The magnitude of the stresses in built-up cylinders is determined by calculation, on the presumption that initial stresses do not exist in the respective layers of the tube and of the hoops which make up the walls of the cylinder.  Nevertheless, Rodman, as early as the year 1857, first drew attention to the fact that when metal is cast and then cooled, under certain conditions, internal stresses are necessarily developed; and these considerations led him, in the manufacture of cast iron guns, to cool the bore with water and to heat the outside of the moulds after casting.  Although Rodman’s method was adopted everywhere, yet up to the present time no experiments of importance have been made with the view of investigating the internal stresses which he had drawn attention to, and in the transition from cast iron to steel guns the question has been persistently shelved, and has only very lately attracted serious attention.  With the aid of the accepted theory relating to the internal stresses in the metal of hooped guns, we can form a clear idea of the most advantageous character for them to assume both in homogeneous and in built-up hollow cylinders.  In proof of this, we can adduce the labors of Colonels Pashkevitch and Duchene, the former of whom published an account of his investigations in the Artillery Journal for 1884—­St. Petersburg—­and the latter in a work entitled “Basis of the Theory of Hooped Guns,” from which we borrow some of the following information.

The maximum resistance of a tube or hollow cylinder to external stresses will be attained when all the layers are expanded simultaneously to the elastic limit of the material employed.  In that case, observing the same notation as that already adopted, we have—­

R — r0
P0 = T --------        (1)
r0

But since the initial internal stresses before firing, that is previous to the action of the pressure inside the bore, should not exceed the elastic limit,[2] the value of R will depend upon this condition.

[Footnote 2:  We must, however, remark that in a built-up hollow cylinder the compression of the metal at the surface of the bore may exceed the elastic limit.  This cannot occur in the case of natural stresses.]

In a hollow cylinder which in a state of rest is free from initial stresses, the fiber of which, under fire, will undergo the maximum extension, will be that nearest to the internal surface, and the amount of extension of all the remaining layers will decrease with the increase of the radius.  This extension is thus represented—­

(r0) squared        (r_x) squared + R squared
(t_x)¹ = P0 ------------ . ------------
R squared — (r0) squared       (r_x) squared

Therefore, to obtain the maximum resistance in the cylinder, the value t_x of the initial stress will be determined by the difference (T — t’x),[*need to check the prime with library or work out the equations] and since P0 is given by Equation (1), then