ACI STRUCTURAL JOURNAL
TECHNICAL PAPER
Title no. 103-S58
Steel Fiber Concrete Slabs on Ground: A Structural Matter
by Luca G. Sorelli, Alberto Meda, and Giovanni A. Plizzari
An extensive experimental investigation with the aim of studying
the structural behavior of slabs on ground made of steel fiberreinforced concrete (SFRC) is presented in this paper. Several fullscale slabs reinforced with different volume fractions of steel fibers
having different geometries were tested under a point load in the
slab center. A hybrid combination of short and long fibers was also
considered to optimize structural behavior. Experimental results
show that steel fibers significantly enhance the bearing capacity
and the ductility of slabs on ground.
The nonlinear behavior of these SFRC structures is well captured
by performing nonlinear fracture mechanics analyses where the
constitutive relations of cracked concrete under tension were
experimentally determined. Finally, from an extensive parametric
study, design abaci and a simplified analytical equation for predicting
the minimum thickness of SFRC slabs on ground are proposed.
Keywords: pavement; reinforced concrete; slabs on ground.
INTRODUCTION
In the last decades, the use of steel fiber-reinforced
concrete (SFRC) has significantly increased in industrial
pavements, roads, parking areas, and airport runways as an
effective alternative to conventional reinforcement (that is,
reinforcing bars or welded mesh). Because heavy concentrated
loads from industrial machinery and shelves may cause
intensive cracking and excessive deformation of pavements,
a diffused fiber reinforcement may help the structural behavior.
Many of these pavements are slabs on ground that are
statically undetermined structures. For this reason, even at
relatively low volume fractions (<1%), steel fibers effectively
increase the ultimate load and can be used as partial (or total)
substitution of conventional reinforcement (reinforcing bars or
welded mesh) of slabs on ground. Fiber reinforcement also
provides a better control of the crack development to improve
the structural durability and to reduce the number of joints.1-3
Moreover, fiber reinforcement enhances the impact and fatigue
resistance of concrete structures and reduces labor costs due to
the amount of time saved in the placement of the reinforcement.
At present, design rules for SFRC structures are not
present in the main international building codes, even though
ACI Committee 544, RILEM Technical Committee 162-TDF,
and the Italian Board of Standardization have recently proposed
recommendations or design guidelines.4-6 Because these guidelines are under development, designers often work under the
usual assumption of elastic behavior of a concrete slab on an
elastic subgrade, according to the Westergaard theory.7 This
assumption is markedly restrictive for SFRC slabs and leads
to a significant underestimation of the actual bearing capacity of
the slab.8 In fact, a linear elastic approach cannot properly
take into account the beneficial effects of fiber reinforcement
which become effective only after cracking of the concrete
matrix when SFRC behavior is significantly nonlinear8,9
(Fig. 1). As a consequence, more appropriate methods based
on the yield line theory have been proposed to predict the
ACI Structural Journal/July-August 2006
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 SFRC
structures with low volume fractions of steel fibers (Vf < 1%)
due the softening behavior of the material (Fig. 1). If a
nonstandard form of the yield line theory is formulated by
assuming an average post-cracking strength, the collapse
load is still underestimated.11
The finite element (FE) method based on nonlinear fracture
mechanics (NLFM)12 appears to be the most accurate tool
for analyzing SFRC slabs on ground because it allows a
reproduction of the actual collapse mechanism and the
development of a new design approach.13
Because a limited number of experiments are currently
available in the literature,8,14,15 several full-scale tests on FRC
slabs were carried out in an extensive research program to
validate the NLFM approach. The experimental model aims
to simulate a zone of pavement included between joints where
one or more concentrated loads may be applied at any point.
Because part of a load close to a joint is transferred to the
adjacent slabs,16 however, it was found that the load placed in
the center of a single slab is particularly significant for design.
For the sake of clarity, it should be noted that other important
phenomena (such as the curling effect) present in concrete
pavement are not considered herein.
