age. Several empirical methods for providing 

 such a correlation have been proposed. The best 

 known and simplest of these is the cumulative- 

 damage law proposed by Miner. 



As an illustration of the manner in which the 

 endurance life of a typical structural component, 

 say a plate-stiffener combination, can be pre- 

 dicted, consider Fig. 22. For the sake of simplicity 

 it will be assumed that the mean stress level in the 

 structure is zero. The histogram of stress varia- 

 tions shown in the left of the figure is also known 

 as a load spectrum. The damage due to each 

 component of the load spectrum is computed and 

 the result is plotted as a damage spectrum. For 

 example, 100,000 applications of a stress variation 

 occur in the range of 4 to 8 per cent of the static 

 ultimate stress. According to the S-N curve 

 approximately 1 million applications of this stress 

 are required to cause failure. Therefore, accord- 

 ing to Miner's criterion, the damage due to the 

 application of 100,000 applications of this stress is 

 100,000/1,000,000, equal to 10 per cent. The in- 

 tegral of the damage spectrum over the stress 

 range experienced in service gives the expected 

 total damage during 1 year. In the illustrative 

 example the expected life of the structure is 5.4 

 yr. Owing to the variability of the endurance 

 strength of nominally identical structural com- 

 f)onents, as well as the variability of the service 

 loads for structures which presumably are sub- 

 jected to the same service, it is necessary to re- 

 duce the expected life of the structure by a factor 

 in order to arrive at a "safe life" which may be 

 expected to be attained with an acceptably high 

 degree of probability. 



The illustrative example was applied to a simple 

 case in which the mean stress level was zero. The 

 method can be extended easily to the more real- 

 istic case where the sea-induced stresses are super- 

 imposed on various mean levels of stress. In the 

 latter case it is only necessary to sum the damage 

 resulting from the load spectra associated with 

 each mean level of stress. 



An interesting exposition of this general ap- 

 proach to the design of airplane structures has 

 been given by W. Tye (23) . 



Prediction of Optimum Operating Conditions 



In ship operations it is often necessary to 

 select a heading of the ship relative to the direc- 

 tion of the waves and a ship speed such that the 

 pitching or rolling motions are below a level be- 

 yond which certain operations would become im- 

 practical. Examples of such operations are fuel- 

 ing at sea and the launching and landing of air- 

 craft. For this purpose it is convenient to have a 

 graphical or numerical presentation of a parameter 



-7^ 



^ 



X 



Head Seas 



Ship Speed, 14 knots 



5 10 15 20 25 30 35 



Significant Wave Height in feet 



USCGC UNIMAK 



Variation of the Statistic E with the Seveiity of the Sea 



(Variations in Pitch Angle) 



g 30P^ 



?=:z:r= 



_Ship Speed 17 knots_ 



/^ 



Sign. Wave Height, 8 feet V 



^=?^ 



Ship Speed, 7)4 knots 



Sign, Wave Height, 21 feet 



?- — 



4^ 



■90-45 45 90 135 180 



Direction From Which Waves Are Coming in degrees 

 (r,1easured from bow of ship) 



USCGC UNIMAK 



Variation of the Statistic E with Ship Heading 



(Variations in Pitch Angle) 



Fig. 23 Typical Presentation of the Statistic E 

 Which Defines the Probability Distributions 



(Statistic R is integral of power spectrum. Maximum expected 

 magnitude of pitch angle is proportional to Vii.) 



which, at a glance, indicates the relative severity 

 of motion to be expected for any given ship speed, 

 heading, and sea condition. 



Accepting the hypothesis that the short-term 

 distributions of ship motions and stresses can be 

 represented by the Rayleigh distribution, then 

 one can plot the parameter E, which defines this 

 distribution, as a function of ship heading, ship 

 speed, and sea state. Fig. 23 illustrates such a 

 plot. The most probable maximum value of the 

 ship motion expected in a given number of oscilla- 

 tions is proportional to \/E. Table 4 gives the 

 number by which \/E is to be multiplied in order 

 to give the magnitude of the expected maximum. 



Tables or graphs could be prepared for any 

 given ship type to cover the range of operating 

 conditions likely to be encountered in service. 



The foregoing examples are illustrations of some 

 useful applications of the statistical distributions 

 discussed in this paper. 



37 



