size N [belonging to a given distribution specified by P (x)] which will have at least one value 

 of a; > a; . The formula is valid only if P (a; ) has a value close to unity, that is, the 



formula is designed to estimate the values which occur rarely. The formula is 



-iV[l-P(« )] 

 f=l-e 1 .. 



or alternatively 



-log, (1-/) 



^ '"l N 



where [1 - P (x )] is the probability of exceeding x^ in the fraction / of all samples of 



size A' each. Knowing [1 - P(x„ )], it is easy to compute the corresponding value of the 



1 

 variable a;„ . 



mi 



In order to estimate the largest values of motion and bending moments for design pur- 

 poses we will use this formula to estimate the value x^ which, on the average, is exceeded 

 by the fraction / of all similar ships during their service life. Thus / represents the risk of 

 exceeding x^ . 



It will be assumed that the worst combination of operating conditions is the most severe 

 of those listed in Tables 1 through 6, viz., a State 5 sea characterized by a significant wave 

 height estimated to be 21 ft. The values of E^ specifying the corresponding Rayleigh distri- 

 butions are listed in Table 12. Assume that the ship will be subjected to these operating con- 

 ditions for a duration of 12 hours, experiencing V variations in this period of time, and that 

 this situation will be repeated n times during the service life of the ship; therefore A' = nV. 

 For the Rayleigh distribution we have 



-X 2 /£ 



[1 -P (0=^^)1 =e 1 



Substitution in the expression for / gives: 



— 1 

 1 - / = exp [- e ^] where y = log^ A' 



Table 1 of Reference 9 tabulates the values of exp [-e"^] as a function of y. Thus, for a 

 specified risk / of exceeding x one may look up the corresponding value of y and then solve 

 for the desired value of x frora the relation 



a? 2 = £■ [y + log A'] 



As an example consider the maximum value for the variation in roll angle. 



From Table 12: E^ = 176 (deg)2 , V = 4600 



If we take / = 0.001 and n = 10, then Reference 9 gives y = 7.0. Therefore: 



X = [176 (7.0 + 10.74)]'/' = 56 deg (port to starboard) 

 25 



