behavior of fiber - reinforced concrete columns under axially and eccentriacally compressive loads

272 ACI Structural Journal/May-June 2010

ACI Structural Journal, V. 107, No. 3, May-June 2010.MS No. S-2008-046.R6 received June 25, 2009, and reviewed under Institute publication

policies. Copyright © 2010, American Concrete Institute. All rights reserved, including themaking of copies unless permission is obtained from the copyright proprietors.Pertinent discussion including author’s closure, if any, will be published in the March-April 2011 ACI Structural Journal if the discussion is received by November 1, 2010.

ACI STRUCTURAL JOURNAL TECHNICAL PAPER

An experimental investigation into the behavior of 16 short,confined, reinforced concrete columns with and without steel fiberswas carried out.

The columns with square sections had a concrete core 165 x 165 mm(6.49 x 6.49 in.) at the midsection and were hunched at the ends toapply eccentric loading and prevent boundary effects. The specimenswere tested to failure at different strain rates under two loadingschemes: concentric compression and eccentric compression with aconstant eccentricity.

The axial load and axial strains were obtained to evaluate theeffects of the presence of steel fibers, the thickness of the coverconcrete, and the eccentricity of the applied axial load. Thecomparative analysis of the experimental results showed that thepresence of steel fibers delayed the spalling of concrete cover andincreased the strain capacity and ductility; the eccentricity of theapplied axial load caused substantial variation in the peak load,ultimate strength, and failure modes. Finally, the structuralresponse of cross sections of normal concrete (NC) and steel fiber-reinforced concrete (SFRC) columns subjected to compressiveconcentric and eccentric loading was numerically modeled tocompare the experimental results. A suitable choice of constitutivelaws for concrete and reinforcing steel bars and a reasonablecalibration criterion of the model allowed for the reproduction ofthe experimental results with a good approximation level in termsof load-axial strain and the moment-curvature curves.

Keywords: columns; compressive tests; confined concrete; moment-curvature diagrams; steel fibers.

INTRODUCTIONInterest in using fiber-reinforced concrete (FRC) for

structural members has increased in recent decades both forscientists and producers, and especially in the field of precastmembers. Significant applications of using FRC, which inthe past were mainly in industrial floors and pipes, todayinclude the use of structural members such as deep members,columns, and beam to column joints.

Several experimental and theoretical investigations high-light the effectiveness of using FRC for members of framedstructures (beams,1 columns,2 beam-column joints,3 andcorbels and brackets4); shells;5 and plates.6 For thesemembers, it is possible to partially or completely substitutethe transverse or the secondary steel reinforcements (mesh,stirrups) by using fibers in adequate percentages andgeometry. All of the information available on this topic inthe literature was used by several countries, such as the U.S.,Japan, Canada, France, Germany, and Italy, to collect a databaseand to develop recommendations and code proposals.7,8

The present paper refers to the compressive behavior ofFRC reinforced columns subjected to axial load with andwithout initial eccentricity. The influence of the coverthickness and the presence of hooked steel fibers on thecompressive and flexural responses of columns is analyzed.

From a theoretical point of view, and on the basis of theavailable analytical models given in the literature, thecomplete load-axial shortening curves and the moment-curvature diagrams were obtained and compared with theexperimental results. The comparison shows good agreementand also highlights the ability of the proposed model tocapture the fundamental phenomena occurring in thebehavior of compressed members, such as the confinementeffects, the buckling of longitudinal bars, and the softeningin compression of cover concrete.

RESEARCH SIGNIFICANCEThis study considers an experimental and theoretical

research into the behavior of reinforced concrete columnswith and without fibers under axial and eccentric loads. Theresults give information about the performance of FRCmaterials used in structural applications, which can also beuseful for seismic and shock mitigation.

The available research on this topic mainly refers to thetype of concrete,1-4 the main damaging and rupturephenomena occurring in the pre-peak and post-peakphases,2-16 the type, and the loading history.3-4 The aim ofthis research was to estimate the influence of the presence offibers and the cover thickness on the compressive responseof reinforced concrete (RC) columns, focusing attentionmainly on the strength and strain capacities of RC members.It was also intended to highlight the influence of the coverspalling process, including the buckling of longitudinal barsand the yielding of transverse stirrups.

EXPERIMENTAL PROGRAMThe experimental program carried out was intended to

study the compressive behavior of ordinary and fibrousconcrete columns with a square transverse cross section andsubjected to concentric and eccentric axial loads. Themechanical description of constituent materials (concreteand steel) was given initially.

SpecimensSixteen columns hunched at the ends (generally known as

dog-bone shape) were tested in compression. Geometry anddetails of steel reinforcements are shown in Fig. 1.

Columns were 1250 mm (49.21 in.) long; the midportion,with a length of 600 mm (23.62 in.), had a square transversecross section and core dimensions of 165 mm (6.49 in.),

Title no. 107-S26

Behavior of Fiber-Reinforced Concrete Columns under Axially and Eccentrically Compressive Loadsby Giuseppe Campione, Marinella Fossetti, and Maurizio Papia

273ACI Structural Journal/May-June 2010

measured in the axis of the stirrup. To study the influence ofthe cover thickness, two different values δ = 10 mm [0.39 in.]and δ = 25 mm [0.98 in.] were adopted. Therefore, theexternal dimensions of the transverse cross section were b =185 and 215 mm (7.28 and 8.46 in.), respectively.

