weremeasured in the plastic hinges, as expected. The curvaturesat the ends are influenced by the localized longitudinalreinforcement bond slip and are not purely due to flexuraldeformation of the plastic hinge.

The maximum ultimate curvatures in Specimens ISL1.0and ISL1.5 were comparable, indicating that the change indistance of the spiral sets did not affect the curvatureperformance. This observation was in agreement with thedisplacement ductility capacities of the two models. Themaximum curvatures in the columns with high shear werealso comparable within the group, but were approximately 2/3of the curvatures of the ISL group. The lower curvatures areconsistent with the smaller displacement ductility capacitiesthat were observed for this group. The peak top and bottomcurvatures in Specimens ISH1.0 and ISH1.25 were comparable,confirming that the loading mechanism to bend the columnsin double-curvature fixed-fixed mode was successful. InSpecimens ISH1.5 and ISH1.5T, the peak top curvatureswere 20 to 25% lower than the bottom curvature due to slightrotation of the loading head that occurred under high loadsand prevented fully fixed response at the top.

Measured strainsThe strain gauges on the longitudinal reinforcement were

placed at the potential plastic hinge regions of all thecolumns and the footings, and in the loading heads of theISH group. In all specimens, the longitudinal bars yielded

Fig. 11—Measured displacement ductility capacity versusaverage shear stress index.

Fig. 12—Measured curvature for ISL group.

Fig. 10—Normalized lateral force-displacement envelopes.

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ACI Structural Journal/July-August 2007 399

extensively and flexural deformations dominated theresponse. Higher strains were measured at or near the base ofall the columns and also at the top of the ISH group. Becausethe response in all the specimens was dominated by flexure,the test variables did not significantly affect the trends in thelongitudinal and spiral bar strains except as noted in thefollowing.

The correlation between the apparent damage and the longi-tudinal bar strains was studied. Five damage states wereselected representing an increasing level of damage: 1) flexuralcracks; 2) first spalling and shear cracks; 3) extensive crackingand spalling; 4) visible spirals and longitudinal bars; and 5)imminent failure. The fifth damage state refers to the case wherecore damage is observed or is about to occur and some of thelongitudinal bars show signs of bending that might lead tobuckling and failure in subsequent runs. This damage statecorresponded to the run before the failure run in the shake tabletests. Figure 14 shows the average of the highest three straindata in the longitudinal bars versus the damage states in eachmodel. The average data for three gauges, rather than themaximum strain, were used because local bar strains areinfluenced by cracks and present erratic patterns. The data forall specimens were averaged and shown on the graph.

It can be seen in Fig. 14 that, within each damage state, thelongitudinal bar strains were generally higher in the ISLgroup. This is because the moment gradient in the high-shearcolumns is relatively high, making the strain more localizedand the average strains lower. The larger distance betweenthe spiral sets in Specimen ISL1.5 led to higher bar strains inthe first three damage states. Within the ISH group, the barstrains did not seem to be sensitive to the distance betweenthe spiral sets.

The average bar strains in all specimens increasedespecially during the first three damage states. Averagestrains of approximately 3.5 times the yield strain wererecorded when flexural cracks were observed in thecolumns. When first spalling and shear cracks were visible,the strain in the longitudinal bars increased to approximately7.5 times the yield strain. An average strain of 14.5 times the

yield strain was recorded when extensive cracking andspalling was observed in the columns. Average strains of 18and 19 times the yield strain were recorded for the last twodamage states.

The correlation between the spiral bar strains and differentdamage states was also reviewed. It was found that spiral barstrains remain small (generally less than 2/3 of the yieldstrain) until the run before failure. These data are presentedand discussed in more detail in Reference 7. It wasdetermined that it would be more useful if the trends in spiralbar strains are studied as a function of displacementductilities. The average of peak spiral strains is plottedagainst displacement ductilities in Fig. 15. It can be seen thataverage strain was below yield until higher ductilities werereached. The larger distance between the spiral sets inSpecimen ISL1.5 led to higher strains than those ofSpecimen ISL1.0 under large ductilities. The higher spiralstrains are attributed to the slight degradation of the loadcapacity (Fig. 10) observed in Specimen ISL1.5. In addition,Fig. 15 shows slightly smaller strains in Specimen ISH1.0compared with the rest of the high-shear specimens until thelast motion. The average maximum spiral strains in

Fig. 14—Longitudinal bars strain versus observed damage.

Fig. 13—Measured curvature for ISH group.

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400 ACI Structural Journal/July-August 2007

Specimens ISH1.25 and ISH1.5T were nearly the same, andthe average maximum spiral strain in Specimen ISH1.5 wasthe highest until a displacement ductility of approximately1.6 was reached.

Plastic hinge lengthThe plastic hinge length (PHL) is used to estimate post-

yield lateral displacements based on the moment curvatureproperties of the plastic hinge while empirically taking intoaccount displacements due to bond slip and sheardeformation. To determine the sensitivity of PHL to thespiral set distance and the level of shear, the PHL for eachcolumn was estimated using the measured plastic curvaturesand displacements. The moment area method was used torelate displacements and curvatures assuming that the plasticrotation θp over the equivalent PHL, lp, is defined by

(1)

where φu equals the ultimate curvature capacity, and φyequals the idealized yield curvature.

The center of rotation was assumed to be at the center ofthe plastic hinge. Equation (2) was assumed to relate plasticrotation and plastic displacements. The PHL was determinedusing this equation. In the ISH group, two plastic hingeswere formed and, hence, the average measured curvatures atthe top and bottom were used.

(2)

where L equals the distance from point of maximum momentto the point of contraflexure.

θp φu φy–( )lp=

Δp θp Llp

2---–⎝ ⎠

⎛ ⎞=

In Eq. (1), the average of the measured curvatures over theextreme two gauge lengths (203 mm [8 in.] in low-shearcolumns and 254 mm [10 in.] in high-shear columns) was usedbecause most of the plastic deformation was concentrated overthat region according to the measured curvatures and strains.

Table 2 lists the data used to determine the measured lp forSpecimens ISL1.0 and ISL 1.5. The values of lp of 0.75 and0.83 times the total depth of the column were found forSpecimens ISL1.0 and ISL1.5, respectively. It can be seen thatthe larger spiral distance in Specimen ISL1.5 led to an increasein the ratio of the PHL over the column depth byapproximately 10%. The aspect ratio (column height dividedby the column section depth in the loading direction) ofSpecimen ISL1.5 was approximately 10% larger than theSpecimen ISL1.0 aspect ratio. Under equal conditions,Specimen ISL1.5 would experience a smaller sheardeformation. The larger spread of the spirals in SpecimenISL1.5, however, appear to have led to higher sheardeformations that necessitated a larger PHL to match themeasured displacement.

The values of lp of 0.98, 0.96, 1.12, and 1.27 times the totaldepth of the columns were found for Specimens ISH1.0,ISH1.25, ISH1.5, and ISH1.5T, respectively. The aspectratios for these columns were nearly the same. In high-shearcolumns, the increase in the distance between the spiralsfrom 1.0R to 1.5R appears to have increased displacementdue to shear, thus increasing the apparent plastic hingelength by approximately 20%.

CONCLUSIONSBased on the observations and the experimental results of

this study, the following conclusions are drawn:1. The seismic performance of columns with relatively

low shear with spiral distance di of 1.0R and 1.5R was similarand satisfactory with displacement ductility capacities ofnear 10. The strength degradation was slightly larger when diwas 1.5R. This degradation began at a displacement ductilityof 7.4, however, which exceeded the target designdisplacement ductility of 5;

2. Because the low-shear column with di of 1.5R did notexperience significant shear cracking, and based on thesatisfactory displacement ductility capacity, it appears thatthe Caltrans provision of allowing a di value of up to 1.5R isappropriate for columns with low shear;

3. The seismic performance of column models with di of1.0R and 1.25R subjected to high shear was similar andsatisfactory. Even though the columns failed in shear/flexuremode, they were ductile and achieved the designdisplacement ductility capacity of 5;

4. Vertical cracks in the interlocking region were observedunder small earthquakes in the column with high shear anddi of 1.5R. The large area of plain concrete in the interlockingzone is susceptible to cracking when di is 1.5R and thecolumn shear is relatively high. The addition of horizontalcrossties connecting the interlocking hoops not only reducedand delayed vertical cracks in the interlocking region, butalso reduced the strength degradation;

5. The measured displacement ductility capacity wasapproximately 4 in columns with high shear and a di of 1.5R.Even though the desired ductility capacity was 5, the columnis considered to be sufficiently ductile for most applications.Crossties are recommended to reduce premature verticalcracking in these columns; and

Table 2—Data for plastic hinge length

Variables

Specimen

ISL1.0 ISL1.5 ISH1.0 ISH1.25 ISH1.5 ISH1.5T

φp , Rad/mm(Rad/in.)

0.204(0.008)

0.159(0.006)

0.124(0.005)

0.116(0.005)

0.101(0.004)

0.074(0.003)

Δy, mm (in.) 16.901(0.67)

18.172(0.72)

21.1(0.83)

21.1(0.83)

32.1(1.26)

26.7(1.05)

Δu, mm (in.) 161(6.34)

188(7.4202)

98.6(3.88)

105(4.15)

127(5.02)

102(4.00)

L , mm (in.) 1473(58)

1828(72)

1473(58)

1600(63)

1753(69)

1753(69)

lp , mm (in.) 351(13.8)

428(16.84)

363(14.3)

384(15.1)

480(18.9)

541(21.3)

Fig. 15—Maximum average strain in the spirals.

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ACI Structural Journal/July-August 2007 401

6. The plastic hinge length to match the measured plasticlateral displacement increased as the distance of the spiralssets increased from 1.0R to 1.5R by 10 to 20%, depending onthe level of shear.

ACKNOWLEDGMENTSThe research presented in this paper was sponsored by the California

Department of Transportation. The dedicated assistance of P. Laplace, J.Pedroarena, and P. Lucas of the University of Nevada-Reno bridgelaboratory is gratefully acknowledged. Specials thanks are expressed to N.Wehbe of South Dakota State University for developing a moment-curvatureanalysis program for interlocking spiral columns.

REFERENCES1. California Department of Transportation, “Bridge Design Specifications,”

Engineering Service Center, Earthquake Engineering Branch, Calif.,July 2000, 250 pp.

2. California Department of Transportation, “Seismic Design Criteria

Version 1.2,” Engineering Service Center, Earthquake Engineering Branch,Calif., Dec. 2001, 133 pp.

3. Tanaka, H., and Park, R., “Seismic Design and Behavior of ReinforcedConcrete Columns with Interlocking Spirals,” ACI Structural Journal, V. 90,No. 2, Mar.-Apr. 1993, pp. 192-203.

4. Buckingham, G. C., “Seismic Performance of Bridge Columns withInterlocking Spirals Reinforcement,” MS thesis, Washington State University,Pullman, Wash., 1992, 146 pp.

5. Benzoni, G.; Priestley, M. J. N.; and Seible, F., “Seismic Shear Strengthof Columns with Interlocking Spiral Reinforcement,” 12th World Conferenceon Earthquake Engineering, Auckland, New Zealand, 2000, 8 pp.

6. Correal, J.; Saiidi, M.; and Sanders, D., “Seismic Performance of RCBridge Columns Reinforced with Two Interlocking Spirals,” Report No.CCEER-04-6, Center for Civil Engineering Earthquake Research, Departmentof Civil Engineering, University of Nevada-Reno, Reno, Nev., Aug. 2004,438 pp.

7. Correal, J., and Saiidi, M., “Lessons Learned from Shake Table Testing ofRC Columns in Relation to Health Monitoring,” IMAC-XXIII—A Conference& Exposition on Structural Dynamics—Structural Health Monitoring,Orlando, Fla., 2005, 9 pp.

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ACI Structural Journal/July-August 2007 503

DISCUSSION

The authors have presented an interesting paper on theshear strength of reinforced concrete T-beams withouttransverse reinforcement. However, the discusser would liketo offer the following comments:

1. The authors have mentioned the basic outline of aderivation of Eq. (1), but Eq. (1) appears to be based on theneutral axis (NA) located at the center of the beam in atypical homogeneous rectangular concrete beam. Theauthors’ Eq. (2) was a simplification of Eq. (1) based on theexperimental database of reinforced concrete beams, whichis inconsistent with the Rankine’s failure criteria of a plainhomogeneous concrete beam.

Based on ft = 6 (or 0.1fc′ ) and assuming the Rankine’sfailure criteria of plain concrete, and by considering variousstrength ratios of flexural stress σm versus concrete compressivestress fc′ (σm/f ′c = 14.2%,10 the flexural stress σm of a plainhomogeneous concrete beam equals to 114.2%10 of the tensilestrength of plain concrete ft . Based on the aforementionedassumptions, the discusser arrived at the authors’ Eq. (2)without considering the experimental database of reinforcedconcrete beams.

Another simplified approach is that Eq. (2) can also bederived from the current ACI Building Code9 (that is,authors’ Eq. (5)) by assuming an average depth of NA equals0.4d11,12 and by substituting c = 0.4d in the authors’ Eq. (5),which would result in authors’ Eq. (2). Based on the afore-mentioned two approaches, the discusser believes that thereis no need to have a reinforced concrete beam database, thatis, Fig. 1 and 2. Is this consistent with the shear strength ofreinforced concrete T-beams without transverse reinforce-ment plain concrete?

2. The authors’ concept on shear funnel (Fig. 8 and 10) issomewhat unclear. Please note that there is no reinforcementwithin the compression and/or flange area. Based on Fig. 8,considering a simplified approach, a portion of the cross-sectional area above NA in the T-beam could be convertedinto an equivalent rectangular section, but not the entiresection of the T-beam when a shear force is computed. Thediscusser has computed over 100 specimens of T-beamsfrom Reference 1 by assuming the flange depth as one unitand the overall depth and web width were transferred into theflange depth units with varying depths of NA (that is, NAwas assumed within the flange and within the web of theT-beams) and found that approximately 20% of the cross-sectional area increases above NA as compared with itsequivalent rectangular section and approximately 10% of thecross-sectional area increases to its equivalent rectangularsection, if the entire beam was compared with the rectangularsection. These values are somewhat inconsistent in theauthors’ Table 2.

REFERENCES10. Kato, K., Concrete Engineering Data Book, Nihon University,

Koriyama-City, Fukushima Prefecture, Japan, 2000.11. Eurocode No. 2, “Design of Concrete Structures, Part 1: General

Rules and Rules of Buildings,” ENV 1992-1-1, Commission of the EuropeanCommunities,1991.

12. British Standard Institution, “Structural Use of Concrete, Part 1:Code of Practice for Design and Construction,” BS 8110:Part 1:1997, London,UK, 1997.

AUTHORS’ CLOSUREThe authors thank the discusser for his interest in this

paper. The comments are addressed in the same order aspresented by the discusser.

