Accepted Manuscript
Material characterisation of macro synthetic fibre reinforced concrete
Ali Amin, Stephen J. Foster, R. Ian Gilbert
PII:
S0958-9465(17)30773-4
DOI:
10.1016/j.cemconcomp.2017.08.018
Reference:
CECO 2897
To appear in:
Cement and Concrete Composites
Received Date: 14 January 2016
Revised Date:
0958-9465 0958-9465
Accepted Date: 23 August 2017
Please cite this article as: A. Amin, S.J. Foster, R.I. Gilbert, Material characterisation of macro
synthetic fibre reinforced concrete, Cement and Concrete Composites (2017), doi: 10.1016/
j.cemconcomp.2017.08.018.
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ACCEPTED MANUSCRIPT
Material Characterisation of Macro Synthetic Fibre
Reinforced Concrete
Ali Amin, Stephen J. Foster & R. Ian Gilbert
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A. Amin*
Associate Lecturer, Centre for Infrastructure Engineering and Safety, School of Civil and
Environmental Engineering, The University of New South Wales, Australia.
S. J. Foster
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Professor, Head of School, Civil and Environmental Engineering, The University of New
South Wales, Australia.
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R. I. Gilbert
Emeritus Professor, Centre for Infrastructure Engineering and Safety, School of Civil and
Environmental Engineering, The University of New South Wales, Australia.
Abstract:
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* Corresponding Author: Phone: +61 2 9385 5033; Email: Ali.Amin@unsw.edu.au
In this paper, the post cracking behaviour of macro synthetic polypropylene fibre reinforced
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concrete is investigated through a series of matched tests that measure tension directly
through uniaxial tension tests and indirectly through prism bending and determinate round
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panel tests. An analytical model previously developed by the authors for the determination of
the residual tensile strength provided by steel fibres in prism bending tests is adapted for the
round panel tests fibres and is shown to correlate well with the collected experimental data.
Keywords:
Macro synthetic, fibre reinforced concrete, bending, uniaxial tension, round
panel, fracture
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1
INTRODUCTION
A considerable growth in the use of fibre reinforced concrete (FRC) in practice has occurred
over the last two decades, or so. Applications of the material are now common practice in the
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Australian infrastructure sector for industrial pavements and sprayed tunnel linings (Foster,
2009). With the advent of ultra-high performance concretes, the material has also been
successfully used in large scale bridge projects (Toutlemonde and Resplendino, 2011; Voo et
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al. 2015). This progression has come as a result of a significant body of research devoted to
the understanding of the composite material to a wide range of loading conditions.
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Fundamentally, when dealing with the design of members manufactured with FRC, the most
important property to consider is its post cracking tensile strength (Shah and Rangan, 1970,
1971; Walraven, 2009; di Prisco et al. 2013). In other words, it is the residual tensile strength
(or toughness) of concrete that is significantly enhanced with the addition of fibres, after the
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matrix has cracked, and quantifying this defines the material for design. Only when reliable
material constitutive relationships for FRC have been developed can critical structural
elements be designed using FRC.
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While most studies over the last five decades have focussed on steel fibres, macro synthetic
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fibres have gained popularity in practice, particularly in applications where susceptibility to
corrosion and alkali attack are significant (Bentur and Mindess, 1990; Zheng and Feldman,
1995). Synthetic polypropylene fibres have also been shown to significantly reduce cracking
induced by plastic shrinkage (Soroushian et al. 1995). With reference to its performance in
uniaxial tension, the addition of polypropylene fibres (as with macro steel fibres) to concrete
does not significantly enhance its tensile strength, i.e. the stress at which cracking occurs is
not significantly increased. After cracking, the force carried across the crack by the fibres
depends on the type and quantity of fibres, with polypropylene fibres requiring a significantly
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larger strain to develop the same stress as occurs in steel fibres. This phenomenon was
observed in the tests of Wang et al. (1990) and Carnovale and Vecchio (2014) and is
attributed to the low elastic modulus of polypropylene relative to steel. After crack
stabilisation, polypropylene fibre reinforced concrete (PPFRC) has been shown to resist an
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increase in residual load, even at large crack widths (Bentur and Mindess, 1990; Buratti et al.
2011; Alani and Beckett, 2013).
The post cracking strength of FRC can most readily be explained by the stress versus crack
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width (σ-w) relationship. The σ-w relationship for FRC can be directly obtained from a
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uniaxial tensile test (Lӧfgren, 2005; Markovic, 2006; Foster, 2014) or, alternatively,
indirectly from a three- or four-point bending test on prism specimens (di Prisco et al. 2013)
or from determinant round panel tests, combined with an inverse analysis (Marti et al. 1999;
Walraven, 2013; Minelli and Plizzari, 2015; Nour et al. 2015). This is summarised in
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Figure 1. To this end, the results of an experimental campaign to obtain the post cracking
behaviour of PPFRC through a series of matched uniaxial tension, prism bending and round
panel tests is presented here. The inverse analysis model of Amin et al. (2015) developed for
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steel fibre reinforced concrete (SFRC) tested in flexure is extended to PPFRC and is adapted
to the round panel test.