Four 12 mm (0.47 in.) longitudinal bars and closed stirrupsof 6 mm (0.24 in.) diameter at pitch s = 65 mm (2.55 in.)were adopted. The end portions of the columns were reinforcedwith eight 12 mm (0.47 in.) longitudinal bars and stirrups of6 mm (0.24 in.) diameter at pitch 25 mm (0.98 in.) to giveadditional reinforcement and to prevent premature failure ofthe end portions of the specimens during the tests.

The total confinement transverse reinforcementpercentage per spacing14,15 was ρ = 2Ast/(sb)100 = 0.53%,with Ast as the area of one leg of the stirrup. The percentagesof longitudinal bars with whole area Afl were ρfl = Afl /b2(100) = 1.32% and 0.98% for δ = 10 mm (0.39 in.) andδ = 25 mm (0.98 in.), respectively.

Experimental studies concerning compressed RCcolumns9,10 showed that higher strength at the end portionsof the specimens is required to prevent premature and brittlefailure in these regions and to concentrate failure in themiddle portion of the specimens. Therefore, in the presentstudy, the dog-bone shape was adopted, considering also thatthis shape, in the cases of eccentrically loaded specimens,allows high values of eccentricity. For eccentrically loadedcolumns, a constant value of eccentricity e = b/2 wasassumed herein.

Several series of columns were prepared to be tested incompression considering the variation in the concrete type(ordinary and fibrous concrete with volume percentage) andcover thickness δ = 10 mm (0.39 in.) and δ = 25 mm (0.98 in.).For each type of column, two identical specimens were prepared.

To estimate the effects of fibers and cover thickness, thefollowing parameters were fixed during the tests: coredimensions, material characteristics (steel and concrete),dimension and arrangement of longitudinal (number of barsand diameter), and transverse steel bars (diameter and pitch).The pitch of the stirrups adopted in the middle portion of thecolumns was chosen in such a way as to give confinementeffects comparable with those obtained in similar experi-mental researches.10-12

Columns tested in uniaxial compression were denotedwith the initial symbol CC, whereas columns under axialeccentric forces were denoted EC. Particularly, CC1 andCC2 had no fiber and a cover of 10 mm (0.39 in.), whereasCC3 and CC4 had no fiber and a cover of 25 mm (0.98 in.).In the case of FRC, the analogous specimens were CC5,CC6, CC7, and CC8. The EC series considers specimens inplain and FRC with a different cover thickness.

Steel molds with a 3 mm (0.12 in.) wall thickness wereadopted to cast the concrete columns. The casting directionof the specimens was horizontal; therefore, mainly two-dimensional alignments of the fibers are expected. Concretevibration was performed through a needle vibrator for thetime necessary to compact concrete and to avoid the formationof balling of fibers or segregation of coarse aggregates. Aftera few days, the concrete specimens were unmolded andcured in a room at a controlled humidity (90% HM) and aconstant temperature of 20°C (68°F) for a period of 28 days.

Material characterizationThe plain concrete used consisted of the following

composition: 450 kg/m3 (760 lb/yd3) of portland cement(ASTM International Type I), 225 kg/m3 (380 lb/yd3) ofwater, 1150 kg/m3 (1940 lb/yd3) of natural gravel (coarseaggregates with a maximum size of 10 mm [0.39 in.]), 850 kg/m3

(1430 lb/yd3) of sand, and 6.75 kg/m3 (11.4 lb/yd3) of a high-range water-reducing admixture.

Considering that a typical maximum grain size for realcolumns is 20 mm (0.79 in.), the use of 10 mm (0.39 in.)gravel in the tested specimens can be assumed to be acceptablein relation to the scale ratio. A similar size of aggregates,dimensions, and number of specimens were used in a recentinvestigation of FRC columns.12

Fibrous concrete was obtained by adding hooked steelfibers in fresh concrete. The fiber had a length of 30 mm(1.18 in.), a diameter of 0.55 mm (0.02 in.), and a tensilestrength (declared by the manufacturer) of 1115 MPa(161.73 ksi). The fiber percentage adopted was 1% byvolume of concrete, corresponding to 80 kg/m3 (4.99 lb/ft3).

For the mechanical characterization of ordinary andfibrous concrete, 100 x 200 mm (3.94 x 7.87 in.) cylinderswere tested in compression and in tension (split tests) after28 days of curing. Referring to compressive tests on concretecylinders, the authors denote the peak stress with f ′c , thecorresponding strain with εco, and the secant modulus ofelasticity measured in the initial branch of the stress-straincurves with Eco. In addition to this, with ε085 the strain corre-sponding to a stress 0.85f ′c is denoted after the peak load(measured in the softening branch). The tensile strength fct

Giuseppe Campione is an Associate Professor in the Department of Structural andGeotechnical Engineering at the University of Palermo, Palermo, Italy, where he alsoreceived his PhD. His research interests include behavior of fiber-reinforced concretemembers in shear and in compression, flexural behavior of composite members, andconfinement effects in reinforced concrete columns due to FRP materials.

