The aim of this study is to identify a PM strategy that will maintain the p_f of an examined bridge element under

the actions of different PM measures, with the minimum possible cost (Equation (2)) lower than a target lifetime p_f (Equation (3)). Within the context of this study, the p_f represents the probability of corrosion initiation on the top level of the steel rebar.

where is the total cost of PM measures; is the cost of PM measure carried out in year; is

the type of PM measure chosen for year; p_f is the probability of failure for PM measure carried out in year; p_f (target) is the maximum acceptable value of p_f; is the expected service life of the bridge element (set here equal to 120 years); t is the bridge element age (years) and is the present age of the bridge element (set here equal to 0).

The key elements of the developed GA based methodology are outlined in this section (Figure 1) while a more detailed description can be found in [2] . The interpretation of Figure 1 shows that initially GA produces a popu- lation of chromosomes; all are representing a possible solution to a given problem (Step 1). During each successive generation every member is tested according to a fitness function (Step 2). Highly “fit” chromosomes are ranked at the top of the pool and can be selected to mate with other highly fit members (form the parents) to produce new chromosomes (offsprings) [10] (Step 3). The offsprings contain characteristics from both of the parent chromosomes. Furthermore, in a few randomly selected individuals some arbitrary changes are made in their characteristics through mutation to add some diversity to the population (Step 4). However, it does provide ac- countable improvement to perform mutation to already good solutions. For this reason elitism selection is often embedded [6] [8] to enforce the preservation of the best individual chromosomes of the current generation to the next. With elitism, a number of good solutions are designated as elite solutions and are destined to propagate unchanged. At this point a new population is formed with chromosomes closer to the optimum solution (Step 5). Therefore, although GA is randomized, it efficiently incorporates information from previous generations to create new search points and explores the most promising areas.

The entire process is repeated for a number of generations until the chromosomes will evolve and ultimately converge to an optimum solution. The convergence in this study is reached when in the parent population the chromosomes are the same. Hence, in this case even after crossover, there is no change in the genes and therefore one optimum solution is proposed (Step 6 in GA Part A, Figure 1). The entire “GA Part A” process shown in Figure 1 is repeated to obtain a number of optimum solutions and to select the solution that fits best the objec- tive/fitness functions.

Unfortunately, the “GA Part A” may fail because of a convergence to an unacceptable local optimum. To overcome this limitation, an improved two-stage GA methodology (Part A + Part B), outlined in Figure 1, is proposed. The purpose of the “GA Part B” is twofold: firstly, to aid the finding of the optimum solution and secondly, to ascertain whether there are flaws in the selection of key parameters of the GA process in order to revise them and achieve convergence more efficiently.

As Figure 1 illustrates, “GA Part A” is executed and the collected optimum solutions form the genome which is treated as the initial population in “GA Part B” (Step 1 in GA Part B). The simulations in “GA Part B” stop when all the chromosomes of the genome are the same, thus one optimum solution (optimum PM strategy) is proposed. The termination of the simulations is based on the logic that “optimum” parents (optimum solutions) cannot produce “more optimum” offspring (optimum solutions). For the implementation of the proposed methodology a computer program was developed using Visual Basic environment.

A Pilot case is examined first in which all the proactive and reactive measures shown in Table 2 are included as options. In this application it is assumed that the discount rate is 3% [18] , the target p_f is 0.1 [19] -[21] , the initial p_f is 0 (representing either a new element or one where chloride ingress has not commenced) and the service life of the element is 120 years.

A WLC of ?00.7/m² is obtained for the optimum strategy. Table 5 presents the different actions selected at 5-year intervals as established by the GA methodology. A combination of surface treatments and “do nothing” option are selected initially until p_f reaches the target value at which point a reactive measure is used. Based on

^*The maintenance cost includes maintenance of the anodes, application of sealer and inspection costs for every 5 years.

Decoded Actions: 1: Do nothing; 2: Silane; 3: Sealer; 4: P-m Coating; 6: ECE; -: P-m coating still effective due to its service life (10 years).

the input data for this case ECE is identified as the best reactive PM measure to use as opposed to CP or CR when critical concentration of chloride is predicted on the rebar. This makes sense since the use of ECE has a dramatic impact on the p_f profile at little cost compared to the other reactive options. The cycle of proactive PM followed by a reactive PM is repeated throughout the lifetime. The combination of proactive measures is not the same throughout the lifetime of the element since the costs are discounted and cheaper solutions are sought at the beginning of the strategy (0 - 20 years)

The element performance when this recommended optimum PM plan is applied can be seen in Figure 2 which shows the p_f profile with time. At no time frame the p_f exceeds the target p_f failure.