STANDARD TESTING OF FRC
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In principle, the uniaxial tension test is the ideal test for measuring the post cracking tensile
strength of FRC (Kooiman, 2000; Ng et al. 2012; Amin et al. 2015). Despite this, the great
majority of tests describing the post cracking behaviour of FRC reported in the literature have
been conducted on flexural prism samples tested in three- or four point bending. These
specimens are easy to manufacture and the tests are easy to perform (di Prisco et al., 2009).
Some researchers prefer notched specimens as this allows for easy measurement of the
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opening of the crack during testing. However, the presence of a notch can significantly
influence the behaviour of prism bending specimens by forcing the crack to form along a
predefined plane that may not be the weakest cross section of the specimen. As a result, a
large scatter in results is often associated with these tests (Dupont and Vandewalle, 2004; di
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Prisco et al. 2009; Minelli and Plizzari, 2015). This scatter at the material level represents,
and is a measure of, the variability of fibre spacing and orientation within the matrix and may
significantly affect the characteristic values adopted in design practice which can be a hurdle
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in promoting economical solutions using FRC.
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The round determinant panel test has been shown to provide more repeatable results with low
scatter (Marti et al. 1999; Bernard, 2000). This is primarily due to the formation of a number
of distinct fracture lines, as opposed to one failure plane that forms in the prism bending test
when testing strain softening FRC. The major drawback of the ASTM C1550 (2012) round
panel test is that the specimens are large (800 mm in diameter), heavy (~90kg) and difficult to
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handle. This has recently been addressed by Minelli and Plizzari (2015).
The residual tensile strength of round panel test specimens has traditionally been expressed in
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Joules, represented by the area under the load versus central displacement curve (ASTM
C1550, 2012). However, over the last decade, or so, design methodologies have moved from
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defining the post cracking strength of FRC from the area under the load versus displacement
curve to defining the toughness of FRC in terms of the residual tensile strength resisted by
specimens at various stages of a material characterisation test (RILEM TC 162, 2003; fib
MC2010, 2013; DR AS5100.5, 2014). Inverse analysis techniques can then be deployed to
infer these results. This represents a significant step forward in physically quantifying the
residual tensile strength offered by FRC.
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3
EXPERIMENTAL PROGRAM
In this study an experimental program was conducted to investigate the post cracking
behaviour of two PPFRC mix designs through a series of matched direct uniaxial tension tests
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and tension obtained indirectly through prism bending and determinant round panel tests. Six
specimens were manufactured for each test set up. The polypropylene fibres used in this
study were the Reoshore58 fibre, supplied by TEXO Australasia Pty Ltd. The fibres were
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58 mm long and had a measured cross sectional area of 0.64 mm . Two fibre dosages were
used in this study: 4 kg/m (FRC4) and 8 kg/m (FRC8), corresponding to fibre volumetric
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dosages (ρf) of 0.44% and 0.88%, respectively.
The specimens were cast in two separate pours from the same concrete mix design obtained
commercially from a local ready mix supplier. The concrete had a prescribed characteristic
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compressive strength of 50 MPa and the aggregate used was basalt with a maximum particle
size of 10 mm. The workability of the fresh PPFRC was measured by means of a slump test
and was found to be 250 mm for the FRC4 batch and 200 mm for the FRC8 batch.
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The uniaxial tension tests were conducted on hour glass shaped “dogbone” specimens in
accordance with DR AS5100.5 (2014). Details of the specimens are illustrated in Figure 2.
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The sample geometry, being approximately 40% narrower in the midsection than at its ends,
permits for failure to occur within a reasonably well defined region. Furthermore, no notches
are required and thus the dominant crack does not pass through a pre-determined plane. To
measure strain (and crack opening displacement), transducers (LVDT or LSCT) were fixed to
each of the four sides over a gauge length of 230 mm. Load was applied to the specimen
using ram displacement control, initially at a rate of 0.12 mm/min until the formation of the
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dominant crack. After cracking, the rate was increased to 0.2 mm/min until the measured
COD reached 2 mm, with additional increases in the rate introduced as the test progressed.
The prism bending tests were conducted on samples 150 mm square in cross-section and
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600 mm long. The specimens were notched 25 mm using a diamond blade saw at their midspan and tested in a three point arrangement spanning 500 mm, as per EN 14651 (2007). The
specimens were tested under a closed loop test system by fixing a clip gauge to the underside
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of the beam at the notch to measure and control the crack mouth opening displacement
(CMOD) at the extreme tensile fibre. Testing was conducted by increasing the CMOD at a
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rate of 0.05 mm/min until the CMOD reached 0.10 mm; this rate was then increased to
0.20 mm/min until the CMOD reached 4 mm. Additional increases in the testing rate were
introduced until the CMOD reached 12 mm at which point testing was concluded.
The determinate round panel test was conducted on 800 mm diameter by 80 mm thick
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samples and tested to ASTM C1550 (2012). The panels were loaded at their centres and were
supported on three symmetrically arranged pivot points giving a loading radius of 750 mm.