The detailed derivation of Eq. (1) and its simplificationinto Eq. (2) are presented in Reference 2 of the paper. Thisderivation was not based on the neutral axis located at thecenter of a beam, but rather based on the location of theneutral axis as calculated based on a cracked section analysis.The discusser is referred to Reference 2 for further clarification.

As noted in Reference 2, Eq. (1) was derived consideringthat failure initiates when the principal stress in the compressionzone reaches the tensile strength of concrete ft. It was shownthat this equation could be simplified for an assumed tensilestrength (6 ) and considering the flexural stress σm. Theexperimental results, however, were considered so that acomplete perspective of the performance of the simplifiedexpression could be accessed.

The discusser notes that Eq. (2) can be derived from theACI code. It appears that the discusser is referring to ACIEq. (11-3) rather than (11-5). Perhaps a better view is thatEq. (11-3) is a subset of Eq. (2). For k = 0.4, Eq. (2) simplifiesit to 2 bwd. The neutral axis depth, c = kd varies accordingto the flexural reinforcement ratio ρ and the modular ratio n.Therefore, Eq. (2) accounts for the reinforcement ratio andthe concrete compressive strength, whereas ACI 318 Eq. (11-3)is insensitive except with respect to its inclusion in the term .

Unfortunately, the discusser’s question “Is this consistentwith the shear strength of reinforced concrete T-beamswithout transverse reinforcement plain concrete?” is unclearand cannot be addressed.

The results presented in Fig. 10 are based on an angledapproach using a 45-degree angle. Simplification can beachieved using an effective flange width approach. Based onthe area achieved from the 45-degree shear funnel, an effectiveoverhanging flange width of 0.5 times the flange depth oneach side of the web can be considered for shear. If theneutral axis falls within the thickness of the flange, thiseffective width approach is conservative. It should be notedthat in either the shear funnel or equivalent flange widthapproach, the neutral axis depth should be computed usingan effective flange width that is based on flexural behavior

fc′

fc′

fc′

fc′

Disc. 103-S67/From the Sept.-Oct. 2006 ACI Structural Journal, p. 656

Shear Strength of Reinforced Concrete T-Beams without Transverse Reinforcement. Paper by A. KorayTureyen, Tyler S. Wolf, and Robert J. Frosch

Discussion by Himat SolankiProfessional Engineer, Building Dept., Sarasota County Government, Sarasota, Fla.

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504 ACI Structural Journal/July-August 2007

and that is different from the flange width considered effectivefor shear.

Table 2 presents a statistical comparison of the performanceof the various design methods considering the ratio of Vtest/Vcalc. Therefore, it is unclear what inconsistencies thediscusser is referring to. However, as emphasized in thepaper, for the evaluation of the shear area when the flangeswere ignored, the neutral axis depth was calculated ignoringthe flanges while the shear funnel approach computed the

neutral axis depth with the flanges considered. This mayexplain the perceived inconsistencies in the discusser’s anal-ysis if he was directly comparing the results provided inTable 2. Regardless, the main premise is that additionalshear area beyond that bounded by the web can be consideredas effective in shear transfer. The percentage of flange areaconsidered will vary depending on the section considered andthe location of the neutral axis.

Disc. 103-S71/From the Sept.-Oct. 2006 ACI Structural Journal, p. 693

Shear Strength of Reinforced Concrete T-Beams. Paper by Ionanis P. Zararis, Maria K. Karaveziroglou, andProdromos D. Zararis

Discussion by Himat SolankiProfessional Engineer, Building Dept., Sarasota County Government, Sarasota, Fla.

The authors have presented an interesting concept in theirpaper on shear strength of reinforced concrete T-beams.However, the discusser would like to offer the followingcomments:

1. The authors have considered εco = 0.002 and fct =0.30f ′c

2/3 (Eq. (4)) based on Reference 10, but no considerationwas given to the depth of compression zone equals 0.8cvalue as suggested in Reference 10. Also, the authors havenot thoroughly explained the assumption of 0.667c value (inthe Appendix) other than the test result values versus theirtheoretical values. Please note that BS 8110:Part 1:199715

considers the depth of compression zone equal a value of0.9c.

2. Based on Eq. (7) and Fig. 7, the authors assumption fora 45-degree projection angle from web to flange appears tobe inconsistent with Fig. 6(b) and other researchers. The45-degree projection angle was a simplified assumptionbased on the depth of compression equals the depth offlange, that is, the neutral axis (NA) is located at the interfaceof the bottom of flange and the top of web.

3. In conclusion, the authors’ statement “An increase ofstirrups does not give any advantage to T-beams over therectangular beams” is a little confusing without thoroughexplanation, because the authors have converted a T-beaminto a rectangular beam with bef web width in lieu of bw webwidth. Let’s consider beam pairs from Table 1: Beam PairTA11-TA12 of Reference 2; Beam Pair T2-T3 and BeamPair T15-T16 of Reference 4; Beam Pair T3a-T3b ofReference 5; and Beam Pair A00-A75 of Reference 7. Thesebeam pairs all have test parameters such as concrete strengthfc′ , longitudinal reinforcement ρ%, and shear span-to-depth ratio a/d approximately identical, except for theshear reinforcement ρv fvy; but the shear strength increaseswith an increase in shear reinforcement ρv fvy. This meansthe shear reinforcement ρv fvy does have some influence onthe shear strength.

4. The authors’ Eq. (9) and the calculated values of A ′s ofthe depth of compression block in Fig. A (of the authors’Appendix) are unclear. It appears that the authors haveconsidered a routine rectangular beam with compressionreinforcement but have not considered the reinforcementwithin the flange width when a T-beam section wasconverted into a rectangular beam section above NA. The

reinforcement in the flange would improve the value of c(depth of NA) as well as the value of Vcr in Eq. (8) and Vu inEq. (10).

5. The discusser has calculated all T-beams exceptBeam ET1, which is a rectangular beam from References 1 and2, as outlined in the authors’ Table 1, by considering thereinforcement in the flange width and by using authors’Eq. (10) for a calculation of NA, c, and then Vcr and Vu werecalculated. Based on this concept, a mean value of Vu,exp/Vu,thof 1.006 and a standard deviation value of 0.05 were found.It was also noticed from Table 11,2,4-8 that the thinner webwidth with higher reinforcement ratios (both longitudinaland shear reinforcement ratios) do not have any advantage overwider web width with lower reinforcement ratio in T-beams.

REFERENCES15. British Standard Institution, “Structural Use of Concrete, Part 1:

Code of Practice for Design and Construction,” BS 8110:Part 1:1997,London, UK, 1997.

AUTHORS’ CLOSUREThe authors would like to thank the discusser for his

interest in the paper and his kind comments. The authorswould like to reply to his comments in the order they are asked.

In the case of rectangular or T-section beams, the truedistribution of stresses in the compressive zone is usuallyreplaced for simplification by an equivalent rectangularstress block. In the ultimate limit state, that is, when thecompressive strain in concrete at extreme fiber is εc = 0.0035,the true distribution of stresses in the compressive zonefollows a parabola-rectangular diagram. Then, the compressiveforce of concrete, as a resultant of stresses, is Fc = 0.81bcfc′ .Thus, the equivalent rectangular stress distribution has anapproximate height equal to 0.8c. In this case, however, theauthors choose a state where the strain of concrete at extremefiber is εco = 0.002. This strain corresponds in a true, exactlyparabolic distribution of stresses in the compressive zone. Inthis case, the corresponding compressive force of concrete isFc = 0.667bcfc′ . Thus, the equivalent rectangular stressdistribution (shown in Fig. A) has a height equal to 0.667c.

There has never been made a 45-degree projection angle bythe authors. As it is written in the text of the paper, the failureoccurs due to a splitting of concrete that takes place in thecompression zone of the T-beam. Taking into account Fig. 2

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and 6, the splitting takes place in an inclined area, the projectionof which, on a cross section of the beam, is approximatelydefined from the shaded part of the section in Fig. 7.Equation (7), giving the effective width, results simply fromthe area of this shaded part of cross section of the T-beam.

This statement means that the contribution of stirrups in theshear strength is the same for T-beams and rectangular beams,as it results from the second part of Eq. (10). The increase inthe strength of the beams that the discusser has mentioned isdue to an increase of the first part of Eq. (10).

The compression reinforcement As′ within the flangewidth has been considered and takes part in Eq. (9) with the

ratio ρ′ = As′ bwd. Nevertheless, an increase of As′ does notincrease the shear strength of a beam; on the contrary, itdecreases the shear strength, exactly because the reinforcementAs′ improves the value of c. Equations (8) and (10) show thata decrease of the depth c decreases the strength. This hasbeen observed both in T-beams and rectangular test beams.

The compression reinforcement A ′s has not been consideredin the calculations of Table 1, because of the lack of dataregarding this reinforcement for all the test beams. As itresults from the discusser’s calculations, however, the smallratios of ρ′ have only a small effect on the shear strength.

Disc. 103-S76/From the Sept.-Oct. 2006 ACI Structural Journal, p. 736

Effect of Reinforced Concrete Members Prone to Shear Deformations: Part I—Effect of Confinement.Paper by Suraphong Powanusorn and Joseph M. Bracci

Discussion by Himat SolankiProfessional Engineer, Building Dept., Sarasota County Government, Sarasota, Fla.

Though the authors have presented an interesting concepton shear deformations in their paper, they have not fullyexplained all necessary assumptions other than the use ofMander et al.’s methodology. Also, the authors have notprovided the details as outlined by Mander et al. (1988).Without a detailed explanation and information, particularlyof the test specimens supplemented by the associatedassumptions, it is very difficult to verify the author’s resultsas well as published results available elsewhere; therefore, thediscusser has the following comments:

1. The discusser has tried to understand the authors’methodology, and has described the authors’ methodologyto the best as follows. In the following concept, there areseveral assumptions that were neither mentioned by Manderet al. (1988), nor by the authors.

The authors’ Eq. (12) is expressed as

where

in which

ε1 = εs + (εs + 0.002)cot2α

Ferguson (1964) suggested that the stress in steel developsfrom 1.15fy to 1.20fy. Therefore, an average value of 1.175fywas considered. That is, εs = 1.175fy/Es, where Es = 29,000 ksi.

Furthermore, it was assumed that the tensile strain iscausing approximately a 35-degree skew angle crack to thestrut’s axis. The 35 degrees falls within the range from 25 to45 degrees, and this angle is consistent with Cusson andPaultre (1994) and Fig. 5 and 13 of Ferguson (1964): ε1 =

0.0024 + (0.0024 + 0.002) cot2 35 degrees = 0.0114. Now,εcc = εco[1 + R((f ′cc/f ′co) – 1)].

Based on the test results of Mander et al. (1988) andScott et al. (1980), f ′cc/f ′co ≈ 1.75 and εco ≈ 0.002 (Richartet al. 1928).

In the previous equation, the R value varies from 3 to 6(Park and Paulay 1990). Based on the authors’ Fig. 1 and 2,the transverse reinforcement details with respect to the longi-tudinal reinforcement, R = 5, as suggested by the authors intheir Eq. (14) appears to be on the low side. Therefore, R= 6 was appropriate and was assumed in the aforementionedequation by the discusser. That is, εcc = 0.002 [1 + 6((1.75) – 1)]= 0.011.

Based on the Mander et al. (1988) and Scott et al. (1980)test results, εcc ≈ 0.0115.

Based on an average value of εcc = 0.01125

Also, based on an average value of εcc = 0.01125 and εc ≈0.0048 was chosen due to lateral expansion (biaxial tension-compression)

x = εc/εcc = 0.0048/0.01125 = 0.425

Esec = f ′co/εcc = 1.75f ′co /3.52εco

≈ 0.5Ec

Now, r = = = 2.0Ec

Now σc =

σcβf ′cc xr

r 1– xr+----------------------=

β1

0.8 0.34 ε1

εcc⁄( )+--------------------------------------------- 1≤=

β1

0.8 0.34 ε1

εcc⁄( )+--------------------------------------------- 1≤=

Ec

Ec Esec–---------------------

Ec

Ec 0.5Ec–-------------------------

βf ′cc xr

r 1– xr+----------------------

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506 ACI Structural Journal/July-August 2007

Substituting β, x, and r values in the previous equation

Because f ′cc ≈ 1.75fc′

σc = 1.1008fc′ or ≈ 1.10fc′

This means approximately 10% compressive stressincreases due to the confinement. This value is consistentwith Vecchio’s (1992) concept as well as the authors’ testsresults as shown in Tables 1 through 3.

Based on Vecchio’s study (Vecchio 1992), an averagestress in shear panels was increased by approximately 5.6%,while an average stress in shearwalls was increased byapproximately 13.4%, that is, an overall average valueincreased in stress would be 9.5%. Is this consistent with themethodology/concept/logic used in this paper?

2. Based on Fig. 1(a), the authors have considered asymmetrical loading case, but the symmetrical loading casemay not be the case for all structures in the practice. Becauseasymmetrical loading conditions would create unbalancedloading, it would require some additional reinforcement pertruss analogy in the dark area, as shown by the authors inFig. 9(a) and (b), depending on the unbalanced load due tothe asymmetrical loading condition.

3. It is unclear how the theoretical values stated in Tables 1through 3 were calculated. Was any correction for variabledepth considered? Or was a uniform depth considered?Though the authors stated the advantage of overlapping stirrupsversus single stirrups, the effectiveness of stirrups ascompared with the longitudinal reinforcement was unclearfrom Table 1 through 3.

4. The discusser would like to point out that because theshear strength and shear deformations relate to the strengthof concrete, a simplified method proposed by Muttoni(2003) could be extended to the authors’ specimens.

5. Using the aforementioned concept outlined in thisdiscussion and Muttoni’s (2003) methodology, the discusserhas also analyzed other test specimens available in the literatureelsewhere (Rodrigues and Muttoni 2004; Fukui et al. 2001;Ferguson 1964). The results are found to be in good agreementwith the test results. Due to brevity, the results are notincluded in the discussion.

ACKNOWLEDGMENTSThe discusser gratefully appreciates S. Unjoh, Leader, Earthquake

Engineering Team, Public Works Research Institute, Tokyo, Japan; A.Muttoni, Institut de Structures, Ecole Polytechnique Fédérale de Lausanne,Lausanne, Switzerland; and N. Pippin and A. Wards, TTI, Texas A&MUniversity, College Station, Tex., for providing publications related to theshear strength of beams.

REFERENCESFukui, J.; Shirato, M.; and Umebara, T., 2001, “Study of Shear Capacity

of Deep Beams and Footing,” Technical Memorandum No. 3841, PublicWorks Research Institute, Tokyo, Japan. (in Japanese)

Cusson, D., and Paultre, P., 1994, “High Strength Concrete ColumnsConfined by Rectangular Ties,” Journal of Structural Engineering, ASCE,V. 120, No. 3, Mar., pp. 783-804.

