APPENDIX C: STRESS-RANGE HISTOGRAM DATA AND REGRESSION

Transcription

1 APPENDIX C: STRESS-RANGE HISTOGRAM DATA AND REGRESSION C-1

2 To determine the appropriate fatigue load for infinite life design, the fatigue-limit-state load with a probability of exceedance of 1:10,000 must be known. The 1:10,000 probability is based on established fatigue tests of steel structures and is the basis for the AASHTO fatigue loads for highway bridges. Statistical analysis was used along with the stress-range histogram data to find the stress-range with a return period of 1:10,000 for each HMLT in the study. The stress range associated with that return period is the maximum threshold for infinite life, or the fatigue-limitstate stress-range (S Rfls). Viewing plots of the stress-range histogram data, it is easy to see that the distribution is positive skewed or right-tailed. Five different types of positive skewed distributions were evaluated to establish which best represents the stress-range data. Those include: 1. Single parameter exponential distribution 2. Single parameter Rayleigh distribution 3. Single parameter gamma distribution 4. Two-parameter Weibull distribution 5. Two-parameter log-normal distribution Each of the distributions is commonly used in reliability engineering, time-to-failure estimates or to model variable-load spectrums. Each of the listed distributions were fit to sample data sets and evaluated using a chi-square goodness-of-fit test. The chi-square values for an expected distribution can be approximated by: 2 χ K j = 1 ( O ) 2 j E j E j where K is the number of bins in the sample distribution, O j and E j are the number of observed and expected occurrences respectively. In addition, E j must also be greater than five for the approximation to yield good results. The best possible fit is achieved when the chi-square value is minimized. Further discussion of the chi-square goodness-of-fit test can be found in any elementary statistics textbook. A spreadsheet was used to calculate the expected distributions fit to the sample data sets. Histogram bin values were set to the maximum value in the range to return the most conservative result. For example, the bin containing values from ksi was set at 2.00 ksi. Mean and variance values were calculated for both the observed and expected distributions, and the chisquare values were minimized using an iterative solve routine so that the mean value was not exceeded and the variance was within a reasonable tolerance. Expected values less than five were eliminated from the chi-square analysis, and in some cases, the lower bins (0.50 and 1.00 ksi) were eliminated as well. Of the five distributions considered, the log-normal appeared to be the most flexible and yielded the best results using the chi-square analysis. The exponential function also yielded promising results, but proved to be unreliable. It is noted that when the stress-range histograms are created, all bins are equally sized at 0.5 ksi, and all values less than 0.25 ksi are disregarded. C-2

3 Since the exponential distribution is heavily weighted toward the left end, a good fit was difficult to achieve based on the higher stress-range bins. A summary of the regression data is presented in Table C-1. Included in the table are, S Rfls values, ratio of observed to estimated effective stress-range values (S Reff), ratio or S Rfls to S Reff values, and the distribution parameters used for each regression. Note there is little variation between the lognormal distribution parameters. Following the summary table, histogram and regression data is presented for each HMLT in the long-term study. ID GAGE S Rfls (S Reff ) obs /(S Reff ) est S Rfls /S Reff µ σ (ksi) scale shape CA-A CH_ CA-X CH_ IAN-A (MT) CH_ IAN-X (MT) CH_ IAS-A CH_ IAS-X CH_ KS-A CH_ KS-X CH_ ND-A CH_ ND-X CH_ OKNE-A CH_ OKNE-X CH_ OKSW-A CH_ OKSW-X CH_ PA-A CH_ PA-X CH_ SD-A CH_ SD-X CH_ CJE-A (FR) CH_ CJE-X (FR) CH_ CJE-A (MT) CH_ CJE-X (MT) CH_ CJW-A (FR) CH_ CJW-X (FR) CH_ CJW-A (MT) CH_ CJW-X (MT) CH_ MEAN SD Table C-1: Summary of histogram regression data C-3