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An LSCT was placed beneath the loading point to measure the central deflection. The
samples were loaded via ram displacement control initially at a rate of 0.3 mm/min, until the
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formation of the three fracture lines and then increased to 3 mm/min. Testing concluded at a
central displacement of 50 mm and 60 mm for the FRC4 and FRC8 mixes, respectively. A
photograph and schematic of the round panel test is presented in Figure 3.
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TEST RESULTS
The raw experimental results for the uniaxial tension test, three-point notched prism bending
specimens and the round panel tests are presented in Figures 4 to 6 and summarised in Table
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1. The uniaxial tension test results are expressed in terms of nominal stress (taken at the most
narrow cross-section of the specimen) versus crack opening displacement, w (COD), with the
point marked on the vertical axis representing the matrix tensile strength, fct. The average raw
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uniaxial post cracking strength f1.5 (taken at a COD of 1.50 mm), are presented in Table 1. It
is noted that the results presented in Figure 4 and Table 1 have not been compensated for any
wall/boundary effects (see below). The prism bending results are expressed in terms of load
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versus CMOD. The residual flexural strength, fR,j, of the PPFRC is calculated at four key
points, namely at a CMOD of 0.50 mm, 1.50 mm, 2.50 mm and 3.50 mm; and the limit of
f
fct,L
, represents the maximum stress attained in the test prior to a
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proportionality,
CMOD = 0.05 mm. The round panel test results are plotted in terms of load versus central
deflection. The ASTM C1550 (2012) measure for this test (W) is defined as the area under
the force-deflection curve up to a central deflection of 40 mm, and measured in Joules.
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Plotted too in Figures 4 to 6 are the mean and 95th percentile characteristic responses
associated with each test. The characteristic values are determined assuming infinite samples
and a normal distribution.
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The behaviour of the PPFRC specimens in uniaxial tension followed a similar trend to that
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observed by others (Carnovale 2013; Carnovale and Vecchio, 2014). After the peak load was
reached, a sharp reduction in load and a significant opening of the dominant crack occurred.
Significant elastic strain energy was stored in the testing frame and the specimen, and hence
no displacement data is available between matrix cracking and stabilized cracking. Soon after
crack stabilization, the capacity of the section increased with increased applied strain over
crack widths of interest (w∈(0, 2.5)mm). For larger crack widths w∈(2.5, 5.0)mm, it can be
seen that the capacity of the direct tension specimens remained relatively constant.
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Following testing, the number of fibres crossing the failure plane was recorded; these results
are presented in Table 2. For fibres randomly orientated in three dimensions, Aveston and
Kelly (1973) showed that the number of fibres (Nf) crossing a plane of unit area is ρf /2Af,
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where ρf is the volumetric ratio of fibres and Af is the cross sectional area of an individual
fibre. In the context of the specimens used in this study where the wall effect can be
pronounced (see below), the expression of Aveston and Kelly for the total number of fibres
ρf Ac
2kt Af
(1)
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Nf =
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crossing a certain failure plane can be expressed as:
where Ac is the cross sectional area of the failure crack and kt is a fibre orientation correction
factor, described below.
The presence of a boundary restricts a fibre from being freely orientated and influences the
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number of fibres that cross a failure plane. Lee et al. (2011) mathematically described the
influence of the wall effect on the fibre orientation and its contribution to strength. Based on
the expression of Lee et al., Ng et al. (2012) developed a simplified approximate expression
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to adjust the results of the uniaxial tension tests to an equivalent 3D fibre distribution free of
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influence from the boundaries; this was further simplified by Amin et al. (2015) for square
section specimens in tension to:
k t = 0.5 ≤
1
≤ 1
0.94 + 0.6 lf b
(2)
where lf is the length of the fibres and b is the width of the cross-section.
Presented alongside the number of fibres crossing the failure plane of the dogbones in
Table 2 are the theoretical fibre dosages calculated from Equation 1, with Ac taken at the
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most narrow cross section of the specimen. It is seen that the theoretical fibre dosage crossing
the failure crack (ρfc) is consistently lower than the average supplied fibre dosage (ρf). It is
apparent that the fibre distribution influences the location of the failure plane and that the
numbers of fibres at the crack are statistically lower than the prescribed mean value. This is
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similarly observed for steel fibre reinforced concrete (Foster et al., 2013; Amin, 2015).
On the behaviour of the PPFRC prisms, it was observed that after cracking a significant drop
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in strength occurred for both series and over increasing CMODs of interest (up to 4.0 mm)
the capacity of the specimens increased. In the round panel tests, three dominant radial cracks
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formed in all specimens and this coincided with a reduction in capacity of the panels. It was
observed, however, that the initial load after cracking had dropped well below that of the
peak residual strength of the PPFRC specimens. The long tail of the curves represents the
progressively smooth residual capacity of the specimens.
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Theoretically the cracks of the round panel specimens should form midway between the
supports where the applied moment is the greatest; however in testing significant deviation,
γ (refer to Figure 3), was recorded (see Figure 8a). Plotted in Figure 8b is a frequency
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distribution of γ for all panel specimens. It can be seen that the majority of cracks developed
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within 12 degrees of the theoretical failure plane. This deviation from theory is due to the
variation of concrete matrix strength within the specimen and the sensitivity of the fracture
processes to this variation.