A further goal of the research concerns the possibility of
enhancing structural performance by combining steel fibers
of different dimensions and geometries (hybrid fiber reinforced
concrete [HyFRC]). In fact, fibers start activating after
cracking (not visible microcracks) of the concrete matrix.
Because fibers of different sizes become efficient at different
stages of the cracking process, however, a hybrid combination
of short and long steel fibers may enhance the concrete
Fig. 1—Different fiber activation with respect to crack
development in tensile test.20
ACI Structural Journal, V. 103, No. 4, July-August 2006.
MS No. 05-026 received December 17, 2005, and reviewed under Institute publication
policies. Copyright © 2006, American Concrete Institute. All rights reserved, including
the making of copies unless permission is obtained from the copyright proprietors. Pertinent
discussion including author’s closure, if any, will be published in the May-June 2007
ACI Structural Journal if the discussion is received by January 1, 2007.
551
Luca G. Sorelli is investigating ultra-high-performance concrete (UHPC) structural
implications by micromechanics and chemo-plasticity approaches at Massachusetts
Institute of Technology, Cambridge, Mass. He received his doctorate from the University
of Brescia, Brescia, Italy.
Alberto Meda is an Associate Professor of structural engineering, Department of
Engineering Design and Technology, University of Bergamo, Bergamo, Italy. He
received his degree in environmental engineering from the Milan University of
Technology, Milan, Italy, in 1994. His research interests include concrete fracture
mechanics, fiber-reinforced concrete, and fire design of reinforced concrete structures.
ACI member Giovanni A. Plizzari is a Professor of structural engineering, Department
of Civil Engineering, University of Brescia. His research interests include material
properties and structural applications of high-performance concrete, fiber-reinforced
concrete, concrete pavements, fatigue and fracture of concrete, and steel-to-concrete
interaction in reinforced concrete structures.
Table 1—Composition of concrete matrix
Mixture 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
Aggregate (4 to 15 mm)
450 kg/m3 (28.09) lb/ft3
Aggregate (8 to 15 mm)
450 kg/m3 (28.09) lb/ft3
Table 2—Geometrical and mechanical properties
of steel fibers
Fiber
code
Lf ,
mm (in.)
φf , mm (in.)
Lf /φf
fft , MPa (ksi)
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 (1.18) 0.60 (0.0236)
50.0
1100 (159.5)
20/0.4 20 (0.79) 0.40 (0.0157)
12/0.18 12 (0.47) 0.18 (0.0071)
50.0
66.6
1100 (159.5)
1800 (261.1)
30/0.6
Fiber shape
—
Table 3—Volume fractions of steel reinforcement
(steel fibers or welded mesh)
Steel fibers
Slab no.
50/1.0
%vol
30/0.6
%vol
20.04
%vol
12/0.18
%vol
Vf,tot,
%vol
S0
S1
—
—
—
0.38
—
—
—
—
0.00
0.38
S3
0.38(a)
—
—
0.19
0.57
S4
(a)
—
—
—
0.38
0.38
—
—
0.38
0.38(b)
—
—
—
0.38
(a)
—
—
—
0.57
(b)
—
0.19
—
0.57
S5
0.38
—
S8
S11
S14
0.57
0.38
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 HyFRC
appears to be a promising application for pavements
subjected to aggressive environments.
RESEARCH SIGNIFICANCE
Whereas the structural behavior of plain concrete and
conventionally reinforced slabs on ground is well known,
552
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.
there is still a lack of design rules for steel fiber reinforced
concrete slabs in building codes. Due to this lack, conventional
design methods, based on the elastic theory, are used for fiber
reinforced slabs whose behavior is significantly nonlinear.
The behavior of slabs on ground with steel fibers was experimentally studied by performing full-scale tests; a design
approach based on nonlinear fracture mechanics is also proposed.
To enhance the structural response, the use of HyFRC
systems, combining shorter and longer steel fibers, was also
considered.
