70 On the Strength of Cylindrical Steam Boilers. 



7. Let x= the diameter of the cylinder; f= the force or pressure 



per unit of surface, (pounds per square inch, for example;) T=lhe 



tenacity of metal, which with the diameter x and the force/ will be 



required in the linear unit of the circumference, in order to hold on 



the head. Then, the whole quiescent force will be 3.1416ltT, while 



the divellent will be ,7854^=/; consequently .7854x^f=SA4:l6xT, 



as above stated. Dividing by .7854a?, we have ^=4T; and we de- 



4T 4T «/ . . ^ 



rive immediately x=—c' f~ — ' T=-r' That is, the tenacity of 

 / a? 4 



the longitudinal bar of the assumed unit in width, will be one fourth 

 of the product of the diameter into the pressure, measuring the tena- 

 city by the same standard as the pressure, whether in pounds or 

 kilogrammes. 



8. Now assuming the tenacity required in the circular hand of the 

 same width to be t, we shall, agreeably to what has already been said, 

 have the divellent force expressed by xf and the quiescent by 2t, so 



xf 2t 2t 



that xf=2t and t = :^ ; also/==— j and x--^- Havmg thus obtam- 



J. X J 



ed tvi^o expressions for each of the quantities x and f, we may by 



comparing them, readily discover the relative values of T and i; 



4T 2t 4T 2t ■*" 



thus, x=-7- and x=ji hence ^=-7;*-and 4T=2^ or t—2T:. 



From which it follows, that, under a known diameter, and with a 

 given force or pressure, the tenacity of metal in a cylindrical boiler 

 of uniform thickness, ought to be twice as great in the direction of 

 the curve as in that of the length of the cylinder, and thai if this 

 could be the case the boiler ivoidd still have equal safety in both di- 

 rections. In whichever direction, therefore, the rolling of the metal 

 gives the greatest tenacity, in the same direction must the sheet al- 

 ways be bent in forming the convexity of the cylinder. It follows 

 that if we suppose the tenacity precisely equal in both directions, 

 the liability to rupture, by a mere internal pressure, ought to be tivice 

 as great along the longitudinal direction as at the juncture of the 

 head. This supposes the strain regular and the riveting not to 

 weaken the sheet. 



9. To know how large we may safely make a cylindrical boiler, 



having the absolute tenacity of the metal, in the strongest direction, 



and with a known thickness, we have only to revert to the formula 



2t . . . , 



x=^^' That is, the diameter will be found by dividing twice the 



