Page 33 - vol5_n4

446 IBRACON Structures and Materials Journal • 2012 • vol. 5 • nº 4
Reliability algorithms applied to reinforced concrete durability assessment
5. Applications
In order to illustrate the applicability of the proposed model, pro-
babilistic analyses were performed considering different scenarios
and material properties aiming to evaluate the influence of the en-
vironment aggressiveness, water/cement ratio and cover depth on
the probability of structural failure due chloride penetration. Four
applications are considered and in each of them some specific as-
pect of the problem are explored.
5.1 Example 1
In this example, the probabilistic model is applied at modelling
the uncertainties related to chloride ingress in reinforced concre-
te structures. The results were obtained using direct method and
Monte Carlo’s simulation approaches. Considering direct method,
the tolerance for convergence was verified in terms of reliability
index and design point coordinates change, which ones have to
be smaller than 10-4. It is worth to mention that considering this
approach, only 65 limit state function runs were required in order
to achieve the convergence. Moreover, this number of mechanical
model runs is the highest observed among all scenarios conside-
red. It confirms the good performance of direct method approach,
which is capable to achieve the convergence with low computa-
tional work even in reliability problems involving non-Gaussian
random variables and non-linear mechanical model. The analyses
performed using Monte Carlo’s simulation considered a sampling
of 105 values for each random variable. It leads to 105 simulations
of the limit state function. Table 1 presents the statistical parame-
ters adopted for all random variables considered in this application,
which are based on the work of [12,16].
According [26], categories of aggressiveness (C.A.) II (moderate)
and III (high) represent, respectively, urban regions and coast/in-
dustrial areas. For each one of these C.A., there is a recommen-
ded concrete cover depth, p, which was adopted for the probabilis-
tic analyses of each category.
The evolutions of the probability of corrosion initiation along the
time, for both categories of aggressiveness considered in this ap-
plication, are presented in figure 5 and figure 6. The results illustra-
te a considerable growth of the probability of corrosion initiation as
the time increases. Moreover, it was also verified a strong depen-
dency between the probability of corrosion initiation and the w/c
ratio. This dependency was expected, because as higher be the
w/c ratio, higher will be the concrete permeability and, consequen-
tly, easier is the chloride ingress.
Considering the horizon of time analyzed, for concretes with w/c
ratio values higher than 0.50, after 15 years of structural service
life, the probability of failure tends to increase slower than in the
first 15 years. Therefore, the derivative of the probability of corro-
sion initiation after 15 years is smaller than in the first 15 years.
This behaviour reflects the chloride concentration evolution along
time, which tends to saturation after 15 years. It is important to
stress that this behaviour was observed in both C.A. analyzed.
5.2 Example 2
This application aims to study the influence of cover depth values
on the probability of corrosion initiation considering two different
scenarios of environment aggressiveness. The analyses were per-
formed considering concretes composed by w/c ratio varying from
0.40 to 0.70. The analyses involved cover depth in the range 10-
60mm. For each value of w/c, the mean value of cover depth was
also varied but its coefficient of variation was kept constant. This
procedure was adopted because this study aims at describing the
dependency of the probability of failure with respect to w/c and
cover depth parameters. The time was assumed as deterministic
in this analysis. In order to consider all the combinations, 240 nu-
merical reliability analyses were carried out. The direct method and
Monte Carlo’s simulation were adopted for the evaluation of proba-
bility of failure and reliability indexes. Considering direct method,
the convergence is assumed when the difference, in two succes-
sive iterations, in terms of reliability index and design point coor-
dinates is smaller than 10-4. Considering Monte Carlo simulation,
a sampling of 105 values for each random variable was adopted.
The computational time consumed in this analysis is relatively lo-
wer. Considering direct method, the most expensive case required
65 limit state function calls, corresponding to less than 1 second of
computational work. The Monte Carlo analyses required 105 limit
state function evaluation, due the sampling range adopted. Howe-
ver, it took less than 3 seconds of computational time process. The
Figure 5 – Probability of failure versus time
for C.A.II
0,0
0,2
0,4
0,6
0,8
1,0
5 10 15 20 25
Time (years)
Probability of failure
w/c=0,4 w/c=0,5 w/c=0,6 w/c=0,7
Figure 6 – Probability of failure versus time
for C.A.III
0,0
0,2
0,4
0,6
0,8
1,0
5 10 15 20 25
Time (years)
Probability of failure
w/c=0,4 w/c=0,5 w/c=0,6 w/c=0,7