EXPERIMENTAL PROGRAM
Full-scale slabs on ground were tested under a point load
in the center. The experimental model aimed to reproduce a
square 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 identify
the fracture behavior of SFRC. The slab tests presented in
this paper are part of an extensive research campaign whose
results are published elsewhere.18,21
Materials
The concrete matrix was made with cement CEM II/A-LL
42.5R (UNI-ENV 197-1) and natural river gravel with a
rounded 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 this
research, as reported in Table 2 where geometrical and
mechanical properties of fibers are shown; the fiber code is
conventionally defined by the fiber length and the fiber
diameter (Lf /φf , in millimeter unit). Two straight shorter
fibers (12/0.18 and 20/0.4) and three longer fibers with
hooked 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 ranging
between 50 and 66, and a Young modulus of approximately
210 GPa (30456.9 ksi).
Seven SFRC slabs (S1, S3, S4, S5, S8, S11, and S14), with
a volume fraction Vf of fiber smaller than 0.6% and a reference
slab made of plain concrete (S0) are reported in this paper
(Table 3). Figure 2(a) shows the slab with a hydraulic jack
placed in its center.
ACI Structural Journal/July-August 2006
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
S5
1.79 (260)
NA
35.3 (5120)
36.1 (5236)
NA
NA
23,446 (3400)
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
S14
1.63 (236)
NA
33.1 (4801) 21,486 (3116) 17,335 (2517)
32.3 (4685)
NA
NA
Note: NA = data not available.
Table 4 reports the mechanical properties of concrete of
the different slabs, as determined on the day of the test; in
particular, Table 4 shows the tensile strength fct from cylinders
(φc = 80 mm [3.15 in.], L = 250 mm [9.84 in.]), the compressive
strength from cubes fc,cube of 150 mm side (5.91 in.); the
Young’s modulus as determined from both compression
tests 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 always
greater 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 under
four-point bending according to the Italian Standard22
(Fig. 2(b)). The notch was placed at midspan and had a depth
of 45 mm (1.77 in.) (Fig. 2(b), (d)). These tests were carried out
with a closed loop hydraulic testing machine by using the crack
mouth opening displacement (CMOD) as a control parameter,
which was measured by means of a clip gauge positioned
astride the notch. Additional linear variable differential
transformers (LVDTs) were used to measure the crack tip
opening displacement (CTOD) and the vertical displacement at
the beam midspan and under the load points (Fig. 2(b)).
Test setup and instrumentation
The slabs were loaded by a hydraulic jack placed in the
center by using the load frame shown in Fig. 2(a); the
average loading rate was 2.5 kN/min (0.56 kips/min).
To reproduce a Winkler soil, 64 steel supports were placed
under the slab at centers of 375 mm (14.76 in.) in both directions
(Fig. 3(a)). These supports are steel plates on a square base
having a side of 100 mm (3.94 in.; Fig. 3(b)). Previous
numerical simulations showed that the experimental subgrade
provides a good approximation of a continuous Winkler
soil.16 Because of the curling of the concrete slabs due to
shrinkage and the thermal effect, a layer of high-strength
mortar a few millimeters thick was placed on each spring to
ensure the contact with the bottom face of the slab. The
average spring stiffness was determined by compression
tests performed on each spring with results approximately
equal to 11.0 kN/mm (2.47 kips/mm). By considering the
influence area of each spring (375 x 375 mm [14.76 x
14.76 in.]), the average Winkler constant kw was equal to
0.0785 N/mm3 (289.2 lb/in.3), which corresponds to a
uniform graded sand soil according to ACI classification.23
During the tests, the vertical displacements of 12 points on
the top surface of the slab were continuously monitored;
furthermore, four inductive transducers were placed on the
bottom surface of the slab to detect the width of possible
ACI Structural Journal/July-August 2006
(a)
Fig. 3—(a) Positioning of slab on uniform grid of steel
supports; and (b) three-dimensional view of steel support.
cracks (that were expected to form along the medial lines of
the slabs; Fig. 2(a)).