Mander, J. B.; Priestley, M. J. N.; and Park, R., 1988, “Observed Stressand Strain Behavior of Confined Concrete,” Journal of Structural Engineering,ASCE, V. 114, No. 8, Aug., pp. 1827-1849.

Muttoni, A., 2003, “Schubfestigkeit und Durchstanzen von Platten ohneQuerkraftbewehrung,” Beton und Stahlbetonbau, V. 98, No. 2, Feb., pp. 74-84.

Park, R., and Paulay, T., 1990, “Bridge Design and Research Seminar:

V. I, Strength and Ductility of Concrete Substructures of Bridges,” RRBulletin 84, Transit New Zealand, Wellington, New Zealand.

Richart, F. E.; Brandtzaeg, A.; and Brown, R. L., 1928, “A Study of Failureof Concrete under Combined Compressive Stresses,” Bulletin 185, Universityof Illinois Engineering Experimental Station, Champaign, Ill.

Rodrigues, R. V., and Muttoni, A., 2004, “Influence des DéformationsPlastiques de l’Armature de Flexion sur la Résistance a l’Effort Trenchantdes Pouters sans étriers: Rappart d’essai,” Laboratoire de Construction enBéton (IS-BETON), Istitut de Structures, Ecole Polytechnique Fédérale deLausanne, Oct.

Scott, B. D.; Park, R.; and Priestley, M. J. N., 1980, “Stress-StrainRelationships for Confined Concrete: Rectangular Sections,” ResearchReport 80-6, Department of Civil Engineering, University of Canterbury,Christchurch, New Zealand, Feb.

AUTHORS’ CLOSUREThe authors would like to express a sincere gratitude to the

discusser for comments that give the authors an opportunityto clarify certain issues in the article. The authors’ responseto the discusser is as follows:

GeneralThe purpose of the article under discussion was to present

an alternative method that incorporates the effects ofconfinement into the constitutive equations of the ModifiedCompression Field Theory (MCFT), first proposed byVecchio and Collins (1986). In essence, the extension of theMCFT proposed by the authors is based on two-dimensionalstress and strain analysis. All necessary assumptions werestated at the beginning of the article under the sectionProposed analytical model.

Response to discusser comments1. The discusser demonstrates the application of Eq. (12)

on the constitutive relationship of concrete in compressiontaken into account the effect of confinement given in thepaper with assumptions on a few parameters shown in theequation. It was concluded that the results from applyingEq. (12) led to an approximate 10% increase in compressivestrength of concrete, which was compared with a study byVecchio (1992) on shearwalls and panels and also by theauthors’ reinforced concrete (RC) bent cap tests. From theauthors’ point of view, however, the application of Eq. (12)alone to obtain an increase in strength is only part of thecomparative study. It is the force-deformation behavior thatis important for comparative purposes, especially formembers prone to shear deformations near ultimate loading.

MCFT is generally developed on the basis of: 1) two-dimensional states of stress and strain; 2) the superpositionof stresses in the concrete and reinforcing steel as shown inEq. (1); and 3) the compatibility of strains in the concrete andreinforcing steel as shown in Eq. (2). The model can becategorized into the so-called rotating crack model to maintainthe coaxiality between the concrete principal stresses andprincipal directions. For two-dimensional states of stress andstrain, three components of stresses and strains, which are εx,εy, and γxy and σx, σy, and τxy, are required to define a stateof stress and strain at a given point within the member. Theconstitutive relationships under the context of MCFT,however, have been defined in the principal stress and straincomponents (σ1, σ2) and (ε1, ε2). The general state of stressand strain, εx, εy, and γxy and σx, σy, and τxy, are related to theprincipal stress and strain components (σ1, σ2) and (ε1, ε2)using Mohr’s circle of stress and strain. The concreteconstitutive equation in compression defined in the principalstress and strain directions are given in Eq. (4) through (8)and (11) through (13). The special emphasis of the article is

σc0.8737( )f ′cc 0.425( ) 2.0( )

2.0 1– 0.425 2.0( )+------------------------------------------------------------ 0.629f ′cc= =

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ACI Structural Journal/July-August 2007 507

on the incorporation of the beneficial effects of lateralconfinement of the transverse reinforcement on the concretestress-strain relationship in the principal compressive directionusing an approach adopted by Mander et al. (1988) using thefive-parameter failure surface derived by Willam andWarnke (1974). Due to space limitations, the authors did notinclude the complete development of five-parameter failuresurface in the article. Interested readers should consult theoriginal paper by Willam and Warnke (1974) or books byChen (1982), Chen and Han (1988), and Chen and Saleeb(1982) for further details.

Regarding the discusser’s comments on the R value fordetermining the peak strain corresponding to the peakconcrete stress, additional studies by the authors have shownthat the use of R = 6 led to only a marginal change in thestrength prediction.

2. The MCFT was formulated on the basis of threefundamental principles of structural mechanics, which are:1) equilibrium; 2) compatibility; and 3) material constitutiverelationships. The rationality and generality of the MCFTshould make the theory applicable to any loading pattern.The case of unsymmetric loading, however, was not considered

in this work and would require further experimental andanalytical research to justify recommendations.

3. To justify the proposed model, the authors implementedthe proposed model into a finite element code using a user-defined material subroutine. It is the results from FEM analysisthat are summarized in Tables 1 through 3.

4 and 5. The authors agree with the discusser that the shearstrength and deformation are related to the compressivestrength of concrete and would like to look into furtherdetails on the article by Muttoni (2003).

REFERENCESChen, W.-F., 1982, Plasticity in Reinforced Concrete, McGraw-Hill, New

York, 474 pp.Chen, W.-F., and Han, D. J., 1988, Plasticity for Structural Engineers,

Springer-Verlag, New York, 606 pp.Chen, W.-F., and Saleeb, A. F., 1982, “Constitutive Equations for Engineering

Materials,” Elasticity and Modeling, V. 1, John Wiley & Sons, New York. Mander, J. B.; Priestley, M. J. N.; and Park, R., 1988, “Theoretical Stress-

Strain Model for Confined Concrete,” Journal of Structural Engineering,ASCE, V. 114, No. 8, pp. 1804-1826.

Willam, K. J., and Warnke, E. P., 1974, “Constitutive Model for the TriaxialBehavior of Concrete,” Concrete Structures Subjected to Triaxial Stresses,Paper III-1, International Association of Bridge and Structural EngineersSeminar, Bergamo, Italy, pp. 1-30.

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ACI Structural Journal, V. 104, No. 4, July-August 2007.MS No. S-2006-214 received May 27, 2006, and reviewed under Institute publication

policies. Copyright © 2007, American Concrete Institute. All rights reserved, includingthe making of copies unless permission is obtained from the copyright proprietors. Pertinentdiscussion including author’s closure, if any, will be published in the May-June 2008ACI Structural Journal if the discussion is received by January 1, 2008.

ACI STRUCTURAL JOURNAL TECHNICAL PAPER

An alternative approach for predicting the punching shear strengthof concentrically loaded interior slab-column connections using fuzzylearning from examples is presented. A total of 178 experimentaldatasets obtained from concentric punching shear tests of reinforcedconcrete slab-column connections from the literature are used intraining and testing of the fuzzy system. The fuzzy-based model isdeveloped to address the interaction between various punchingshear modeling parameters and the uncertainties between them,which might not be properly captured in classical modelingapproaches. The model is trained using 82 datasets and verifiedusing 96 datasets that are not used in the training process. Thepunching shear strength predicted by the fuzzy-based model iscompared with those predicted by current punching shear strengthmodels widely used in the design practice, such as ACI 318-05,Eurocode 2, CEB-FIP MC 90, and CSA A23.3-04 codes. It is notedthat the fuzzy-based model yields a significant enhancement in theprediction of the punching shear strength of concentrically loadedinterior slab-column connections while still respecting the funda-mental failure mechanisms in punching shear of concrete.

Keywords: fuzzy systems; punching shear; slab-column connections.

INTRODUCTIONFlat plates consist of slabs directly supported on the

columns without beams. For this simple appearance, flatplate systems have various economic and functional advantagesover other floor systems such as fast construction, low storyheight, and irregular column layout. From a viewpoint ofstructural mechanics, however, flat plates are structures ofcomplex behavior. Moreover, flat plates usually fail in abrittle manner by punching at the slab-column connectionswithin the discontinuity region known as the D-region.1,2 Atthese connections, three-dimensional stresses are developeddue to the combined high shear and normal stresses creatinga stress state that is complex to analyze accurately.3

For the last three decades, a significant amount of researchhas been performed to investigate this complex problem ofconcentric punching shear of reinforced concrete flat platesby using various methods ranging from mechanical modelsup to purely empirical models. In early models includingYitzhaki4 and Long and Rankin,5 punching shear strengthwas defined considering the flexural capacity of reinforcedconcrete slabs. This was based on the experimental observationthat the punching shear strength was close to the flexuralcapacities of the concrete slabs. Pralong6 and Nielsen7

derived lower bound and upper bound values for punching shearstrength based on the theory of plasticity. These formulationsdid not consider the effect of flexural reinforcement on thepunching shear strength. Kinnunen and Nylander8 developedthe first mechanical model for punching shear strength usingfailure criteria based on the observation of shear cracks in theexperiments. In this model, the failure criteria were definedby the inclined radial compressive stress and the tangential

compressive strain at the shear crack. Even though Kinnunenand Nylander’s model8 did not provide high accuracy inpunching shear strength predictions, it significantly contributedto a better understanding of the failure mechanism of theslab-column connections and enabled visualizing a rationalflow of forces in such connections. Alexander andSimmonds2 proposed a strut-and-tie model with concrete tiesto describe the load transfer in the slab-column connections.Bažant and Cao9 developed a punching shear strength modelconsidering size effect of concrete based on principles offracture mechanics. The size-effect model was able toexplain the experimental observations of decreasingpunching failure shear stresses of slab-column connectionswithout reinforcement with increasing slab thickness.

Numerous models suggested modifications to these generaldirections outlined previously (flexure, combined stress-strength criteria, plasticity, strut and tie, and size effect). Arecent review of such models can be found elsewhere.10 Inspite of the importance of these models in understanding thefailure mechanism of slab-column connections, there isconsiderable difficulty in using these models in the dailydesign practice. Moreover, the level of complexity encounteredin using these models for design might be difficult to justifygiven the fact that most of these models do not usually showhigh accuracy in the prediction of punching shear strength.11

To develop simple strength equations, most design codesuse the so-called control perimeter approach12-15 depicted inFig. 1. The applied punching shear stress is calculated at adefined critical perimeter and compared with an allowedvalue based on the calibration of existing test results. Thevarious design codes show significant difference in definingthe location of the critical section as well as the allowedpunching shear stress. It becomes apparent that the complexityof the punching problem and the dependence of the punchingshear strength on a number of interacting variables necessitatethe use of empirical modeling approach to estimate the punchingshear strength. While classical empirical techniques used bymany design codes show limited accuracy, a more robustempirical modeling technique that respects fundamentalfailure mechanisms of the punching shear is needed.

RESEARCH SIGNIFICANCEThe present study introduces a new approach for predicting

the punching shear strength of concentrically loaded interiorslab-column connections using fuzzy learning from examples.The proposed approach incorporates the control perimeter

Title no. 104-S42

Simplified Punching Shear Design Method for Slab-Column Connections Using Fuzzy Learningby Kyoung-Kyu Choi, Mahmoud M. Reda Taha, and Alaa G. Sherif

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approach and targets predicting the punching shear strengthof the slab-column connections based on various geometricand material parameters. The proposed fuzzy-based modelpresented in a simple form respects the failure mechanics ofpunching shear by learning its rules from the experimentaldatabase with the ability to address the interaction betweenthe modeling variables and the uncertainty in these variables.The fuzzy-based model shows high accuracy in predictingpunching shear strength.

FUZZY LEARNING OF PUNCHINGSHEAR DATABANK

Fuzzy systems have been widely used in the last decadefor modeling complex engineering systems (for example,modeling robots16 and in assessing concrete durability17)and their feasibility as universal approximators has beenproven.18 The capability of the fuzzy systems to modelcomplex systems is attributed to their inherent ability toaccommodate a tolerance for uncertainty in the modelingparameters.19,20 While probabilistic empirical models arelimited to random uncertainties, fuzzy systems have the abilityto consider random and nonrandom types of uncertaintiesthat arise due to vagueness and/or ambiguity in the modelingparameters/process.18-20

The fundamental concept in modeling complex phenomenausing fuzzy systems is to establish a fuzzy rule-base that iscapable of describing the relationship between the inputparameters and the output parameters while consideringuncertainty bounds.19 This fuzzy rule-base captures individualand group relationships that distinguish the internal complexrelations between the system parameters.20 As such, systemnonlinearity is not recognized by using nonlinear equationsbut through establishing a number of fuzzy rules (that coulduse linear relations) such that the fuzzy system becomescapable of describing the phenomena to a pre-specified levelof accuracy.20 A group of successful techniques to establish afuzzy rule-base using exemplar observations was recentlydeveloped.20,21

Here, the use of the fuzzy set theory to model the punchingshear strength of a slab-column connection is demonstrated.Preliminary investigations using Bayesian analysis ofsignificance22 have been performed to identify the mostprimary input parameters that have a significant influence onthe punching shear strength. Possible parameters included

concrete compressive strength, slab thickness and effectivedepth, span length, column geometry, punching shearperimeter, and compression and tension reinforcementratios. Assuming the geometry of punching shear perimeterto be known a priori, the Bayesian analysis showed that forcircular and square columns (c1/c2 ratio equals to 1.0), themost significant parameters that affect the punching shearstrength are concrete compressive strength fc′, slab thicknessh, and tension reinforcement ratio ρ. The assumption of thepunching shear perimeter to be known a priori is based on thefact that the punching shear databank does not includedetailed information about the failure pattern and thepunching shear perimeter. This hinders the ability to learnthe failure patterns of slab-column connections as part of thenew model. It is also noted that the results of Bayesian analysisshowed that the compression reinforcement does not have asignificant effect on the maximum punching shear strength.This finding is in agreement with the literature8,23 showingthat the primary effect of compression reinforcement is onpost-punching behavior providing a membrane action.Hereafter, these three parameters have been used as inputparameters to the fuzzy-based model for predicting thepunching shear strength. By considering these three parameters,the fuzzy-based model considers the major criteria on punchingshear examined by many researchers.24-28 These include shearstrength and cracking capacity conventionally representedby the cubical or square root of the compressive strength,6,24,27

size effect related to slab thickness,9 and membrane effect28

represented by the flexural reinforcement ratio.While the ratio of the column dimensions of rectangular

columns and the perimeter-to-depth ratio (bo/d) have beenreported to affect the punching shear strength of slab-columnconnections,24,29 the experimental database for rectangularcolumns or for slabs with significantly large perimeter-to-

ACI member Kyoung-Kyu Choi is a Research Assistant Professor at the University ofNew Mexico, Albuquerque, N. Mex. He received his BE, MS, and PhD in architecturefrom Seoul National University, Seoul, Korea. He is an associate member of ACICommittees 440, Fiber Reinforced Polymer Reinforcement; 548, Polymers inConcrete; and Joint ACI-ASCE Committee 445, Shear and Torsion. His researchinterests include shear strength and seismic design of reinforced concrete structuresand application of artificial intelligence in structural engineering.

