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IBRACON Structures and Materials Journal • 2012 • vol. 5 • nº 4
Reliability algorithms applied to reinforced concrete durability assessment
The solution of this problem converges to the failure point nearest
to the space origin, known as design point or the most probable fai-
lure point
*
U
. In standard space, the distance between this point
and the origin is the reliability index, as shown in figure 2. The re-
liability index
b
can be achieved by applying any optimization algo-
rithm. A particular algorithm, which is efficient in this case, to solve
reliability problems, is the HLRF algorithm [24]. This optimization
algorithm can be coupled directly to the mechanical model. As the
time for corrosion initiation is known point-by-point, the resistance
R
t
is known. Consequently, the limit state function is determined
point-by-point. Then, the gradient of the limit state function can be
determined using any numerical procedure. In this paper, it was
used the forward finite differences technique for this proposal.
Some difficulties may arise from equation (8), particularly its gradients
evaluation, due the presence of the error complementary function. A
natural barrier that can be remarked in this type of approach is the
numerical error due to finite difference procedure, which may affect
the convergence of the coupled procedure, as well as the precision of
the solution, especially for nonlinear transient phenomena. However,
for all problems studied in this paper, numerical problems related to
finite differences method were not observed. Moreover, it was verified
that the direct coupling procedure gives accurate results and stable
convergence rate with a reasonable number of mechanical analyses.
3.3 Monte Carlo simulation
Monte Carlo’s method is a numerical simulation approach wide-
ly used in reliability problems [25]. In this method, a sampling of
random variables is used to construct a set of values aiming to
describe the failure and safe spaces and calculate equation (5).
The sampling is constructed based on the statistical distribution
assigned for each random variable in the problem. As this method
deals with the simulation of the limit state function, as bigger be
the sampling adopted more accurate will be the description of the
space and more accurate will be the probability of failure achieved.
The kernel of this method consists on the construction of a sampling
for each random variable involved in the problem. Then, the domain
of safety and failure points are prospected by simulating equation
(7), as described in figure 3. The probability of failure is calculated,
for Monte Carlo’s simulation, using the following expression:
(10)
( )
( ) ( )
( ) [ ]
i
G
i
i
X i
G
i
i
X
f
xIE dx x f xI
dx x f
P
=
=
=
ò
ò
£
£
0
0
The function
( )
i
xI
is a discrete operator to compute failures and
it can be written as:
(11)
( )
î
í
ì
>®
£®
=
0
0
0
1
G
G
xI
i
Figure 2 – Reliability index and
design point definitions
limit state function
G(U)>0
G(U)<0
G(U)=0
U
2
U
1
P*
b
normal standard space
0
failure domain
safety domain
Figure 3 – Monte Carlo's sampling
for two random variables
safety domain
failure domain
G(U)>0
G(U)<0
U
2
U
1
U*
Figure 4 – Corrosion process of the
reinforcement cross-section area
start of the corrosion process
Time
A
s
A
s
0
t
0
t
d
reinforcement initial
cross-section area