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ACI Structural Journal/July-August 2006 551

ACI Structural Journal, V. 103, No. 4, July-August 2006.MS No. 05-026 received December 17, 2005, and reviewed under Institute publication

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ACI STRUCTURAL JOURNAL TECHNICAL PAPER

An extensive experimental investigation with the aim of studyingthe structural behavior of slabs on ground made of steel fiber-reinforced concrete (SFRC) is presented in this paper. Several full-scale slabs reinforced with different volume fractions of steel fibershaving different geometries were tested under a point load in theslab center. A hybrid combination of short and long fibers was alsoconsidered to optimize structural behavior. Experimental resultsshow that steel fibers significantly enhance the bearing capacityand the ductility of slabs on ground.

The nonlinear behavior of these SFRC structures is well capturedby performing nonlinear fracture mechanics analyses where theconstitutive relations of cracked concrete under tension wereexperimentally determined. Finally, from an extensive parametricstudy, design abaci and a simplified analytical equation for predictingthe minimum thickness of SFRC slabs on ground are proposed.

Keywords: pavement; reinforced concrete; slabs on ground.

INTRODUCTIONIn the last decades, the use of steel fiber-reinforced

concrete (SFRC) has significantly increased in industrialpavements, roads, parking areas, and airport runways as aneffective alternative to conventional reinforcement (that is,reinforcing bars or welded mesh). Because heavy concentratedloads from industrial machinery and shelves may causeintensive cracking and excessive deformation of pavements,a diffused fiber reinforcement may help the structural behavior.

Many of these pavements are slabs on ground that arestatically undetermined structures. For this reason, even atrelatively low volume fractions (<1%), steel fibers effectivelyincrease the ultimate load and can be used as partial (or total)substitution of conventional reinforcement (reinforcing bars orwelded mesh) of slabs on ground. Fiber reinforcement alsoprovides a better control of the crack development to improvethe structural durability and to reduce the number of joints.1-3

Moreover, fiber reinforcement enhances the impact and fatigueresistance of concrete structures and reduces labor costs due tothe amount of time saved in the placement of the reinforcement.

At present, design rules for SFRC structures are notpresent in the main international building codes, even thoughACI Committee 544, RILEM Technical Committee 162-TDF,and the Italian Board of Standardization have recently proposedrecommendations or design guidelines.4-6 Because these guide-lines are under development, designers often work under theusual assumption of elastic behavior of a concrete slab on anelastic subgrade, according to the Westergaard theory.7 Thisassumption is markedly restrictive for SFRC slabs and leadsto a significant underestimation of the actual bearing capacity ofthe slab.8 In fact, a linear elastic approach cannot properlytake into account the beneficial effects of fiber reinforcementwhich become effective only after cracking of the concretematrix when SFRC behavior is significantly nonlinear8,9

(Fig. 1). As a consequence, more appropriate methods basedon the yield line theory have been proposed to predict the

ultimate load.10 The upper bound method of limit analysis,which assumes a flexural mode of failure and perfect plasticity,however, is not a straightforward application of the SFRCstructures with low volume fractions of steel fibers (Vf < 1%)due the softening behavior of the material (Fig. 1). If anonstandard form of the yield line theory is formulated byassuming an average post-cracking strength, the collapseload is still underestimated.11

The finite element (FE) method based on nonlinear fracturemechanics (NLFM)12 appears to be the most accurate toolfor analyzing SFRC slabs on ground because it allows areproduction of the actual collapse mechanism and thedevelopment of a new design approach.13

Because a limited number of experiments are currentlyavailable in the literature,8,14,15 several full-scale tests on FRCslabs were carried out in an extensive research program tovalidate the NLFM approach. The experimental model aimsto simulate a zone of pavement included between joints whereone or more concentrated loads may be applied at any point.Because part of a load close to a joint is transferred to theadjacent slabs,16 however, it was found that the load placed inthe center of a single slab is particularly significant for design.

For the sake of clarity, it should be noted that other importantphenomena (such as the curling effect) present in concretepavement are not considered herein.