ACI member Mahmoud M. Reda Taha is an Assistant Professor in the Departmentof Civil Engineering at the University of New Mexico. He received BSc and MSc fromAin Shams University, Cairo, Egypt, and his PhD from the University of Calgary,Calgary, Alberta, Canada, in 2000. He is a member of ACI Committees 209, Creepand Shrinkage in Concrete; 235, Electronic Data Exchange; 440, Fiber Reinforced PolymerReinforcement; 548, Polymers in Concrete; and E803, Faculty Network CoordinatingCommittee. His research interests include structural monitoring, using artificial intelligencein structural modeling, and fiber-reinforced polymers.

ACI member Alaa G. Sherif is an Associate Professor in the Civil EngineeringDepartment, Helwan University, Mataria-Cairo, Egypt. He received his BSc fromCairo University, Cairo, Egypt, and his MSc and PhD from the University of Calgary.He is an associate member of Joint ACI-ASCE Committee 352, Joints and Connections inMonolithic Concrete Structures. His research interests include the behavior and service-ability of reinforced concrete structures and systems for multi-span cable-stayed bridges.

Fig. 1—Current design codes for punching shear.

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440 ACI Structural Journal/July-August 2007

depth ratio (bo/d > 15) is insufficient to develop the knowledgerule base that is necessary for the fuzzy-based model toconsider both effects on the punching shear strength. There-fore, first, the fuzzy-based model is trained by using theexperimental data with square and circular columns only andwith perimeter to depth ratio (bo/d) < 15. Based on this fact,prediction of the fuzzy-based model will be modified toconsider the effect of rectangularity of columns or highperimeter-to-depth ratios in excess of that used in the training(bo/d > 15) as shown in the Results and discussion section.

In the present study, the punching shear failure load ofslab-column connections without shear reinforcement Vc isdefined as

Vc = vcbod (1)

where Vc equals the punching failure load and bo equals thecritical perimeter at a distance d/2 from the column face;bo = (2c1 + 2c2 + 4d) for a square column and bo = π(D + d)for a circular column. The values c1 and c2 equal the shortand long sizes of a rectangular column, D equals the diameterof a circular column, and vc represents the average ultimatepunching shear strength, which is defined with respect todefective depth. Equation (1), although simplified, has beenadopted by almost all current design codes and respects thefundamental mechanics governing the slab-column punching

failure as observed by many researchers.6-9 The choice of thecritical perimeter to be considered at a distance d/2 from thecolumn face is attributed to the possible use of this locationto estimate the average ultimate shear strength vc for usuallyintersecting most plausible failure planes, as shown in Fig. 2,which is similar to the value (h/2) proposed in Nielsen.7

The modeling is started by defining N fuzzy sets over thedomain of each input parameter x. This definition provideseach value of the parameter x with N membership valuesrepresenting its level of belonging to the N fuzzy set . Theconcept of membership or degree of belonging represents thebasis in the formulation of fuzzy set theory.18,20,21 Themembership denoted μ (x) ranges between 0.0 and 1.0.μ (x) does not express probability of x but characterizes theextent to which x belongs to fuzzy set .20 Several methodsfor establishing membership functions with different levelsof complexity exist. While simplified methods can be usedaccording to expert opinion, complex automated methodsusing artificial neural networks or inductive reasoning areusually considered to be efficient for modeling complexphenomena.20,30 A technique is adopted herein that is basedon providing an initial definition of the fuzzy sets usingk-means clustering31 followed by the automated update ofthe fuzzy sets during the learning process.20,21

The modeling process depends on fuzzifying all threeinput domains and constructing a fuzzy rule-base, whichdescribes the relationship between the fuzzy sets defined onthe input domains and the punching shear strength using agroup of linear equations. Exemplar rule in the fuzzy rule-base can be defined as

If f ′c∈ , h ∈ , and ρ∈ , (2)

then vi = ai f ′c + bih + ciρ + di

where , , and are the k-th fuzzy set (k = 1, 2, … Nj)defined on the fuzzy domains of compressive strength f ′c,slab thickness h, and tension reinforcement ratio ρ, respectively.The value of Nj is the total number of fuzzy sets defined overthe j-th input parameter. In the present study, ρ is definedwith respect to effective depth. Equation (2) represents the i-thrule in the fuzzy rule-base. The values ai, bi, ci, and di areknown as the consequent coefficients that define the outputside of the i-th rule in the fuzzy rule-base.

A bell-shape membership function is employed to representthe fuzzy sets defined on the input domains. The use of othermembership functions (for example, gaussian and triangular) ispossible, but constrained by having a differentiable membershipfunction.21 The bell-shape membership function to representthe k-th fuzzy set of the j-th input parameter xj can bedescribed as (x).

(3)

where x kcj , w k

j , and q kj represents the center, the top width,

and the shape parameters of the membership functiondefining the k-th fuzzy set defined over the j-th input parameter.A pictorial representation of the bell-shaped membershipfunction is shown in Fig. 3. By considering the T-norm(product) operator (Π) to capture the influence of the interaction

A˜ A

˜

A˜ f

k A˜ h

k A˜ p

k

A˜ f

k A˜ h

k A˜ p

k

μA˜

k

μA˜

k xj( ) 1

1 xj xcjk–

wjk

----------------2qj

k

+

-----------------------------------=

Fig. 2—Cross section of slab-column connection showingcritical section at distance d/2 from column face to intersectmost plausible failure planes (angle θ ranges between 30and 45 degrees). Choice of d/2 allows obtaining good estimateof average ultimate punching shear strength vc.

Fig. 3—Pictorial representation of bell-shaped membershipfunction used to represent fuzzy sets defined over inputdomains.

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ACI Structural Journal/July-August 2007 441

between the input parameters32 on the output, the weight ofthe i-th rule (λi) in the fuzzy rule-base can be computed as

for i = 1...R (4)

Factors affecting the choice of the implication operator arediscussed in the following. The value T represents the totalnumber of input parameters (herein, T = 3). The number offuzzy rules R is a function of the number of input variables T andthe number of fuzzy sets Nj defined over each input domain.The punching shear strength vc can then be computed as

(5)

where vi is the output of the i-th rule in the fuzzy rule-baseand λi represents the weight of the i-th rule in the fuzzy rule-base as computed using Eq. (4).

The process for learning from example aims at extractinga knowledge rule-base from a group of input-output datasets.This knowledge rule-base can be used later to model thebehavior of the system (herein the punching shear of slab-column connections) for input datasets not used in thetraining process. While other techniques capable of buildingsimilar learning systems were reported in the literature (forexample, artificial neural networks), the advantage of fuzzysystems is being able to consider nonrandom uncertainty in themodeling process and thus yields robust modeling systems.20

The learning process starts by initializing the premiseparameters (parameters describing the membership functionsx k

cj , wkj , and qk

j ) using the k-means clustering technique.31

This is followed by computing the consequence coefficients(ai, bi, ci, and di) using least square techniques33 such that theroot mean square prediction error E of the punching shearstrength does not exceed a target root mean square predictionerror, herein 1.0 × 10–5. The root mean square predictionerror E is defined as

(6)

where vpn is the predicted punching shear strength for the n-thdataset, vdbn is the punching shear of the n-th dataset fromthe database, and Nd is the total number of training datasets.As the target mean square prediction error will not beachieved from the first learning trial (using the initial fuzzysets and consequence coefficients), the premise parametersdescribing the fuzzy sets can be updated using the gradientdescent method as

(7)

λi

Πj 1=T 1

1 xj xcjk–

wjk

----------------2qj

k

+

-----------------------------------

Σi 1=R Π j 1=

T 1

1 xj xcjk–

wjk

----------------2qj

k

+

-----------------------------------------------------------------------------------------------=

vc λivi

i 1=

R

∑⎝ ⎠⎜ ⎟⎛ ⎞

λi

i 1=

R

∑⎝ ⎠⎜ ⎟⎛ ⎞

⁄=

E

vpn vdbn–( )2

n 1=

Nd

Nd

----------------------------------------=

xcjk m( ) xcj

k m 1–( ) η ∂E m( )∂xcj m( )-------------------+=

(8)

(9)

where x kj (m), w k

j (m), and q kj (m) are the center, the top width,

and the shape of the membership function, respectively,defining the k-th fuzzy set defined over the j-th input parameterin the m-th learning epoch (trial). The values x k

j (m – 1),w k

j (m – 1), and q kj (m – 1) are the center, the top width, and

the shape of the membership function, respectively, definingthe k-th fuzzy set defined over the j-th input parameter in the(m – 1) learning epoch. The value η is the learning rate and∂E(m)/∂xj(m), ∂E(m)/∂wj(m), and ∂E(m)/∂qj(m) are components

wjk m( ) wj

k m 1–( ) η ∂E m( )∂wj m( )------------------+=

qjk m( ) qj

k m 1–( ) η ∂E m( )∂qj m( )-----------------+=

Table 1—Dimensions and properties of specimens

Investigator*†No. of specimens f ′c,

MPa h, mm ρ, %Training Verification

Hallgren and Kinnunen (1993a), Hallgren and

Kinuunen (1993b), Hallgren (1996)

3 3 79.5 to 108.8

239 to 245

0.6 to1.2

Tomaszewicz (1993) 7 6 64.3 to 119.0

120 to 320

1.5 to 2.6

Ramdane (1996), Regan et al. (1993) 4 4 28.9 to

74.2 125 1.0 to 1.3

Marzouk and Hussein (1991) 6 8 30.0 to 80.0

90 to 150

0.4 to 2.1

Lovrovich and McLean (1990) 2 2 39.3 100 1.7

Tolf (1988) 4 3 20.1 to 25.1

120 to 240

0.4 to 0.8

Regan (1986) 11 11 8.4 to 37.5

80 to 250

0.8 to 2.4

Swamy and Ali (1982) 1 1 37.4 to 40.1 125 0.6 to

0.7

Marti et al. (1977),Pralong et al. (1979) 1 1 23.1 to

30.4180 to

1911.2 to

1.5

Schaefers (1984) 1 1 23.1 to 23.3

143 to 200

0.6 to 0.8

Ladner et al. (1977),Schaeidt et al. (1970),

Ladner (1973)2 3 24.6 to

29.5110 to

2801.2 to

1.8

Corley and Hawkins (1968) 1 1 44.4 146 1.0 to 1.5

Bernaert and Puech (1996) 9 9 14.0 to 41.4 140 1.0 to

1.9

Manterola (1966) 4 4 24.2 to 39.7 125 0.5 to

1.4

Yitzhaki (1966) 5 6 8.6 to 19.0 102 0.7 to

2.0

Moe (1961) 7 7 20.5 to 35.2 152 1.1 to

2.6

Kinnunen and Nylander (1960) 6 6 21.6 to

27.7149 to

1580.5 to

2.1

Elstner and Hognestad (1956) 8 9 9.0 to

35.6 152 1.2 to 3.7

Hawkins et al.34 0 6 25.9 to 32.0

138 to 142

0.77 to 1.12

Teng et al.29 0 4 33.0 to 40.2 150 1.24

Criswell35 0 1 35.4 146 1.24

Total 82 96 8.4 to 119.0

80 to 320

0.4 to 3.7

*Reference to investigators work, unless otherwise noted, can be found in Reference 3.†Properties and dimensions of these test specimens were collected from fib Bulletin 12.3

Note: 1 MPa = 0.145 ksi; 1 mm = 0.04 in.

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of the gradient vector of the mean square prediction errorwith respect to the premise parameters of the j-th inputparameter evaluated at the m-th learning epoch. The updatedpremise parameters are then used to recompute a new set ofconsequence parameters and a new root mean square predictionerror. The process continues and the fuzzy rule-base parameters(premise and consequent parameters) are updated in eachtraining epoch until the target root mean square prediction error

or a maximum number of training epochs is reached. Theupdate process therefore allows the fuzzy-based model toreduce the root mean square prediction error and thus learnfrom examples in a much more robust manner compared withany other empirical techniques.

For training and testing of the fuzzy-based model, 178 testspecimens performed by 21 researchers as reported in the fibbulletin3 and other reports in the literature29,34,35 were used.Only specimens that were reported to fail in pure punchingshear (no flexural shear failure) were considered. A specimenreported by Lovrovich and McLean36 was excluded in thisstudy because its span length was extremely short (l1/c1 = 2).Also, six specimens by Yitzchaki,4 Elstner and Hognestad,37

and Tolf38 were also excluded because their tension rein-forcement ratios were extremely beyond practical designrange (ρ ≥ 6.9%). The specimens had two types of boundarygeometries (circular and rectangular flat plates) and two types ofcolumn shapes (circular and square columns). The dimensionsand properties of the specimens are summarized in Table 1.The test specimens had a broad range of design parameters:8.4 ≤ f ′c ≤ 119.0 MPa (1.2 ≤ f ′c ≤ 17.3 ksi), 80 ≤ h ≤ 320 mm(3.1 ≤ h ≤ 15.6 in.), 0.4 ≤ ρ ≤ 3.7%, and 5.5 ≤ bo/d ≤ 24.These data cover a wide range of the material and geometricproperties of slab-column connections. Eighty-two specimenswere used for training of the fuzzy-based model while 96specimens were used for testing the model. All specimens used inthe testing were not used in training the fuzzy-based model.

All modeling parameters were normalized to their maximumvalues determined from the database (178 data sets). Thenormalization process is necessary to avoid the influence ofnumerical weights on the learning process.39 The fuzzy rule-base that achieved the lowest root mean square error duringtraining was used for testing and verification of the model capa-bility to predict punching shear strength in slab-column connec-tions. The optimum number of fuzzy sets for each modelingparameter was developed using the k-means clustering tech-nique.31 The number of membership functions defined on thedomain of any variable x can be used to indicate the sensitivityof the model to this variable x. The higher the sensitivity of themodel to the variable x, the larger the number of membershipfunctions used to describe the variable x. It is worth noting,however, that increasing the number of membership functionsdoes not guarantee enhancing the model accuracy.20,21

It was found that the best learning represented by thelowest root mean square prediction error was achieved whileusing two fuzzy sets to represent the compressive strengthand the tension reinforcement ratio. Three fuzzy sets werenecessary for describing the slab thickness (N1 = N3 = 2,N2 = 3). The initial and final fuzzy sets, as established by thelearning algorithm, are shown in Fig. 4 and Table 2. The totalnumber of rules in the rule-base can be computed bymultiplying the number of membership functions of thethree variables as R = N1N2N3. Thus, 12 rules (R = 12) wereneeded to describe the relationship between the inputparameters: concrete compressive strength, slab thickness,tension reinforcement ratio, and the punching shear strength.While reduction of the total number of rules in the fuzzyrule-base is possible for limiting combinatorial explosion,20

researchers showed that the efficient reduction of the numberof rules shall be performed considering both accuracy androbustness of the model. Exemplar methods for rule reductionin the fuzzy rule-base include the Combs and Andrews40

method and the method suggested by Lucero41 but arebeyond the scope of this work.