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5
DETERMINING THE σ-w RELATIONSHIP OF PPFRC
FROM INDIRECT TENSION TESTS
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Amin et al. (2015) developed a sectional model based on a cracked SFRC prism in bending.
In this paper, that model is adapted to account for the behaviour and performance of macro
synthetic fibres in flexure and observed in the round panel tests.
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Consider the cracked section at the notch in a prism specimen in bending illustrated in
Figure 9a, where D is the total depth of the prism, hsp is the depth minus the notch depth, dn
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is the depth from the extreme compressive fibre to the neutral axis and b is the width of the
prism. From moment equilibrium, the average stress carried by the fibres after cracking, fw, is
k k k Fa
fw = 1 2 b
2
hsp
b
(3)
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(Amin et al. 2015):
In Equation 3, kb is the boundary influence factor (see below), F is the applied load, a is the
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shear span of the prism and k1 is a factor that controls the steepness of the linear post
cracking relationship in tension. For macro-synthetic fibres, in the earlier stages of testing the
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stress in the tensile concrete at the crack increases with increasing strain (as shown in Figure
9c); after a critical crack opening displacement, the stress decreases with strain. Assuming a
rigid plastic model for the concrete on the tensile side on the neutral axis:
k1 =
3
[3.9 − 0.85β ] β
(4)
where β = 1 − dn hsp . For PPFRC tested in flexure, given that the neutral axis is high in the
section for crack widths of practical interest β may be taken as 0.95 and k1 = 1.0. The
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coefficient k2 in Equation 3 accounts for the influence of the notch on defining the critical
crack path and the resulting influence on the measured tensile strength. In Table 2, the ratio
of the number of fibres crossing the failure plane and the number of fibres supplied is
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presented and shown to consistently be lower than unity. The parameter k2 accounts for the
average uniformity of fibre dispersion relative to crack location. In research on SFRC by
Foster et al. (2013) and Amin (2015), k2 was found to be 1.0 for a crack freely located. For a
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crack where the location is forced (e.g. through provision of a significant notch), k2 = 0.82.
For this study on PPFRC, it was observed in the fibre counts across the failed sections in the
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uniaxial tension tests specimens that, on average, the fibre dispersion factor was 0.88 for the
3
specimens with 4 kg/m of fibres and 0.78 for the specimens with 8 kg/m of fibres (refer to
Table 2), with an average of 0.83 taken across the two series. Thus the fibre dispersion factor
may also be taken as k2 = 0.82 for a crack where the location is forced, such as inclusion of a
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significant notch for PPFRC.
The COD (w) for the proposed σ-w curve is obtained from the measured crack mouth
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opening displacement (CMOD) by assuming rigid body rotations about the crack tip and by
assuming that failure of the specimen occurs along a single dominant crack:
CMOD hsp − d n
×
2
D − dn
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w=
(5)
A similar approach is adopted for the determination of the σ-w relationship from round panel
tests; before proceeding, a misnomer in the literature needs correcting. Some researchers
(Tran, 2003, Tran et al., 2005) equate lines of fracture formed in the determinant round panel
test of strain softening FRC to that of yield lines in plasticity and impart the view that yield
line theory may be used to determine the moments of resistance. This implies that should
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fibre orientations, dosages or other parameters be varied, other yield (or fracture) line modes
are possible – this is not the case.
In strain softening FRC, it is the post cracking response in tension that is of interest. After
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cracking of the cementitious matrix, the residual tension over the range of structural interest
(0.5 to 2.5 mm) may be reasonably approximated as rigid plastic. That is, the material shows
some “plastic like” behavior after it has cracked and the load reduced (see Figure 4);
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however, to the point where the crack is arrested by the presence of fibres, it remains a quasibrittle material. In the round panel test, the location of the cracks are based on the rules of
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fracture mechanics (not plasticity) and can be described as fracture lines – once these lines
are established, work equations are used to determine the internal moments for a given
applied load (in a similar way to that of yield line theory).
From the fracture line analysis, assuming that the angles between lines are 120° and
is:
P r
⋅
3 3 R
(6)
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m=
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considering small rotations (θ) and hence displacements, the resisting moment per unit length
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where r is the radius from the center of the panel to the support, R is the radius of the panel
(see Figure 3a) and P is the applied central load. The distance between the neutral axis and
the centroid of the compressive stress block is between 0.60dn and 0.667dn (Amin et al.,
2015) and may be taken as 0.64dn without introducing any significant error. From geometry,
the distance between the neutral axis and the centroid of the tensile stress block is evaluated
as 0.5 ( t − d n ) . From sectional stress blocks (see Figure 10) and noting d n = (1 − β ) t , we
write:
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m = f w β ( 0.64 − 0.14 β ) t 2
(7)
where t is the thickness of the panel and dn is the depth to the neutral axis and β is the neutral
fw =
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axis depth parameter (β = dn/t). Combining Equations 6 and 7 gives:
K kb k3 P r
⋅
R
t2
(8)
where kb is the boundary influence factor (see below), K is an empirical coefficient to account
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for several limitations that occur due to the testing arrangements and analysis and is discussed
k3 =
1
β ( 3.3 − 0.73β )
Taking β = 0.95, k3 = 0.40.