A further goal of the research concerns the possibility ofenhancing structural performance by combining steel fibersof different dimensions and geometries (hybrid fiber reinforcedconcrete [HyFRC]). In fact, fibers start activating aftercracking (not visible microcracks) of the concrete matrix.Because fibers of different sizes become efficient at differentstages of the cracking process, however, a hybrid combinationof short and long steel fibers may enhance the concrete

Title no. 103-S58

Steel Fiber Concrete Slabs on Ground: A Structural Matterby Luca G. Sorelli, Alberto Meda, and Giovanni A. Plizzari

Fig. 1—Different fiber activation with respect to crackdevelopment in tensile test.20

ACI Structural Journal/July-August 2006552

toughness at small crack opening displacements17-19 (Fig. 1).Moreover, due to the better control of the cracking process,shorter fibers reduce the material permeability20 and HyFRCappears to be a promising application for pavementssubjected to aggressive environments.

RESEARCH SIGNIFICANCEWhereas the structural behavior of plain concrete and

conventionally reinforced slabs on ground is well known,

there is still a lack of design rules for steel fiber reinforcedconcrete slabs in building codes. Due to this lack, conventionaldesign methods, based on the elastic theory, are used for fiberreinforced slabs whose behavior is significantly nonlinear.The behavior of slabs on ground with steel fibers was experi-mentally studied by performing full-scale tests; a designapproach based on nonlinear fracture mechanics is also proposed.

To enhance the structural response, the use of HyFRCsystems, combining shorter and longer steel fibers, was alsoconsidered.

EXPERIMENTAL PROGRAMFull-scale slabs on ground were tested under a point load

in the center. The experimental model aimed to reproduce asquare portion of pavement, limited by joints, with a side (L)of 3 m (118.11 in.) and a thickness (s) of 0.15 m (5.91 in.).Additional tensile and bending tests were carried out to identifythe fracture behavior of SFRC. The slab tests presented inthis paper are part of an extensive research campaign whoseresults are published elsewhere.18,21

MaterialsThe concrete matrix was made with cement CEM II/A-LL

42.5R (UNI-ENV 197-1) and natural river gravel with arounded shape and a maximum diameter of 15 mm (0.59 in.);its composition is summarized in Table 1.

Five different types of fibers were considered in thisresearch, as reported in Table 2 where geometrical andmechanical properties of fibers are shown; the fiber code isconventionally defined by the fiber length and the fiberdiameter (Lf /φf , in millimeter unit). Two straight shorterfibers (12/0.18 and 20/0.4) and three longer fibers withhooked ends (30/0.6, 50/1.0(a), and 50/1.0(b)) were adopted.All the fibers have a rounded shaft, an aspect ratio rangingbetween 50 and 66, and a Young modulus of approximately210 GPa (30456.9 ksi).

Seven SFRC slabs (S1, S3, S4, S5, S8, S11, and S14), witha volume fraction Vf of fiber smaller than 0.6% and a referenceslab made of plain concrete (S0) are reported in this paper(Table 3). Figure 2(a) shows the slab with a hydraulic jackplaced in its center.

Luca G. Sorelli is investigating ultra-high-performance concrete (UHPC) structuralimplications by micromechanics and chemo-plasticity approaches at MassachusettsInstitute of Technology, Cambridge, Mass. He received his doctorate from the Universityof Brescia, Brescia, Italy.

Alberto Meda is an Associate Professor of structural engineering, Department ofEngineering Design and Technology, University of Bergamo, Bergamo, Italy. Hereceived his degree in environmental engineering from the Milan University ofTechnology, Milan, Italy, in 1994. His research interests include concrete fracturemechanics, fiber-reinforced concrete, and fire design of reinforced concrete structures.

ACI member Giovanni A. Plizzari is a Professor of structural engineering, Departmentof Civil Engineering, University of Brescia. His research interests include materialproperties and structural applications of high-performance concrete, fiber-reinforcedconcrete, concrete pavements, fatigue and fracture of concrete, and steel-to-concreteinteraction in reinforced concrete structures.

Table 1—Composition of concrete matrixMixture component Quantity

Cement 42.5R (ENV 197-1) 345 kg/m3 (21.54) lb/ft3

Water 190 kg/m3 (11.86) lb/ft3

High-range water-reducing admixture (melamine-based) 0.38%vol

Aggregate (0 to 4 mm) 621 kg/m3 (38.77) lb/ft3



Table 2—Geometrical and mechanical properties of steel fibers

Fiber code

Lf , mm (in.) φf , mm (in.) Lf /φf fft , MPa (ksi) Fiber shape

50/1.0(a) 50 (1.97) 1.00 (0.0394) 50.0 1100 (159.5)

50/1.0(b) 50 (1.97) 1.00 (0.0394) 50.0 1100 (159.5)

30/0.6 30 (1.18) 0.60 (0.0236) 50.0 1100 (159.5)

20/0.4 20 (0.79) 0.40 (0.0157) 50.0 1100 (159.5)

12/0.18 12 (0.47) 0.18 (0.0071) 66.6 1800 (261.1) —

Table 3—Volume fractions of steel reinforcement (steel fibers or welded mesh)

Slab no.