Experimental results
Experimental results are initially presented in terms of
vertical load versus the deflection of the slab center. The
curves are plotted up to the collapse only. The structural
response of reference Slab S0 and the Slabs S1, S5, S4, and
S8, reinforced by an equal volume fraction (Vf = 0.38%) of
fibers 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 conventionally
assumed in correspondence of the loss of linearity that occurs
between a load level of 100 kN (22.48 kips) and 150 kN
(33.72 kips), the fracture behavior of the SFRC slabs is
remarkably different from the plain concrete slab. The
steel fibers effectively enhance the bearing capacity of the
slab up to a maximum load higher than 260 kN (58.45 kips);
moreover, fiber reinforcement assures a ductile failure while
the reference slab (made of plain concrete) showed a brittle
failure when a maximum load equal to 177 kN (26.30 kips)
was reached.
Slabs S1 and S5, reinforced with fibers 30/0.6, performed
slightly better than the Slabs S4 and S8, reinforced with
longer fibers (50/10(a) and 50/10(b)).
The effect of the fiber content on the structural response
is shown in Fig. 4(b), which compares the response of Slabs S4
and 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%) of
fibers 50/10(a), respectively. The experimental curves show that
the higher fiber content slightly increases the structural ductility
while the ultimate load does not seem significantly influenced.
553
Fig. 6—Final crack patterns of slabs.
Fig. 7—Bilinear approximation of stress versus crack-opening
curve of cracked concrete.
Fig. 4—(a) Experimental load-displacement curve for slabs
with fiber content of 30 kg/m3 (Vf = 0.38%); and (b) with
different contents of Fiber 50/1.0(a).
Fig. 5—(a) Experimental load-displacement curve; and
(b) experimental load-crack opening displacement curve for
slabs with same volume fraction (Vf = 0.57%) with fiber
having one or two different geometries.
Figure 5(a) shows the curves obtained from all the slabs
with a volume fraction of fibers equal to 0.57%; the reference
Slab S0 (of plain concrete) is also shown. In particular, Slab S11
is made of a single type of fibers (50/1.0(a)) while Slabs S3 and
S14 are made of a combination of longer and shorter fibers
(refer to Table 3). Although slabs with hybrid fibers have
slightly higher maximum loads, the main contribution of this
554
reinforcement concerns the crack opening (measured on the
slab side at the bottom of a median line) that is significantly
smaller than in Slabs S11, which has a single type of fiber
(Fig. 5(b)). This can be explained by the better efficiency of
shorter fibers to bridge smaller cracks delaying their
coalescence in localized and large cracks.19
The steel fibers did not substantially affect the final crack
patterns of the slabs which are characterized by four major
cracks started from the slab center and developed along the
median lines or, in a few cases, along the diagonals (Fig. 6).
NUMERICAL MODELING
Numerical analyses based on NLFM were performed by
adopting MERLIN24; the experimental crack patterns shown
in Fig. 6 justify a discrete crack approach with the cracks
located along the median or the diagonal lines. Interface
elements in predefined discrete cracks initially connect the
linear elastic subdomains (as rigid links) and start activating
(that is, the crack starts opening) when the normal tensile
stress (at the interface) reaches the tensile strength of the
material. Afterwards, the crack propagates and cohesive
stresses are transmitted between the crack faces according to
a stress-crack opening (σ-w) law (which is given as input for
the interface elements).
Inverse analyses25 of the bending tests were performed to
determine the best fitting softening law of cracked concrete
(σ-w) that was assumed as bilinear (Fig. 7) where the steeper
branch can be associated with (unconnected) microcracking
ahead of the stress-free crack whereas the second part represents
the aggregate interlocking or the fiber bridging.
The beam was modeled by triangular plain stress elements
(Fig. 2(d)) with Young’s modulus Ec experimentally
determined from the cylindrical specimens and Poisson’s
ratio ν assumed equal to 0.15.