Fig. 4—Fuzzy sets used to describe concrete compressivestrength, slab thickness, and tension reinforcement ratio.Before training (left) and after training (right): MF1(Membership Function 1), MF2 (Membership Function 2),and MF3 (Membership Function 3).

Table 2—Parameters describing premise parameters (membership functions)*

Compressive strength f ′c

xc, MPa (ksi) w, MPa (ksi) q

–23.83 (–3.40) 29.9 (4.34) 1.98

78.30 (11.40) 78.2 (11.30) 2.02

Slab thickness h

xc, mm (in.) w, mm (in.) q

42.05 (1.66) 68.0 (2.68) 1.982

127.07 (5.00) 89.4 (3.52) 2.011

272.58 (10.73) 126.7 (4.99) 1.994

Tension reinforcement ratio ρ

xc w q

-0.001 0.012 1.997

0.035 0.018 2.005

*For compressive strength f ′c , slab thickness h, and reinforcement ratio ρ.

A˜ f

1

A˜ f

2

A˜ h

1

A˜ h

2

A˜ h

3

A˜ ρ

1

A˜ ρ

2

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ACI Structural Journal/July-August 2007 443

RESULTS AND DISCUSSIONThe fuzzy-based model was trained using test results with

specific geometrical limits: circular and square columns andslabs with perimeter-to-slab-depth ratio (bo/d) ranging between5.8 and 14.9. Therefore, the punching shear strength of anyslab-column connection within the geometrical limitationslisted previously can be computed using Eq. (10) to (12).Equation (10) can be used to compute the weight λ for eachrule in the rule-base using the premise parameters listed inTable 1. Equation (11) presents the 12 rules forming thefuzzy knowledge rule-base.

for i = 1...12 (10)

(11)

where vi, f ′c, and h are in MPa, MPa, and mm, respectively.The punching shear strength vcf can be computed using Eq. (11)and (12)

(12)

It is important to emphasize the fact that several implicationoperators exist.42 The selection of the implication operator isgoverned by three main issues: the needed logical implication

λi

Πj 1=3 1

1 xj xcjk–

wjk

----------------2qj

k

+

-----------------------------------

Σi 1=12 Π j 1=

3 1

1 xj xcjk–

wjk

----------------2qj

k

+

-----------------------------------------------------------------------------------------------=

R 1: if fc ′ A˜ f

1 h A˜ h

1 and ρ A˜ ρ

1∈,∈,∈=

then v1 0.247fc ′ 0.008h 153.7ρ 4.90+ + +=

R 2: if fc ′ A˜ f

1 h A˜ h

1 and ρ A˜ ρ

2∈,∈,∈=

then v2 0.506fc ′– 0.026h 835.4ρ 11.42–+ +=

R 3: if fc ′ A˜ f

1 h A˜ h

2 and ρ A˜ ρ

1∈,∈,∈=

then v3 0.174fc ′ 0.028h 63.9ρ 8.12–+ +=

R 4: if fc ′ A˜ f

1 h A˜ h

2 and ρ A˜ ρ

2∈,∈,∈=

then v4 0.149fc ′ 0.031h 136.65ρ– 3.49–+=

R 5: if fc ′ A˜ f

1 h A˜ h

3 and ρ A˜ ρ

1∈,∈,∈=

then v5 0.248– fc ′ 0.001h 236.32ρ– 3.91+ +=

R 6: if fc ′ A˜ f

1 h A˜ h

3 and ρ A˜ ρ

2∈,∈,∈=

then v6 0.243fc ′ 0.006h– 53.35ρ– 3.16+=

R 7: if fc ′ A˜ f

2 h A˜ h

1 and ρ A˜ ρ

1∈,∈,∈=

then v7 0.005– fc ′ 0.031h– 84.38ρ– 3.14+=

R 8: if fc ′ A˜ f

2 h A˜ h

1 and ρ A˜ ρ

2∈,∈,∈=

then v8 0.006fc ′ 0.116h 136.57ρ– 2.67–+=

R 9: if fc ′ A˜ f

2 h A˜ h

2 and ρ A˜ ρ

1∈,∈,∈=

then v9 0.006fc ′ 0.002h– 30.73ρ– 0.05+=

R 10: if fc ′ A˜ f

2 h A˜ h

2 and ρ A˜ ρ

2∈,∈,∈=

then v10 0.021fc ′ 0.043h 19.19ρ 7.58–+ +=

R 11: if fc ′ A˜ f

2 h A˜ h

3 and ρ A˜ ρ

1∈,∈,∈=

then v11 0.001fc ′ 0.006h– 49.86ρ 1.96+ +=

R 12: if fc ′ A˜ f

2 h A˜ h

3 and ρ A˜ ρ

2∈,∈,∈=

then v12 0.004fc ′ 0.018h 36.19ρ 6.36–+ +=

vcf λivi

i 1=

12

∑⎝ ⎠⎜ ⎟⎛ ⎞

λi

i 1=

12

∑⎝ ⎠⎜ ⎟⎛ ⎞

⁄=of information, the influence of the fused output on themodel prediction, and the effect of the fusion method on thecomputational efficiency of the learning algorithm. Theproduct implication Π was selected herein for three reasons.First, to perform the fuzzy and operation as indicated byEq. (4). Second, the product implication tends to dilute theinfluence of joint membership values that are small and

Table 3—Testing to predicted punching shear strength ratio using existing design codes and fuzzy-based model

Investigator*

ACI 318-05 VTest/

Vpredicted†

CEB-FIP VTest/

Vpredicted†

Eurocode 2 VTest/

Vpredicted†

CSA A23.3-04

VTest/

Vpredicted†

Fuzzy-based model VTest/

Vpredicted†

Hallgren and Kinnunen

(1993a), Hallgren and Kinuunen

(1993b), Hallgren (1996)

0.88 to 0.98

0.83 to 1.00

0.86 to 0.97

0.93 to 1.02

0.96 to 1.07

Tomaszewicz (1993)

1.39 to 1.64

0.80 to 1.17

0.94 to 1.29

1.41 to 1.64

0.70 to 1.23

Ramdane (1996), Regan et al.

(1993)

1.46 to 1.66

1.15 to 1.31

1.20 to 1.37

1.27 to 1.47

1.25 to 1.41

Marzouk andHussein (1991)

0.71 to 1.61

1.13 to 1.84

0.97 to 1.64

0.63 to 1.40

0.91 to 1.42

Lovrovich and McLean (1990)

1.18 to 1.26

0.73 to 0.78

0.79 to 0.85

1.02 to 1.10

0.87 to 0.94

Tolf (1988) 0.88 to 1.21

0.92 to 1.34

0.82 to 1.15

0.77 to 1.05

0.91 to 1.02

Regan (1986) 1.17 to 1.78

0.97 to 1.47

1.04 to 1.47

1.02 to 1.54

0.60 to 1.29

Swamy and Ali (1982) 1.10 1.19 1.12 0.96 1.00

Marti et al. (1977), Pralong et al.

(1979)1.32 0.97 1.00 1.15 0.77

Schaefers (1984) 1.19 1.14 1.05 1.04 1.00

Ladner et al. (1977), Schaeidt

et al. (1970), Ladner (1973)

1.48 to 1.79

1.22 to 1.34

1.26 to 1.47

1.29 to 1.56

0.89 to 1.26

Corley and Hawkins (1968) 0.87 0.85 0.85 0.75 0.72

Bernaert and Puech (1996)

0.88 to 1.93

0.80 to 1.28

0.81 to 1.43

0.76 to 1.68

0.70 to 1.45

Manterola (1966) 0.88 to 1.36

0.81 to 0.96

0.85 to 0.98

0.76 to 1.18

0.65 to 0.92

Yitzhaki (1966) 1.51 to 1.98

1.01 to 1.54

1.01 to 1.53

1.31 to 1.72

0.80 to 1.16

Moe (1961) 1.24 to1.65

0.70 to 1.38

0.83 to 1.40

1.07 to 1.43

0.68 to 1.12

Kinnunen and Nylander (1960)

0.83 to1.75

0.93 to 1.23

0.92 to 1.23

0.72 to 1.52

0.85 to 1.36

Elstner and Hognestad (1956)

1.19 to2.23

0.88 to 1.20

1.05 to 1.30

1.03 to 1.94

0.79 to 1.27

Hawkins et al.34 0.90 to1.05

0.87 to 1.10

0.89 to 1.05

0.78 to 0.91

0.88 to 1.19

Teng et al.29 0.88 to1.15

0.89 to 1.15

0.92 to 1.19

0.76 to 1.00

1.02 to 1.49

Criswell35 0.94 0.89 0.96 0.82 0.86

Mean 1.375 1.098 1.139 1.219 1.019

Standard deviation 0.314 0.207 0.198 0.280 0.189*Reference to investigators work, unless otherwise noted, can be found in Reference 3.†Strength ratio (= VTest/Vpredicted), where VTest equals actual strengths (test results),and Vpredicted equals predicted strengths by current design methods (ACI 318-05,CEB-FIP, Eurocode 2, and CSA A23.3-04) or fuzzy-based model, respectively.

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444 ACI Structural Journal/July-August 2007

therefore magnify the contribution of the rules associatedwith high membership values in computing the shearstrength (Eq. (4) and (5)). This fact promoted the use of the

product operator in artificial neural networks as an efficientHebbian-type learning algorithm.20 Finally, the choice of theproduct implication was also controlled by the need to

Fig. 5—Strength prediction by current design method and fuzzy-based model. (Note: 1 MPa =0.145 ksi; 1 mm = 0.04 in.)

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ACI Structural Journal/July-August 2007 445

produce a continuous and differentiable error function (Eq. (6))to enable efficient computation of the error gradients duringthe learning process.

Table 3 presents a summary of punching shear strength ofthe specimens predicted by the fuzzy-based model. In theverification, the 96 specimens, which were not used in thelearning process, were used. Figure 5(a) shows the ratiosbetween the actual test to the fuzzy-based model predictedstrength (Vtest/Vpredicted) to have a mean value 1.019 and astandard deviation of 18.9%. Figures 5(b) to (e) show the ratiosbetween actual to predicted strength (Vtest/Vpredicted) usingthe CEB-FIP MC 90,12 the Eurocode 2,13 ACI 318-05,14 andCSA A23.3-04,15 to have mean values of 1.098, 1.139,1.375, and 1.219, respectively, with standard deviations of20.7, 19.8, 31.4, and 28.0%, respectively (refer to Table 3).The results show that the fuzzy-based model can be used topredict the punching shear strength of slab-column connectionswith various slab thicknesses, reinforcement ratios, andcircular and square columns. Moreover, higher predictionaccuracy of the fuzzy-based model can be observed comparedwith predication accuracies for all existing design codes.

It is interesting to note that, except for Eurocode 2,13 currentdesign methods show a considerable scatter represented byhigh standard deviations of test-prediction ratios. Moreover,observing Fig. 5(c), the CEB-FIP MC 90 code underestimatesthe punching shear strength of specimens with low tensionreinforcement ratios while it overestimates the punchingshear strength of specimens with high tension reinforcementratios. The Eurocode 213 shows good accuracy in predictingthe punching shear strength at different reinforcement ratios.Finally, ACI 318-0514 and CSA A23.3-0415 underestimate thepunching shear strength of specimens with high reinforcementratios while they overestimate the punching shear strength ofspecimens with low reinforcement ratios. This is attributedto the fact that ACI 318-05 and CSA A23.3-04 codes do notaccount for the effect of the tension reinforcement ratio on thepunching shear strength. It is also evident from Fig. 5(a) thatthe fuzzy-based model predicts punching shear strength atboth low and high reinforcement ratios with consistent accu-racy. It is worth noting that the slab thickness and the tensionreinforcement ratio in addition to the compressive strength arefound to have a significant influence on modeling punchingshear strength using the fuzzy-based model. These parametershave also been promoted by other researchers before because oftheir influence on the size effect43 and their possible role indeveloping shear friction.44

To consider other rectangularity ratios c2/c1 (>1) and highperimeter to depth ratios bo/d (>15.0), a design approachbased on the fuzzy-based model is proposed as

(13)

where βc = c2/c1, c1 and c2 equal the short and long sizes ofrectangular columns, vcf is the fuzzy-based shear strengthestimated using Eq. (12), and n is a power coefficient.Equation (13) is modeled in a format similar to that of theACI equation for predicting the punching shear strength. If

vc min

vcf

0.5 1

βcn

-----+⎝ ⎠⎛ ⎞ vcf

0.5 10bo d⁄------------⎝ ⎠

⎛ ⎞ n+⎝ ⎠

⎛ ⎞ vcf⎩⎪⎪⎪⎨⎪⎪⎪⎧

=

n = 1.0 as similar to the ACI equation is used, the model willsignificantly overestimate the punching slab-columnconnections with rectangular columns and with bo/d higherthan 15. A mean value and a standard deviation of thestrength-prediction ratios (Vtest/Vpredicted) of the specimens(Table 3) using n = 1 are 0.977 and 0.193, respectively, whilethose using n = 2 are 1.019 and 0.189. Therefore, the authorsrecommend the use of n = 2. The model prediction with n = 2for a wide range of bo/d and for rectangular columns areshown in Fig. 6 and 7. The choice of n = 2 for the second andthird components of Eq. (13) was based on examining eachcomponent separately. It has become evident that refinementin the value of n for each part would not yield any enhancementin the prediction accuracy of the model.

Figure 6 demonstrates the fact that the modified fuzzy-based model using a modification factor (Eq. (13)) canaccurately predict the punching shear strength of slab-column connections with various bo/d (5.8 ≤ bo/d ≤ 24.0)even though the fuzzy-based model (Eq. (12)) was developedwithin the geometrical limits (5.8 ≤ bo/d ≤ 14.9) due to thelack of test data. This is attributed to the fact that the fuzzy-based model was developed by using the average ultimateshear strength vc considering bo and d (Eq. (1)). It is evidentthat the modified fuzzy-based model can properly considerthe interaction between bo/d and vc in its strength equation(Eq. (13)). In Fig. 7, the fuzzy-based model also accuratelypredicts the punching shear strength of slab-column connectionswith rectangular columns (c2/c1 > 1). From this result, it isnoted that the modified fuzzy-based model properly considersthe effect of rectangularity of columns in practical design range(1 ≤ c2/c1 ≤ 5). It is worth noting that, if enough experimentaldata with high bo/d ratios and rectangular columns wereavailable in the literature, the use of modification factors for

Fig. 6—Variation of strength-prediction by fuzzy-basedmodel according to bo/d.

