Basic HTML Version
445
IBRACON Structures and Materials Journal • 2012 • vol. 5 • nº 4
C. G. NOGUEIRA | E. D. LEONEL | H. B. CODA
By simulating the limit state function for a convenient number of
sampling, the mean value (
E
) of
( )
i
xI
will be an estimator for the
probability of failure. Then:
(12)
( )
[ ]
( )
å
=
=
=
N
i
i
i
f
xI
N
xIE P
1
1
The disadvantage of this method is related to the large amount
of simulations
N
required to compute accurately the probability of
failure. Normally, in order to estimate accurately the probability of
failure of
10
n
−
, the number of simulations must be higher than
2
10
n
+
or
3
10
n
+
. It means, in civil engineering structures, where
the probability of failure is in between
3
10
−
to
6
10
−
, it is required,
at least,
5
10
to
9
10
realizations of the limit state function. When
complex numerical mechanical models are involved, which lead to
high computational work, this method may be not reliable. Howe-
ver, theoretically, this method leads to the real probability of failure
when the sampling range becomes infinity.
4. Methodology of analysis
The corrosion phenomenon modelling in reinforced concrete struc-
tures has to take into account two different stages, as illustrated
in figure 4. The first stage is related to chloride ingress into con-
crete porous. In this stage, the chloride concentration, along the
cover depth, increases as the time passes. Then, the corrosion
starts when the chloride concentration surrounding the reinforce-
ments reaches a threshold level, leading to the loss of the con-
crete chemical passive protection. At the end of this stage the
steel of reinforcements remains undamaged. The second stage,
called propagation stage, is characterized by the reduction of the
reinforcement’s cross sections, which generates the loss of struc-
tural strength along the time.
Compared with the first stage, the propagation period is relatively
short. Therefore, the time for corrosion initiation has been widely
adopted for structural durability and safety assessments. In this
regard, the objective of this paper is to assess the probability of
failure considering the failure scenario predicted by initiation stage.
Therefore, the failure is observed when the chloride concentration
at the reinforcement’s depth reaches the threshold level. In this
regard, the main parameters considered for all reliability analyses
performed are:
n
Chloride concentration threshold at the interface between con-
crete and reinforcement bars, which defines the beginning of
the corrosion process,
( )
txC
,
. This parameter was studied
experimentally by [12];
n
Chloride concentration at the structural surface,
0
C
. This pa-
rameter is related to the environment aggressiveness and its
reference’s values may be determined by experimental obser-
vations, as presented by [12], or defined using an international
concrete standard design code [26];
n
Concrete diffusion coefficient,
0
D
, which has been studied
by [27];
n
Structural depth, which in this study, is defined as the reinfor-
cement concrete cover,
p
.
It is worth to stress that initial cracks due concrete cure and/or ben-
ding/shear effects and longitudinal cracking have not been consi-
dered in the formulation presented in this paper. These phenome-
na affect the corrosion process and its modelling can be accurately
performed using numerical methods as finite element method and
boundary element method [28].
According [12], the chloride concentration at the structural surfa-
ces is a function of the atmosphere (environment) where these
elements are located. The cover depth is also defined according
the environment, which is stated by international concrete stan-
dard design in categories of aggressiveness. In particular, the va-
lues shown in [26] have been used in this paper. The coefficient
of diffusion of concrete, which represents the material resistance
against chloride ingress, depends on the water/cement ratio. As
higher be the proportion of water, higher will be the empty volumes
inside the matrix due the cure process of the concrete. Conse-
quently, higher will be the material permeability and lower will be
the material resistance against the chloride penetration. Therefore,
in regions close to the coast it is strongly recommended to cons-
truct reinforced concrete structures with lower water/cement ratio
and/or large covers.
The proposed probabilistic model allows the evaluation of the
probability of structural failure taking into account the random va-
riables previously presented. Moreover, this model is capable to
describe the dependency relation between the probability of failure
and time. Then, the proposed model can be used to solve an inte-
resting structural problem which is related to structural maintenan-
ce plans based on structural safety.
In order to apply the model in this problem a given safety
level must be defined. The mentioned safety level may be de-
termined using [29], where the prevention against structural
failures is measured based on a target reliability index. As the
chloride concentration at the structural cover increases along
time, the safety against this failure mode reduces along time.
Therefore, the intervals of time for periodic structural mainte-
nance procedures, based on structural safety, are achieved
when the reliability index calculated using the proposed mo-
del reaches the target reliability index stated by the analyst.
It is worth to mention that the repair procedures are assumed,
in this case, as perfect, i.e, after the maintenance the struc-
ture recover its initial integrity conditions without chlorides.
These intervals are determined as long as the parameters
related to material, cover depth and environment aggressive-
ness be defined a priori.
Another application of the proposed model relates the definition
of values for cover depth and concrete properties, as for example
water/cement ratio (w/c), based on a given safety level and the ex-
pected structural life-time. When the intervals of time for structural
maintenance and/or the expected structural life-time are stated a
priori, the w/c ratio and the cover depth values can be obtained
by simulating the proposed model in order to define the couple of
these values which lead the structure to maintain a safety level at
least equal to the target reliability index during the specified period
of time. In this application the values of w/c ratio and cover depth
are achieved when the reliability index given by the proposed mo-
del is equal to the target reliability index.
It is important to emphasize that, these procedures do not take into
account the costs involved neither in the maintenance procedure
nor in the concrete production. However, these applications of the
proposed model can be performed if the analysts are interested in
design considering, exclusively, safety criterion.