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AS = 93.54 cm2 v = 0.0089 cm/year Exposure time = 25 years

FIGURE 36.23 Optimized concrete T-girder designed for a 25-year service life and a minimum acceptable reliability index of 3.0 while sustaining a deterioration rate of 0.0089 cm/year (0.0035 in./year) on corroding reinforcing steel (beam dimensions are in centimeters). (Frangopol et al. 1997a. Reprinted with permission from the American Society of Civil Engineers.)

0.0045 in./year) were considered for the steel reinforcement once corrosion was initiated. The Reliability Based Structural Optimization (RBSO) included 21 design variables such as flange width, number of stirrups, reinforcement ratio, and height of the web. In Frangopol et al. (1997a), the objective of the optimization process was to minimize the total cost of steel, concrete, and failure. Figure 36.23 shows the optimized concrete T-girder that sustains a corrosion deterioration rate of 0.0089 cm/year (0.0035 in./year). The minimum area of steel is 93.54 cm2 (14.5 in.2) and the dimensions of the beam are designed to ensure a minimum acceptable reliability of b = 3.0 throughout its expected 25-year service life. The analysis was completed for different deterioration rates, service lives, and acceptable reliabilities. Figure 36.24 shows that the optimum reliability index is b* = 2.5 for a T-girder with an expected 75-year life and a deterioration rate of 0.0064 cm/year (0.0025 in./year) based on the minimum total life cycle cost. The cost of the steel was assumed to be 50 times as great as the cost of the concrete, and the failure cost was 10,000 times the cost of the concrete.

Next, Frangopol et al. (1997b) included preventive maintenance costs, which increased linearly over time. Inspection costs were included, and four inspection methods with varying costs and capabilities were available. The cost of repair was a function of the degree of damage, and an applied aging factor dictated the effect of the postrepair reliability. With multiple lifetime inspections, an event tree accounted for the repair/no repair decisions made after the inspection. The total expected life cycle cost was based on the consequences and probability associated with each branch on the event tree (Frangopol et al. 1997b). Figure 36.25 shows the reliability of a concrete T-girder over a 75-year life using a life cycle strategy that applies four inspections (m = 4) and three repairs (n = 3). The corrosion rate is

FIGURE 36.24 Optimum design of a concrete T-girder based on minimum total life cycle cost. The corrosion rate is 0.0064 cm/year (0.0025 in./year) due to corrosion of the reinforcement, and the expected service life is 75 years. (Frangopol et al. 1997a. Reprinted with permission from the American Society of Civil Engineers.)

FIGURE 36.24 Optimum design of a concrete T-girder based on minimum total life cycle cost. The corrosion rate is 0.0064 cm/year (0.0025 in./year) due to corrosion of the reinforcement, and the expected service life is 75 years. (Frangopol et al. 1997a. Reprinted with permission from the American Society of Civil Engineers.)

v = 0.0064 cm/year (0.0025 in./year), and the inspection technique has a 50% probability of detection when there is 10% damage (z0.5 = 0.10). The number of inspections, inspection technique, and number of repairs were varied until an optimal life cycle cost solution was obtained. The study included both uniform and nonuniform inspection intervals. While a lot of the information was hypothetical, the study was one of the first to incorporate so many key aspects of life cycle analysis and design.

### Case 36.2

Estes and Frangopol (1999) treated an existing Colorado highway bridge as a structural system and developed an optimal repair strategy based on minimum life cycle cost. The bridge in Figure 36.26 consisted of three simple spans where each span contained nine girders. The girders were classified as interior, exterior, and interior-exterior based on their load and exposure condition. Using 13 failure modes and 24 random variables, the bridge was modeled as a series-parallel system. After eliminating the nonrelevant failure modes and taking advantage of symmetry and assumed failure mode correlation, the simplified series-parallel model shown in Figure 36.26 was developed assuming that any three adjacent girders must fail for the bridge superstructure to fail. Both shear and moment failure modes were considered for the girders. The girders were corroding over time, and both the concrete slab and the pier cap were deteriorating due to exposure to chlorides from road salts. Meanwhile the live load was increasing over time.

Time, years

FIGURE 36.25 The reliability of a concrete T-girder over its 75-year life under a life cycle plan that implements four inspections (m = 4) and three repairs (n = 3). The corrosion rate of the reinforcement is 0.0064 cm/year (0.0025 in./year), and the capability of the inspection technique is a 50% probability of detection when there is 10% damage. (Frangopol et al. 1997b. Reprinted with permission from the American Society of Civil Engineers.)

Time, years

FIGURE 36.25 The reliability of a concrete T-girder over its 75-year life under a life cycle plan that implements four inspections (m = 4) and three repairs (n = 3). The corrosion rate of the reinforcement is 0.0064 cm/year (0.0025 in./year), and the capability of the inspection technique is a 50% probability of detection when there is 10% damage. (Frangopol et al. 1997b. Reprinted with permission from the American Society of Civil Engineers.)

A repair was required any time the reliability of the bridge system fell below bsystem = 2.0. Five repair options that included replacing the deck, replacing exterior girders, replacing exterior girders and deck, replacing superstructure, and replacing the entire bridge were considered along with their associated costs using a discount rate of 2%. Figure 36.27 shows the lifetime reliability of the relevant individual failure modes and the bridge system when the repair strategy is to continually replace the slab. The slab gets replaced at year 50 and year 94 when the system reliability falls below acceptable levels. At year 106, replacement of the slab is no longer sufficient to raise the system reliability above the minimum requirement, and some other repair option is necessary. The optimal solution was obtained by applying various combinations of the discrete repair alternatives.

Figure 36.27 indicates that some failure modes (shear in the exterior and interior-exterior girders) were allowed to fall below the minimum threshold b = 2.0 due to the parallel nature of the structural model. Due to different deterioration rates of various bridge components and effects of repair on reliability of these components, the most relevant failure mode in the beginning of the life of the structure was not necessarily the most critical mode later on. Different repair strategies were developed for different series-parallel models, different deterioration rates, and different correlations among girder resistances.

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