_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_x
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 r0 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 P0, the stress on the radius r_x is determined by the
following expression deduced from the well-known fundamental formulæ of
Lame:[1]
r0² R² + (r_x)²
T = P0 ------- · -------------
R²-r0² (r_x)²
From which we see that T is a maximum when r_x = r0, i.e., for
the layer immediately next to the bore of the cylinder. Calling t0
the internal stress in this layer, and T0 the stress resulting from
the action inside the bore of the pressure P0, and allowing that the
sum of both these quantities must not exceed the elastic limit U of the
material, we have--T0 = U - t0. And for this value of T0, the
corresponding pressure inside the bore will be
R² - r0²
P = (U - t0) ----------.
R² + r0²
This pressure increases with the term (U - t0). With t0 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 t0
has the greatest numerical value. Such stresses are termed injurious
or detrimental stresses. With t0 negative, the strength of the
cylinder increases with the numerical value of t0, 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.
[Footnote 1: Lame holds that in a homogeneous tube subjected to
the action of two pressures, external and internal, the
difference between the tension and the compression developed at
any point of the thickness of the tube is a constant quantity,
and that the sum of these two stresses is inversely proportional
to the square of the radius of the layer under consideration. Let
r0, R, and r_x be the respective radii, p0, p¹, and p_x the
corresponding pressures, and T0, T¹, and T_x, the tensions, then
we have:
T0 - p0 = T_x - p_x (1)
(T0 + p0) r0² = (T_x + p_x) (r_x)² (2)
T_x - p_x = T¹ - p¹ (3)
(T_x + p_x)(r_x)² = (T¹ + P¹)R² (4)
if the radii are known and p and p¹ be given, then deducing from
the above equations the values T0 and T¹, and also the variable
pressure p_x, we determine--
p0 r0²(R² + (r_x)²) - p¹ R²((r_x)² + r0²)
T_x = ------------------------------------------
(R² + r0²) (r_x)²
This is the formula of Lame, from which, making p¹ = 0, we obtain
the expression in the text.]
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--
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)² (r_x)² + R²
(t_x)¹ = P0 ------------ . ------------
R² - (r0)² (r_x)²
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
/ r0 (r_x)² + R² \
t_x = T ( 1 - ---------- · ------------- ) (2)
\ R0 + r0 (r_x)² /
The greatest value t_x = t0 corresponds to the surface of the bore
and must be t0 = -T, therefore
r0² + R²
--------------- = 2
r0 (R + r0)
whence P0 = T sqrt(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.