assumed distribution. However they felt that no 

 valuable (statistical) conclusions could have been 

 drawn from their results since they were not really 

 in a form suitable for vigorous significance tests. 

 Nevertheless, the short-term distribution used by 

 them^'''^ probably would be theoretically more 

 accurate than the Rayleigh distribution. 



Professors Pierson and Korvin-Kroukovsky 

 have indicated that the fit of the proposed dis- 

 tribution to the data is "unusually" good whereas 

 Messrs. Lewis and Press note occasional appreci- 

 able departures of the data from the theoretical 

 distributions. The fit is really not "too good to be 

 true." Dr. Lieblein's discussion bears directly on 

 this question. 



It should have been stated in the paper that 

 not every one of the hundred or so distributions 

 was actually subjected to statistical tests. Rather, 

 all these distributions were examined visually and 

 then several typical distributions as well as sev- 

 eral relatively poorly fitting distributions were 

 selected for statistical testing. This seemed to be 

 a reasonable procedure. Also the confidence 

 limits fitted to the cumulative distribution are 

 applicable individually to each experimental frac- 

 tUe but they do not by their nature extend to very 

 small or very large fractiles, unless very large sam- 

 ples are used. Thus nothing can be said about 

 the fit at these extremes and it is therefore quite 

 possible that occasional significant deviations 

 could indeed be present at these extremes, with- 

 out being detected. Of course from the stand- 

 point of practical application all that is desired 

 is a fit good enough to permit making reasonable 

 estimates. The latter criterion has been satis- 

 fied, it is believed, to a good degree for the ship's 

 response to the sea. The author cannot conclude 

 that the deviations from the proposed distribu- 

 tions, which were noted by Messrs. Lewis and 

 Press, are statistically significant. 



Messrs. Lewis, Comstock, and Vasta have em- 

 phasized the importance of being able to estimate 

 the extreme value of hull girder stress, raising 

 the question as to how this could be done ade- 

 quately. The long-term distribution would not 

 be suitable for this purpose, as stated in the paper. 

 The following method is suggested as reasonable: 



Determine £„ for the Rayleigh distribution of 

 stress corresponding to the worst combination of 

 sea state, ship speed and heading that a ship is 

 likely to experience, utilizing either model tests, 

 full-scale data, or theory. Estimate the longest 

 duration that the ship will operate continuously 

 under this combination of steady conditions and 

 note the number N, of variations experienced in 

 this length of time. Next, estimate the number 

 of times n, that these extreme operating condi- 



tions (storms) will be experienced during the serv- 

 ice life of the ship. Thus (nN) stress variations 

 are expected corresponding to a basic distribu- 

 tion defined by Em- Then, utilizing Formula 

 [2] of the paper, we can estimate the fraction / 

 of all similar ships which, on the average, will ex- 

 ceed (Tm over their service life. Thus / represents 

 the risk of exceeding Cm- Since for the Rayleigh 

 distribution 



[1 -P{am)] =6-"^''"'^ 



we have from Formula [2] 



1 -/= expf-e-"] 



where 



y = 



loge {nN) 



Table 1 of Bureau of Standards bulletin "Ap- 

 plied Math Series No. 22" tabulates the value of 

 exp [ — «""] as a function of y. Thus for a 

 specified risk / of exceeding the stress Om one may 

 look up the corresponding value of y and then 

 solve for the desired stress from the relation 



<tJ = Emly + loge{Nn)] 



Another way of making the estimate is to 

 utilize the approximation that if /i is the proba- 

 bility of exceeding Cm in a single storm then 

 w/i is the probability of exceeding ffm in m storms. 

 That is / = w/i. The formulas given here thus 

 far are valid only if / is much smaller than unity 

 which will be true if we are looking for extreme 

 values. The exact expression for / is 



/= 1 - (i-Zi)-^ 



As an example consider an aircraft carrier with 

 Em = 16, N = 1000, n = 100, and take / = 

 0.001; i.e., one chance in a thousand. Then y = 

 7.0 and 



am = [16(7.0 -f 11.5) ]'/2 = 17.2 kips/sq in. 



The method of Hazen and Nims referred to by 

 Mr. Lewis is likely to result in an overestimate of 

 the stress but otherwise appears to offer an at- 

 tractive solution. These approaches may be 

 utilized to establish an upper limit to the long- 

 term distribution also. 



The short-term Rayleigh distributions fitted by 

 Mr. Lewis to experimental model test data of wave 

 height, ship motions and bending moments (Figs. 

 24, 25) do show encouraging agreement. It is the 

 author's expectation that, once agreement be- 

 tween model tests and full-scale tests has been 

 established, model tests will be used to define the 

 distribution patterns. Mr. Lewis' fear that an 



55 



