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21
IBRACON Structures and Materials Journal • 2012 • vol. 5 • nº 1
A. D. de Figueiredo | A. de la Fuente | A. Aguado | C. Molins | P. J. Chama Neto
load
F
min,pos
for a displacement
v
c
of 3 mm as in the previous cases.
However, it can be asserted that the load
F
min,pos
from these pipes
exceeds the value of 94.5 kN, since the pipes with 20 kg/m
3
of
fibres showed values 14.3% higher than this (108 kN) according to
the numerical model.
Table [2] gathers the average experimental and numerical values for
F
c
,
F
u
and
F
3mm
. The parameter ξ
is the relative error of the numerical
value with regard to the experimental data. Positive values of ξ indicate
that the experimental data exceeds the numerical one, and vice versa.
With regard to the results illustrated in Table [2], it is deduced that:
n
The load
F
cr
obtained numerically is independent from the
C
f
,
since tension
σ
1
is related exclusively to the concrete matrix
(see Figure [3b] and Table [1]).
F
cr
depends exclusively on
σ
1
,
h
and
D
i
(see de la Fuente
et al
. [3]). The model tends to over-
estimate
F
cr
with regard to the average experimental results
between a 4.3% (pipe with 10 kg/m
3
) and a 6.5% (pipe with 40
kg/m
3
). This can be due to the incorporation, during the mixing,
of additional water (the bigger the amount of fibres, the bigger
the quantity added) into the concrete used for the contrasted
pipes (see Figueiredo
et al
. [19]).
n
The model underestimates the maximum load
F
u
with regard to
the experimental values between a 3.7% (pipe with 40 kg/m
3
)
and a 13.6% (pipe with 10 kg/m
3
). For this load level, the matrix
of concrete has already cracked and the fibres work with the
maximum efficiency; then, the reason for this difference in the
results can be due to the fact that the values of
σ
i
used in the
constitutive equation might be too conservative for low levels of
displacement. In this regard, de la Fuente
et al
. [38] proves that
the fibres in concrete pipes manufactured with traditional sys-
tems (for example, turbo-compression) work oriented towards
the stress flow, practically guaranteeing their maximum efficien-
cy. One of the ways to consider this fact is by using an amplifi-
cation coefficient of the parameters
σ
i
. So, the use constitutive
equations incorporating this effect, in order to take into account
the preferential orientation of the fibres could be considered as a
good alternative, as suggested by Laranjeira [39].
n
As regards the load
F
3mm
, the numerical values obtained are
6.4% (10 kg/m
3
) and a 6.9% (20 kg/m
3
) lower in comparison
with the experimental data. This could also be due to the un-
derestimation of the parameters
σ
i
adopted to model the ten-
sile response of SFRC in this regime of
v
c
.
To sum up, the MAP model adjusts satisfactorily to the experimen-
tal results, even considering that the input parameters of the con-
stitutive equation used to model the tension response of SFRC are
calibrated from concretes with a
f
ck
ranging between 25 and 30
MPa (see Barros
et al
. [36]), as opposed to the reported 50 MPa.
Similarly, the equation used does not take into account the effect of
the preferential orientation of the fibres within the wall of the pipe.
Therefore, the MAP model tends to underestimate the experimen-
tal results in most of the cases, yet these differences do not exceed
the 13.6%. Nevertheless, the results can be considered a success,
taking into account the multitude of variables involved in the prob-
lem, their uncertainty and the difficulties for the direct experimental
determination of some of them.
5. Example of the optimal design
of the amount of fibres
The following example of MAP model application to a pipe with
D
i
of
400 mm,
h
of 67 mm and a total length of 2500 mm is purposed to
illustrate the methodology of the design of the optimal
C
f
in SFRCP.
This diameter was chosen for two reasons: (1) because it is within
the range of diameters for which the hypotheses from the MAP
model are appropriate, and (2) because it is a commercial diameter
nd
Table 2 – Average experimental and numerical values of F obtained for the 2 series of tests
Fibre dosage
3
(kg/m )
F
cr
F
u
F
3mm
Exp.
(kN)
MAP
(kN)
e
e
e
(%)
Exp.
(kN)
MAP
(kN)
(%)
Exp.
(kN)
MAP
(kN)
(%)
10
94
98
-4.3
132
114
13.6
94
88
6.4
20
93
98
-5.4
142
123
13.4
116
108
6.9
40
92
98
-6.5
162
156
3.7
-
-
-
Table 3 – Parameters to simulate the tensile behavior of the SFRC for the pipe of D = 400 mm
i
C
f
3
(Kg/m )
f
R,1
(MPa)
f
R,4
(MPa)
E
cm
(MPa)
s
1
(MPa)
e
1
(mm/m)
s
2
(MPa)
e
2
(mm/m)
s
3
(MPa)
e
3
(mm/m)
0
0.702
0.650
36000
3.795
0.105
0.316
0.205
0.241
25.000
10
1.647
1.525
36000
3.795
0.105
0.741
0.205
0.564
25.000
20
2.592
2.400
36000
3.795
0.105
1.166
0.205
0.888
25.000
30
3.537
3.275
36000
3.795
0.105
1.592
0.205
1.212
25.000