PSI - Issue 11

Pablo Benítez et al. / Procedia Structural Integrity 11 (2018) 60–67 Benítez et al./ Structural Integrity Procedia 00 (2018) 000 – 000

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inspection technique and the net discount rate that allows obtaining the Net Present Value of total cost during the service life of the structure. Thus, the individual cost of an inspection technique can be expressed as (Frangopol, Lin, and Estes 1997):   20 min 1 insp insp C     (5) (6) Where is a constant value that represents a fraction of the initial total cost ( = 1000 ) of a structure and is associated with the inspection method quality; and is the net discount rate of money equal to 5.5%. Although a small number of inspections supposes a lower expected total cost of the life cycle, decreasing the number of inspections can lead to a failure in the structure that was not be detected in time, which may increase the failure cost. This expected failure cost ( ) can be related with the probability of failure and the cost of this failure ( = 2000 ), where the former can be expressed as the complementary of the probability of damage detection before failure (Frangopol, Lin, and Estes 1997). Thereby, the failure cost is estimated as follow: * F f BF C P   (7) The cost associated with a failure must include all the expected costs related to this event which are not easily estimated. These failure costs can be the replacement of damaged elements, cost due to lost use, fatalities, and so on. Hence, a maintenance strategy must be oriented to reduce costs preserving an admissible structural reliability. A numerical example regarding the methodology applied for the inspection planning is presented in this section, where the expected total cost of this inspection planning is considered as the sum of the inspection cost and the failure cost. The context of this research is highly influenced by uncertainties. Thus, several parameters of the equations considered are governed by a stochastic process that requires large experimental investigation which is outside of the aims of this paper. In civil engineering, the widespread use of the normal distribution is very common to define real valued random variables whose continuous probability distribution is unknown. For an illustrative purpose, some of these variables have been assumed with an estimated value and are summarized in Table 3.         1 1 1 inspi n INSP insp t i C C r       4. Results and Discussion

Table 3. Values of random variables and deterministic parameters Variable Units Mean Value

0 .5 ; 0 .5 ; 0 .5 0 .5 ; 0 .5 ; 0 .5 ; 0

Standard Deviation

Distribution Lognormal Lognormal Lognormal Deterministic Deterministic Deterministic

cm

1.6

0.032 0.003

cm/year

0.0089

year

3.2

1.2

- - -

0.225 ; 0.175 ; 0.125 0.0225 ; 0.0175 ; 0.0125

- - -

5; 8; 9

Source: (Frangopol, Lin, and Estes 1997; Kim and Frangopol 2011) A failure condition does not necessarily have to involve a collapse of the structure. Regarding carbonation-induced corrosion, some authors assume the failure conditions when the carbonation front comes into contact with the rebar reinforcement. Hereinafter, a critical section loss has been assumed when the damage degree reached in main structural elements is greater than or equal to 25%. So, by setting the variable as the critical damage degree ( = 0.25) , the time to failure and its probability is founded. After using 100.000 iterations in the MCS, the PDF for the time to failure has shown a mean value of 14.76 years and a standard deviation equal to 9.48 for a 6.48% of probability. The PDF for the time to failure is quite sensitive to the parameters involved in the limit state function. Mainly, the corrosion rate is the most influential parameter in the analysis. A slight increase in this value could increase the probability of early structural failure, which means that structures placed in more aggressive environments must carefully schedule its inspections, as well as the cover thickness.