Marinella Fossetti is a PhD Student in the Department of Structural and Geotech-nical Engineering at the University of Palermo. Her research interests include the useof fibrous concrete for structural applications.

Maurizio Papia is a Full Professor of structural engineering in the Department ofStructural and Geotechnical Engineering at the University of Palermo. His researchinterests include probabilistic dynamic analysis of multi-story framed systems in seismicareas, nonlinear seismic response of asymmetric spatial systems, dynamic behavior ofmasonry structures and stability checks, bond stress-slip between steel and concrete,theoretical and experimental investigation of the cyclic behavior of fiber-reinforcedconcrete elements, and reinforcement of masonry structural elements with FRP.

Fig. 1—Construction details of tested columns. (Note:dimensions in mm; 1 mm = 0.0394 in.)


was obtained from split tests. In Table 1, values of f ′c , εco,ε085, Eco, and fct are given for both plain and fibrousconcrete. Values given in Table 1 are an average of thevalues obtained by the three specimens tested for each seriesinvestigated (same batch).

For a mechanical characterization of longitudinal (dl =12 mm [0.47 in.]) and transverse (dst = 6 mm [0.24 in.]) steelbars, direct tensile tests were carried out. The yield stress fy andthe corresponding strain εy, the maximum stress fsmax and thecorresponding strain εsmax, and the ultimate stress atrupture fsu and the corresponding strain εsu are given inTable 2. An average value of the modulus of elasticity of Es =206,690 MPa (29,981.14 ksi) was estimated from measurements.

Test setupA press with a 4000 kN (899.28 kip) load carrying

capacity was used to test the columns. The testing machinewas used in a displacement controlled mode. Tests werecontrolled by an electronic device that recorded the reactionforce and applied displacements.

The testing machine was equipped with a stiff steelhead-plate having a spherical joint at the top and connectedto four stiff steel columns. The head plate moved with relationto the fixed bottom part of the testing machine by means ofa hydraulic pump, controlled by a suitable software.

The vertical load in the concentric compressive tests wasapplied to the columns by interposing two steel plates of20 mm (0.79 in.) thickness between the specimens and thetesting machine. Columns were placed with their verticalaxis corresponding to the vertical axis of the testing machine.

For compressive tests under eccentric loads, two additionalsteel cylinders having 50 mm (1.97 in.) diameter and twoadditional steel plates with 30 mm (1.18 in.) thickness wereplaced between the column and the testing machine (one onthe top and one on the bottom). Columns were placed vertically

and eccentrically with respect to the vertical axis of thetesting machine.

Vertical displacements were recorded by means of fourlinear voltage displacement transducers (LVDTs) placed inthe middle portion of the columns and operating with agauge length equal to the side of the cross section of eachcolumn. This length was chosen in accordance with indica-tions given in the available literature.9 The selected lengthwas sufficient to demonstrate typical failure phenomenaoccurring in the RC columns and short enough to providerepresentative shortening under eccentric testing for calcula-tion of moment-curvature diagrams.10

RESULTS AND DISCUSSIONTest under axial loads

Figure 2 shows the load-axial deformation (P-ε) curvesrecorded during the compressive tests. Each graph showsresults obtained by the two columns for each series investigated.In all of the values of Fig. 2, the actual load recorded by theload cell of the testing machine is denoted as P, whereas ε isthe axial strain estimated on the gauge length equal to thebase of the LVDTs placed in the middle portion of the columns.

For FRC columns, no significant variation in the peak loadwas observed compared to that of ordinary concrete columns(for the same values of cover thickness), but higher ductilitywas observed. It was perhaps because a high percentage offibers was used and a poor compaction of FRC concreteoccurred with respect to the RC columns.

First cracks form at the end portions of all specimens.When their stabilization occurs due to the presence of localsteel reinforcements, new additional cracks form and mainlypropagate into the middle portion of the columns.

At the reaching of peak load, vertical and diagonal cracksform in the middle portion of the columns and the coverspalling process occurs. Cover spalling starts at the cornersof the cross section; therefore, cover spalling occurs alongthe flat portion of the members. In the case of FRC concretecolumns, similar effects are observed but only after the peakload is reached.

After the concrete cover is spalled off (suddenly in ordinaryconcrete and in a more progressive manner for FRCmembers), a loss of load carrying capacity is observed,accompanied by a loss of ductility of the members and, inmany cases, by the overall instability of the longitudinalreinforcement bars.

Figures 3(a) and (b) show a typical failure mode observedfor ordinary and fibrous concrete columns. This mode ischaracterized by the formation of an inclined fracture planecrossing two successive stirrups followed by the rupture of a

Table 1—Mechanical properties of plain and fibrous concrete

fc′ εco ε085 Eco, MPa fct, MPa

Vf = 0% 29.00 0.0019 0.0025 17,679 2.76

Vf = 1% 32.19 0.0023 0.0055 19,464 3.52

Note: 1 MPa = 0.145 ksi.

Table 2—Mechanical properties of main bars

Diameter, mm fy, MPa εy fs max, MPa εsmax fsu, MPa εsu

6 485 0.0025 562 0.0800 472 0.0940

12 461 0.0031 546 0.1278 457 0.1383

Note: 1 MPa = 0.145 ksi; 1 mm = 0.0394 in.