Steel fibersVf,tot,%vol

50/1.0 %vol

30/0.6 %vol

20.04 %vol

12/0.18 %vol

S0 — — — — 0.00

S1 — 0.38 — — 0.38

S3 0.38(a) — — 0.19 0.57

S4 0.38(a) — — — 0.38

S5 — 0.38 — — 0.38

S8 0.38(b) — — — 0.38

S11 0.57(a) — — — 0.57

S14 0.38(b) — 0.19 — 0.57

Fig. 2—Test setup for: (a) slab on ground; (b) small beams;(c) deformed FE meshed for numerical simulation of slab;and (d) notched beams under bending.


Table 4 reports the mechanical properties of concrete ofthe different slabs, as determined on the day of the test; inparticular, Table 4 shows the tensile strength fct from cylinders(φc = 80 mm [3.15 in.], L = 250 mm [9.84 in.]), the compressivestrength from cubes fc,cube of 150 mm side (5.91 in.); theYoung’s modulus as determined from both compressiontests on cylinders Ec (φc = 80 mm [3.15 in.]; L = 200 mm[7.87 in.]) and from core specimens Ec,core (φc = 76 mm[2.99 in.]; L = 150 mm [5.91 in.]) drilled out from the slab(after the test). The slump of the fresh concrete was alwaysgreater than 150 mm (5.91 in.).

Fracture properties were determined from six notched beams(150 x 150 x 600 mm [5.91 x 5.91 x 23.62 in.]) tested underfour-point bending according to the Italian Standard22

(Fig. 2(b)). The notch was placed at midspan and had a depthof 45 mm (1.77 in.) (Fig. 2(b), (d)). These tests were carried outwith a closed loop hydraulic testing machine by using the crackmouth opening displacement (CMOD) as a control parameter,which was measured by means of a clip gauge positionedastride the notch. Additional linear variable differentialtransformers (LVDTs) were used to measure the crack tipopening displacement (CTOD) and the vertical displacement atthe beam midspan and under the load points (Fig. 2(b)).

Test setup and instrumentationThe slabs were loaded by a hydraulic jack placed in the

center by using the load frame shown in Fig. 2(a); theaverage loading rate was 2.5 kN/min (0.56 kips/min).

To reproduce a Winkler soil, 64 steel supports were placedunder the slab at centers of 375 mm (14.76 in.) in both directions(Fig. 3(a)). These supports are steel plates on a square basehaving a side of 100 mm (3.94 in.; Fig. 3(b)). Previousnumerical simulations showed that the experimental subgradeprovides a good approximation of a continuous Winklersoil.16 Because of the curling of the concrete slabs due toshrinkage and the thermal effect, a layer of high-strengthmortar a few millimeters thick was placed on each spring toensure the contact with the bottom face of the slab. Theaverage spring stiffness was determined by compressiontests performed on each spring with results approximatelyequal to 11.0 kN/mm (2.47 kips/mm). By considering theinfluence area of each spring (375 x 375 mm [14.76 x14.76 in.]), the average Winkler constant kw was equal to0.0785 N/mm3 (289.2 lb/in.3), which corresponds to auniform graded sand soil according to ACI classification.23

During the tests, the vertical displacements of 12 points onthe top surface of the slab were continuously monitored;furthermore, four inductive transducers were placed on thebottom surface of the slab to detect the width of possible

cracks (that were expected to form along the medial lines ofthe slabs; Fig. 2(a)).