The material parameters identified from the bending tests
are summarized in Table 5; as typical examples, the numerical
and experimental load-displacement curves for the materials
used in Slabs S4 and S8 are compared in Fig. 8. These material
ACI Structural Journal/July-August 2006
Fig. 9—Conventional assumption of experimental collapse
load for slab on ground.
Table 5—Fracture properties of concrete
σct,
w1,
σ1,
wcr ,
GF ,
Slab Ec,core,
no. MPa (ksi) MPa (psi) mm (in.) MPa (psi) mm (in.) 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)
24,786
(3594.8)
3.3
(479)
3.0
(435)
0.023
(0.00091)
0.028
(0.00110)
0.88
(128)
1.20
(174)
20.00
(0.78740)
10.00
(0.39370)
8.84
(1.817)
6.04
(1.242)
20,989
(3044.1)
19,964
(2895.4)
3.0
(435)
3.5
(508)
0.035
(0.00138)
0.022
(0.00087)
0.75
(109)
0.67
(97)
0.25
(0.00984)
20.00
(0.78740)
0.15
(0.031)
6.74
(1.386)
17,355
S11 (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)
Fig. 8—Comparison between numerical (dotted) and
experimental load-CTOD curves as obtained from bending
tests on SFRC used for: (a) Slab S4; and (b) Slab S8.
parameters fct, σ1, w1, and wcr were adopted for the numerical
simulations of the slab specimens.
The slab was modeled by 4432 four-node tetrahedral
elements for the elastic subdomains linked by 576 interface
elements (Fig. 2(c)), along the cracks. The elastic soil
(Winkler soil) was modeled by 616 linear elastic truss
elements, simulating bidirectional springs connected to the
slab bottom nodes, and with global stiffness equal to the
experimental value (kw = 0.0785 kN/mm3 [289.2 lb/in.3]).
Although no-tension springs were used in the tests, only a
reduced area of the slab corner was observed to uplift during
the experiments.
Numerical results
The failure of a SFRC slab on ground is neither sudden nor
catastrophic and the slab continues to carry further load even
after a collapse mechanism occurs. The ultimate load was
conventionally defined as corresponding to a sudden change
of the monitored displacements (Fig. 9) that evidence the
formation of a collapse mechanism (fully developed crack
surface along the medians or the diagonals, depending on the
ratio between the slab and the soil stiffness).
The numerical and experimental load-displacement curves
are compared in Fig. 10 (the displacement is measured on the
top surface of the slab center). In addition, the numerical
development of the crack pattern is displayed. The cracks
begin to develop on both the median and the diagonal
surfaces and the slab collapse occurs when two cracks (either
the median or the diagonal ones) develop up to the slab
border. It can be noticed that, in all cases, the overall structural
behavior is well captured by the numerical analyses based on
NLFM. These results further confirm the opportunity of
using an NLFM approach to analyze SFRC structures.
Table 6 reports the values of the maximum load obtained
from both the experiments and the numerical analyses: one
should note that the numerical predictions are in good agreement
with the experimental values (the average discrepancy is
approximately 7.9% with a maximum of 14.2%).
ACI Structural Journal/July-August 2006
S5
S6
S8
*
*
Assumed equal to 20 GPA (2900.7 ksi).
Table 6—Experimental and numerical collapse
loads for tested slabs
Slab
no.
