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IBRACON Structures and Materials Journal • 2012 • vol. 5 • nº 4
C. G. NOGUEIRA | E. D. LEONEL | H. B. CODA
cal model at the First Order Reliability Method (FORM). The second
approach used is the classical Monte Carlo’s simulation. In both ca-
ses, the models are used to evaluate the probabilities of failure con-
sidering the chloride ingress process, in reinforced concrete struc-
tures. These approaches will be discussed in the following sections.
3.2 Direct approach – mechanical model and FORM
The basic procedure in this model consists in directly coupling the
reliability model FORM with the mechanical model given by equa-
tion (4). This approach has demonstrated be accurate and robust
for analysis of many complex engineering problems, as discussed
by [21-23].
As described in the previous section, the limit state function defi-
nes the boundary between safe and failure domains. Considering
chloride penetration problem, the limit state function can be written
in terms of the time for corrosion initiation:
(7)
( ) ( )
a
R
t X t
XG
- =
in which:
R
t
is the time for corrosion initiation that depends of the
set of random variables
X
;
a
t
is the structural life-time expected
in design which was considered as a deterministic parameter or it
can be the suggested time for inspections.
The time
R
t
is evaluated from equation (4) assuming the chloride
concentration
( )
tpC
,
as known at a given position
p
inside the
concrete. Actually,
( )
tpC
,
is assumed to be equal to the chloride
concentration threshold value, over which the steel corrosion is
triggered, in failure condition. The range of
p
position, in this case,
is the concrete cover, in which it assumes zero at the external sur-
face of the structural member and the cover value at the reinforce-
ment surface inside concrete. In this regard, the time for corrosion
initiation can be determined explicitly from equation (4) as:
(8)
( )
[
]
2
0
0
,
2
1
þ
ý
ü
î
í
ì
=
CtpC erfc
p
D
t
R
In order to include invariance measure of safety, the random variables,
defined in the physical space, are transformed into independent standard
Gaussian variables [20], by using appropriate probabilistic transforma-
tions. The limit state function
( )
0
=
XG
, defined in the physical space,
is transformed into
( )
0
=
UH
in the standard normalized space with
[
]
1 2
, ,...,
T
n
U u u u
=
, where U is the set of normalized randomvariables.
In this standard space, the reliability index
b
is given by the mi-
nimum distance between the failure domain and the origin of the
standard space. The reliability index can be evaluated by solving a
constrained optimization problem, as described below:
Find:
U
, which minimizes:
UU
T
⋅
=
b
and subject to:
(9)
Find:
U
, which minimizes:
UU
T
×
=b
and subject to:
( )
0
=
UH
known as the limit state itself. It is worth to mention that an explicit
expression of the limit state function is usually not possible. When
numerical mechanical models are used, only at a desired number
of points it can be computed. In this paper, the limit state is defined
using the critical failure mode calculated by equation (4).
The probability of failure is evaluated by integrating, over the failure
domain, the joint density function as presented by [19]:
(5)
(
)
1 2
1 2
0
, , ,
,
,
f
X
n
n
G
P
f x x x dx dx dx
£
=
ò
K
L
in which:
( )
X f
X
is the joint density function of the random va-
riables
X
.
As the evaluation of integral defined by equation (5) is almost im-
possible in practice, alternative procedures have been developed
on the basis of reliability index concept [20]. This parameter is
defined by the distance between the mean point and the failure
point placed at the limit state function
( )
0
=
XG
in the norma-
lized space of random variables. The reliability index allows us to
calculate the probability of failure, using the First Order Reliability
Method (FORM) as follow:
(6)
( )
b-F=
f
P
in which:
( )
⋅Φ
is the standard Gaussian cumulated distribution
function,
b
is the reliability index.
There are some alternative procedures, available in the reliability
theory, which allow the probabilities assessment of structural failure.
These procedures are based on numerical simulation techniques.
The most important approach, among them, is the Monte Carlo’s
simulation method. However, when numerical mechanical models
expensive in terms of computational work are adopted, this approa-
ch may be unreliable, due the large sampling required for simulation.
In this study, two reliability approaches are adopted. The first is kno-
wn as direct approach, as it is the result of direct coupling a mechani-
Figure 1 – Domains of failure and
safety for two random variables
safety domain
limit state function
failure domain
G(X)>0
G(X)<0
G(X)=0
X
2
X
1