Fig. 7—Variation of strength-prediction by fuzzy-basedmodel according to c2/c1 higher than 1.

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446 ACI Structural Journal/July-August 2007

addressing these issues can be completely omitted. Thisindicates the fact that a refined fuzzy-based model wouldalways be possible to develop, once experimental databeyond these geometrical limitations becomes available.

PROPOSED DESIGN CHARTFor design purposes, the direct use of the fuzzy-based

model as an empirical method using Eq. (10) to (13) and thepremise parameters from Table 2 might not be feasible fordesigners. To avoid such complexity and to make use of thedemonstrated ability and relative high accuracy of the fuzzy-based model in design of slab-column connections withoutshear reinforcement, the authors suggest a simplified designmodel that is developed based on a set of design charts thatare developed using the fuzzy-based model. Following aformat similar to that used in ACI 318-05, the designstrength for punching shear of slab-column connections isdefined as

φVn = φvcbod (14)

where vc is calculated according to Eq. (13) using n = 2, andφ is the strength reduction factor taken equal to 0.6. Thepunching shear strength vcf can be estimated using Fig. 8.Figures 8(a) to (d) show a group of design charts to estimatethe punching shear strength vcf of slab-column connectionsusing the fuzzy-based model. The design charts are developed

for a wide range of primary design parameters: 20 ≤ f ′c ≤100 MPa (2.9 ≤ f ′c ≤ 14.5 ksi), 100 ≤ h ≤ 300 mm (3.9 ≤ h ≤11.8 in.), and 0.8 ≤ ρ ≤ 2.0%. For space limitations, only fourdesign charts are developed herein covering the aforemen-tioned range of parameters. Additional design charts can bedeveloped using the model equations described previously.The φ factor of 0.6 corresponds conservatively to the lowestbound shown in Fig. 5(a). Obtaining a refined shear strengthreduction factor (higher than 0.6) can be done using principlesof load and resistance factor design (LRFD),45 but is beyondthe scope of this study.

It can be observed from Fig. 8(a) to (d) that the punchingshear strength decreases as slab thickness increases, whichrespects previous findings of the size effect by Bažant and Cao9

and Eurocode 2.13 In cases with high reinforcementratios, however, this size effect is disturbed by the combinedeffect of size and membrane force generated by the tensionreinforcement. As observed in Fig. 8(c) and (d), for hightension reinforcement ratios and low concrete compressivestrength, the punching shear strength increases as the slabthickness increases. This can be attributed to the possibilitythat the increase in the slab thickness with high reinforcementratios results in an increase in the axial membraneforce,24,26,27 which contributes to punching shear strengthdue to the increase in the shear friction effect.44 This possibleshear friction contribution to the punching shear strength hasbeen argued by other researchers in shear analysis.44,46

This phenomenon is due to the combined effect of theprimary parameters (compressive strength, slab thickness,and tension reinforcement ratio) and can be also observed inprevious test results from the punching shear database.3

Figure 9 shows the punching shear strength reported inexisting test results. For this study, Elstner and Hognestad,37

Shaeidt el al.,47 Regan,48 Marzouk and Hussein,25 Hallgrenand Kinnunen,49 and Tomaszewicz’s50 specimens wereused. Each data set itself has similar dimension and property.The dimensions and properties of the specimens aresummarized in Table 1. As expected, for all data sets withhigh concrete compressive strength, the punching shearstrength of thick slabs is always less than that of thin slabsdue to the size effect24,38,43 (see Fig. 9(a)). In Fig. 9(b),however, for low concrete compressive strength and highreinforcement ratios (ρ ≥ 0.012), the punching shear strengthof thick slabs may be greater than that of thin slabs, whichindicates the trade-off between size effect and shear frictioneffect. These combined effects can be successfully describedby the fuzzy-based model.

CONCLUSIONSA new alternative design method and a set of design charts

based on fuzzy learning from examples are proposed. Thenew method can accurately predict the punching shearstrength of simply supported interior slab-column connectionswithout shear reinforcement. One hundred and seventy eighttest specimens from the punching shear databank were usedfor training and testing the proposed model (82 for trainingand 96 for testing). The training and testing data sets cover awide range of the material and geometric properties. Thetesting data set was not used in the training process. Investi-gations for developing a model with good accuracy showedthat concrete compressive strength, slab thickness, andtension reinforcement ratio are the primary parameters thatdominate the punching behavior of slab-column connections.This finding is limited to circular and rectangular columns

Fig. 8—Design chart for punching shear strength usingfuzzy-based model. (Note: 1 MPa = 0.145 ksi; 1 mm = 0.04 in.)

Fig. 9—Strength variation according to primary designparameters.25,37,44-47

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447ACI Structural Journal/July-August 2007

and slabs with perimeter-to-slab-depth ratios (bo/d) rangingbetween 5.8 and 24.0 and column size ratios (c2/c1) rangingbetween 1.0 and 5.0. The fuzzy-based model demonstrateshigher prediction accuracy compared with all current designcodes including ACI 318-05, Eurocode 2, CEB-FIP MC 90,and CSA A23.3-04 in predicting the punching shear strengthof slab-column connections. The proposed model, whileaddressing uncertainty and interactions between modelingparameters, was shown to respect the fundamental mechanicsof punching shear as described by many researchers.

ACKNOWLEDGMENTSThe financial support by the Defense Threat Reduction Agency (DTRA)

University Strategic Partnership to the University of New Mexico is greatlyappreciated.

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32. Gupta, M. M., and Qi, J., “Theory of T-Norms and Fuzzy InferenceMethods,” Fuzzy Sets and Systems, V. 40, No. 3, 1991, pp. 431-450.

33. Fan, J. Y., and Yuan, Y. X., “On the Convergence of a New Levenberg-Marquardt Method,” Technical Report, AMSS, Chinese Academy of Sciences,2001, 11 pp.

34. Hawkins, N. M.; Fallsen, H. B.; and Hinojosa, R.C., “Influence ofColumn Rectangularity on the Behaviour of Flat Plate Structures,” Cracking,Deflection and Ultimate Load of Concrete Slab Systems, SP-30, AmericanConcrete Institute, Farmington Hills, Mich., 1971, pp. 127-146.

35. Criswell, M. E., “Strength and Behaviour of Reinforced ConcreteSlab-Column Connections Subjected to Static and Dynamic Loading,”Technical Report N-70-1, U.S. Army Engineer Waterways Experiment Station,Vicksburg, Miss., Dec. 1970.

36. Lovrovich, J., and McLean, D., “Punching Shear Behaviour of Slabwith Varying Span-Depth Ratios,” ACI Structural Journal, V. 87, No. 5,Sept.-Oct. 1990, pp. 507-511.

37. Elstner, R. C., and Hognestad, E., “Shearing Strength of ReinforcedConcrete Slabs,” ACI JOURNAL, Proceedings V. 53, No. 7, July 1956,pp. 29-58.

38. Tolf, P., “Plattjocklekens Inverkan På Betongplattors Hållfasthet vidGenomstansning,” Försök med Cikulåra Plattor, TRITA-BST Bull, 146,KTH Stockholm, Sweden, 1988, 64 pp.

39. Berenji, H. R., and Khedkar, P., “Learning and Tuning Fuzzy LogicControllers Through Reinforcements,” IEEE Transactions on NeuralNetworks, V. 3, No. 5, 1992, pp. 724-740.

40. Combs, W. E., and Andrews, J. E., “Combinatorial Rule ExplosionEliminated by a Fuzzy Rule Configuration,” IEEE Transactions on FuzzySystems, V. 6, No. 1, pp. 1-11.

41. Lucero, J., “Fuzzy Systems Methods in Structural Engineering,” PhDdissertation, University of New Mexico, Department of Civil Engineering,Albuquerque, N. Mex., 2004.

42. Yager, R., “On a General Class of Fuzzy Connectives,” Fuzzy Setsand Systems, V. 4, 1980, pp. 235-242.

43. Bažant, Z. P., Fracture and Size Effect in Concrete and Other QuasiBrittle Materials, CRC Press, New York, 1997, 280 pp.

44. Loov, R. E., “Review of A23.3-94 Simplified Method for ShearDesign and Comparison with Results Using Shear Friction,” CanadianJournal of Civil Engineering, V. 25, No. 3, 1998, pp. 437-450.

45. Nowak, A. S., “Calibration of LRFD Bridge Code,” Journal ofStructural Engineering, ASCE, V. 121, No. 8, 1995, pp. 1245-1251.

46. Kani, G. N. J., “The Riddle of Shear Failure and Its Solutions,” ACIJOURNAL, Proceedings V. 61, No. 4, Apr. 1964, pp. 441-468.

47. Schaeidt, W.; Ladner, M.; and Rösli, A., “Berechnung von Flach-decken auf Durchstanzen,” Eidgenössische Materialprüfungs-und Versuch-sanstalt, Dübendort, 1970.

48. Regan, P., “Symmetric Punching of Reinforced Concrete Slabs,”Magazine of Concrete Research, V. 38, 1986, pp. 115-128.

49. Hallgren, M., and Kinnunen, S., “Punching Shear Tests on CircularHigh Strength Concrete Slabs,” Utilization of High Strength Concrete,Proceedings, Lillehammer, 1993.

50. Tomaszewicz, A., “High-Strength Concrete: SP2-Plates and Shells—Report 2.3,” Punching Shear Capacity of Reinforced Concrete Slabs,Report No. STE70 A93082, SINTEF Structures and Concrete, Trondheim,1993, 36 pp.

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ACI Structural Journal, V. 104, No. 4, July-August 2007.MS No. S-2006-232 received June 6, 2006, and reviewed under Institute publication

policies. Copyright © 2007, American Concrete Institute. All rights reserved, includingthe making of copies unless permission is obtained from the copyright proprietors. Pertinentdiscussion including author’s closure, if any, will be published in the May-June 2008ACI Structural Journal if the discussion is received by January 1, 2008.

ACI STRUCTURAL JOURNAL TECHNICAL PAPER

This paper presents test results for large cast-in-place anchorbolts in concrete. The tests were performed to evaluate the tensileperformance of large anchors, that is, anchors with a diametergreater than 2 in. (50 mm) or an embedment depth greater than 25 in.(635 mm), which are not addressed by ACI 318, Appendix D, andACI 349, Appendix B. The tests were also intended to investigatethe safety of such anchors for use in nuclear power plants and theeffects of regular (conventional) and special reinforcement on thestrength of such anchors. The test results are used to assess theapplicability of existing design formulas valid for smaller anchorsto large anchors. Suggestions are made for incorporating theeffects of deep embedment or large diameter in existing designprovisions for cast-in-place tensile anchor bolts under tension load.

Keywords: anchor; anchor bolt; cast-in-place; embedment; tension test.

INTRODUCTIONCurrent anchorage designs for nuclear power plants in

Korea use large anchor bolts with diameters exceeding 2 in.(50 mm), embedment depths exceeding 25 in. (635 mm), aspecified yield strength of 140 ksi (980 MPa), and a specifiedultimate strength of 155 ksi (1085 MPa). Whereas the tensilebehavior of smaller anchors has been studied extensively,large anchors have not been adequately addressed. In theresearch described herein, large anchors were tested intension to develop design criteria for anchors that are notaddressed by ACI 318-05, Appendix D,1 or ACI 349-01,Appendix B,2 and to evaluate the applicability ofcapacity-prediction methods developed for smaller anchors.

To evaluate the tensile behavior of anchors with large diametersand embedment depths, various anchors, with diameters from2.75 to 4.25 in. (69.9 to 108 mm) and embedment depths from25 to 45 in. (635 to 1143 mm) were tested.

RESEARCH SIGNIFICANCEThe research described herein is the first experimental

information on the tensile behavior of very large headedanchor bolts (hef ≥ 21 in. [525 mm]). It is important becausealthough such anchor bolts are commonly used in powerplants and for the anchorage of tanks, no design provisionsvalidated by tests exist for them.

EXISTING FORMULAS FOR PREDICTING TENSILE CAPACITY OF ANCHOR BOLTS IN CONCRETEPresuming the head of the anchor is large enough to

prevent pull-out failure (refer to ACI 318, Appendix D), thetensile capacity of large anchor bolts is governed by tensileyield and fracture of the anchor steel or by tensile breakoutof the concrete in which the anchor is embedded. Steel yieldand fracture are well understood. The breakout formulas ofcurrent U.S. design provisions (ACI 318-051 and ACI349-012) are based on the concrete capacity design (CCD)

method (CCD method),3 which is a derivative of the Kappamethod4 described in Reference 5.

According to the CCD method, the average concrete breakoutcapacity of headed anchors in uncracked concrete is given byEq. (1). This equation is valid for anchors with a relativelysmall head (mean bearing pressure at breakout load ofapproximately 13fc′ ).

3 In ACI 318, Appendix D,1 the 5%-fractileof the concrete cone breakout loads are predicted, which isassumed as 0.75 times the mean value. This leads to Eq. (2).ACI 318-05, Appendix D,1 allows the use of Eq. (4) forcalculating the nominal breakout capacity of headed anchorswith an embedment depth hef ≥ 11 in. (279 mm) in uncrackedconcrete. Equation (4) modifies the CCD method slightly bychanging the exponent on the embedment depth hef from 1.5to 1.67. The mean concrete capacity may be calculatedaccording to Eq. (3). In ACI 349-97,6 a 45-degree conemodel is used to calculate the concrete breakout capacity(Eq. (5)). Because Eq. (5) was used in design, it may beconsidered to predict approximately the 5%-fractile of testresults. A summary of the proposed predictors are given as

DESCRIPTION OF EXPERIMENTAL PROGRAMTest specimens

To evaluate the effects of embedment depth, anchor diameter,and supplementary reinforcement patterns on the tensilecapacity of large anchors, five different test configurationswere selected and four test replicates with each configuration

Equationnumber Predictor Remark

(1) Mean breakout strength,

CCD-method withexponent 1.5 on hef

(2) Nominal breakout strength,ACI 318-05, Appendix D

(3)

Mean breakout strength for anchors with hef ≥ 10 in.(254 mm), CCD-methodwith exponent 1.67 on hef

(4)

Nominal breakout strengthfor anchors with

hef ≥ 10 in. (254 mm)according to ACI 318-05,

Appendix D

(5)Nominal breakout strength,

ACI 349-97(45-degree cone model)

Note: fc′ = specified concrete compressive strength (psi); hef = effective embedment (in.); and db = diameter of anchor head (in.).