Experimental resultsExperimental results are initially presented in terms of

vertical load versus the deflection of the slab center. Thecurves are plotted up to the collapse only. The structuralresponse of reference Slab S0 and the Slabs S1, S5, S4, andS8, reinforced by an equal volume fraction (Vf = 0.38%) offibers having different geometries but the same aspect ratio(50/1.0(a), 50/1.0(b), and 30/0.6), is compared in Fig. 4(a).Beyond the first cracking point, which can be conventionallyassumed in correspondence of the loss of linearity that occursbetween a load level of 100 kN (22.48 kips) and 150 kN(33.72 kips), the fracture behavior of the SFRC slabs isremarkably different from the plain concrete slab. Thesteel fibers effectively enhance the bearing capacity of theslab up to a maximum load higher than 260 kN (58.45 kips);moreover, fiber reinforcement assures a ductile failure whilethe reference slab (made of plain concrete) showed a brittlefailure when a maximum load equal to 177 kN (26.30 kips)was reached.

Slabs S1 and S5, reinforced with fibers 30/0.6, performedslightly better than the Slabs S4 and S8, reinforced withlonger fibers (50/10(a) and 50/10(b)).

The effect of the fiber content on the structural responseis shown in Fig. 4(b), which compares the response of Slabs S4and S11. These slabs were reinforced with 30 kg/m3 (1.87 lb/ft2;Vf = 0.38%) and 45 kg/m3 (2.81 lb/ft2; Vf = 0.57%) offibers 50/10(a), respectively. The experimental curves show thatthe higher fiber content slightly increases the structural ductilitywhile the ultimate load does not seem significantly influenced.

Table 4—Mechanical properties of concrete

Slab no.fct ,

MPa (psi)fc,cube,

MPa (psi)Ec,

MPa (ksi)Ec,core,

MPa (ksi)

S1 2.01 (292) 35.3 (5120) NA 24,463 (3548)

S3 2.18 (316) 33.9 (4917) NA 22,486 (3261)

S4 1.79 (260) 35.3 (5120) NA 23,446 (3400)

S5 NA 36.1 (5236) NA 24,786 (3595)

S6 1.84 (267) 35.9 (5207) 21,438 (3109) 20,989 (3044)

S8 1.40 (203) 30.4 (4409) 24,790 (3595) 19,964 (2895)

S11 1.63 (236) 33.1 (4801) 21,486 (3116) 17,335 (2517)

S14 NA 32.3 (4685) NA NA

Note: NA = data not available.

Fig. 3—(a) Positioning of slab on uniform grid of steelsupports; and (b) three-dimensional view of steel support.

(a)

554 ACI Structural Journal/July-August 2006

Figure 5(a) shows the curves obtained from all the slabswith a volume fraction of fibers equal to 0.57%; the referenceSlab S0 (of plain concrete) is also shown. In particular, Slab S11is made of a single type of fibers (50/1.0(a)) while Slabs S3 andS14 are made of a combination of longer and shorter fibers(refer to Table 3). Although slabs with hybrid fibers haveslightly higher maximum loads, the main contribution of this

reinforcement concerns the crack opening (measured on theslab side at the bottom of a median line) that is significantlysmaller than in Slabs S11, which has a single type of fiber(Fig. 5(b)). This can be explained by the better efficiency ofshorter fibers to bridge smaller cracks delaying theircoalescence in localized and large cracks.19

The steel fibers did not substantially affect the final crackpatterns of the slabs which are characterized by four majorcracks started from the slab center and developed along themedian lines or, in a few cases, along the diagonals (Fig. 6).

NUMERICAL MODELINGNumerical analyses based on NLFM were performed by

adopting MERLIN24; the experimental crack patterns shownin Fig. 6 justify a discrete crack approach with the crackslocated along the median or the diagonal lines. Interfaceelements in predefined discrete cracks initially connect thelinear elastic subdomains (as rigid links) and start activating(that is, the crack starts opening) when the normal tensilestress (at the interface) reaches the tensile strength of thematerial. Afterwards, the crack propagates and cohesivestresses are transmitted between the crack faces according toa stress-crack opening (σ-w) law (which is given as input forthe interface elements).

Inverse analyses25 of the bending tests were performed todetermine the best fitting softening law of cracked concrete(σ-w) that was assumed as bilinear (Fig. 7) where the steeperbranch can be associated with (unconnected) microcrackingahead of the stress-free crack whereas the second part representsthe aggregate interlocking or the fiber bridging.

The beam was modeled by triangular plain stress elements(Fig. 2(d)) with Young’s modulus Ec experimentallydetermined from the cylindrical specimens and Poisson’sratio ν assumed equal to 0.15.

