13th International Conference on Fracture June 16–21, 2013, Beijing, China -9- Table 5 Variances for fatigue crack growth with 1.6 η= 1RΔ 2RΔ 3RΔ 4RΔ 5RΔ 6RΔ 7RΔ 100.2 86.6 65.7 34.2 13.2 84.2 189.4 Depicted in Figs. 8 and 9 are the plots of non-monotonic variations of the data that can be found in Tables 2-5 for crack length. The minima are shown for both of the curves. The curves display the feature of decreasing and then increasing. Distinctive minimum are found to be around 6 years and 4.5 years, respectively. According to the principle of least variance, compare Fig.6 for the reliable portions of the fatigue crack growth. With 1.4 η= , the time range before fast fracture is approximately between 5 and 7 years. Increase of tightening coefficients from 1.4 to 1.6 reduces the reliable portion to be around between 4 and 5 years. It is noticeable that variance for the range of fast fracture deviates tremendously from the average. This means that it is least reliable. The initial stage of crack initiation also possesses substantial deviation from the average. It is, however, relatively difficult for detection at such an early stage. The region of high reliability for the fatigue crack growth can be identified with the span of 5-7 years and 4-5 years, respectively. It should be pointed out that time rates and time intervals also affect the variances in a modest manner. Nevertheless, similar conclusions can be made based on the theory of least variance. Fig. 8 Variance for crack depth with 1.4 η= Fig. 9 Variance for crack depth with 1.6 η= 6. Concluding remarks Long-span bridge cables usually consist of thousands of high strength steel wires that are measured in millimeters. Wire breakages cause the eventual failure of cables. Fatigue degrades the life of wires and cables as well as material aging and stress corrosion. It has become one of the biggest concerns in the maintenances of bridges. Clearly, uncertainty of cable force variation in the operating period can greatly affect the fatigue crack growth behavior. One problem emerges with regard to the inspection of cables. That is, each of the cables is unique on the aspects of loads and configuration. During the operating time, some of the cables may need replacement at a time when other cables of the same bridge may still possess ample remaining life. This poses great challenge for the assessment of reliability of wires and cables. The theory of least variance offers one of the possible solutions by proposing significant variables including length, velocity, mass and energy. Minimizing the oscillation of 1R, 2R …, nR has been assumed to be a measure of system reliability. The principle of least variance identifies the 0 1 2 3 4 5 6 7 0 20 40 60 80 100 120 140 160 180 200 Traffic load tightening cable no.3 Variance ΔR Time t (year) η=1.6 0 2 4 6 8 10 0 40 80 120 160 200 Variance ΔR Time t (year) η=1.4 Traffic load tightening cable no.3
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