# Reliability Analysis for Selected Bridges

The reliability indices for individual girders were calculated using the procedure developed for calibration of the AASHTO LRFD Code (Nowak 1999). For each combination of span length and girder spacing, three design cases were considered: (1) with individual girders designed according to the code and thus providing reliability indices close to the target pT = 3.5, (2) with under­designed girders, with the girder reliability indices close to pT = 2.0, and (3) with over-designed girders and girder reliability indices close to pT = 4.5. The ratio of the actual girder resistance and the minimum required resistance for the target P, is shown in Figures 6 and 7 for Pt = 3.5, for moments and shear forces, respectively. The ratios for shear are very large because the design is governed by the moment capacity, so the girders are over-designed with regard to shear. Therefore, the shear capacity was not considered in the system reliability analysis.

In the system reliability analysis, the two main random variables are bridge system resistance and bridge live load. Both are represented in form of the gross vehicle weight (GVW) of two identical trucks placed side-by-side, each truck with axle configuration of the design truck
(AASHTO 2004). The system reliability index was calculated for various truck positions within the roadway width, and the final system reliability index was determined as an expected value using the weighting factors based on the curb distance distributions (Figure 3).

Figures 8a to 8c show the relationship between the girder reliability index and the corresponding system reliability index for three different girder spacings, and span length of 18, 30, and 42 m, respectively.

Figure 8 Girder Reliability Index versus System Reliability Index for Different Girder Spacings and
Span Length of (a) 18 m, (b) 30 m, and (c) 42 m

The results indicate that the ratio of system reliability to girder reliability decreases with increasing girder reliability. This is because for over-designed bridges (Pt = 4.5), the ratio of system capacity to girder capacity is smaller than that for under-designed bridges (Pt = 2.0). Moreover, the system reliability indices increase with number of girders. This is due to the increased redundancy of the system. It is observed that not only does the ratio of system reliability to girder reliability decreases with increasing girder reliability, but it also decreases with the increase of span length or girder spacing.

Figures 9a to 9c show the girder and system reliability indices as a function of span length, for girder spacing of 3.0 m, for the three considered cases of the target girder reliability index, pT, of 2.0, 3.5, and 4.5. Also shown are wide flange sizes for the selected main girders.

The next step was to investigate the effect of correlation between the resistances of individual girders represented by the yield stress, Fy, of structural steel. Four cases are considered: (1) no correlation, with coefficient of correlation p = 0, (2) full correlation with p=l,

(3) and (4) partial correlation, with different values of p depending on the number of girders. The correlation between girder strengths resulted in a reduction of the number of different random variables considered in system reliability analysis.

For a 6-girder bridge with girder spacing of 1.8 m, it was assumed that a partial correlation of p = 0.33 can be represented by the case of two adjacent girders having identical strength, and p = 0.66 is represented by four adjacent girders being identical. In addition, also considered were cases with randomly distributed correlated girders (two or four out of six). A similar approach was used in case of 5-girder bridges, with girder spacing of 2.4 m. However, as this time bridges had only 5 girders, so the global correlation between the strength of girders was p = 0.40 and p = 0.60 for cases when two and four girders were correlated, respectively. For 4- girder bridges, with girder spacing of 3.0 m, it was assumed that two or three girders were
correlated resulting in the global correlation between the strength of girders of p = 0.50 and p = 0.75 for cases when two and three girders were correlated, respectively.

 Figure 9 Girder Reliability Indices and Corresponding System Reliability Indices for Different Spans and Girder Spacing of 2.4 m, Target Girder Reliability Index of (a) pT=2.0, (b) pT=3.5, and (c) pT=4.5

Examples of the effect of correlation on the system reliability indices for composite steel girder bridges are shown in Figure 10. The target reliability index for the girders is pT = 3.5.

 Figure 10 System Reliability Index for Different Degree of Correlation between Girder Resistance and Different Span Length, Girder Spacing of (a) 1.8 m, (b) 2.4 m, and (c) 3.0 m