The material parameters identified from the bending testsare summarized in Table 5; as typical examples, the numericaland experimental load-displacement curves for the materialsused in Slabs S4 and S8 are compared in Fig. 8. These material

Fig. 4—(a) Experimental load-displacement curve for slabswith fiber content of 30 kg/m3 (Vf = 0.38%); and (b) withdifferent contents of Fiber 50/1.0(a).

Fig. 5—(a) Experimental load-displacement curve; and(b) experimental load-crack opening displacement curve forslabs with same volume fraction (Vf = 0.57%) with fiberhaving one or two different geometries.

Fig. 6—Final crack patterns of slabs.

Fig. 7—Bilinear approximation of stress versus crack-openingcurve of cracked concrete.


parameters fct, σ1, w1, and wcr were adopted for the numericalsimulations of the slab specimens.

The slab was modeled by 4432 four-node tetrahedralelements for the elastic subdomains linked by 576 interfaceelements (Fig. 2(c)), along the cracks. The elastic soil(Winkler soil) was modeled by 616 linear elastic trusselements, simulating bidirectional springs connected to theslab bottom nodes, and with global stiffness equal to theexperimental value (kw = 0.0785 kN/mm3 [289.2 lb/in.3]).Although no-tension springs were used in the tests, only areduced area of the slab corner was observed to uplift duringthe experiments.

Numerical resultsThe failure of a SFRC slab on ground is neither sudden nor

catastrophic and the slab continues to carry further load evenafter a collapse mechanism occurs. The ultimate load wasconventionally defined as corresponding to a sudden changeof the monitored displacements (Fig. 9) that evidence theformation of a collapse mechanism (fully developed cracksurface along the medians or the diagonals, depending on theratio between the slab and the soil stiffness).

The numerical and experimental load-displacement curvesare compared in Fig. 10 (the displacement is measured on thetop surface of the slab center). In addition, the numericaldevelopment of the crack pattern is displayed. The cracksbegin to develop on both the median and the diagonalsurfaces and the slab collapse occurs when two cracks (eitherthe median or the diagonal ones) develop up to the slabborder. It can be noticed that, in all cases, the overall structuralbehavior is well captured by the numerical analyses based onNLFM. These results further confirm the opportunity ofusing an NLFM approach to analyze SFRC structures.

Table 6 reports the values of the maximum load obtainedfrom both the experiments and the numerical analyses: oneshould note that the numerical predictions are in good agreementwith the experimental values (the average discrepancy isapproximately 7.9% with a maximum of 14.2%).

As a further comparison, Fig. 11(a) shows the numericaland experimental displacements monitored on Slab S5 at themaximum load; the model response is stiffer because of thenumerical assumption of bilateral behavior of the springsupports, whereas tractions cannot be transmitted by theexperimental springs. As previously mentioned, however, onlya small slab portion is subjected to uplift at collapse. Thecomparison between the numerical and the experimental crackopening, measured at the bottom surface of Slab S5, is displayedin Fig. 11(b); once again, a good agreement between thenumerical and the experimental response was noted.

Fig. 8—Comparison between numerical (dotted) andexperimental load-CTOD curves as obtained from bendingtests on SFRC used for: (a) Slab S4; and (b) Slab S8.

Table 5—Fracture properties of concrete

Slab no.

Ec,core, MPa (ksi)

σct, MPa (psi)

w1,mm (in.)

σ1,MPa (psi)

wcr,mm (in.)

GF ,N/mm (lb/yd)

S1 24,463 (3547.9)

3.1 (450)

0.025 (0.00098)

0.93(135)

13.00 (0.51181)

6.08 (1.250)

S3 22,486 (3261.2)

3.3 (479)

0.019 (0.00075)

1.23 (178)

20.00 (0.78740)

12.33 (2.535)

S4 23,446 (3400.4)

3.3 (479)

0.023 (0.00091)

0.88 (128)

20.00 (0.78740)

8.84 (1.817)

S5 24,786 (3594.8)

3.0 (435)

0.028 (0.00110)

1.20 (174)

10.00 (0.39370)

6.04 (1.242)

S6 20,989 (3044.1)

3.0 (435)

0.035 (0.00138)

0.75 (109)

0.25 (0.00984)

0.15 (0.031)

S8 19,964 (2895.4)

3.5 (508)

0.022 (0.00087)

0.67 (97)

20.00 (0.78740)

6.74 (1.386)

S11 17,355 (2517.0)

3.3 (479)

0.020 (0.00079)

0.90 (131)

26.00 (1.02362)

11.73 (2.411)

S14 * 3.1 (450)

0.026 (0.00102)

1.02 (148)

18.00 (0.70866)

9.22 (0.031)

*Assumed equal to 20 GPA (2900.7 ksi).

