SPHERES AND CYLINDERS 



131 



From Lame's formulas the stresses p l and p z may be obtained in 

 terms of the unit pressure between the surfaces in contact. Thus, 

 from formula (183), the stress on the inside of the hole is 



(191) 



_ 

 ft 



where Z> 3 denotes the outside diameter of the ring, while, by substi- 

 tuting r = a and b = in the equations of article 78, the stresses 

 on the outer surface of the shaft are found to be 



and consequently 



Eliminating w between these expressions for p l and p 2 , we have 

 (192) Pl=1 



Now, to simplify the solution, let the coefficient of p 2 be denoted by 

 H; that is, let 



77 

 -" = 



in which case 



-A 2 



Eliminating p 1 between this relation and the above expression for 

 the allowance K, we have finally 



K 



, Z> 2 



(193) 



In applying these formulas the constant H is first computed from 

 the given dimensions of the parts. If the allowance K is given, 

 the unit stresses p 1 and p 2 in ring and shaft are then found from the 

 above. If K is to be determined, a safe value for the stress in the 

 ring p 1 is assigned, and p 2 is calculated from the second equation. 

 This value is then substituted in the first equation, and K is 

 calculated. 