Fig. 2—Load-axial strain curves (P-ε) for columns underuniaxial compression. (Note: 1 kN = 0.2248 kip; 1 mm =0.0394 in.).

Fig 3—Failure condition of compressed specimens with δ =25 mm: (a) RC; and (b) FRC. (Note: 1 mm = 0.0394 in.)


stirrup in tension, the latter occurring in a position near thecorner zone of the cross section. When FRC was used, amore dissipative mechanism was observed (fibers bridgingthe main inclined cracks) and a higher value of axial straincorresponding to the failure of the stirrup was reached.Longitudinal bars buckled in all cases examined.

Test under eccentric loads In this section, results of compressive tests under eccentric

load at a constant eccentricity are shown both in terms ofload-axial strain curves and moment-curvature diagrams.

Figure 4 shows the load axial-strain curves for ordinaryand FRC columns and for the two different values of coverthickness examined. In each diagram, the strain values referto the face in compression (positive strain) and the face intension (negative strain).

An initial linear-elastic branch was observed up to a loadvalue of approximately 11% for δ = 10 mm (0.39 in.) and Vf =0%; 17% for δ = 25 mm (0.98 in.) and Vf = 0%; 23% for δ =10 mm (0.39 in.) and Vf = 1%; and 26% for δ = 25 mm (0.98 in.)and Vf = 1%. After this stage, a nonlinear ascending branchwas observed. This branch was characterized by a loss ofinitial stiffness, mainly due to the formation and propagationof horizontal cracks in the face in tension of the columns(they were approximately spaced as the pitch of the stirrups),and vertical cracks formed initially at the corners of the crosssection of the compression face.

With the increase in vertical load, additional cracksappeared in the tension face with spacing approximatelyequal to half of the pitch of the stirrups. In the compressionface, vertical cracks appeared mainly at the corner zones andalso propagated along the flat portion of the section. Maincracks in the tension face propagated up to the neutral axisposition. When the peak load was reached, the cracks in thetension face were wide open and were between 2 and 6 mm(0.08 and 0.24 in.) wide in RC members and less so in FRC

members. In the compression face, the formation of the mainvertical cracks correspond to the cover spalling process.

In all tests carried out, the complete failure of the specimenscorresponded to the rupture of the main bars in tension, aphenomenon observed in the stress strain curves of Fig. 4when the complete loss of the load carrying capacity occurred.

Results obtained highlighted that, for FRC, the cracksincreased in number and width up to the peak load, but theywere finer with respect to ordinary concrete. After cracklocalization, the softening response began, and the presenceof fibers mitigated the cover spalling process and controlledthe crack opening in the tensile zone, ensuring moreductile behavior.

The experimental derivation of the moment-curvature (M-φ)diagram is of particular interest in the study. For each axialshortening value imposed by the testing machine, the appliedmoment is considered constant along the axis of the memberand its value is M = P⋅e (where e is the eccentricity of theexternal load). This assumption is justified as the secondorder effects can be considered negligible due to the reducedslenderness of the columns (e is constant). Further studieswill be addressed to investigate the second order effects forthe slender FRC column.

Denoting εt and εc the axial strains estimated in the facesin tension and in compression, respectively, and H the horizontaldistance between the two LVDTs, the corresponding curvatureis expressed by

(1)

Figures 5(a) and (b) show the moment-curvature diagramsfor columns with cover thicknesses of 10 and 25 mm (0.39 and0.98 in.), respectively. Curves referring to ordinary concretereinforced columns are indicated in the graphs with thedashed line, while those referring to fibrous concretecolumns are indicated with the continuous line. An initial

φεc εt–

H--------------=

Fig. 4—Load-axial strain (P-ε) for columns under eccentric loading. (Note: 1 kN =0.2248 kip; 1 mm = 0.0394 in.)


elastic branch common to both ordinary and FRC columnswas apparent from the trend of the moment-curvature curves;therefore, it followed a nonlinear branch characterized mainlyby the cracking of concrete in tension.

The main contribution of fibers observed during the testswas the control in crack opening and the reduction in crackthickness. Fibers also controlled the cover spalling processand reduced the risk of buckling of longitudinal bars. Allthese phenomena reflected on the softening response of RCcolumns, ensuring more ductile behavior in the case of FRCmembers. The positive effect of fibers on the stability of thelongitudinal compressed bars was of fundamental importancefor the structural behavior of RC columns as a reduction in theload carrying capacity of the columns occurs if bars buckle.Also, a consequent significant reduction in the confinementpressures occurred and was reflected in less ductility, asconfirmed experimentally. When fibers were used, a lowerslope in the softening branch was observed, but, in any case,the complete failure of columns occurred when stirrups andmain bars failed in tension at the same time.

ANALYTICAL MODELS FOR COMPRESSIVE AND FLEXURAL RESPONSE

The analytical procedures adopted consider bucklingphenomena of longitudinal bars, biaxial state of stresses inconcrete cover, and confinement effects of concrete core.The theoretical model adopted for moment-curvature curvesis the conventional layered method.

The main focus of the analytical section was the choice ofappropriate constitutive laws in compression and in tensionto be adopted for FRC and to describe the behavior in tensionand in compression of longitudinal bars, including thesecond-order effects. Although, in the following section, themain mechanical properties of material will be assumed bythe experimental values, indications for their analytical calcu-lation are also given. It was done to validate the proposedprocedure and to highlight the important role of the behavior ofconstituent materials on the whole response of the columns.

