Basic HTML Version
14
IBRACON Structures and Materials Journal • 2012 • vol. 5 • nº 1
Steel fibre reinforced concrete pipes. Part 2: Numerical model to simulate the crushing test
stitutive model to each material and integrates the stresses result-
ing from a given strain plane (see Figure [3a]). The total concrete
strain
ε
c
(t,t
0
)
, assessed at an instant of time
t
, is considered to
be the sum of the mechanical strains
ε
c
m
(
t
0
)
, produced instanta-
neously at
t
o
, and the non-mechanical strains
ε
c
nm
(
t,t
0
)
(see de la
Fuente
et al
. [21] and Marí
et al
. [23])
.
In this paper,
ε
c
nm
(
t,t
0
)
are
not considered since the test only takes a few minutes to be ex-
ecuted, insufficient time to present non-mechanical strains due to
the concrete creep. Likewise, according to Heger [24], shrinkage
hardly influences the stress state of the pipe cross section; hence,
it is also disregarded in this model.
For the simulation of the concrete compressive behavior the
diagram suggested by Thorenfeldt
et al
. [25] is used, since
it could be adjusted correctly to a wide range of concrete
strengths and suitably simulates the post-failure response. On
the other hand, the tensile behavior and concrete stiffening be-
tween cracks is described by means of the equation proposed
in Collins
et al
. [26].
According to Bencardino
et al
. [27], the inclusion of metallic fi-
bres modifies the SFRC compressive behavior depending on
the volume of fibres used. In this respect, the expression sug-
gested by Barros
et al
. [28] fits properly the uniaxial compres-
sive behavior in the post-failure regime of SFRC. On the other
hand, the simulation of its tensile behavior has been dealt by
means of the
σ
c
-ε
c
model (see Figure [3b]) proposed in (Vande-
walle
et al
. [16]), because it has already been used in several
numerical-experimental contrasting tests (see Pujadas [29]),
guaranteeing good results.
3. Model for the simulation of the
crushing test
The required subroutine for the simulation of the CT up to high
displacement levels should take in to account paramount aspects
as the cracking the post-failure response of the materials and the
modeling of the SFRC behavior. In that sense, the Analysis of Evo-
lutive Sections (AES) introduced in de la Fuente
et al.
[21] was
used in order to deal with these aspects.
On the other hand, the MAP routine, which includes the AES
model, was also developed. The bases for the structural model
were already suggested by Pedersen [22] for the analysis of
pipes with a small diameter. However, for this work, several
changes were made as regards the behavior at the sectional
level, the constitutive equations of SFRC and the possibility of
considering the coexistence of steel rebars and structural fibres
as reinforcement.
This section puts forward the main foundations of the AES model,
highlighting the modifications introduced for this work, as well as the
analytical equations of the MAP model and the calculation algorithm.
3.2 Sectional analysis model
3.2.1 Modeling the materials
The AES model discretizes the concrete in 2-D differential ele-
ments (
dA
c
), and the steel rebars in elements with concentrated
area
A
s,i
in its gravity center
y
s,i
. Then, it assigns the suitable con-
Figure 2 – Crushing test: (a) cross section and (b) longitudinal section
=
=
l
/2
F
l
Supports
Ridge
Invert
h
Springline
F
Spreading beam
2β
F/
2
F/
2
D
O
D
i