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below; and:
(9)
The COD (w) for the proposed σ-w curve derived from the round panel test has previously
(2012):
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been evaluated by Tran et al. (2005) and successfully deployed by de Montaignac et al.
(10)
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 3∆ 
3
w = tan −1 
∆ ( t − dn )
 × ( t − d n ) ≅
2r
 2r 
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For the determination of the depth of the neutral axis, dn, as is required to evaluate
Equations 3 to 10, Amin et al. (2015) and Amin and Foster (2016) showed that the location of
dn had little influence on the w/CMOD ratio. For this reason, we fix the depth of the
compressive stress block and take dn = 0.1hsp and dn = 0.1t for the prisms and panels,
respectively.
To determine the uniaxial strength of the composite for a given COD (w), the contribution of
the concrete (σ(w) = fw + fc) must be included. For plain concrete, the tensile softening stress
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can be modelled using an exponentially decaying function as explained by Foster et al.
(2006):
f c = f ct e −c1w
(11)
(12a)
c1 = 20 (1 + 100 ρ f ) …. for concrete with dg > 10 mm
(12b)
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c1 = 30 (1 + 100 ρ f ) …. for mortar and concrete with dg ≤ 10 mm
where dg is the maximum size of the aggregate particles.
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In Equation 11, c1 is a function of the maximum aggregate size and fibre dosage (ρf):
MODEL VALIDATION
Before comparing the results from the inverse analysis of the bending and panel tests, the test
data needs to be compensated for the boundary/wall effect (Stroeven, 2009; Lee et al. 2011).
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In the case of the prism and panel tests the wall effect can reasonably be approximated as a
2D problem. For these tests, provided that the length of the fibre does not exceed the shortest
dimension of the specimen, the boundary influence factor may be taken as:
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π
3.1 + 0.6 lf b
≤ 1
(13)
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kb =
The results of the inverse analysis are presented in Figure 11. When compensated for the
boundary effect, the proposed model for the prism bending test is in reasonable agreement
with the data obtained from the uniaxial tensile test data for the region of interest (that is
w∈(0, 2.5) mm); however, independent data is needed to validate the model. At a COD of
w = 1.5 mm the uniaxial tension test gives f1.5 = 0.52 MPa and f1.5 = 0.68 MPa for mixes
FRC4 and FRC8, respectively. For the prism tests by the model proposed in Equation 3, the
predicted stresses at w = 1.5 mm are 0.68 MPa and 1.15 MPa for mixes FRC4 and FRC8,
respectively and the test to predicted ratios are 0.76 and 1.01, respectively. Similarly for the
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panel tests with the model proposed in Equation 8, the predicted stress at w = 1.5 mm are
0.59 MPa and 1.14 MPa for mixes FRC4 and FRC8, respectively and the test to predicted
ratios are 0.88 and 1.02, respectively.
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From the plots of Figures 4 to 6, it is seen that the panel test provides the lowest scatter for
the mix FRC4, compared to the prism bending and uniaxial tension tests. For mix FRC8, the
uniaxial tension tests displayed the lowest variation, followed by the round panel tests and
prism bending tests. At a crack opening displacement of w = 1.5 mm, the coefficient of
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variation (COV) for the tests on FRC4 were 0.33, 0.22 and 0.06 for the uniaxial tension test
and the back calculated values determined for prism bending and round panel tests,
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respectively; for mix FRC8, the COVs were 0.07, 0.19 and 0.08, respectively.
On study of the round panel test, several limitations are evident in back determining
fundamental material laws. Besides the assumptions of a rigid plastic σ-w model to describe
the post cracking response of PPFRC and the assumed depth to the neutral axis, described
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above, errors may occur due to: 1) assumption that lines of fracture coincide with the 120
degree theoretical arrangement; 2) tensile membrane actions that occur due to friction
imposed by supporting arrangements are ignored; 3) the influence on the location of the
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centroid of the tensile force in the cross-section due to the boundary (wall) effect; and 4)
alignment of fibres that may result from flow of the concrete during casting. These are
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discussed, in-turn.
Bernard and Xu (2008) concluded deviation of fracture lines from theory (γ1, γ2, and γ3 in
Figure 3a) leads to small errors only. Deviations in γι (i = 1, 2, 3) influence the internal work
calculations and, as the internal work is a minimum on the 120 degree theoretical pattern, the
error leads to a non-conservative solution. The variation of angles γι for the 12 panels tested
is shown in Figure 8 and presented in Table 3, together with the resulting sum of the internal
work (ΣWI) relative to theory. An accurate calculation for the work (internal and external)
based on the experimentally observed fracture lines, is demonstrated in Figure 12. The angles
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αi (i = 1...6) are dependent on the location of the origin ( O ′ ) and the deviation of the fracture
lines from their theoretical values. In these analyses, the value of α are determined from the
equations outlined in Tran (2003) assuming that O ′ coincides with O. An example is shown
in Figure 12b. The results show a maximum error of 8%, with an average error of just 2%.
theory do not significantly influence the derived results.
