By General Nicholas Kalakoutzky.

No. I. Determination Of The Influence Of Internal Stresses On The Strength Of Materials

We call internal stresses those which exist within the mass of any hollow cylinder or other body, when it appears to be in a state of repose, or not under the influence of external forces. When pressure is applied to a hollow cylinder, either externally or internally, the interior layers into which its walls may be conceived to be divided are subjected to a new series of stresses, the magnitude of which is independent of those already existing. These additional stresses combine with the former in such a manner that at every point of the thickness of the cylinder they have common resultants acting in various directions. Thus, if we call t the internal stress existing at a distance r from the axis of the cylinder, and in a direction tangential to its cross section, and T the additional stress due to pressure inside the cylinder acting at the same point and in the same direction, then the newly developed stress will be t + T.

If R and r be the external and internal radii of the cylinder, and if we suppose the external pressure nil, then, if the pressure inside the bore be P, the stress on the radius r is determined by the following expression deduced from the well-known fundamental formulae of Lame:1

 T = P r2 - - - - R2-r2 · R2 + r2 - - - - r2

From which we see that T is a maximum when r = r, i.e., for the layer immediately next to the bore of the cylinder. Calling t the internal stress in this layer, and T the stress resulting from the action inside the bore of the pressure P, and allowing that the sum of both these quantities must not exceed the elastic limit U of the material, we have - T = U - t. And for this value of T, the corresponding pressure inside the bore will be

 P = (U - t) P R2 - r2 - - - - R2 + r2

This pressure increases with the term (U-t). With t positive, i.e., when the internal stresses in the thickness of the hollow cylinder are such that the metal of the layers nearest to the bore is in a state of tension and that of the outer layers in a state of compression, then the cylinder will have the least strength when t has the greatest numerical value. Such stresses are termed injurious or detrimental stresses. With t negative, the strength of the cylinder increases with the numerical value of t, and those stresses which cause compression in the layers nearest to the bore of the cylinder and tension in the outer layers are termed beneficial or useful stresses.

For these reasons, and in order to increase the power of resistance of a cylinder, it is necessary to obtain on the inner layer a state of initial compression approaching as nearly as possible to the elastic limit of the metal. This proposition is in reality no novelty, since it forms the basis of the theory of hooped guns, by means of which the useful initial stresses which should be imparted to the metal throughout the gun can be calculated, and the extent to which the gun is thereby strengthened determined. The stresses which arise in a hollow cylinder when it is formed of several layers forced on one upon another, with a definite amount of shrinkage, we call the stress of built-up cylinders, in order to distinguish them from natural stresses developed in homogeneous masses, and which vary in character according to the conditions of treatment which the metal has undergone. If we conceive a hollow cylinder made up of a great number of very thin layers - for instance, of wire wound on with a definite tension - in which case the inner layer would represent the bore of the gun, then the distribution of the internal stresses and their magnitude would very nearly approach the ideally perfect useful stresses which should exist in a homogeneous cylinder; but in hollow cylinders built up of two, three, and four layers of great thickness, there would be a considerable deviation from the conditions which should be aimed at.

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 -

 P = T R - r - - - r (1)

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.

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 -

 t1 = P r2 - - - - R2 - r2 · r2 + R2 - - - - r2

Therefore, to obtain the maximum resistance in the cylinder, the value t of the initial stress will be determined by the difference T - t', and since P is given by Equation (1), then

 t = T ( 1 - r - - - R + r · r2 + R2 - - - - r2 ) (2)

The greatest value t = t corresponds to the surface of the bore and must be t =-T, therefore

 r2 + R2 - - - - r (R + r) = 2

whence P = T √2 = 1.41 T.

From the whole of the preceding, it follows that in a homogeneous cylinder under fire we can only attain simultaneous expansion of all the layers when certain relations between the radii obtain, and on the assumption that the maximum pressure admissible in the bore does not exceed 1.41 U.