S0
Fu,exp ,
kN (kips)
177.0 (39.79)
Fu,num ,
kN (kips)
174.0 (39.12)
F u, num – F u, exp
err = --------------------------------,%
F u, exp
S1
S3
265.0 (59.58)
274.9 (61.80)
240.5 (54.07)
236.0 (53.06)
–9.2
–14.2
S4
S5
238.6 (53.64)
252.3 (56.72)
245.3 (55.15)
247.0 (55.53)
2.8
–2.1
S8
S11
246.2 (55.35)
231.9 (52.14)
215.4 (48.43)
255.7 (57.49)
–12.5
10.3
S14
273.0 (61.38)
244.7 (55.01)
–10.7
–1.7
As a further comparison, Fig. 11(a) shows the numerical
and experimental displacements monitored on Slab S5 at the
maximum load; the model response is stiffer because of the
numerical assumption of bilateral behavior of the spring
supports, whereas tractions cannot be transmitted by the
experimental springs. As previously mentioned, however, only
a small slab portion is subjected to uplift at collapse. The
comparison between the numerical and the experimental crack
opening, measured at the bottom surface of Slab S5, is displayed
in Fig. 11(b); once again, a good agreement between the
numerical and the experimental response was noted.
555
Fig. 10—Numerical (dotted lines) and experimental load versus center
displacement of slab.
Design abaci based on nonlinear fracture mechanics
A parametric study, based on approximately 1000 FE
simulations, has been carried out to develop design abaci for
FRC slabs on ground,13 by considering the following variables:
subgrade modulus kw (0.03, 0.06, 0.09, 0.12, 0.15, 0.18, and
0.21 kN/mm3 [110.5, 221.0, 331.6, 442.1, 552.6, 663.1, and
773.7 lb/in.3]), slab thickness s (150, 200, 250, 300, and
350 mm [5.91, 7.87, 9.84, 11.81, and 13.78 in.]), loading area
a (400, 14,400 mm2 [0.62, 223.20 in.2]), concrete compressive
strength fc (25, 30, and 40 MPa [3.6, 4.4, and 5.8 ksi]) and
fiber 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 as
input for the NLFM analyses, the 15 materials considered in
the parametric study were cast and tested under tensile and
bending tests and, eventually, the σ-w laws of cracked
concrete were determined by performing inverse analyses.
Figure 12 shows a typical abacus that, once the soil stiffness
and the design load is known, easily provides the minimum
slab thickness. This aims to help professional engineers,
whose offices often are not equipped with NLFM programs,
to powerfully apply NLFM in practice.13
Fig. 11—(a) Numerical and experimental vertical displacements
along diagonal of Slab S5 at collapse; and (b) curve load
versus crack opening displacement for Slab S5.
556
Simplified model
To further simplify the design approach, the numerical
curves of the abaci could be approximated by closed-form
equations, which also provide the minimum slab thickness
ACI Structural Journal/July-August 2006
Fig. 12—Design abacus from NLFM model (circled lines) and
prediction with proposed simplified equation (dashed lines).
once the design load and the material and soil properties
are known.
By considering the main geometrical and mechanical
parameters governing the slab behavior, the load-carrying
capacity of FRC slabs on ground can be written in the
following form
A
F u = c 1 ⋅  -----L-
 2
L
α1
⋅B
α2
α3
α4
α5
⋅ k w ⋅ f If ⋅ f res + c 2
Fig. 13—(a) Typical load-CTOD curve determination from
Italian Standard; and (b) crack opening ranges used for
calculation of equivalent strength.22
(1)
where AL is the loading area; B is the slab stiffness defined as
3
Ec ⋅ t
-----------------------2
12 ( 1 – ν )
where t is the slab thickness and ν is the Poisson’s ratio
(assumed equal to 0.18); fIf is the first cracking strength and
fres is an average residual strength that should represent the
post-cracking behavior of SFRC for smaller crack openings.
The latter two parameters are defined by the Italian
Standard22 as (in Fig. 13, fres is indicated as feq(0-0.6) for
smaller crack opening and feq(0.6-3) for larger crack opening)
P If s
f If = -----------------------2
b ( h – a0 )
(2a)
CTOD 0 + 0.6
F(w)
------------ ⋅ dw
W
CTOD 0
= -----------------------------------------------0.6
∫
f res
(2b)
where PIf is the (total) first-crack load, b is the beam width, h is
the beam depth, and a0 is the notch length.22 The International
Standards on SFRC characterization usually provide residual
strength values corresponding to both smaller and larger
crack openings. The latter is useless in slabs on ground because
the collapse occurs with small crack openings (Fig. 5(b)).
