130 



STRENGTH OF MATERIALS 



Problem 133. In a fire-tube boiler the tubes are of drawn steel, 2 in. internal 

 diameter and | in. thick. What is the factor of safety for a working gauge pres- 

 sure of 200 lb./in.2 ? 



110. Thick cylinders ; Lame's formulas. Consider a thick circular 

 cylinder of external radius a and internal radius />, which is subjected 

 to the action of either internal or external uniform pressure, or t. 

 both. Suppose a section is cut out of the cylinder by two plane- 

 pendicular to the axis at a unit's distance apart, and consider a small 

 sector ABCD of angle a cut out of the ring so obtained, as shown 

 in Fig. 96. Let j? A denote the tangential stress, or hoop stress, acting on 

 this infinitesimal element, p r the radial stress acting on the inner Bur- 

 face AD, and p r + dp r the radial stress acting on the outer surface BC. 



P, d P, 



FIG. 90 



Then the internal and external radii being r and r + dr resp.-. -tiveh -. 

 the length of AD is ra and of BC is (r + ,/,- 

 of the piece is unity, the resultant radial force acting on the piece, or 

 the difference between the pressure on the inner and outer surfaces, 

 is (p r + dp r )(r + dr)a-p r ra. Therefore, rinoe the resultant 01 

 hoop stress in a radial direction is (p k a)dr, in order that the radial 

 stresses shall equilibrate, 



(Pr + dp r ) (r + dr)a- p r ra = p k ,l,' : 

 or, neglecting infinitesimals of an order higher than the second, 



which may be written 

 (74) 