Constitutive laws for unconfined concreteThe stress-strain curve assumed for concrete cover

(considered as unconfined concrete) in both the presence andabsence of fibers, is a rearranged version of that originallyproposed by Hognestad.17 The application of this modelonly requires the knowledge of three mechanical parametersto be determined experimentally (f ′c , εco, and ε085) and isalso a model that several authors used with minor changes todescribe the behavior of confined concrete, obtaining goodagreement with the available experimental data.

The current value of stress is linked to the correspondingvalue of strain by the following expressions

(2a)

(2b)

where f ′c , εco, and ε085 were already defined in Table 1 withreference to ordinary and fibrous concrete, reflecting mainparameters governing the behavior of constituent materials.

According to these values, Eq. (2) defines the pre-peakand the post-peak branches. The post-peak branch wasstopped at strain values εu = 0.0059 and εu = 0.0158 forordinary and fibrous concrete, respectively, because thesevalues correspond to the negligible post-peak strength ofmaterial (observed experimentally).

It was observed that εu is a very important parameter to beestimated because it controlled the post-peak behavior ofconcrete, influencing the overall post-peak response of theRC column. In this paper, the values assumed for εu werederived experimentally; they were very close to thosedetermined by using Eq. (2b), assuming σ = 0.1 f ′c and ε = ε085(refer to Table 1). Recent studies also give analyticalexpression to calculate values for FRC material withhooked steel fibers.18

When referring to the compressive behavior of concretecover, Eq. (2) was used, but compressive peak stress andcorresponding strain were reduced considering the softeningcoefficient proposed by Zhang and Hsu,19 which depends onthe compressive strength of concrete and on the maximumreached lateral strain εt

(3)

with εt related to compressive strain ε by the Poisson coefficientνc (εt = νcε). The Poisson coefficient was assumed to be variablewith the axial strain ε in accordance with the expression20

(4)

The softening coefficient ς that affects f ′c and εco takesinto account the biaxial state of stresses (compression-tension) induced in concrete cover by the compressive loaddirectly carried out and by the tensile stresses arising for thereduced lateral expansion of the concrete core (additionalconfinement due to cover).

Referring to the behavior in tension, the analytical modeladopted is that proposed by Mansur and Ong.21 For ordinaryconcrete, a linear elastic behavior up to the peak stress wasadopted, followed by a linearly decreasing branch to thepoint of zero stress and strain εs1 equal to twice the yieldingstrain of longitudinal bars.

The elastic modulus of concrete in tension was assumed tobe half of the initial modulus in compression, the latter assuggested by Mander et al.22 Therefore, by assuming

(5)

fc fc′ 2 εεco------- ε

εco------⎝ ⎠

⎛ ⎞ 2–⋅= 0 ε εco≤ ≤

fc fc′ 1 0.15ε εco–

ε085 εco–----------------------–⎝ ⎠⎛ ⎞= εco ε εu≤ ≤

ς 5.8

fc′--------- 1

1 400 εt⋅+------------------------------- 0.9

1 400 εt⋅+--------------------------≤=

ν ν0 1 1.3763 εεco------ 5.36 ε

εco------⎝ ⎠

⎛ ⎞ 2

8.586 εεco------⎝ ⎠

⎛ ⎞ 3

⋅+⋅–⋅+⋅=

Ect5000

2------------= fc′ fct 0.7 fc′= εct

fct

Ect-------=

Fig. 5—Experimental moment-curvature diagrams (M-φ): (a)δ = 10 mm; and (b) δ = 25 mm. (Note: 1 kN⋅m = 0.74 kip·ft;1 mm = 0.0394 in.)

ACI Structural Journal/May-June 2010 277

the stress-strain constitutive law in tension for ordinaryconcrete was expressed by

(6a)

(6b)

For FRC, too, an initial linear elastic stress-strain curvewas assumed up to the peak stress. After that, a reduced slopewas assumed with a linear branch connecting the peak stresswith the point of stress equal to the post-peak strength ofcomposite ft,res and corresponding to the previously definedstrain value ε = –εs1. The post-peak strength, as suggested byCampione et al.,1 was assumed to be variable with thevolume percentage of fibers Vf ; the aspect of the ratio offibers (length/diameter); the efficiency factor, depending onfiber orientation; and the pull-out resistance of fiber.

Referring to the current investigation, it shows that1

(7)

Finally, for FRC, the stress-strain curve in tension isexpressed by means of

(8a)

(8b)

(8c)

In the present model, a limit in the maximum strain wasassumed equal to –2εs1. This value does not reflect the realproperties of FRC in tension, but it is appropriate to take intoaccount all of the main phenomena occurring in a crackedreinforced member.

Equations (2) to (8) define the stress-strain curves adoptedherein for ordinary and fibrous concrete, both in tension andin compression.