The use of appropriate discount rate is important as this affects the sequence of application of PM and therefore the WLC profoundly (Equation (4)). The discount rates are quite variable between different agencies and are normally within the range of 2% - 8% [22] . Here, in addition to the Pilot case in which the discount rate is taken as 3%, other cases with 0%, 3.5%, 6% and 8% are investigated.

Cases 2-5 examine the sensitivity of the optimum strategy outcome to changes in the discount rate. As shown in Figure 3 the highest WLC is obtained when the discount rate is 0% while the lowest cost is obtained when the rate is 8%. It is also clear that the sensitivity of the WLC results to the discount rate is significantly higher within the lower range of values.

The optimum maintenance plans proposed for different discount rates can be seen in Table 6, which shows that when the discount rate is 0% P-m coating is utilized for most PM intervals. However, Table 5 and Table 6 show that when the discount rate is higher than 0% (e.g. 3%, 3.5%, 6%, 8%) the optimization method selects less costly actions for the first 20 years, such as “do nothing”, followed by the use of ECE and surface treatment at different PM intervals for the remaining life of the element. Where the higher discount rates are used (6% and 8%) there is a bigger reduction in costs with time (through discounting) compared to the lower discount rates (Equation (4)). This result in the increased use of ECE which is employed almost twice as many times compared to the lower discount rate cases (0%, 3%, 3.5%). The same optimum strategy is obtained for the two higher discount rates (6%, 8%) as shown in Table 6 and Figure 5. While there is some difference in the discounting between these two cases this is not sufficiently high to change the optimum strategy solution. This is due to the number of available PM options investigated. A wider range of PM measures may result in different PM strategies between the higher discount rates. Figure 4 and Figure 5 illustrate the p_f profile under the action of the proposed PM plans. At no point in time does the p_f exceed the target p_f.

Decoded Actions: 1: Do nothing; 2: Silane; 3: Sealer; 4: P-m Coating; 6: ECE; -: P-m coating still effective due to its service life (10 years).

The target probability of failure is an important parameter in any probabilistic approach, within the context of in- tegrity management of structures. Acceptable levels of probability of failure are not easy to define and depend on complex issues such as acceptable levels of risk and social impact. Some guidelines are provided in this area within Eurocode BS EN 1990 (2002) Annex B [20] .

While a value of 0.1 was assumed for the Pilot study based on recommendations from codes of practice BS EN 1990 [20] and JCSS [21] a range of p_f values are examined within the sensitivity study to quantify the effect of this parameter. Case studies 6-13 adopt values within ±5%, ±10%, ±15% and ±20% of the assumed target p_f used in the pilot case. From the results of these cases it is clear that the target p_f influences highly the outcome of the WLC. Figure 6 shows that when the target p_f increases the cost decreases. This is to be expected since increased reliability (lower p_f) would require more frequent use of PM measures and hence higher overall cost.

It is interesting to note that while on the whole the WLC decreases with increased target p_f the reduction in WLC is not continuous. This is due to the availability of the options to fit a strategy to ensure that the p_f remains below a target p_f. As more options are available the difference in WLC between the cases is expected to be more uniform. While the WLC is sensitive to changes in p_f the percentage change in WLC is lower than the corres- ponding change in p_f. A ±20% change in p_f results in only +8.1% and −8.3% change in WLC.

The optimum plans obtained from the GA methodology for a selection of case studies which their target p_f value provides minimum, maximum and average WLC are given in Table 7.

The p_f profiles of these cases are shown in Figure 7. Despite some differences in the choice of PM actions, a similar general pattern can be identified from these figures. The element is left to deteriorate with no action tak- en, approximately for the first 20 years. Then some surface treatments are selected for a period of 20 - 30 years

Decoded Actions: 1: Do nothing; 2: Silane; 3: Sealer; 4: P-m Coating; 6: ECE; -: P-m coating still effective due to its service life (10 years).

followed by the use of ECE. The latter pattern is repeated throughout the lifetime of the bridge element.