4 CH_3 CA CH_ ,709,572 5,693, ,941,887 6,946, ,425,372 2,354, ,278,048 2,245, , , , , , , , , ,163 87, ,795 43, ,153 34, ,163 14, ,827 14, ,194 5, ,154 6, ,437 1, ,703 2, , ,714 1, N = 9,214,499 9,180,161 N = 9,962,977 9,962,004 MEAN = MEAN = VAR = VAR = scale, µ = scale, µ = shape, σ = 0.59 shape, σ = :10,000 S R = :10,000 S R = 4.51 Table C-2: Histogram data for CA HMLT C-4

5 CH_9 IA-N (MT) CH_ ,076,392 1,088, ,384,229 1,445, , , , , , , , , ,506 59, ,150 60, ,266 22, ,999 19, ,246 9, ,273 7, ,720 3, ,901 2, ,775 1, ,117 1, , N = 1,851,318 1,861,185 N = 2,321,984 2,396,250 MEAN = MEAN = VAR = VAR = scale, µ = scale, µ = shape, σ = 0.59 shape, σ = :10,000 S R = :10,000 S R = 5.21 Table C-3: Histogram data for IA-N HMLT C-5

6 CH_2 (A) IA-S CH_1 (X) ,242,007 1,378, ,737,889 1,974, , , , , ,405 44, ,762 43, ,727 8, ,446 6, ,052 1, ,403 1, N = 1,715,148 1,712,706 N = 2,322,935 2,361,431 MEAN = MEAN = VAR = VAR = scale, µ = scale, µ = shape, σ = 0.50 shape, σ = :10,000 S R = :10,000 S R = 2.70 Table C-4: Histogram data for IA-S HMLT C-6

7 CH_2 (A) KS CH_6 (X) ,304,897 8,668, ,588,207 7,588, ,707,481 3,597, ,028,591 3,051, ,150,277 1,277, ,677 1,109, , , , , , , , , ,619 89, ,408 86, ,554 42, ,138 42, ,475 21, ,767 22, ,991 11, ,648 12, ,557 6, ,965 6, ,121 3, ,477 4, ,574 2, ,594 2, ,161 1, ,985 1, , , N = 15,125,738 14,407,558 N = 13,549,087 12,556,526 MEAN = MEAN = VAR = VAR = C-7

8 scale, µ = scale, µ = shape, σ = 0.65 shape, σ = :10,000 S R = :10,000 S R = 7.73 Table C-5: Histogram data for KS HMLT C-8

9 CH_1 (A) ND CH_5 (X) ,892,012 7,486, ,049,652 7,834, ,356,556 2,185, ,010,680 2,790, , , , , ,859 80, , , ,029 17, ,187 32, ,287 4, ,023 8, ,044 1, ,555 2, N = 9,592,205 10,182,658 N = 10,713,385 11,402,184 MEAN = MEAN = VAR = VAR = scale, µ = scale, µ = shape, σ = 0.48 shape, σ = :10,000 S R = :10,000 S R = 3.87 Table C-6: Histogram data for ND HMLT C-9

10 CH_3 (A) OK-NE CH_5 (X) ,588,444 3,772, ,180,501 4,470, ,540,060 1,498, ,703,737 1,644, , , , , , , ,156 95, ,793 31, ,710 25, ,365 10, ,693 7, ,282 3, ,638 2, ,650 1, N = 5,599,228 5,815,012 N = 6,331,421 6,635,610 MEAN = MEAN = VAR = VAR = scale, µ = scale, µ = shape, σ = 0.53 shape, σ = :10,000 S R = :10,000 S R = 4.19 Table C-7: Histogram data for OK-NE HMLT C-10