Constitutive laws for confined concreteThe choice of the constitutive law for confined plain

concrete and FRC is complex mainly for the followingreasons: the stress-strain relationships proposed in theliterature are numerous, but they are strictly influenced bythe hypothesis of the distribution of confinement pressures inthe plane of the stirrups and in the space between twosuccessive stirrups; the stress-strain curves proposed in theliterature are deduced on the basis of concentric compressivetests, for which confinement effects are uniform; and noclear indications are available for FRC confined members.

When referring to the confinement effects of concrete inthe compressed region of RC members subjected to axialforce and bending moment, it must be observed that differentlevels of confinement pressure occur in the different strips ofthe compressed zone. Particularly, strips near the neutral axisare lightly confined, while strips closer to the corners of thetransverse cross section are more confined.

fc Ectε= ε– ct ε 0≤ ≤

fc fctε εs1+

εct εs1–------------------= ε– s1 ε ε– ct≤ ≤

ft res,

0.2 0.6 fc′×=

fc Ectε= ε– ct ε 0≤ ≤

fc f– t res,

fct ft res,

–( )ε εs1+

εct εs1–-------------------+= ε– s1 ε ε– ct≤ ≤

fc f– t res,

= ε ε– s1≤

For confined concrete, the model used herein is the oneproposed by Saatcioglu et al.,14 which was also experimentallytested in the cases of eccentric loading.15 This model wasadopted to describe the confinement effects in both RC andin FRC members.

From zero stress up to the maximum compressive stressf ′cc , the stress-strain curve is expressed by

(9)

The softening branch of the response is linearly decreasingup to the strain εc,02, beyond which a residual strength of theconstant value fc = 0.2f ′cc is assumed. Therefore, the softeningbranch was expressed by Eq. (2b) replacing f ′c with f ′cc andε085 with εc,085, and considering the followingrelationships14

(10)

The effective confinement pressure fle depends on thegeometrical ratio of stirrups ρ, the yielding stress of stirrups,the distance between the main bars, and the dimension of theconfined core. Of course, when fibers are used, εco, f ′c , andε085 reflect the properties of FRC in compression.

By setting fle = 0 and ρ = 0 in Eq. (10), Eq. (9) degeneratesin Eq. (2a) and the softening branch proves to be expressedby Eq. (2b). The application of Eq. (9) up to the peak strainis strictly related to the fact that longitudinal bars in compressiondo not buckle before this strain value23; therefore, the modeladopted considers that, if main bars buckle, the confinementpressure is reduced to zero value at this stage.

Constitutive laws for longitudinal barsThe stress-strain σ-ε behavior for longitudinal steel bars in

tension was assumed to be constituted by three linearbranches. The first one was an elastic branch up to theyielding stress σf = fy, the second was a plastic horizontalbranch with a constant stress fy up to the strain εs, and thethird linear branch was a hardening branch with a reducedslope and with modulus Eh = 0.0033Es.

The use of the previous model (elastic-plastic with strainhardening) for compressed bars is also strictly related toverifying that the longitudinal bar does not buckle after thecover is spalled off and the bars have to be yielded.

The approach proposed by Papia et al.23 was used to verifythe stability condition. The critical load and the buckledlength are obtained according to the equivalent stiffness ofthe system constituted by the longitudinal bars and thestirrups. This stiffness is measured by the parameter γ = αs3/(ErIf), where If is the moment of inertia of the longitudinalbar; Er is the reduced modulus of elasticity of steel bars; andα = EsAst/bc, where bc is the effective length of one leg ofthe stirrup.

By calculating Er and γ with the procedure shown inReference 23, the values Er = 2778 MPa (402.96 ksi) andγ = 3557.8 are found. Depending on this value of theequivalent stiffness, the following corresponding values ofcritical stress and critical length of the bar are calculated:σcr = 188 MPa (27.27 ksi), L = 1.2, and s = 78 mm (3.07 in.).

fc ′fcc 2 εεcco

--------- εεcco

---------⎝ ⎠⎛ ⎞ 2

–

11 2k+---------------

= 0 ε εcco≤ ≤

f ′cc fc′ 6.7fle0.83+= k 6.7fle

0.83 fc′⁄=

εcco εco 1 5k+( )= εc 085,

ε085 260ρεcco+=


It emerges that when longitudinal bars yield in compression,the sudden reduction in the modulus of elasticity (from Es toEr values) determines the buckling of the longitudinal barsthemselves, occurring in a space involving approximatelytwo successive stirrups. In this case, the stress-strain (σ-ε)curve has to be modified by assuming the three branchesproposed by Dhakal and Maekawa24: from the yieldingstress up to a particular point of coordinates (ε*, σ*) a firstlinear reduction in the maximum stress has to be adopted;therefore, a further linear branch with a negative slope equalto –0.02Es has to be used up to the value σ = 0.2fy; finally, ahorizontal branch characterized by this residual stress occurs.

The point of coordinate (ε*, σ*) depends24 on the parameterλb = 0.1Lfy

0.5/dl, that in the cases herein examined is λb =13.956. Moreover, because in the current case σ*= 324 MPa(47.00 ksi) for σ* = 0.0512, the second branch of the curvedescribing the behavior of the buckled bars was not considered.

When the stress-strain curve σ-ε of the buckled longitudinalbar is defined, it is necessary to establish the range of axialstrain for which the bar is effectively involved in the flexuralresponse of the transverse cross section; otherwise, the valueof axial strain of the concrete cover becomes ineffective inpreventing the overall buckling of the longitudinal bars.