The unknown parameters αi and ci of Eq. (1) were
determined by adopting the least square method, which
minimizes the square of the differences between the collapse
ACI Structural Journal/July-August 2006
Fig. 14—Correlation between collapse loads predicted with
NLFM model and with proposed approximated equation.
loads calculated by the FE analyses and the ones predicted
by Eq. (1). In addition, for the sake of safety, it is imposed
that the approximated maximum load is always smaller
than the corresponding value determined by means of FE
analyses based on NLFM. Although this additional condition
reduces the accuracy of the approximated solution, it is
better used in structural design to obtain a safer evaluation
of the minimum thickness. Hence, the equation parameters
involving 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 calculated
by NLFM analyses and by Eq. (3) is shown in Fig. 12; note
the satisfactory agreement that is characterized by a correlation
coefficient of 0.94 (Fig. 14). The mean absolute value of the
prediction error is approximately 29.7% with a standard
deviation of 16.9%.
557
From Eq. (2), the minimum slab thickness can be easily
determined (within the variable ranges here adopted), as a
function of the design parameters, in a closed form
3.648
0.087
( FS ⋅ F u – c 2 )
⋅L
s = k ⋅ -----------------------------------------------------------------------------0.043
0.333
0.227
0.406
0.271
⋅ k w ⋅ f If ⋅ f res
AL ⋅ Ec
(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 of
safety. Figure 12 shows the values of the minimum slab
thickness by using the abacus with NLFM (s = 230 mm
[9.06 in.]) and with Eq. (3) (s = 260 mm [10.24 in.]) for an
ultimate load of 600 kN (134.89 kips).
CONCLUDING REMARKS
The 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 ground
and makes the structural response more ductile; volume
fraction of steel fibers higher than 0.38% slightly
improve the ultimate load but remarkably enhance the
slab ductility;
• The analyses of FRC slabs based on NLFM predict the
slab response with appreciable accuracy. Extensive
parametric studies based on NLFM determine abaci
useful for design;
• A simplified closed-form equation is proposed to provide
an approximated value of the minimum slab thickness
by considering the main physical parameters governing
the structural behavior of slabs on ground; and
• Preliminary results showed higher energy dissipation at
small crack openings for hybrid systems of fibers
(cocktail of fibers having different lengths) and encourage
further research on this topic.
ACKNOWLEDGMENTS
The research project was financed by Officine Maccaferri S.p.A., Bologna,
Italy, whose support is gratefully acknowledged. The authors are indebted
to V. E. Saouma for his kind agreement to use the finite element software
MERLIN. A special acknowledgment goes to engineers P. Martinelli and
L. Cominoli for their assistance in carrying out the experiments and in the
data reduction. This research project was supported jointly by the Italian
Ministry of University and Research (MIUR) within the project, “Fiber
Reinforced Concrete for Strong, Durable, and Cost-Saving Structures and
Infrastructures” (2004-2006).
NOTATION
a0
=
b
=
=
Ec
Ec,core =
=
Es
fc,cube =
=
fct
fct,core =
=
fft
=
GF
h
=
=
kw
L
=
=
Lf
s
=
t
=
w
=
=
φc
=
φf
ν
=
558
notch length of beam for fracture test
width of beam for fracture test
Young’s modulus of concrete
Young’s modulus measured from concrete cylindrical cores
Young’s modulus of steel fiber
concrete compressive strength measured from cubes
concrete tensile strength
concrete tensile strength measured on cylindrical cores
steel tensile strength
specific fracture energy
depth of beam for fracture test
subgrade modulus
side of the square slab
fiber length
span of beam for fracture test
slab thickness
crack opening displacement (COD)
diameter of concrete cylinder
fiber diameter
Poisson’s ratio
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ACI Structural Journal/July-August 2006
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