ON THE STRENGTH OF MATERIALS FOR IRON SHIPS. 255 



The results of experiments closely confirm the truth of this deduction. In 

 wrought-iron girders, for example, the bottom section is about fths of the 

 top section ; and in cast-iron beams the section of the bottom flange is about 

 6| times that of the top flange, -which is approximately, in the case of cast 

 iron, the ratio of the resistance of compression to that of extension. 



The next condition of maximum strength determines generally the relative 

 amount of the material on each side of the neutral axis. The principle 

 upon which this distribution of material depends is this : when the material 

 at the upper edge of the beam is upon the point of yielding by compression, 

 the material at the lower edge must at the same time be upon the point of 

 yielding by extension ; hence 



S S. S h ..-.,. 



*=V or s;=*- ; (1) 



that is, the distances of the neutral axis from the upper and loiver edges of the 

 beam should be in the ratio of the ultimate resistances of the material at these 

 parts respectively. 



o 



If r=o-, and D=the whole depth of the beam, then 



£=-^-...(2), and^— (3) 



In the case of wrought iron, r=% nearly; that is, the ultimate resistance 

 of wrought iron to compression is about £ of its ultimate resistance to ex- 

 tension. 



24. To determine the area, a t , of the material in the transverse section of 

 the upper deck (art. 22) so as to have a maximum strength with a given 

 amount of material. 



The strains on a ship afloat are somewhat different from the strains on an 

 ordinary fixed girder ; for it is found that the upper part of the transverse 

 section of a ship when afloat is sometimes subjected to compression and at 

 other times to that of extension. But of these two strains the former is 

 generally the more violent. It will therefore be expedient, in calculating the 

 distribution of the material in the section of an iron ship, to regard the upper 

 portion of the section as that which is subjected to extension ; and therefore 

 in this case we should have 



A= — —T' 

 r-fl 



Making this substitution in equation (7), art. 22, and putting a -f a 1 + a 2 -f a 3 

 for K, we get 



«i=^- {.(K+«.) D + «*«» } -(«»+«>+««) (1) 



If we assume r=l, then this expression becomes 



In one of our most approved iron ships, ffl =440 square inches, a 2 =156, 

 a 3 =490, D=23| feet, a 2 =7| feet, required a v 



Substituting these values m equation (1), we find a v or the area of the 

 upper deck, to exceed 300 square inches. Now in this iron ship the equivalent 

 area of the upper deck is only about 230 square inches ; hence it appears 

 that this portion of the ship should be about one-half stronger than it is, in 

 order to have a proper distribution of the material. 