In the case of ordinary concrete, confirmed experimentallyherein, the contribution to the lateral stiffness given by theportion of unconfined concrete can be assumed ineffectivewhen the axial strain reaches the value ε085. For this value,as suggested in the original formulation,17 the linearresponse in terms of the stress-strain curve given by Eq. (2b)must be stopped (if the high degradation of the concretematrix at this stage of strain is taken into account).

In the case of FRC, it must otherwise be verified that thecompressed bar does not buckle before the complete coverspalling process by considering the effective contributionof the fibers to the cover strength. In this case, the wholeresponse in the compression of longitudinal bars without abuckling effect is attained up to ultimate strain ε = εu =0.0158. To do this, the formulation given by Russo andTerenzani25 is adopted, in which the longitudinal bar ismodeled through an elastic beam on elastic continuoussprings. The stiffness of these springs was calculated byspreading the concentrated stiffness of the single stirrups inthe pitch s giving βst = α/s and adding to this value a furthercontribution due to the fibers, which can be assumed inthe form26

(11)

where Efb is the elastic modulus of fiber, lfb is the length, b isthe side of the transverse cross section, and nb is the numberof longitudinal bars in the transverse cross section subject tosimultaneous buckling. In the absence of fibers (Vf = 0),Eq. (11) gives zero stiffness and only the stiffness due tosteel stirrups has to be considered in the calculation of crit-ical load.

In the more unfavorable case examined herein, that is,when b = 185 mm (7.28 mm), Eq. (11) provides βfb =14,173 MPa (2055.85 ksi), whereas the distributed stiffnessof stirrups is βst = 536 MPa (77.75 ksi). Neglecting βst, thecritical load of the longitudinal bars turns out to be Pcr =(12ErIfβfb)0.5 = 707.12 kN (158.97 kip), and therefore σcr =

βfbEfb

λfb

------- bnb

-----Vf

3-----⎝ ⎠

⎛ ⎞23---

=

6252 MPa (906.87 ksi). This result confirms that if the cover iseffective, the compressed bars are not affected by buckling effects.

ANALYTICAL PROCEDURE FOR LOAD-AXIAL STRAIN CURVES IN COMPRESSION

To determine the complete axial load-axial strain curve foreach shortening value, it was assumed that the full load P isthe sum of the three different strength contributions constitutedby Pcover due to the concrete cover area in a biaxial state ofstresses (Eq. (2a) and (2b)); Pcore due to the concrete corearea in a triaxial stress state of stresses (Eq. (9) and followingbranches); and Psl due to the longitudinal bars, includingbuckling phenomena.

The procedure is based on the following steps: an initialvalue of axial shortening ε is assumed, the lateral strain εt =νcε is computed assuming a fixed variation law of ν with ε(Eq. (4)), the axial stresses in concrete cover and core aredetermined by using the constitutive laws shown previously,and the steel contribution is calculated including bucklingeffects. Repeating this procedure for all possible values ofaxial strain, the complete load-strain curve is plotted.

The experimental and analytical compressive responses oftested columns obtained by this procedure are shown inFig. 6(a) and (b). The experimental responses are theaverage curves of the two tests for each series investigated.

The graphs clearly show that the analytical model fits theexperimental results with acceptable agreement. Moreover,it can be observed that the addition of fibers producessignificant improvement in the performance of the columns,especially referring to strength reduction occurring in theconcrete cover after the peak load. This reduction is moreprogressive for FRC columns with respect to that of RCcolumns with a beneficial influence on the stability conditionof longitudinal bars.

ANALYTICAL PROCEDUREFOR MOMENT-CURVATURE DIAGRAMS

The analytical model adopted herein is able to determinethe moment-curvature diagrams based on the discretizationof the generic cross section into strips. The moment-curvature diagrams for the columns examined refer to anysection in the middle portion of the columns because theeffects of the self weight of the column and the second ordereffects are negligible.

For each value of assigned curvature φ, the section is aswholly compressed if, in the more stressed fiber of thetransverse cross section, the axial strain εc assumes valueεc,max = φb. By assuming the previously mentioned value forεc and denoting di and dfi as the distances of the barycenterof the strip and the reinforcement area from the barycenter ofthe section, the characteristic strain value of each strip is εi =εc – φ(b/2 – di) and for longitudinal bars is εfi = εc – φ(b/2 –dfi). Therefore, by knowing the constitutive laws of constituentmaterials, the corresponding stresses are calculated and thestrength contributions of each strip are included in the equi-librium conditions. The sum of these strength contributionsgives two M, P values characterized by the ratio e = M/P thatis certainly lower than the effective value e = b/2.

Therefore, maintaining φ constant, the axial strain εc isreduced by a fairly small quantity Δεc, and the previouslymentioned numerical procedure is repeated for a furtherreduction of εc to the following condition is verified.

ACI Structural Journal/May-June 2010 279

(12)

In Eq. (12), the j – 1 and j indexes correspond respectivelyto the axial force and to the bending moment calculated forεc1 = εc,max – (j – 1)Δεc and εc2 = εc,max – jΔεc. The effectivepair of M, P values corresponding to the given values ofcurvature φ and eccentricity b/2 is determined by assumingthat the effective value of εc is linearly varying witheccentricity e in the range from ej – 1 to ej. The whole curveis obtained by increasing the curvature value from zero up toa preset maximum value.

