32 



Mr. J. H. Cotterill on Elliptic Ribs. 



an equation which gives the bending moment on any section 

 whose eccentric angle is </>'. 



On account of the thinness of the tube the strength is in- 

 versely as the maximum bending moment, which is always at the 

 extremity of the major axis; so that the strength varies inversely 

 as M . Suppose p x the maximum radius of curvature, that is, 



0COS 



-5, then a=p l cos/3 ; 



.'. M =— %pap } cos /3 sin 2 /3 



E, 



The annexed Table gives the values of cos /3 sin 2 /3 ^- for 



values of /3 between 45° and 65°, differing by 2 j° ; whence it 



P 



appears that the mean of the values of cos /3 sin 2 ft ^- for values 



of /3 between 47 J° and 65° is '212, and that the extreme deviation 

 from this mean value is not more than 5 per cent, of the whole. 



/3. 



p 



cos /3 sin 2 /3— . 

 ^1 





 45 



•192 



50 



•201 

 •209 



521 



•214 



55 



•218 



57i 



60 



62| 



•219 

 •217 

 •213 



65 



•206 



Whence it appears that between these limits of /3 (which 

 correspond to the values §rds and j%ths nearly of the ratio 

 of the axes) M (xap 1 nearly, and the strength inversely as ap v 

 Now Mr. Fairbairn states that the strength of elliptical tubes 

 varies inversely as the length, and inversely as p x ; so that, for 

 tubes of the same proportion of breadth (a) to length, the strength 

 varies inversely as ap v Thus, within the limits specified, the 

 relative strength of tubes of indefinite length is the same as 

 that of tubes of a given ratio of breadth to length*. 



5. Next let it be required to find the stress at any point of a 

 semielliptic rib, with major axis horizontal, fixed at the spring- 

 ing, and sustaining the pressure of a depth of water so great 

 that the pressure at different points is sensibly constant. 



* For reasons previously referred to, the solution is inapplicable when 

 the tube is nearly circular. 