Another important parameter is the initial p_f of the examined element. This parameter represents the probability of corrosion initiation on the reinforcement of the element at the time examined. For example, if the initial p_f is zero then it means that the element is either new or chloride propagation has not commenced. The sequence of application of PM will be influenced by the initial p_f value.

Therefore the following case studies are examining the effect of initial p_f on the WLC. As expected with higher initial p_f the WLC also increases (Figure 8). This is due to the fact that more money is needed to restore the initial condition of the bridge element to safe levels (concerning its probability of failure) at an early stage. It can be seen that there is a significant increase in cost (85%) with an increase of the p_f from 0% to 10%. This relatively large increase in the WLC is due the cost of the PM actions required to restore the initial condition of the element. As these actions are applied at the beginning of its life they are not discounted, hence they have a marked effect on the overall cost of the strategy.

As shown in Table 8 the first use of a reactive PM measure within each strategy is brought forward as the initial p_f increases. This is then followed by various cycles which combine either “do nothing” with sealer and ECE or “do nothing” with P-m coating and ECE. The use of P-m coating generally seems to increase in later years due to the discounting.

The optimum plans for different initial p_f are given in Table 8.

The service life of an element is defined as the period of time during which it meets the standards and/or the target specified [23] . In the Pilot case the time period is set to 120 years to comply with the service life span of

Decoded Actions: 1: Do nothing; 2: Silane; 3: Sealer; 4: P-m Coating; 6: ECE; -: P-m coating still effective due to its service life (10 years).

bridges designed before Eurocode requirements [24] . Additional analyses are performed to examine the effect of the service life on the optimum strategy selection. The service life values examined in Cases 17-18 are 60 years and 100 years [19] .

Figure 9 illustrates the corresponding p_f profiles of these cases. The outcome of the analyses, which are the optimum strategies, the WLC and the corresponding p_f profile are presented in Table 9 and Figure 10. The PM strategy is the same for all three cases for the first 30 years and combines the use of surface treatments and “do nothing” until the target p_f is reached when the reactive measure ECE is applied. In the subsequent cycles the P-m coating which is the most effective and expensive proactive measure becomes more beneficial and is se- lected more often. This is justified due to the combination of the longer life considered and the effect of the dis- counted cost. For the cases considering 60 and 100 years service life, two applications of ECE are required while in the case of 120 years, four applications are scheduled. When considering the WLCs for the three cases (Figure 11) it can be seen that as expected the WLC increases as the service life period increases although the rate of increase reduces with larger service life periods. The average cost per year reduces approximately linearly with increasing service life. The latter is partly due to the effect of discounting and also due to the fact that a longer service life plan provides more flexibility with the combination of options which can result in reduced overall cost.

So far the optimization methodology and its sensitivity were demonstrated with respect to a bridge beam. The methodology, however, is generic and can be applied to different bridge elements or even different deteriorating structures. In this example the methodology is demonstrated for a bridge deck instead of a beam. In the case of bridge decks the use of waterproofing system (WS) is mandatory in the UK since 1965 [13] therefore the WS is applied at year zero (0). Other actions that can be applied to bridge decks are the cathodic protection (CP) and concrete replacement (CR) hence these are also included as PM options in the analysis. Surface treatments with inadequate depth of penetration are not recommended since they can quickly wear when exposed to traffic abrasion [11] therefore they are excluded from this example.

Decoded Actions: 1: Do nothing; 2: Silane; 3: Sealer; 4: P-m Coating; 6: ECE; -: P-m coating still effective due to its service life (10 years).

The WS is assumed to have a service life of 25 years [13] and the cost is taken as ?1/m² [12] . Following the initial application of WS which is effective for 25 years the methodology is free to select any of the PM options included to maintain the p_f profile within acceptable limits and minimize the WLC.

In this case the use of WS is selected throughout the 120 years instead of CP or CR (Table 10). The WS is applied every 25 years (service life of WS) with the exception of a 5-year gap (at the 25th year) during which the deck can be left to deteriorate with no action taken (Figure 12). The repeated application of WS is sufficient to ensure that p_f remains below the critical p_f of 0.1 for the entire 120 years period without the need for any of the reactive PM measures CP and CR. The WLC is equal to ?5.5/m² which is significantly less than the beam element WLC of ?00.7/m². This suggests that the WS applied on bridge decks is a very effective PM measure.

Decoded Actions: 1: Do nothing; 8: waterproofing system; -: waterproofing system still effective during its service life (25 years).