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IBRACON Structures and Materials Journal • 2012 • vol. 5 • nº 1
Steel fibre reinforced concrete pipes. Part 2: Numerical model to simulate the crushing test
merically obtained with the AES model and
M
s
is deduced impos-
ing external bending moment equilibrium (Eq. 15).
(13)
φ
s crk
= 2l
cb
hE
cm
(
F h + σ
1
)
(14)
F = π 2R
m
M
c
+ E
cm
I 2R
m2
(
φ
c
- φ
c crk
- φ
s
+ φ
s crk
)
(15)
M
s
(
(
F, φ
s
l
cb
)
)
= FR
m
- M
c
0, φ
c
l
cb
The vertical displacement at the ridge
(
v
c
) and the horizontal dis-
placement at the springline (
u
s
) (see Figure [5d]) are calculated as
a composition of the elastic strain of the pipe (
v
c
e
and
u
s
e
) and a
plastic one (
v
c
p
and
u
s
p
) due to the rotation of the rigid body after
the appearance of cracks in the critical sections.
The displacement
v
c
e
is calculated by means of Eq. 16 and Eq. 17.
(16)
v
c e
= R
m2
E
cm
I
(
M
c
- 1 2 FR
m
)
if
φ
c
=
φ
c
crk
(17)
v
c e
= R
m2
E
cm
I
(
(
M
c
- 1 2 FR
m
)
)
+ 1 2 R
m
φ
c
- φ
c crk
if
φ
c
>
φ
c
crk
The displacement
u
s
e
is deduced by means of Eqs. 18 and 19.
(18)
u
s e
= R
m2
E
cm
I
[
M
c
(
(
π 2 - 1
)
)
+ FR
m
]
π 4 - 1
if
crk
φc
≤
φc
(19)
u
s e
= R
m 2
E
cm
I
[
M
c
(
π 2 - 1
)
+ FR
m
(
π 4 - 1
) ]
+ 1 2 R
m
(
φ
c
- φ
c crk
)
if
φ
c
>
φ
c
crk
The displacements
v
c
p
and
u
s
p
are expressed by means of Eqs.
20-22.
(20)
v
c p
= u
s p
= 0
if
φ
c
≤
φ
c
crk
(21)
v
c p
= u
s p
=
1 2
R
m
(
φ
c
- φ
c crk
)
if
φ
c
>
φ
c
crk
y
φ
s
≤
φ
s
crk
(22)
v
c p
= u
s p
= 1 2 R
m
(
φ
c
- φ
c crk
)
+ 1 2 R
m
(
φ
s
- φ
s crk
)
if
crk
crk
φc > φc y φs > φs
Relevant results are obtained with application of the MAP model,
including the curves
F
–
v
c
from the CT. These curves allow under-
standing the behavior of the structure at each of the stages.
3.3.3 Solution procedure
The process is initiated with zero values for the rotations at the
ridge (
φ
c
) and spring line (
φ
s
). The control variable is
φ
c
, which
increases with variable steps depending on the behavior stage.
Establishing
N
= 0 (simple bending) for each value of
φ
c
at the
ridge, the value of
M
c
is obtained by means of the AES model (Eq.6
for elastic regime, State I, and Eq. 10 for cracked regime, States II
and III). After that, the values of
M
s
and
F
are calculated with the
expressions previously presented:
n
At Stage I:
F
is obtained with Eq. 7 and
M
s
, by means of Eq. 8.
At this stage the whole pipe, even the critical sections, works in
a linear regime.
n
At Stage II:
F
is obtained with Eq. 11 and
M
s
, by means of Eq.
12. At this stage, a degree of non-linearity is introduced due
to the cracking of section C (
φ
c
> φ
c
crk
), whereas section S still
works in a linear regime (
φ
s
≤ φ
s
crk
). Therefore, due to the de-
gree of hyperstatism of the system, a redistribution of moments
from C to S takes place.
n
At Stage III:
F
is obtained with Eq. 14 and
M
s
, by resolving
Eq. 15. At this stage, unlike at the previous ones, the balance
condition (Eq. 15) is non-linear due to the fact that both sec-
tion C and section S have cracked. For its solution, an iterative
Newton – Raphson schema was implemented (see Yang
et al
.
[33]). In this sense, it has to be noticed that the section S works
under a bending - compression state during the whole load
process (
N
=
F
and
M
=
M
s
).
The algorithm stops either when the maximum strain is reached
at any of the two critical sections, or when the displacement
v
c
exceeds the fixed value pre-established by the user
v
c,max
.
Once the
F
and
M
values have been obtained, the displacements
in sections C and S can be assessed with Eqs. 16-22.
This procedure guarantees good results in concrete pipes with a
predominantly rigid behavior: pipes with a small-medium diam-
eter (300 - 1000 mm) and with moderate reinforcement densi-
ties. With these hypotheses, it can be guaranteed that, in most
cases, the cracks are concentrated in sections C and S, while
the rest of the pipe works with its entire section (de la Fuente
et al
. [3]). In the opposite case, cracks appear intermediately
and the model deviates from the experimental results and it is
necessary to resort to other models capable of considering the
distributed cracking, such as the one presented in de la Fuente
et al
. [12].