Nu m, 40 fc′ h1.5ef lb( )=

Nu 30 fc′ h1.5ef lb( )=

Nu m, 26.7 fc′ h1.67ef lb( )=

Nu 20 fc′ h1.67ef lb( )=

Nu 4 fc ′πh2ef= 1 dk hef⁄+( ) lb( )

Title no. 104-S46

Tensile-Headed Anchors with Large Diameter and Deep Embedment in Concreteby Nam Ho Lee, Kang Sik Kim, Chang Joon Bang, and Kwang Ryeon Park

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ACI Structural Journal/July-August 2007480

were performed giving 20 specimens in total. The testprogram is summarized in Table 1. The test specimens areshown in Fig. 1. All anchors were fabricated of ASTM A540Gr. B23 Class 2 steel (equivalent to ASME SA 549 Gr. B23Class 2 used in Korean nuclear power plants) with fy = 140 ksi(980 MPa) and fu = 155 ksi (1085 MPa). The anchor headconsisted of a round thick plate which was fixed to the boltby clamping nuts (Fig. 2). The diameter of the round platewas dh = 6 in. (152.4 mm) (db = 2.75 in. [69.9 mm]), dh =8.5 in. (215.9 mm) (db = 3.75 in. [95.3 mm]), and dh = 10 in.(254.0 mm) (db = 4.25 in. [108.0 mm]). The size of theconcrete test block was large enough to avoid splittingfailure. The concrete volume (width/length/depth) availablefor each anchor is shown in Table 1. Furthermore, to

minimize the width of eventual shrinkage cracks, the top andbottom of the test member were reinforced in both directionswith No. 10 bars at 16, 10, and 10 in. (406.4, 254, and 254 mm)spacing for Specimens T1, T2, and T3, respectively. Thissurface reinforcement does not significantly influence theconcrete breakout load. As shown in Fig. 1, wooden and steelframes were constructed to suspend the cast-in-placeanchors in the correct position and at the correct embedmentdepth. The concrete mixture for the test specimens is shownin Table 2(a). The concrete used in the test specimens wascomparable to the concrete used in the Korean Nuclear Plant,except that 20% by weight of the Type I cement was substitutedby fly ash and 1 in. (25 mm.) crushed aggregate was usedinstead of 3/4 in. (19 mm). The target concrete strength at42 days was fc′ = 5500 psi (37.9 MPa). The actual concretestrength at the time of testing is given in Table 2(b). Theconcrete for the specimens of one test series was placed fromone batch. Whereas in test Series T1 to T3, no special reinforce-ment was used to resist the applied tension load, in test Series T4and T5, supplementary reinforcement (refer to Fig. 3) wasused to increase the ultimate load. The supplementaryreinforcements consisted of vertical stirrups (eight No. 8 barsand 16 No. 8 bars for test Series T4 and T5, respectively), asshown in Fig. 3.

ACI member Nam Ho Lee is a Senior Research Engineer in the Civil EngineeringDepartment of the Korea Power Engineering Co. He received his BS from SeoulNational University and his MS and PhD from the Korea Advanced Institute ofScience & Technology. He is a member of ACI Committees 349, Concrete NuclearStructures, and 355, Anchorage to Concrete, and Joint ACI-ASME Committee 359,Concrete Components for Nuclear Reactors. His research interests include thenonlinear behavior of concrete structures and anchorage to concrete.

Kang Sik Kim is a Senior Researcher, Environment and Structure Laboratory, KoreaElectric Power Research Institute, Daejeon, Korea. His research interests include thebehavior of concrete-filled steel plate structures and anchorage to concrete.

Chang Joon Bang is a Project Engineer at Korea Hydro & Nuclear Power Co. Ltd.,Seoul, Korea. He is currently a Graduate Student of civil engineering, LehighUniversity, Bethlehem, Pa.

Kwang Ryeon Park is a Research Engineer, Civil Engineering Department, KoreaPower Engineering Co.

Fig. 1—Tension test Specimens T1, T2, T3, T4, and T5.

Table 2(b)—Concrete strength at time of testingTest specimen Curing ages, days Compressive strength, psi (MPa)

T1-A/B/C/D 58/50/44/42 5771 (39.8)/5630 (38.8)/5508 (38.0)/5464 (37.7)

T2-A/B/C/D 41/45/47/49 5177 (35.7)/5248 (36.2) / 5291 (36.5)/5320 (36.7)

T3-A/B/C/D 61/56/54/50 5448 (37.6)/5348 (36.9) / 5305 (36.6)/5220 (36.0)

T4-A/B/C/D 57/55/54/50 5945 (41.0)/5917 (40.8)/ 5903 (40.7)/5817 (40.1)

T5-A/B/C/D 71/70/69/68 6144 (42.4)/6130 (42.3)/6130 (42.3)/6116 (42.2)Fig. 2—Details of anchor head.

Table 1—Description of tension test specimens

SpecimenReinforce-

ment

Anchordiameter,

db, in. (mm)

Diameterof anchor

head,dh , in. (mm)

Effective embedment

hef , in.(mm)

Concretevolume

available for each anchor

(width/length/depth)

T1-A,B,C,D None 2.75(69.9)

6.0(152.4)

25(635)

5.9hef /5.0hef /2.9hef

T2-A,B,C,D None 3.75(95.3)

8.5(215.9)

35(889)

5.4hef/4.7hef/2.0hef

T3-A,B,C,D None 4.25(108.0)

10.0(254.0)

45(1143)

5.0hef /3.6hef /2.0hef

T4-A,B,C,D Supp. No. 1 2.75(69.9)

6.0(152.4)

25(635)

5.9hef /5.0hef /2.9hef

T5-A,B,C,D Supp. No. 2 2.75(69.9)

6.0(152.4)

25(635)

5.9hef /5.0hef /2.9hef

Table 2(a)—Concrete mixture proportioningNominal strength,

psi, at42 days

W/(C + FA)

S/a, %

W, lb C, lb

FA, lb S, lb G, lb

WRA,*

mLAEA,†

mL

5500 0.44 44 525 514 128 1257 1617 474 26*Water-reducing admixture.†Air-entraining admixture.

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ACI Structural Journal/July-August 2007 481

Test setupThe test setup consisted of a loading frame, loading plate,

jack assembly, load cell, and other items, as shown in theschematic and photo in Fig. 4. The load was applied to theanchor under force-control in an increment of approximately3.5% of ultimate steel strength of the anchor bolt (Fu = 925,1683, and 2192 kips [4114.6, 7486.4, and 9750.5 kN], forbolts with a diameter of 2.75, 3.25, and 4.25 in. [69.90,82.55, and 107.95 mm], respectively), that is, 30, 60, 77,68, and 48 kips (133.4, 266.9, 342.5, 302.5, and 213.5 kN)for Series T1, T2, T3, T4, and T5, respectively. It wasreacted in two directions by a stiff frame to minimize thebending moment in the test specimen. The clear distancebetween the supports was 4.0 hef for Specimens T1 throughT5, thus allowing for an unrestricted formation of a concretecone. The applied load was measured by a load cell. Additionally,the strain along the embedment length of the anchor bolt wasmeasured (Fig. 5). Furthermore, the displacement of the topend of the anchor was measured by LVDTs (Fig. 5).

TEST RESULTSFailure loads, failure modes and load displacement behavior

The average failure loads are summarized in Table 3(a)(Series T1 to T3) and Table 3(b) (Series T4 and T5). Thevalues given in the tables are normalized to fc′ = 5500 psi(37.9 MPa) by multiplying the measured peak load of eachtest with the factor (5500/fc,test)

0.5. In test Series T1 to T3,failure was caused by concrete cone breakout well below theanchor bolt steel capacities (Fu = 925, 1683, and 2192 kips[4114.6, 7486.4, and 9750.5 kN] for bolts with diameters of2.75, 3.25, and 4.25 in. [69.90, 82.55, and 107.95 mm],respectively). The cracking patterns in the specimen after thetest are depicted in Fig. 6(a). Generally, one major longitudinalcrack was observed, centered approximately on the sides ofthe block, in combination with a horizontal crack and sometransverse cracks. On the top surface of the block, the cracksformed a circular pattern around the anchor. To identify theinternal crack propagation defining the roughly conical breakoutbody, one replicate of each specimen type was selected, andthe concrete was cored on two orthogonal planes whoseintersection coincided with the axis of the anchor. The coresconfirmed a breakout cone whose angle with the concrete

surface varied from α = 20 to 30 degrees, following thetypical crack profiles shown in Fig. 6(b).

In general, test Specimens T4 and T5, with supplementaryreinforcement (Fig. 3), were not tested to failure. At theapplied peak load, the measured steel strains exceeded theyield strain and because of safety concerns a sudden ruptureof the bolt was avoided. Only Specimen T4-A was tested to

Fig. 4—Tension test setup: (a) schematic; and (b) photo.

Fig. 3—Supplementary reinforcement in Specimens T4 and T5. Fig. 5—Location of LVDTs and strain gauges (Specimen T1).

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482 ACI Structural Journal/July-August 2007

failure. Failure of this specimen was caused by forming aconcrete cone. From the load-displacement curves (Fig. 7), itcan be concluded that in test Series T4, the applied maximumloads were almost identical with the failure loads. In testSeries T5, however, the failure load of the anchors was notreached. Because Specimens T4 and T5 showed no crackingat the concrete surface, no cores were taken to check whethera cone had begun to form.

The load-displacement curves for Specimens T1, T2, T3,T4, and T5 are shown in Fig. 7(a) through 7(e), using thedisplacement measured at the top of each anchor. Theload-displacement relationship for each test replicate variedbased on the concrete strength at the time of testing. Theprojecting lengths of the anchor shafts from the concretesurface to the top of the anchor for Specimens T1, T2, T3,T4, and T5 were 41.7, 48.6, 53.1, 41.7, and 41.7 in. (1059,1234, 1348, 1059, and 1059 mm), respectively. Because themeasured displacements shown in Fig. 7 include the steelelongation of the projecting anchor length, the actual anchordisplacements at the top of the concrete surface, which areaccumulated along the embedded portion of the anchor, aremuch smaller than shown in Fig. 7. In Fig. 8, the relationshipbetween load and anchor displacement at the surface of theconcrete (calculated from the displacements measured at theanchor top end subtracting the steel elongation of theprojecting length) are plotted for test Series T1 to T5. Insome tests, the calculated displacements at the concretesurface are negative for low loads. It is believed that this is

Table 3(a)—Tension test results and predictionsfor unreinforced Specimens T1, T2, and T3

Classification Reference

Concrete breakout capacities, kips (kN),by embedment

Specimen T125 in.

(635 mm)

Specimen T2,35 in.

(889 mm)

Specimen T3,45 in.

(1143 mm)

Predictions

ACI 349-97, Eq. (5) 676 (3006) 1305 (5804) 2138 (9510)

ACI 318-05, Eq. (4) 320 (1423) 562 (2499) 855 (3803)

CCD method with

Eq. (1)371 (1650) 614 (2731) 895 (3981)

CCD method with

Eq. (3)428 (1903) 750 (3336) 1142 (5079)

Tests

Mean 509 (2264) 744 (3309) 1242 (5524)

COV, % 5.8 2.8 6.1

5%-fractile 393 (1748) 662 (2944) 944 (4199)

5%-fractile/mean 0.77 0.89 0.76

h1.5ef

h1.67ef

Classi-fication

Sym-bol in Fig. 9

Com-parison

Ratios of observed to predicted capacities

Specimen T125 in.

(635 mm)

Specimen T2,35 in.

(889 mm)

Specimen T3,45 in.

(1143 mm) Mean

5% fractile of test results

(I)Nu,5%/ Eq. (5)

0.58 0.51 0.44 0.51

(II)Nu,5%/ Eq. (4)

1.24 1.19 1.12 1.18

Mean of test results

(III) Mean/ Eq. (1) 1.37 1.21 1.39 1.32

(IV) Mean/ Eq. (3) 1.19 0.99 1.09 1.09

Table 3(b)—Tension test results and predictions for reinforced Specimens T4 and T5

Classification Reference

Concrete breakout capacities, kips (kN),by embedment

Specimen T425 in.

(635 mm)

Specimen T5,25 in.

(635 mm)

Specimen T1,25 in.

(635 mm)

Predictions

ACI 349-97, Eq. (5) 676 (3006) 676 (3006) 676 (3006)

ACI 318-05, Eq. (4) 320 (1423) 320 (1423) 320 (1423)

CCD method with

Eq. (1)371 (1650) 371 (1650) 371 (1650)

CCD method with

Eq. (3)428 (1903) 428 (1903) 428 (1903)

Tests

Mean 733 (3260) 725 (3224) 509 (2264)

COV, % 1.7 3.5 5.8

5%-fractile 685 (3047) 625 (2780) 393 (1748)

5%-fractile/mean 0.93 0.86 0.77

h1.5ef

h1.67ef

Fig. 6—(a) Cracking pattern for four test replicates (A, B,C, and D) of Specimens T1, T2, and T3; and (b) typicalinternal crack profile in Specimen T1.

(a)

(b)

Classification

Sym-bol in Fig. 9 Comparison

Ratio of observed to predictions(hef = 25 in. [635 mm])

T4 T5 T1 T4/T1

5% fractile of test results

(I) Nu,5%/Eq. (5) 1.01 0.92 0.58 1.74

(II) Nu,5%/Eq. (5) 2.16 1.97 1.24 1.74

Mean of test results

(III) Mean/Eq. (1) 1.98 1.96 1.37 1.45

(IV) Mean/Eq. (3) 1.71 1.70 1.19 1.44

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ACI Structural Journal/July-August 2007 483

caused by bending of the anchors if they were not installedperfectly perpendicular to the concrete surface. It can be seenthat the anchor displacements at peak load of Specimens T1to T3 (concrete cone failure) are rather small. This can beexplained by the rather large anchor heads that, due to thelow concrete stresses, did not slip much. For head sizesallowed by ACI 318-05, Appendix D, the breakout failureloads increase approximately proportional to hef

1.5. Withmuch larger heads, the power on the embedment depths isgreater than 1.5.7 In the present tests, at failure, the relatedpressure under the head was on average p/fc′ = 4.37, 3.36,and 5.31 for test Series T1, T2, and T3. It was much smallerthan the pressure allowed by ACI 318-05 for uncrackedconcrete (pn = 10fc′ ).

Comparison of predicted and tested tensile breakout capacities

In Table 3(a), tension test results for unreinforcedSpecimens T1, T2, and T3, and results in Table 3(b) forreinforced Specimens T4 and T5, are compared with predictedcapacities. The measured mean failure loads are comparedwith the predicted mean capacities according to Eq. (1) and(3), respectively, and the 5%-fractiles of the measuredfailure loads calculated by assuming an unknown standarddeviation are compared with the values according to Eq. (4)and (5). In Fig. 9, the ratios of measured capacities topredicted values are plotted. Figure 10 shows the measuredfailure loads of each test compared with the values predictedaccording to Eq. (5), Fig. 10(a); Eq. (1), Fig. 10(b); and Eq. (3),Fig. 10(c), as a function of the embedment depth. In Fig. 11,the measured concrete breakout loads, as well as the failureloads according to best fit equations using the current testresults and Eq. (1), (2), (3), and (5), are plotted as a functionof the embedment depth.

