SPHERES AND CYLINDERS 



125 



Problem 130. A marine boiler shell is 16 ft. long, 8 ft. in diameter, and 1 in. thick. 

 What is the stress in the shell for a working gauge pressure of 1601b./in. 2 ? 



Problem 131. The air chamber of a pump is made of cast iron of the form 

 shown in Fig. 93. If the diameter of the air chamber is 10 in. and its height 24 in., 

 how thick must the walls of the air chamber be made to stand a pressure of 

 600 lb./in. 2 with a factor of safety of 4? 



* 108. Differential equation of elastic curve for circular cylinder. 

 A cylindrical shell subjected to internal pressure is in a condition of 

 stable equilibrium, for the internal pressure tends to preserve the 

 cylindrical form of the shell, or to restore it to this form if, by any 

 cause, the cylinder is flattened or otherwise deformed. A cylindrical 

 shell which is subjected to external pressure, however, is in a con- 

 dition of unstable equilibrium, for any deviation from a cylindrical 

 form tends to be increased rather than 

 diminished by the stress. In tlu's respect 

 thin hollow cylinders under external 

 pressure are in a state of strain similar 

 to that in a column, and the method of 

 finding tin* critical pressure just preced- 

 ing collapse is similar to that for finding 

 the critical Inad fora column, as explained 

 in the derivation of Euler's formula. 



Consider a thin hollow cylinder which 

 is subjected to a uniform external pres- 

 sure of amount w per unit of area, and suppose that in some way the 

 cylinder has been compressed in one direction so that it assumes the 

 flattened form shown in Fig. 94. The first step in the solution of 

 the problem is to find the differential equation of the elastic curve in 

 curvilinear coordinates, or, in other words, the differential equation 

 of the elastic curve of the flattened cylinder referred to its original 

 circular form. 



In polar coordinates let be the origin and OA the initial line. 

 Also, let a denote the radius of the circular cylinder, and r the radius 

 vector of the flattened or elliptical form. Now suppose that the cir- 

 cular wall of the cylinder is considered as a piece which was origi- 

 nally straight and has been made to assume a circular form by a 



FIG. 94 



* For a brief course the remainder of this chapter may be omitted. 