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This reinforces the conclusions of Bernard and Xu that the variations of the crack angles from
Missing from the above calculation in terms of work is errors in the external work calculation
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due to the location of the crack junction being away from the centre of the panel. The
locations of the crack junction ( O ′ in Figure 3a) for the panels tested in this study are
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presented in Table 3. The external work is WE = P k ∆, where k equals the length of a line
drawn normal to the nearest axis of rotation (Figure 12a) divided by r = 375 mm. The
maximum error in the external work assumption is 15%, with an average of 6%. When the
errors in the internal work are combined with those of the external work, noting that the
errors are multiplicative, the maximum error in the determined moment for the panels tested
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is +22% and the average is +10%.
The second source of error in determining fundamental materials laws from the round panel
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tests comes from tensile membrane effects. This effect is described in Bernard (2005); with
significant panel deflections friction in the support arrangements prevent the segments freely
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moving resulting in tensile membrane forces. The error due to friction is quantified by
Bernard as 15–20%.
The third source of error is derived from the influence of the wall effect on the lever arm (z in
Figure 10b); noting that the wall effect is included in determining the tension T through
Equation 13. At a distance lf /2 from the panel soffit the fibres are assumed to be randomly
distributed in three-dimensions. At the boundary, the fibre distribution is two dimensional.
The fibre orientation factor, Kf is determined by probability of the fibre crossing the fracture
plane and is affected by the shape of the domain over which it is influenced (Ng et al., 2012).
16
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The fibre orientation factor for 2D distribution Kf = 2/π (≈0.64); whereas, for a 3D
distribution Kf = 0.5. The influence of the wall effect on the lever arm diminishes as the ratio
lf / t reduces or when a notch is present, as for the case of the notched prism test. For the panel
tests in this study lf / t = 0.73; the error induced by ignoring the wall effect on the internal
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lever arm between the compressive and tensile forces is estimated to be 4%.
The fourth of the identifiable potential errors is effect of concrete flow in cast concrete panels
from where the mix is placed in the moulds. The more the concrete is moved through the
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casting and vibration processes, the more likely the fibre distribution is biased towards the
flow direction (Stähli et al., 2008). It is hypothesised that that this effect is more significant in
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the round panel test than the flexural prism or uniaxial tension tests due 1) to the size of the
specimen and the length that the concrete is moved from where it is placed in the form and
that the fracture lines include regions distant from the panel centre. No data is currently
available to quantify this effect or assess its significance.
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The modelling assumption of post cracking rigid-perfectly plastic behaviour needs
examination. In the case of PPFRC it was noticed during the uniaxial testing at immediately
after cracking the load dropped substantively during crack stabilisation, the load then
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increased until the peak residual was reached at a COD of about 2.5 mm (see Figure 4), after
which the load reduced gradually. This effect was significantly greater than that observed in
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the SFRC tests of Amin et al. (2015) where the peak was generally observed to be less than
0.5 mm, depending on fibre type and dosage. Thus, in the region of interest for the strength
limit state for structures (w = 0.5 to 2.5 mm), the assumption of rigidly-perfectly plastic
requires some assessment on its potential for error. The draft of the Australian Bridge
Standard AS5100.5 (2014) adopts an approach for conversion of results obtained from
notched prism bending tests to determine the σ-w law with a calibration factor that
corresponds to a COD (w) of 1.5 mm. This is assessed by considering the value of K in
Equation 8 for increasing w; a near constant value would affirm the reasonableness, or
17
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otherwise, of the fundamental assumption. For mix FRC4, the back calculated value was
calculated and varies approximately linearly from 0.39 at w= 1.0 mm to 0.54 at w = 2.5 mm;
for mix FRC8, the value of K varies from 0.58 at w = 1.0 mm and to 0.69 at w = 2.5 mm. It is
seen that K is not constant; this is attributed to the increasing error from the assumption of
approach of DR AS5100.5 (2014), a value of K = 0.5 is selected.
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rigid-perfect plastic behaviour as the crack width at the slab soffit increases. Adopting the
As independent data for PPFC is currently unavailable, the model is validated using SFRC
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test data of de Montaignac et al. (2012), who performed matched direct (uniaxial) tensile and
indirect (round panel) tensile tests. The uniaxial tension tests were conducted on 6 × 85 mm
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diameter cored samples taken from cast prisms and notched at mid-height for each mix. The
round panel tests were tested to ASTM C1550 (2012). As the uniaxial test specimens were
notched, an adjustment is needed to convert the results to an equivalent unnotched condition.
This is done by multiplying the tensile stress by kn = 0.82, as recommended in Foster et al.
(2013). The tensile stress versus crack opening displacement predicted by the model
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developed above for round panel tests is compared to the direct tension data in Figure 12. It
can be seen that model proposed reasonably captures the post cracking behaviour of the
SFRC with the predicted stresses given by Equation 8 giving 1.49 MPa and 1.88 MPa for
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Mixes F35-1.0 and F60-1.0, respectively. At w = 1.5 mm, the test to predicted ratios are 0.90
7
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and 0.96, respectively.
