100-S27_(2003) Analysis_of_fiber_reinforced_polymer_composite_grid_RC_beams.pdf

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

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.

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.

ACI Structural Journal/March-April 2003 253

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

254 ACI Structural Journal/March-April 2003

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.

ACI Structural Journal/March-April 2003 255

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.

256 ACI Structural Journal/March-April 2003

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.

ACI Structural Journal/March-April 2003 257

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.

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

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