June, 1920] Springs of Minimum Inertia 321 



SPRINGS DUE TO BENDING. 



All springs of this type take the form of some sort of a 

 beam and we will then have the following formulae from the 

 fundamental principles of strength of materials. 



PT PL^ 



(1) f-W (2)A^ 



ZN ^ ' KEI 



In the above equations N and K are constants which 

 determine the method of loading and the manner in which the 

 beam is supported. 



BEAMS OF RECTANGULAR SECTION. 



In this case — 



bd2 bd^ 



Z = -^ I = T^ and W=wbdL 



These give when substituted in equations (1) and (2), 



/o^ ^ 6PL /,N A l^PL^' 



^^) ^ = Nbd-^ (^) ^ = KEbd^ 



Square equation (3) and divide the result b}^ equation (4) 

 and we have, 



,., f2 3PKE , ,^. ... 3PKE A 



In equation (6) bdL is simply the volume of the spring and 

 hence if we multiply this by its weight per cubic inch, we will 

 .have its total weight, whence 



(7) W = wbdL = 3w (^\ (-^\ PA 



In either of the cases of a beam fixed at one end and loaded 

 at the other, supported or fixed at both ends and loaded in the 

 middle, the quantity (K/N2)=3, whence the weight will be 

 given by 



(8) W = 9 (^)PA 



It will be noticed that the dimensions of the beam have 

 entirely disappeared from equation (8) or in other words the 

 weight is entirely independent of the section of the beam. To 