CONCLUSIONS
The post cracking behaviour and response of polypropylene macro synthetic fibre reinforced
concrete (PPFRC) has been investigated by conducting a series of matched direct (uniaxial)
tension tests and indirect tension (notched prism and determinate round panel) tests on
softening PPFRC.
A significant loss in capacity soon after cracking was observed in the experiments for the
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fibres considered in this study. However, over crack widths of interest, the stress resisted by
the polypropylene fibres increased with increased displacement. Of the three test methods,
the round panel test displayed the lowest degree of variability. From a design perspective, this
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should be given some consideration, particularly when characteristic values are adopted.
To analyse the results derived from indirect tension tests, inverse analysis techniques may be
deployed to convert force-displacement data to a generalised stress-crack width law. In this
paper, a model previously developed by the authors for the determination of the stress-crack
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opening displacement relationship for SFRC based on prism bending tests has been adapted
for PPFRC tested in flexure and for determinate round panels tested to ASTM C1550 (2012).
study and by others.
Acknowledgements
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The model has been shown to reasonably capture the response of specimens tested in this
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The fibres used in this study were kindly supplied by TEXO Australasia Pty Ltd. Their
contribution to this study is acknowledged with thanks. The writers would also like to thank
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Mr J. Gilbert for his technical assistance throughout the experimental program.
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List of Tables
Table 1 Average raw mechanical properties of PPFRC mixes.
Table 2 Number of Fibres crossing failure plane of uniaxial tests.
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Table 3 Cracking angles, cracking centre and comparison of calculated internal and external
work relative to theory by fracture line analysis.
List of Figures
Approaches to determine the tensile properties of PPFRC.
Fig. 2
Details of uniaxial tension test specimens
Fig. 3
Round panel test: (a) specimen dimensions and failure mechanism; (b) photograph
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Fig. 1
of test set up.
Raw uniaxial tension test results: (a) FRC4; (b) FRC8.
Fig. 5
Prism bending test results: (a) FRC4; (b) FRC8.
Fig. 6
Round panel test results: (a) FRC4; (b) FRC8.
Fig. 7
Deviation between theoretical failure plane and actual failure plane in round panel
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Fig. 4
tests: (a) definition; (b) results.
σ-w model for PPFRC prisms.
Fig. 9
σ-w model for PPFRC round panels.
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Fig. 8
Fig. 10 Comparison of predicted σ-w curves obtained from inverse analysis of notched
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prisms and round panel tests.
Fig. 11 Definitions of fracture line pattern after the establishment of cracks in round panel;
fracture lines for Panel FRC 4-1.
Fig. 12 Application of inverse analysis of SFRC panels to de Montaignac et al. (2012)
specimens: (a) F60-1.0; (b) F35-1.0.
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Series
fct (MPa)
f1.5 (MPa)
FRC4
66
4.15
0.63
FRC8
63
3.87
1.25
fR1
fR2
(MPa)
(MPa)
fR3
fR4
f
fct,L
(MPa)
(MPa)
(MPa)
Round Panel Test
(ASTM C1550, 2012)
W (Joules)
1.96
2.46
2.74
2.85
5.37
701
2.75
3.62
4.09
4.24
5.91
1075
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fcm (MPa)
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(DR AS5100.5, 2014)
Notched 3 Point Bending Test (EN 14651, 2007)
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Direct Tension Test
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Table 1. Average raw mechanical properties of PPFRC mixes.
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Table 2. Number of Fibres crossing failure plane of uniaxial tests.
No. of fibres
crossing fracture
plane
Specimen ID
No. of fibres
crossing fracture
plane
FRC4-1
62
FRC8-1
104
FRC4-2
-
FRC8-2
104
FRC4-3
49
FRC8-3
FRC4-4
51
FRC8-4
FRC4-5
61
FRC8-5
FRC4-6
67
FRC8-6
Mean (ρfc)
58
Mean (ρfc)
102
Theory (ρf)
66
Theory (ρf)
131
ρfc
0.39
ρfc
0.69
ρfc/ρf
0.88
ρfc/ρf
0.78
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Specimen ID
106
102
99
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Table 3. Cracking angles, cracking centre and comparison of calculated internal and
external work relative to theory by fracture line analysis.
Panel
γ2
γ3
(deg.)
(deg.)
(deg.)
FRC4-1
0
0
FRC4-2
10
FRC4-3
ΣWI
ΣWI .theory
ΣWE
ΣWE.theory
Overall
Error
(%)
y (mm)
-8
11.5
-8.8
1.003
0.962
5
-16
9
48.6
16.7
1.023
0.911
16
-6
-13
4
-19.1
0.0
1.012
0.956
8
FRC4-4
36
4
11
21.0
-10.8
1.078
0.937
19
FRC4-5
1
-11
15
FRC4-6
6
-26
0
FRC8-1
-8
-16
0
FRC8-2
-21
-3
-12
FRC8-3
-2
0
4
FRC8-4
-8
0
FRC8-5
6
-12
FRC8-6
0
-10
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x (mm)
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γ1
Coords. of crack
junction (Fig 3)
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Cracking angles (Fig 3)
0.4
1.021
0.851
22
6.5
-19.4
1.037
0.959
14
-1.1
-0.8
1.017
0.996
4
2.8
-9.1
1.032
0.981
8
-19.3
-2.4
1.001
0.952
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-64.9
-1.2
-41.1
1.003
0.942
7
12
27.7
-42.1
1.018
0.880
18
-4
7.3
0.0
1.006
0.983
3
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0
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F
F
CMOD
P
σ
P
w
P
F
w
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F
SC
Direct
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Inverse
Analysis
Inverse
Analysis
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Fig 1. Approaches to determine the tensile properties of PPFRC.
