-Ib- 



es beyond the proportional limit unless the corrected value of E be used. It 

 should also be noted that in this region, the values of //a must necessarily be 

 small. Hence 9 becomes fairly large and its square cannot be neglected. Therefore 

 the more exact formula (B) must be used Instead of the approximate formula (D) 

 which was obtained from (B) by neglecting 9 ^. The two formulas may give values 

 differing by 20^ or more as shown in the above table. 



It is evident, therefore, that formula (D) does not apply to short frame 

 spaces. The attempts of the Germans( Johow-Foerster,''Hilf sbuch ftlr den Schiffbau" 

 Berlin, I928, p. 929) to apply this formula to short frame spaclngs by using coef- 

 ficients of 0,4 to 0.6 depending upon the shell thickness were based on the as- 

 sumption that the discrepancies v^ere due to variations from circular form. It 

 should be noted also that these coefficients were applied to the incorrect form- 

 ula (D) which gives values at least 20^ below those given by the exact formula 

 (B) in this region. 



If it is desired to apply these instability formulas to submarine de- 

 sign, the exact formula (B) should be used and also the corrected value of the 

 elastic modulus. This latter value is, of course, very difficult to obtain. It 

 requires an accurate knowledge of the elastic curve and the stresses at the time 

 of collapse, since Ei varies rapidly as the yield point of the material is ap- 

 proached. 



On the other hand, the application of a constant multiplying factor de- 

 pendent upon the shell thickness only is but a makeshift device for extracting 

 reasonable answers from a formula in a region where it was never intended to be 

 used and where it can not possibly give reliable results. If any multiplying fac- 

 tor is to be used, it should at least be a function of the stresses at the time of 

 collapse and not of thickness only. 



Tests conducted at the U.S. Experimental Model Basin show that formula 

 (D) gives very good results for models whose length is equal to or greater than 

 the radius, erring slightly on the side of safety. For shorter models, formula 

 (D) gives values which are considerably too high.) 