11 CH_8 (A) OK-SW CH_6 (X) ,453,571 6,822, ,172,707 11,686, ,728,795 2,650, ,681,966 4,475, , , ,041,721 1,166, , , , , ,288 50, ,529 94, ,075 15, ,070 30, ,541 5, ,749 10, ,368 1, ,628 4, , ,177 1, N = 9,979,764 10,395,012 N = 17,245,973 17,787,576 MEAN = MEAN = VAR = VAR = scale, µ = scale, µ = shape, σ = 0.52 shape, σ = :10,000 S R = :10,000 S R = 4.61 Table C-8: Histogram data for OK-SW HMLT C-11

12 CH_6 (A) PA CH_1 (X) , , , , ,713 37, ,817 56, ,001 3, ,085 4, N = 394, ,516 N = 669, ,414 MEAN = MEAN = VAR = VAR = scale, µ = scale, µ = shape, σ = 0.43 shape, σ = :10,000 S R = :10,000 S R = 2.23 Table C-9: Histogram data for PA HMLT C-12

13 CH_6 (A) SD CH_8 (X) ,140,254 14,312, ,059,585 15,065, ,969,213 5,313, ,480,006 5,650, , , ,478 1,175, , , , , ,507 34, ,015 57, ,132 7, ,067 14, ,365 1, ,621 4, , ,242 1, N = 18,971,633 20,810,538 N = 20,623,451 22,216,759 MEAN = MEAN = VAR = VAR = scale, µ = scale, µ = shape, σ = 0.45 shape, σ = :10,000 S R = :10,000 S R = 3.74 Table C-10: Histogram data for SD HMLT C-13

14 CH_8 (A) WY-CJE (FR) CH_6 (X) ,849,383 12,723, ,537,265 18,961, ,828,050 4,627, ,622,158 7,998, ,983 1,055, ,897,431 2,160, , , , , ,904 64, , , ,127 18, ,303 58, ,920 5, ,832 20, ,230 1, ,721 7, , ,805 3, ,507 1, N = 17,778,845 18,747,100 N = 28,777,530 29,986,573 MEAN = MEAN = VAR = VAR = scale, µ = scale, µ = shape, σ = 0.50 shape, σ = :10,000 S R = :10,000 S R = 4.68 Table C-11: Histogram data for WY-CJE HMLT no strakes C-14

15 CH_4 (A) WY-CJE (MT) CH_6 (X) ,728,506 3,000, ,700,862 3,959, ,415,012 1,294, ,748,643 1,654, , , , , ,554 92, , , ,660 27, ,366 36, ,874 8, ,995 12, ,098 2, ,215 4, ,437 1, ,798 1, N = 4,549,210 4,771,755 N = 5,992,470 6,238,388 MEAN = MEAN = VAR = VAR = scale, µ = scale, µ = shape, σ = 0.52 shape, σ = :10,000 S R = :10,000 S R = 4.67 Table C-12: Histogram data for WY-CJE HMLT - mitigated C-15

16 CH_8 (A) WY-CJW (FR) C-16 CH_6 (X) ,386,927 14,708, ,198,811 16,768, ,689,332 6,275, ,326,427 7,933, ,493,498 1,826, ,222,138 2,489, , , , , , , , , ,189 62, , , ,272 23, ,038 39, ,580 9, ,697 16, ,346 4, ,572 7, ,775 1, ,807 3, , ,776 1, , N = 23,030,758 23,640,706 N = 27,628,271 28,426,213 MEAN = MEAN = VAR = VAR = scale, µ = scale, µ = shape, σ = 0.55 shape, σ = :10,000 S R = :10,000 S R = 5.45 Table C-13: Histogram data for WY-CJW HMLT no strakes

17 CH_1 (A) WY-CJW (MT) CH_3 (X) , , , , , , , , ,568 84, ,531 86, ,152 19, ,416 19, ,689 4, ,006 4, ,731 1, ,620 1, N = 1,248,053 1,339,739 N = 1,320,528 1,416,688 MEAN = MEAN = VAR = VAR = scale, µ = scale, µ = shape, σ = 0.48 shape, σ = :10,000 S R = :10,000 S R = 3.95 Table C-14: Histogram data for WY-CJW HMLT- mitigated C-17