Discussion 



It has been the main objective in this paper to 

 show that the distribution of wave-induced varia- 

 tions in the pitch, roll and heave motions of ships 

 as well as the wave-induced,'^ hull-girder stress 

 variations may be represented by a Rayleigh dis- 

 tribution for steady conditions of the sea, ship 

 speed, and course, whereas these responses of the 

 ship to the sea will be approximated by a loga- 

 rithmically normal distribution when distributions 

 are considered which extend over a period of weeks, 

 months, or longer, during which period the ship is 

 subjected to a wide variety of operating con- 

 ditions. 



A secondary objective has been to show that the 

 foregoing hypothesis is also applicable to the dis- 

 tribution patterns of the heights and lengths of 

 ocean waves. 



Over 100 individual short- terra distributions of 

 ship motions and stresses obtained for a wide 

 range of environmental conditions on the USCGC 

 Uniinak were analyzed. A smaller number of 

 similar short-term distributions of stresses and 

 motions for a large aircraft carrier and for a de- 

 stroyer were analyzed in a similar manner. No 

 significant difference was found between the hy- 

 pothesis and the test data. On the basis of the 

 statistical tests of significance, the quantity and 

 the scope of the data, one may accept confidently 

 the hypothesis that the "short-term" distribution 

 of ship motion and stress approximates to the 

 Rayleigh type. 



The long-term distribution of hull-girder stresses 

 or motions was determined for a wide variety of 

 ship types and operating conditions. The ships 

 studied included an oil tanker in coastwise service, 

 two dry-cargo vessels, a Coast Guard weathership, 

 a destroyer escort, an aircraft carrier, and a de- 

 stroyer. Extensive statistical analysis and tests of 

 significance were applied to the results of the 

 Esso Asheville tests. In this case, it was possible 

 to study the extreme-value distribution as weU 

 as the basic distribution of hull stresses which were 

 arrived at by independent methods. The distri- 

 bution patterns evidenced by these two distribu- 

 tions complement each other; that is, if the basic 

 distribution is log-normal then the extreme-value 

 distribution of the stress should be of the type 



F(x) = exp [-e-y] 



The confidence limits applied to the cumulative 

 distributions as well as the chi-square test applied 

 to the grouped measured data indicate that the 

 hypothesis may be accepted with confidence; 



i.e., the wave-induced stresses and motions, over 

 the long term may be represented by a log-normal 

 distribution. 



Finally, from a utilitarian point of view, one 

 also may accept the hypothesis that the short- 

 terra distribution of wave heights follows the 

 Rayleigh pattern and that the long-term distri- 

 bution of wave heights and wave lengths foUows 

 the log-normal pattern. The latter conclusion 

 cannot be evaluated by the use of standard statis- 

 tical tests of significance since most wave obser- 

 vations were made by eye and no measure of the 

 accuracy of these observations has been deter- 

 mined thus far. Nevertheless the very extensive 

 data obtained by independent observers, the con- 

 sistent accommodation of these data to the log- 

 normal pattern, and the fact that the response of 

 the ship to the waves may be represented by a 

 log-normal distribution, all tend to the conclusion 

 that the log-normal distribution is applicable to 

 the specification and prediction of wave heights 

 and wave lengths. 



The existence of a predictable distribution pat- 

 tern will greatly simplify the collection and analy- 

 sis of service data, as well as furnish a concise and 

 graphic manner for the specification and predic- 

 tion of ship stresses and motions for given ships 

 in a given service. Prediction of the largest 

 stresses and motions to be expected over a speci- 

 fied period of time can be made readily, although 

 the prediction of extreme values is basically less 

 reliable than a prediction of those values which 

 occur more frequently. The method for predict- 

 ing extreme values eventually breaks down by 

 predicting values which are too large. This occurs 

 because the theoretical distribution cannot be re- 

 lied upon at the extreme ranges of the function. 

 For example, the theory could predict stresses in 

 excess of the ultimate strength of a structure, ob- 

 viously an invalid prediction. Rejection of pre- 

 dicted values lying outside the range of experience 

 would provide a proper safeguard against too ex- 

 treme an estimate. 



The parameters computed for the distribution 

 functions discussed in this paper are not affected 

 appreciably by variations or scatter of those 

 values or events which occur relatively infre- 

 quently. For this reason, provided the type of 

 basic distribution pattern has been established, it 

 is possible to obtain a reasonably good approxi- 

 mation to the distribution pattern with a limited 

 amount of experimental data, because it is not 

 necessary to continue measurements until the 

 more extreme and infrequently occurring members 

 of the population have been sampled. Inasmuch 

 as ship motions and huU-girder stresses are of 

 rather small magnitudes during the greater por- 



38 