It must be observed that if, in applying the proposedmodel, a strain limit for εc,max is imposed (in the current caseit was assumed εc,max = 0.03), a high value of curvature φ canlead to the conditions that εc,max < φb and M/P > b/2 for thefirst value of strain assumed. In this case, the procedure isstopped at the last point of the moment-curvature diagram,which corresponds to the previous step.

Figure 7 shows the comparison between the resultsobtained by using the proposed model and the experimentalones. The moment-curvature diagrams M-φ deducednumerically highlight the good agreement with the experi-mental data, confirming the appropriate interpretation ofmain phenomena involved in the structural behavior of RCor FRC members. Regarding the cases of fibrous concretefor Vf = 1% and δ = 10 mm (0.39 in.), the model gives agood prediction of the experimental response, whereas forspecimens in fibrous concrete with δ = 25 mm (0.98 in.), themodel overestimated the experimental response, particularly

Mj 1–

Pj 1–

------------ ej 1–b2---<=

Mj

Pj

------ ejb2--->=

Fig. 6—Comparison between analytical and experimentalaxial load-axial strain diagrams of: (a) RC columns; and (b)FRC columns. (Note: 1 kN = 0.2248 kips; 1 mm = 0.0394 in.)

Fig. 7—Comparison between analytical and experimental moment-curvature diagrams(continuous line = analytical and dashed line = experimental). (Note: 1 kN⋅m = 0.74 kip⋅ft;1 mm = 0.0394 in.)


when referring to the peak load. This scatter is probably dueto an overestimation of the mechanical properties of unconfinedmaterial, especially when referring to the concrete cover.

CONCLUSIONSIn the present paper, experimental and analytical research

was presented that served to explain the compressivebehavior of RC columns with square cross sections underaxial force with and without initial eccentricity. Cases ofordinary and fibrous concrete columns in the presence oflongitudinal and transverse steel reinforcements were examined.

The paper highlighted the influence of the cover thicknessand the presence of hooked steel fibers, and fixed values ofthe other parameters (volumetric ratio, spacing, yieldingstrength, concrete grade and length, diameter and fibervolume, and dimension of the transverse cross section) onthe strength and strain capacity of RC columns.

The experimental research carried out showed that, in thecase of axial concentric load, the use of fibers ensured ahigher post-peak strength and ultimate strain compared toRC columns and a reduction in the brittleness of the mechanisminvolving crushing of compressed cover. For columns underaxial eccentric forces, it was observed that the use of fibersproduced increases in: the maximum curvature of the section;the strain values corresponding to failure of stirrups; and thestrains of longitudinal bars in tension. Moreover, by using fibers,the cover spalling process and, consequently, the buckling of thecompressed reinforcing bars, was delayed.

From a theoretical point of view, the focus of the researchwas on the derivation of the load-axial strain and moment-curvature diagrams, including the constitutive laws of thematerials, the effects of fibers, the confinement induced bytransverse steel, and the buckling of longitudinal bars. Withregard to the accuracy of the model, it must be observed thatit was capable of considering the effect of several parameterssuch as the cover thickness and the buckling phenomena oflongitudinal bars, both aspects strongly influencing thestructural response of the columns.

The comparison between the theoretical and the experimentalresults, both in terms of load-axial shortening and moment-curvature diagrams, highlighted the capacity of the proposedmodel to adequately include the main phenomena involvedin the structural behavior of RC or FRC columns.

To give more general conclusions, further studies with ahigher number of specimens and variation of details will beaddressed for the examination of larger scale testing. Moreover,further studies will be addressed to consider the interactionbetween the behavior of confined and unconfined concrete,including the variation in confinement pressures due to thebuckling of longitudinal bars.

NOTATIONAf = area of longitudinal steel barsAst = area of one leg of stirrupb = side of transverse cross sectiondf = diameter of fiberdl = diameter of longitudinal barsdst = diameter of leg of stirrupEco = initial modulus of concreteEct = elastic modulus of concrete in tensionEfb = elastic modulus of fiberEh = hardening modulus of steelEs = elastic modulus of steele = eccentricityfc′ = compressive strength of unconfined concretefcc′ = compressive strength of confined concretefct = tensile strength of concrete

fs max = maximum stress of steelfsu = ultimate stress of steelfy = yielding stress of steelH = gauge length for evaluation of curvatureL = critical length of longitudinal steel barM = bending momentP = vertical loads = pitch of stirrupsVf = volume percentage of fibersδ = cover thicknessεc = strain in compressed face of columnεco = peak strain of unconfined concrete in compressionεcco = peak strain of confined concrete in compressionεct = peak strain in tensionεs max = strain at maximum stress fs maxεsu = strain at ultimate stress fsuεt = strain in face in tension of columnεu = ultimate strain of concrete in compressionε085 = strain in compression at 0.85fc′φ = curvature of sectionlfb = length of fiberν = Poisson coefficientρ = confinement transverse reinforcement percentageρfl = longitudinal reinforcement percentage

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behavior of fiber - reinforced concrete columns under axially and eccentriacally compressive loads

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