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1. Introduction
Concrete structures, especially reinforced concrete structures, are
one of most used types of structures around world. When it is lo-
cated in non aggressive environments, these types of structures
respect, in general, its structural life predicted, which is given by
the durability criteria. Regarding durability, it can be measured as
a period of time in which the structure maintains, at least, minimal
functional conditions, resistance and external aspects required in
design. However, the structural durability can be strongly affec-
ted by degradation processes of environmental and/or functional
origins [1,2]. Among these processes, it is worth to mention the
penetration of chlorides, carbonation, fatigue and creep. Therefo-
re, reinforced concrete design with reliability analysis for structural
durability as well as prediction of repairs and maintenance has be-
coming increasingly accepted [3-5].
One of the most important degradation processes in reinforced
concrete structures is the reinforcement corrosion phenomenon.
It is directly responsible for the durability and failure of concrete
structures. With respect to the external agents, chloride diffu-
sion is identified as one of the major factors that cause reinfor-
cement corrosion. When corrosion begins, the durability of the
structure is affected by several phenomena like reduction of the
reinforcement’s cross section, bursting of concrete and adherence
loss between steel and concrete [6,7].
In general, the corrosion process can be divided into two stages:
the initiation and the propagation period. During the initiation pe-
riod, chloride ions diffuse through concrete toward the reinforce-
ments. The chloride concentration reaches a threshold value, whi-
ch triggers the corrosion of steel. The propagation period of the
corrosion process is defined as the time from the start of corrosion
to a critical steel loss limitation being reached. Compared with the
first stage, the propagation period is relatively short. Therefore, the
chloride diffusion process is frequently used to indicate the durabi-
lity and service life of concrete structures [8].
Regarding the prediction models for those corrosion times, seve-
ral models have been proposed in the literature to deal properly
with the corrosion phenomenon in reinforced concrete structures.
Most of them propose deterministic approaches in order to mo-
delling this structural problem. However, due the large number of
inherent uncertainties, as the penetration of chlorides in concrete,
these problems can only be accurately analyzed considering pro-
babilistic approaches. Among these models and formulations, it is
worth to mention [9-13], where the corrosion process was studied
considering several conditions and variables. Based on these pro-
babilistic models, life assessment analysis in reinforced concrete
structures exposed to chloride penetration can be accurately per-
formed [14-16]. For example, considering these results, aspects of
maintenance and inspection plans, an optimal structural concrete
cover can be proposed taking into account the randomness inhe-
rent of the variables in each model regarding the involved cost on
each procedure.
In order to achieve the durability requirements in reinforced con-
crete design, a mechanical model to evaluate the chloride concen-
tration at every position inside concrete is needed. Moreover, the
chloride penetration mechanism is controlled by complexes inte-
ractions among physical and chemical mechanisms, which are a
large source of uncertainties. However, this phenomenon is often
simplified, without significant loss, by problems controlled only by
diffusion process.
Fick’s diffusion laws have all the requirements to modelling the
problem and it is based in some hypotheses such as: concrete is
an homogeneous material in space; a direct consequence of this
assumption is a constant value for the coefficient of diffusion along
time; the model supposes that concentration of ions chloride in the
environment is constant and it admits the concrete in saturated
condition. Assuming all these behaviours, probabilistic analyses of
chloride ingress in concrete structures can be performed by cou-
pling Fick’s diffusion laws with reliability algorithms.
In this paper, a coupled mechanical and reliability model is develo-
ped in order to allow probabilistic analyses of reinforced concrete
structures subjected to ions chloride ingress. These analyses aim
at quantifying the probability of corrosion start in reinforced concre-
te structure based on a reliability approach. The mechanical model
is based on the Fick’s second diffusion law, which is capable to si-
mulate the chloride penetration process in porous materials. Then,
the chloride concentration at a given concrete cover depth and the
respective time can be evaluated. In order to determine the pro-
bability of failure, two reliability algorithms were considered: direct
coupling of mechanical model with FORM and Monte Carlo simu-
lation. Both algorithms determine the probability of failure conside-
ring the failure scenarios achieved by the mechanical model based
on Fick’s diffusion law. The probabilistic chloride penetration is
analyzed for some particular cases and the set of values achieved
by the proposed models are discussed. A simplified procedure to
obtain the optimal concrete covers for periodic inspections is also
illustrated by the building of some abacuses.
2. The mechanical model – Fick’s
diffusion law
Corrosion of reinforcements induced by chlorides can take place
in the presence of oxygen and moisture when the chloride build-
-up within the structures exceeds a threshold value. Even for a
carefully constructed concrete, with negligible or practically non
chloride inherited at the construction stage, the gradual build-up
of the required level of chloride content to initiate corrosion of rein-
forcement takes place slowly through ingress of chloride ions from
external sources.
The transport phenomenon associated with the movement of chlo-
ride ions along structures exposed to aggressive environments is
attributed mostly to diffusion of chloride ions into concrete porous
under a concentration gradient. Chloride diffusion coefficient of
concrete, which depends upon the pore structure of the concre-
te, characterizes this flow under a given concentration of chloride
exposure and it is considered as a characteristic property of a har-
dened concrete.
In order to simulate the chloride ingress and its transport into the
concrete porous, Fick’s second law of diffusion [17] has been wide-
ly considered as an acceptable model. The Fick’s laws for diffusion
are applicable for material homogeneous, isotropic and inert [18].
Moreover, the mechanical properties of diffusion process are assu-
med to be the same along all directions and kept constants along
time. Considering concrete, these hypotheses are not completely
satisfied because concrete is well known as heterogeneous, ani-
sotropic and chemically reactive (continued hydration and micro-
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IBRACON Structures and Materials Journal • 2012 • vol. 5 • nº 4
C. G. NOGUEIRA | E. D. LEONEL | H. B. CODA