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P
P
125
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125
215
125
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200
R145
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125
Universal
joint
16 mm dia.
threaded rod
P
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P
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Fig 2. Details of uniaxial tension test specimens
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Support
C
A
x
75
=3
γ2
Support
B
(a)
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Reference frame
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r
O'
O
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R = 400
Support
γ1
y
γ3
(b)
Fig 3. Round panel test: (a) specimen dimensions and failure mechanism; (b) photograph of
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4
3
2
Mean
Characteristic
1
4
3
Mean
2
1
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Tensile Stress (MPa)
5
Characteristic
0
0
0
1
2
3
4
5
0
1
2
3
4
5
Crack Opening Displacement, w (mm)
SC
Crack Opening Displacement, w (mm)
(b)
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Fig 4. Raw uniaxial tension test results: (a) FRC4; (b) FRC8.
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Tensile Stress (MPa)
5
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25
25
20
15
10
5
Mean
15
10
Characteristic
5
Characteristic
0
0
0
2
4
6
8
10
12
0
2
4
6
8
10
12
CMOD (mm)
SC
CMOD (mm)
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Load (kN)
Mean
(b)
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Fig 5. Prism bending test results: (a) FRC4; (b) FRC8.
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Load (kN)
20
33
50
40
40
Mean
20
10
30
Mean
20
10
Characteristic
0
Characteristic
0
0
10
20
30
40
50
60
Central Deflection (mm)
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Load (kN)
50
0
10
20
30
40
50
60
Central Deflection (mm)
(b)
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(a)
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Fig 6. Round panel test results: (a) FRC4; (b) FRC8.
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Support
15
8
5
3
2
8
2
1
28 to 36
20 to 28
12 to 20
-4 to 4
-12 to -4
-20 to -12
-28 to -20
4 to 12
0
SC
0
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(a)
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-36 to -28
Number of Cracks
12
γ (degrees)
(b)
Fig 7. Deviation between theoretical failure plane and actual failure plane in round panel
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tests: (a) definition; (b) results.
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fw
notch
(b) Stresses
at crack
(a) Section
(c) Simplified
Model
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Fig 8. σ-w model for PPFRC prisms.
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36
dn
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C
z
t _ dn
(a) Section
fw
SC
(b) Simplified
Model
T
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t
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Fig 9. σ-w model for PPFRC round panels.
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Data range direct
tension test x k t
1
0
0
1
2
3
COD, w (mm)
2
1
Data range direct
tension test x kt
4
0
0
1
2
3
COD, w (mm)
4
(b)
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(a)
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2
3
SC
3
4
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Tensile Stress (MPa)
4
FRC8
Prism - Mean
Panel - Mean
Prism - Characteristic
Panel - Characteristic
5
Tensile Stress (MPa)
FRC4
Prism - Mean
Panel - Mean
Prism - Characteristic
Panel - Characteristic
5
Fig 10. Comparison of predicted σ-w curves obtained from inverse analysis of notched
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prisms and round panel tests: (a) FRC4 series; (b) FRC8 series..
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α3
Support
α4
662 mm
y
400
α5
O'
O
x
A
α6
30
38
38
5
37
30
-8
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C
Support
Theory
Axis of rotation
SC
α1
mm
α2
717
686
mm
Support
22
Axis of
rotation
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Fig 11.
(b)
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22
(a) Definitions of fracture line pattern after the establishment of cracks in round
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3
4
Steel RC80/60 fibres
l f = 60 mm d f = 0.75 mm ρ f = 1.00%
t = 80 mm R = 400 mm r = 375 mm
K = 0.5 k 3 = 0.82 k n = 0.82
3
2
Equation 8
1
Direct tension
test data × k n
Direct tension
test data × k n
2
Equation 8
Steel RC65/35 fibres
l f = 35 mm d f = 0.55 mm ρ f = 1.00%
t = 80 mm R = 400 mm r = 375 mm
K = 0.5 k 3 = 0.82 k n = 0.82
1
0
0
1
2
3
4
Crack Opening Displacement, w (mm)
1
2
3
4
Crack Opening Displacement, w (mm)
(b)
M
AN
U
(a)
0
SC
0
de Montaignac et al. (2012) - F35-1.0
RI
PT
de Montaignac et al. (2012) - F60-1.0
Tensile Stress (MPa)
Tensile Stress (MPa)
5
Fig 12. Application of inverse analysis of SFRC panels to de Montaignac et al. (2012)
AC
C
EP
TE
D
specimens: (a) F60-1.0; (b) F35-1.0.
40