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IBRACON Structures and Materials Journal • 2012 • vol. 5 • nº 4
Reliability algorithms applied to reinforced concrete durability assessment
and the chloride concentration. In this case, Fick’s second law can
be rewritten in the simpler form:
(3)
2
2
0
p
C D
t
C
¶
¶
=
¶
¶
in which:
D
0
is the constant chloride diffusion coefficient of concre-
te,
t
is the time of the assessment of chloride concentration.
The solution of the differential equation (3), for a semi-infinite do-
main with an uniform concentration at the structural surface, is gi-
ven by:
(4)
( )
ú
ú
û
ù
ê
ê
ë
é
=
tD
p
erfc C tpC
0
0
2
,
in which:
C
0
is the ions chloride concentration at the structural sur-
face supposed constant in the time,
erfc
is the complementary er-
ror function.
In this paper, equation (4) is used to evaluate the chloride concen-
tration
( )
tpC
,
, at a given position
p
and time
t
, and for that rea-
son is also used as the mechanical model in this paper. Based on
the concentration values of chloride ions at a given structural dep-
th, it is possible to assess the structural safety. However, in order
to allow the safety assessment, a coupling among the described
mechanical model and reliability algorithms has been constructed.
These coupling models take into account the inherent randomness
of the variables in the diffusion process. One of the most important
product of these coupling is the possibility to choose the critical
time of structural maintenance based in a given target reliability
index or the determination of the concrete cover depth based on
the structural safety level target.
3. Reliability concepts and methods
of analysis
3.1 General concepts
The reliability analysis aims at computing the probability of failure
regarding a specific failure scenario, known as limit state. The first
step in the reliability assessment is to identify the basic set of ran-
dom variables
[
]
T
n
x xx X
,...,
,
2 1
=
for which uncertainties have
to be considered. For all these variables, probability distributions
are attributed in order to model its randomness. These probabili-
ty distributions can be defined by physical observations, statisti-
cal studies, laboratory analysis and expert opinion. The number
of random variables is an important parameter to determine the
computing time consumed during the reliability analysis. In order to
reduce the size of the random variable space, it is strongly recom-
mended to consider as deterministic all variables whose uncertain-
ties lead to minor effects on the value of probability of failure.
The second step consists in defining a number of potentially critical
failure modes. For each of them, a limit state function, separates the
space into two regions as described in figure 1: the safe domain,
where
( )
0
>
XG
and the failure domain where
( )
0
<
XG
. The
boundary between these two domains is defined by
( )
0
=
XG
,
-cracking process) material. However, the methods commonly
adopted for chlorides transportation modelling in concrete consi-
der this process governed only by ionic diffusion, then, it assu-
mes that the concrete cover is completely saturated. Therefore,
it makes the Fick’s laws hypotheses acceptable for the chloride
ingress modelling, because, in this case, the material is assumed
completely saturated, with unidirectional chloride flux, i.e., from the
exterior surface into the concrete depth. When chloride diffuses
into concrete, a change of chloride concentration occurs at any
time in every point of the concrete, i.e., it is a non-steady state
of diffusion. In order to simplify its analysis, the diffusion problem
is considered as one-dimensional. Many engineering problems of
chloride ingress, as those discussed in this paper, can be solved
considering this simplification.
The assumption of Fick’s diffusion theory is that the transport (gi-
ven by the flux) in concrete of chloride ions though a unit section
area of concrete per unit of time is proportional to the concentration
gradient of chloride ions measured normally to the section. Then:
(1)
p
CD F
¶
¶
-=
in which:
F
is the flux of ions chloride into concrete,
D
is the ge-
neral coefficient of diffusion of the concrete,
C
corresponds to the
chloride concentration at any position inside concrete,
p
is the such
position.
The negative sign in the equation above arises because the diffu-
sion of chloride ions occurs in the opposite direction of the concen-
tration increasing of chlorides ions. In general,
D
is not constant,
but depends on many parameters as the time for which diffusion
has taken place, location in the concrete, composition of the con-
crete, among other factors. If the chloride diffusion coefficient is
constant, equation (1) is usually referred as Fick’s first diffusion
law. If this is not the case, the relation is usually referred as Fick’s
first general diffusion law.
There are some cases where this simple relation should not be
applied. In this regard, it is worth to mention the cases where
the diffusion process may be irreversible or has a history-de-
pendence. In such cases, Fick’s diffusion law is not valid and
the diffusion process is referred as anomalous. However, none
observation so far indicates that the chloride diffusion in con-
crete should be characterized as an anomalous diffusion. The
Fick’s second law can be derived considering the mass balance
principle. Then:
(2)
÷÷
ø
ö
çç
è
æ
¶
¶
-
¶
¶
=
¶
¶
p
CD
p t
C
In order to apply Fick’s second diffusion law, in this form, for con-
crete exposed to chloride during a long period of time, one ought to
know the variation of the chloride diffusion coefficient along time. If
only few observations exist in a specific case, it is possible to esti-
mate upper and lower boundary for the variation of
D
in time. Des-
pite this dependence, an especial case can be considered where
the chloride diffusion coefficient is independent of location, time