Table 8 SS Esso Asheville — Computation of Confidence Limits for Distribution of Stress 

 Variations, Port Side, Shown in Fig. 16 



stress 

 psi 



X 



Log,;, Stress 



y 



y" 



i(^0" 



(l-p)xl02 



v" 





= L0Ex^„ 





Confidence Limits 

 H = 0.67 



psi 



^lower 



psi 



20,000 



4.3010 



-1.3255 



1.757 



1.980 



0.0011 



0.99999 



0.247 



4.5480 



4.0540 



35,300 



11,300 











xlO-* 







(0.494)t 



(4.7950) 



(3.8070) 



(62,400) 



(6,400) 



16,000 



4.2041 



-1.2286 



1.5095 



6.798 



0.005 



0.99995 



0.153 



4.3571 



4.0511 



22,800 



11,100 











xlO^ 







(0.306) 



(4.5101) 



(3.8981) 



(32,400) 



(7,900) 



14,000 



4.1461 



-1.1706 



1.3703 



1.361 



0.009 



0.99991 



0.103 



4.2491 



4.0431 



17,900 



10,900 











xlO-3 







(0.206) 



(4.3521) 



(3.9401) 



(22,500) 



(8,700) 



10,000 



4.0000 



-1.0245 



1.050 



6.718 



0.050 



0.9995 



0.0491 



4.0491 



3.9509 



11,300 



8,900 











xlO-' 







(0.0982) 



(4.0982) 



(3.9018) 



(12,700) 



(8,000) 



8,000 



3.9031 



-0.9276 



0.8604 



1.729 



0.150 



0.9985 



0.0335 

 (0.0670) 



3.9366 

 (3.9701) 



3.8696 

 (3.8361) 



8,600 

 (9,300) 



7,400 

 (6,900) 



4,000 



3.6021 



-0.6266 



0.3926 



0.1781 



2.3 



0.977 



0.0126 

 (0.0252) 



3.6147 

 (3.6273) 



3.5895 

 (3.5769) 



4,100 

 (4,200) 



3,900 

 (3,800) 



1,000 



3.0000 



-O.0245 



0.00060 



1.256 



47.0 



0.53 



0.0060 

 (0.0120) 



3.0060 

 (3.0120) 



2.940 

 (2.9°80) 



1,010 

 fl,040) 



990 

 (970) 



'The quantiles of x^ are distributed normal! 



cumulative distribution function cottespondin 

 \'-= 4,597 observations. 



with the 

 g to Xj^ an 



nean value i^^ 

 if(Xf) is the v 



nd standar 

 alue of the 



1 deviation — ^1/— vtiere p is the theoretical value of the 

 f{x^)\ N 



logarithmic distribution function corresponding to x^. 



"The distribution function of x^ (log of stres 



s on port s 



ide of deck) ha 



s a mean value x^ = 2.9755 and a standi 



rd deviation a^ 



= 0.3167, that is 



^-oX 4'"^}"°^-^=°- 



004-2.16 



50(x^-xj2. 













tThe figures in parentheses are computed on 



the basis 



of tlie assumpti 



an that eve 



ry fourth observation was an 



ndependeni random observation. 



\' = | (4,597). 

















stresses is normally distributed. Statistical tests 

 of significance assume that the data consist of 

 random independent observations. Consecutive 

 cycles of stress and motion are not independent. '* 

 Thus it will not be possible to give accurate prob- 

 abilities for the hypothesis considered. However 

 if it is assumed, falsely, that all data are random 

 and independent then a better fit will be required 

 between the assumed log-normal distribution 

 and the experimental data than is necessary at 

 the given confidence level. 



One rather simple test of significance is Kol- 

 mogorov's statistic (17) which permits one to 

 assert with a specified level of probability that 

 the true continuous probability distribution func- 

 tion is contained within the confidence band 

 everywhere. A more accurate (and tedious) test 

 is to compute the expected variation of the 



" Autocorrelations have been computed for typical stress and 

 pitcli variations: these computations indicate that olaservations sepa- 

 rated by about 6 cycles are "reasonably" independent. 



quantiles of samples taken from the hypothetical 

 population. The latter method has been utilized 

 to fit confidence limits to the distribution of stress 

 variations measured on the port side amidships, 

 see Fig. 16 and Table 8. It is evident from an 

 inspection of Fig. 16 that the experimental data 

 are approximated by the log-normal distribution 

 with a good degree of probability, especially 

 when the accuracy with which the stress meas- 

 urements were obtained is considered. To avoid 

 confusion, confidence bands have not been fitted 

 to those straight hnes in Fig. 16 which represent 

 the cumulative distribution function of the star- 

 board and of the average stresses. However, 

 visual inspection indicates that here too there is 

 good agreement. 



Finally, the hypothesis may be tested by in- 

 quiring into the relationship that exists between 

 the underlying cumulative distribution of all 

 stress variations and the cumulative "extreme 

 value" distribution of the stress variations. 



25 



