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17
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
ing the hypothesis of perfect bond between concrete and steel
(Eq. 5).
(3)
(4)
(5)
ε(y) = ε
c,inf
- y. χ
Eqs. 3-5 leads to a nonlinear system of equations which is solved
by using the Newton-Raphson iterative method (see Yang
et al
.
[33]). After solving the system, the values of the unknown param-
eters
ε
c,inf
and
χ
, which define the strain plane, are obtained.
3.3 Structural analysis model
3.3.1 Basic hypotheses
For the simulation of the CT (Figure [1]), the following hypotheses
have been considered: (1) the structure can be idealized as a me-
dium plane piece with a curved shape and a constant radius
R
m
;
(2) symmetry with regard to the vertical and the horizontal axes,
so only a quarter of the pipe is simulated; (3) the initial curvature
of the piece does not have an influence over the distribution of the
stresses along the piece, nor over its deformed shape; (4) the axial
and shear effects are disregarded in the assessment of the pipe
displacements; (5) the continuous test is considered to be repre-
sentative for the simulation of the FRCP behavior up to post-failure
(Figueiredo [5]); and (6) three stages are considered:
n
Stage 1: linear elastic stage (Figure [5a]).
n
Stage 2: elastic stage with cracking at ridge (point C in Figure [5b]).
n
Stage 3: elastic stage with cracking at ridge and springs (point
S in Figure [5c]).
3.3.2 Behavior equations
The governing equations for the structural problem implemented in
MAP were deduced by Pedersen [22] for the simulation of SFRC
pipes with small diameters. The strategy consists in considering that
the response is represented by the three stages previously described,
and that the pipe behaves elastically throughout the whole test, ex-
cept the section at the ridge and at the spring line. This response
pattern has also been observed in pipes tested by Figueiredo
et al
.
[13]. The behavior of both sections is simulated with the AES model.
This paper presents the final form of the governing equations for the
problem. Their analytical deduction can be found in Pedersen [22]. They
are based on the energy theorems by Castigliano (see Timoshenko [34])
and on other classical considerations about the calculation of structures.
The applied force
F
, and the bending moments at the ridge
M
c
and
at the spring line
M
s
(see Figure [5]) are the determinant param-
eters. They depend on the behavior regime of the pipe and, as a
consequence, the analytical formulation varies depending on the
stress state of the control sections.
In the elastic regime (
Stage I
),
M
c
is assessed by means the linear
equation (Ec. 6). Once
M
c
is known,
F
and
M
s
are obtained using
Eq. 7 and Eq. 8, respectively. This regime ends when a crack is
formed at ridge for a rotation
φ
c
crk
(Eq. 9).
(6)
M
c
= E
cm
I
φ
c
l
bc
(7)
F = π 2 M
c
R
m
(8)
M
s
=
(
1 - 2 π
)
FR
m
(9)
φ
c crk
= 2l
bc
σ
1
hE
cm
At
Stage II
,
M
c
(Eq. 10) is numerically obtained with the AES model
since the formation of the first crack in C leads to a non linear
system (Eqs. 3-5). Then,
F
is calculated by means of Eq. 11 and
M
s
is deduced by imposing external bending moment equilibrium
(Eq. 12).
(10)
M
c
= M
c
(N, χ
c
) = M
c
(
0, φ
c
l
cb
)
(11)
F = π 2R
m
M
c
+ E
cm
I 2R
m2
(
φ
c
- φ
c crk
)
(12)
M
s
= FR
m
- M
c
(
0, φ
c
l
cb
)
Stage III
(Figure [5c]) starts when a crack is formed in the section
S. This crack appears when a rotation
φ
s
crk
(Eq. 13) is reached. At
this stage,
F
is calculated by means of Eq. 14,
M
c
(Eq. 10) is nu-