Fig. 9—Conventional assumption of experimental collapseload for slab on ground.

Table 6—Experimental and numerical collapse loads for tested slabs

Slab no.

Fu,exp ,kN (kips)

Fu,num ,kN (kips)

, %

S0 177.0 (39.79) 174.0 (39.12) –1.7

S1 265.0 (59.58) 240.5 (54.07) –9.2

S3 274.9 (61.80) 236.0 (53.06) –14.2

S4 238.6 (53.64) 245.3 (55.15) 2.8

S5 252.3 (56.72) 247.0 (55.53) –2.1

S8 246.2 (55.35) 215.4 (48.43) –12.5

S11 231.9 (52.14) 255.7 (57.49) 10.3

S14 273.0 (61.38) 244.7 (55.01) –10.7

errFu num,

Fu exp,–

Fu exp,

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

556 ACI Structural Journal/July-August 2006

Design abaci based on nonlinear fracture mechanicsA parametric study, based on approximately 1000 FE

simulations, has been carried out to develop design abaci forFRC slabs on ground,13 by considering the following variables:subgrade modulus kw (0.03, 0.06, 0.09, 0.12, 0.15, 0.18, and0.21 kN/mm3 [110.5, 221.0, 331.6, 442.1, 552.6, 663.1, and773.7 lb/in.3]), slab thickness s (150, 200, 250, 300, and350 mm [5.91, 7.87, 9.84, 11.81, and 13.78 in.]), loading areaa (400, 14,400 mm2 [0.62, 223.20 in.2]), concrete compressivestrength fc (25, 30, and 40 MPa [3.6, 4.4, and 5.8 ksi]) andfiber content Vf (0, 20, 30, 40, 50 kg/m3 [0, 1.25, 1.87, 2.50,and 3.12 lb/ft3]). To identify the material properties, needed asinput for the NLFM analyses, the 15 materials considered inthe parametric study were cast and tested under tensile andbending tests and, eventually, the σ-w laws of crackedconcrete were determined by performing inverse analyses.

Figure 12 shows a typical abacus that, once the soil stiffnessand the design load is known, easily provides the minimumslab thickness. This aims to help professional engineers,whose offices often are not equipped with NLFM programs,to powerfully apply NLFM in practice.13

Simplified modelTo further simplify the design approach, the numerical

curves of the abaci could be approximated by closed-formequations, which also provide the minimum slab thickness

Fig. 10—Numerical (dotted lines) and experimental load versus centerdisplacement of slab.

Fig. 11—(a) Numerical and experimental vertical displacementsalong diagonal of Slab S5 at collapse; and (b) curve loadversus crack opening displacement for Slab S5.


once the design load and the material and soil propertiesare known.

By considering the main geometrical and mechanicalparameters governing the slab behavior, the load-carryingcapacity of FRC slabs on ground can be written in thefollowing form

(1)

where AL is the loading area; B is the slab stiffness defined as

where t is the slab thickness and ν is the Poisson’s ratio(assumed equal to 0.18); fIf is the first cracking strength andfres is an average residual strength that should represent thepost-cracking behavior of SFRC for smaller crack openings.The latter two parameters are defined by the ItalianStandard22 as (in Fig. 13, fres is indicated as feq(0-0.6) forsmaller crack opening and feq(0.6-3) for larger crack opening)

(2a)

(2b)

where PIf is the (total) first-crack load, b is the beam width, h isthe beam depth, and a0 is the notch length.22 The InternationalStandards on SFRC characterization usually provide residualstrength values corresponding to both smaller and largercrack openings. The latter is useless in slabs on ground becausethe collapse occurs with small crack openings (Fig. 5(b)).