EVALUATION OF TEST RESULTS FOR UNREINFORCED SPECIMENS T1, T2, AND T3

According to the 45-degree cone model (Eq. (5)), thebreakout capacities increase in proportion to hef

2 . Thepredicted capacities Nu,calc are much higher than themeasure values Nu,test and the ratio Nu,test /Nu,calc decreaseswith increasing embedment depth (Fig. 10(a)). On average,the 5%-fractiles of the observed capacities are approximatelyhalf the capacities predicted by ACI 349-97 (Table 3(a)).This demonstrates that the 45-degree cone model isunconservative for deep anchors. This agrees with the findingsby Fuchs et al.3 and Shirvani et. al.8 In contrast, the predictionsaccording to the CCD method are conservative. Themeasured average breakout loads are approximately 30%higher than the values predicted according to Eq. (1) (Nuproportional to hef

1.5) with no significant influence of theembedment depth (Fig. 10(b)). On average, the ratio ofmeasured failure loads to the values predicted by Eq. (3) (Nuproportional to hef

1.67) is 1.09 (Table 3(a)). It decreasesslightly with increasing embedment depth (Fig. 10(c)).

In Fig. 10(d) to 10(f), the breakout failure loads of headedanchors with an embedment depth hef ≥ 8 in. (200 mm)measured in the present tests and taken from other sources3,8

are compared with values predicted by the CCD method.According to Fig. 10(d), the prediction according to Eq. (1)is conservative for large embedment depths. The failureloads predicted by Eq. (3) agree quite well with the measuredvalues (Fig. 10(e)). Figure 10(f) shows that the CCD methodchanging the exponent on hef from 1.5 to 1.67 at an effectiveembedment depth of 10 in. (250 mm) predicts the failureloads of anchors with hef ≥ 8 in. (200 mm) best. Only two

Fig. 8—Relation between load and anchor displacement atconcrete surface.

Fig. 7—Measured load-displacement relationships.

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484 ACI Structural Journal/July-August 2007

test points at hef = 8 in. (200 mm) fall below the assumed5%-fractile, which is equal to 75% of the average value.

The 5%-fractiles of the capacities observed in the presenttests average approximately 120% of the values predicted byACI 318-05, Appendix D (Eq. (4)) (refer to Table 3(a)). Thehigher ratio Nu,test/Nu,calc when comparing the 5% fractileswith each other instead of the average values is due to therather low scatter of test results. On average, the coefficientof variation (COV) was approximately 5%. This results in anaverage ratio Nu,5%/Nu,m of 0.81, whereas in ACI 349-01,a ratio of 0.75 is assumed. In actual structures, the concretestrength, and thus the concrete cone resistance, might vary morethan in the present test specimens. Therefore, the ratioNu,5%/Nu,m assumed in ACI 318-05, Appendix D, shouldbe maintained.

Numerical investigations by Ozbolt et al.7 using asophisticated three-dimensional nonlinear finite elementmodel demonstrates that the concrete breakout capacity ofheaded anchors is influenced by the head size, that is, thepressure under the head, related to the concrete compressivestrength as described previously.

Based on the previous evaluations, it is recommended topredict the nominal concrete breakout capacities of anchorswith an embedment depth hef ≥ 10 in. (250 mm) in uncrackedconcrete by Eq. (4). Equation (4) is valid, however, only ifthe head size is large so that the pressure under the head atthe nominal capacity is pn ≤ 3fc′ . This limiting value isdeduced from the results of the test Series T1 to T3. In thesetests, the pressure under the head was pn/fc′ = 3.4 to 5.3, onaverage 4.3. The nominal capacity is approximately 75% of

the mean capacity (compare Eq. (4) with Eq. (3)). Whenapplying this reduction factor, one gets pn/fc′ = 3.2 ~ 3.0.This limiting value is supported also by the numerical analysisresults.7 For smaller heads, for which the nominal pressureunder the head is pn > 3fc′ , the breakout capacities in uncrackedconcrete should be predicted by Eq. (2).

In cracked reinforced concrete, lower breakout capacitiesthan in uncracked concrete are observed.9 Therefore, ACI318-05, Appendix D, reduces the nominal breakout capacities ofheaded anchors in cracked reinforced concrete by a factor0.8 compared with uncracked concrete. Therefore, incracked concrete Eq. (4) with hef

1.67, multiplied by the factor0.8, should only be used for deep anchors if the pressureunder the head is pn ≤ 2.4fc′.

Fig. 10—Ratios of observed to predicted concrete tensilebreakout capacities as function of embedment depth.

Fig. 9—Ratios of test results (5% fractile and mean) topredicted capacities; compare with Table 3. Fig. 11—Test results and comparison with predicted capacities.

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EFFECT OF SUPPLEMENTARY REINFORCEMENTReinforced Specimen T4

Test Specimens T4, with supplementary reinforcement,are shown in Fig. 3. The mean tested failure load (733 kips[3260 kN]) is close to the sum (806 kips [3585 kN]) of thecalculated reinforcement strength (378 kips [1681 kN]) andthe unreinforced concrete strength (428 kips [1904 kN]) byEq. (3). It can be inferred that the adopted reinforcementpattern effectively acted in the anchorage system to resisttension load.

The tested breakout strength of the unreinforced testSpecimen T1 with the same embedment depth as Specimen T4was 509 kips (2264 kN). Comparison of the mean testedstrengths of Specimens T1 and T4 shows that the effectiveincrease in capacity due to supplementary reinforcement isroughly 224 kips (996 kN), or approximately 60% of thecalculated yield strength of the supplementary reinforcement.

The loading on Specimen T4-A was increased to theexpected total yield force of the supplementary reinforcement sothat the load distribution to each of the two reinforcementgroups could be estimated. The load resisted by thesupplementary reinforcement in the inner concentriccircle (4.2 in. [106 mm] from the axis of the anchor) was2.2 times the load resisted by the equal area of supplementaryreinforcement in the outer concentric circle (8.5 in. [216 mm]from the axis of the anchor).

According to the measured strains in the strain gaugesattached to reinforcing bars, the reinforcing bars close to theanchor were more effective in increasing the tensile capacityand their maximum stress was measured close to theanchor head.

Reinforced Specimen T5The mean tested capacity (725 kips [3225 kN]) of the four

replicates of test Specimen T5, with supplementary reinforce-ment as shown in Fig. 3 was much smaller than the sum(1129 kips [5021 kN]) of the calculated reinforcementstrength, 16 x 60 ksi x 0.79 in.2 = 758 kips (3371 kN) andconcrete breakout strength per the CCD method given byEq. (1), 371 kips (1650 kN). These test results indicate thatthis layout of supplementary reinforcement contributes witha low level of effectiveness to the capacity of the anchor.This conclusion is corroborated by measured strains in thegauges attached to the reinforcing bars, which indicates littlestrain in the reinforcement. As noted previously, however,Specimen T5 were not fully loaded up to failure due to safetyconcerns. As a consequence, the results of Series T5 are judgedto not be useful in verifying the absolute effectiveness of thesupplementary reinforcement. By comparing results fromSpecimens T4 with those of Specimens T5, however, it is stillpossible to judge the relative effectiveness of the differentsupplementary reinforcement patterns. For a given applied load,stresses in the supplementary reinforcement of Specimens T5along the outer circles are less than half of those along the innercircle. The relative trends of stress distribution are similar foreach reinforcement in both Series T4 and T5. Therefore, it canbe inferred that the increase in tensile capacity is approximatelyproportional to the amount of supplementary reinforcement.

The load-displacement curves of Series T4 show that thepeak load was nearly reached in the tests. In Series T5, theload could still be increased. In Series T4, the supplementaryreinforcement was not strong enough to resist the concretebreakout load. In Series T5, the loading was stopped beforethe supplementary reinforcement could be fully activated.

Therefore, it is not possible to formulate a general modelfrom the test results. The results, however, show that withsupplementary reinforcement arranged as in Specimens T4and dimensioned for about 80 to 100% of the expectedultimate concrete breakout capacity, the failure load wasincreased by approximately 50% over the unreinforced case.This result can reasonably be used in the calculation ofultimate strength.

SUMMARY AND CONCLUSIONSTensile load-displacement behavior of large anchors without supplementary reinforcement

The test results show that ACI 349-97 (Eq. (5)) significantlyoverestimates the tensile breakout capacity of large anchors.The ratio Nu,test /Nu, calc decreases with increasing embedmentdepth (Fig. 10(a)). Furthermore, the slope of the concretecone was much flatter than 45 degrees. Therefore, theoverestimation of the failure loads would be even larger foranchors at an edge or for anchor groups. For these reasons,this formula in ACI 349-97 should not be used in design.

The CCD method with hef1.5 (Eq. (1)) is conservative for

large anchors (Fig. 10(b)). This is probably due to the factthat this method is based on linear fracture mechanics, whichis valid only for anchors with high bearing pressure, that is,anchors with small heads. The tested anchors, however, hadrather large heads. The test results can best be predicted bythe CCD method with (Eq. (3)) (refer to Fig. 9 and 10(e)). Onaverage, the measured failure loads are approximately 10%higher than the predicted values. If all available results aretaken into account (refer to Fig. 10(f)), however, a change ofEq. (3) seems not to be justified.

It is proposed to calculate the characteristic resistance ofsingle anchor bolts with hef ≥ 10 in. (250 mm) and lowbearing pressure (pressure under the head at nominalbreakout load pn ≤ 3fc′ [uncracked concrete] or pn ≤ 2.4fc′[cracked concrete]) according to ACI 318-05, Appendix D,or ACI 349-01, Appendix B, using the equation with hef

1.67).According to the test results, however, the average coneangle was not 35 degrees (as assumed in the CCD method)but only approximately 25 to 30 degrees. Therefore, thecharacteristic spacing scr,N and characteristic edge distanceccr,N are probably larger than scr,N = 2ccr,N = 3hef as assumedin ACI 318-05. Therefore, it seems prudent to calculate theresistance of anchorages at an edge or corner, or of groupanchorages, according to ACI 318-05, but with scr,N = 4.0hef instead of scr,N = 3.0 hef as given in ACI 318-05.

Tensile load-displacement behavior of large anchors with supplementary reinforcement

In Series T4, the supplementary reinforcement was notstrong enough to resist the applied load. Even in Test T4-A,in which the supplementary reinforcement yielded, onlyapproximately 1/3 (246/759 ≈ 0.33) of the applied peak loadwas resisted by the reinforcement. In Series T5, which had astronger reinforcement, the tests had to be stopped becauseof tensile yielding of the anchors before the supplementaryreinforcement had been fully mobilized. Therefore, theresults of these tests cannot be used to develop a general designmodel for anchors with supplementary reinforcement.

Nevertheless, the results of test Series T4 showed that thepeak load could be increased by approximately 50%compared with the results from test Series T1 withoutsupplementary reinforcement. Therefore, it is proposed toincrease the concrete breakout resistance calculated as described

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previously by a factor of 1.5 if supplementary reinforcementis present around each anchor of an anchor group. Thesupplementary reinforcement must be arranged as in Tests T4(four U-shaped stirrups at a distance ≤ 4 in. (100 mm) or≤ 0.15hef from the anchor) and dimensioned for the character-istic concrete breakout resistance according to Eq. (4)).

In a more general model, the supplementary reinforcementshould be dimensioned to take up 100% of the applied load,thus neglecting the contribution of the concrete. Thesupplementary reinforcement should be designed using astrut-and-tie model. The characteristic resistance of thesupplementary reinforcement is given by the bond capacityof the supplementary reinforcement in the anticipatedconcrete cone, which should be assumed to radiate from thehead of the anchor at an angle of 35 degrees. The bondcapacity should be calculated according to codes of practice(for example, ACI 318-051 or Eurocode 210). The designstrength is limited by the yield capacity of the bars. Thismodel is described in detail in References 11 and 12.

ACKNOWLEDGMENTSThe authors would like to acknowledge the financial and technical help of

Korea Hydro & Nuclear Power Co. Ltd. and Korea Electric Power ResearchInstitute for financing this research work and several on-going researchprojects related to the capacity of anchorage to concrete structures. Theauthors are also grateful for the valuable advice of R. Eligehausen, Universityof Stuttgart, Stuttgart, Germany; R. Klingner, University of Texas at Austin,Austin, Tex.; and members of ACI Committee 355, Anchorage to Concrete.

REFERENCES1. ACI Committee 318, “Building Code Requirements for Structural

Concrete (ACI 318-05) and Commentary (318R-05),” American ConcreteInstitute, Farmington Hills, Mich., 2005, 430 pp.

2. ACI Committee 349, “Code Requirements for Nuclear Safety-RelatedConcrete Structures (ACI 349-01),” American Concrete Institute, FarmingtonHills, Mich., 2001, 134 pp.

3. Fuchs, W.; Eligehausen, R.; and Breen, J. E., “Concrete CapacityDesign (CCD) Approach for Fastening to Concrete,” ACI StructuralJournal, V. 92, No. 1., Jan.-Feb. 1995, pp. 73-94.

4. Rehm, G.; Eligehausen, R.; and Mallée, R., “Befestigungstechnik”(Fastening Technique), Betonkalender 1995, Ernst & Sohn, Berlin,Germany, 1995.

5. Comité Euro-International du Beton, “Fastening to ReinforcedConcrete and Masonry Structures,” State-of-the-Art Report, CEB, ThomasTelford, London, 1991, pp. 205-210.

6. ACI Committee 349, “Code Requirements for Nuclear Safety RelatedConcrete Structures (ACI 349-97),” American Concrete Institute, FarmingtonHills, Mich., 1997, 123 pp.

7. Ozbolt, J.; Eligehausen, R.; Periskic, G.; and Mayer, U., “3D FEAnalysis of Anchor Bolts with Large Embedment Depths,” Fracture Mechanicsof Concrete Structures, V. 2, No. 5, Apr. 2004, Vail, Colo., pp. 845-852.

8. Shirvani, M.; Klingner, R. E.; and Graves III, H. L., “Behavior of TensileAnchors in Concrete: Statistical Analysis and Design Recommendations,”ACI Structural Journal, V. 101, No. 6, Nov.-Dec. 2004, pp. 812-820.

9. Eligehausen, R., and Balogh, T., “Behavior of Fasteners Loaded inTension in Cracked Reinforced Concrete,” ACI Structural Journal, V. 92,No. 3, May-June 1995, pp. 365-379.

10. Eurocode 2, “Design of Concrete Structures, Part 1: General Rulesand Rules for Buildings,” 2004.

11. Technical Committee CEN/TC 250, “Design of Fastening forUse in Concrete, Part 2: Headed Fasteners,” Final Draft, CEN TechnicalSpecifications, 2004.

12. Comité Euro-International du Beton (CEB), Design Guide forAnchorages to Concrete, Thomas Telford, London, 1997.

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