The unknown parameters αi and ci of Eq. (1) weredetermined by adopting the least square method, whichminimizes the square of the differences between the collapse

Fu c1AL

L2------

α1

Bα2 kw

α3 fIf

α4 fres

α5 c2+⋅ ⋅ ⋅ ⋅ ⋅=

Ec t3⋅

12 1 ν2–( )-------------------------

fIfPIf s

b h a0–( )2------------------------=

fres

F w( )W

------------ wd⋅

CTOD0

CTOD0 0.6+

∫

0.6------------------------------------------------=

loads calculated by the FE analyses and the ones predictedby Eq. (1). In addition, for the sake of safety, it is imposedthat the approximated maximum load is always smallerthan the corresponding value determined by means of FEanalyses based on NLFM. Although this additional conditionreduces the accuracy of the approximated solution, it isbetter used in structural design to obtain a safer evaluationof the minimum thickness. Hence, the equation parametersinvolving the best fitting analytical curve are determined by:c1 = 1.894 × 105 mm0.466 × N0.661 (1.5642 × 104 in.0.466 ×lb0.661); c2 = –1.316 × 106 N (2.959 ×105 lb); α1 = 0.012;α2 = 0.091; α3 = 0.062; α4 = 0.111; and α4 = 0.074.

The comparison between the minimum thickness calculatedby NLFM analyses and by Eq. (3) is shown in Fig. 12; notethe satisfactory agreement that is characterized by a correlationcoefficient of 0.94 (Fig. 14). The mean absolute value of theprediction error is approximately 29.7% with a standarddeviation of 16.9%.

Fig. 12—Design abacus from NLFM model (circled lines) andprediction with proposed simplified equation (dashed lines).

Fig. 13—(a) Typical load-CTOD curve determination fromItalian Standard; and (b) crack opening ranges used forcalculation of equivalent strength.22

Fig. 14—Correlation between collapse loads predicted withNLFM model and with proposed approximated equation.

ACI Structural Journal/July-August 2006558

From Eq. (2), the minimum slab thickness can be easilydetermined (within the variable ranges here adopted), as afunction of the design parameters, in a closed form

(3)

where k is a constant equal to 1.261 × 10–19 mm–1.700 × N–2.412

(1.128 × 10–15 in.–1.700 × lb–2.412) and FS is the factor ofsafety. Figure 12 shows the values of the minimum slabthickness by using the abacus with NLFM (s = 230 mm[9.06 in.]) and with Eq. (3) (s = 260 mm [10.24 in.]) for anultimate load of 600 kN (134.89 kips).

CONCLUDING REMARKSThe results presented herein lead to the following

concluding remarks:• A relatively low content of steel fibers effectively

enhances the load-carrying capacity of slabs on groundand makes the structural response more ductile; volumefraction of steel fibers higher than 0.38% slightlyimprove the ultimate load but remarkably enhance theslab ductility;

• The analyses of FRC slabs based on NLFM predict theslab response with appreciable accuracy. Extensiveparametric studies based on NLFM determine abaciuseful for design;

• A simplified closed-form equation is proposed to providean approximated value of the minimum slab thicknessby considering the main physical parameters governingthe structural behavior of slabs on ground; and

• Preliminary results showed higher energy dissipation atsmall crack openings for hybrid systems of fibers(cocktail of fibers having different lengths) and encouragefurther research on this topic.

ACKNOWLEDGMENTSThe research project was financed by Officine Maccaferri S.p.A., Bologna,

Italy, whose support is gratefully acknowledged. The authors are indebtedto V. E. Saouma for his kind agreement to use the finite element softwareMERLIN. A special acknowledgment goes to engineers P. Martinelli andL. Cominoli for their assistance in carrying out the experiments and in thedata reduction. This research project was supported jointly by the ItalianMinistry of University and Research (MIUR) within the project, “FiberReinforced Concrete for Strong, Durable, and Cost-Saving Structures andInfrastructures” (2004-2006).

NOTATIONa0 = notch length of beam for fracture testb = width of beam for fracture testEc = Young’s modulus of concreteEc,core = Young’s modulus measured from concrete cylindrical coresEs = Young’s modulus of steel fiberfc,cube = concrete compressive strength measured from cubesfct = concrete tensile strengthfct,core = concrete tensile strength measured on cylindrical coresfft = steel tensile strengthGF = specific fracture energyh = depth of beam for fracture testkw = subgrade modulusL = side of the square slabLf = fiber lengths = span of beam for fracture testt = slab thicknessw = crack opening displacement (COD)φc = diameter of concrete cylinderφf = fiber diameterν = Poisson’s ratio

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s kFS Fu c2–⋅( )3.648

L0.087⋅

AL0.043 Ec

0.333 kw0.227 f If

0.406 fres0.271⋅ ⋅ ⋅ ⋅

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