R r_x - Rr
t_x = T -------- . ---------- (3)
R + r0 (r_x)²
Substituting successively r_x = r0 and r_x = R, we obtain
expressions for the stresses on the external and internal radii--
R - r0 R R - r0
t_R = T -------- and t_r0 = -T ---- --------
R + r0 r0 R + r0
Therefore, in a homogeneous hollow cylinder, in which the internal
stresses are theoretically most advantageous, the layer situated next to
the bore must be in a state of compression, and the amount of
compression relative to the tension in the external layer is measured by
the inverse ratio of the radii of these layers. It is further evident
that the internal stresses will obey a definite but very simple law,
namely, there will be in the hollow cylinder a layer whose radius is
sqrt(R r0), in which the stress is _nil_; from this layer the
stresses increase toward the external and the internal radii of the
cylinder, where they attain a maximum, being in compression in the
internal layers and in tension in the external ones.
The internal pressures corresponding to these stresses may be found by
means of very simple calculations. The expression for this purpose,
reduced to its most convenient form, is as follows:
R / R \ / r0 \
p_x = T -------- ( --- - 1 ) ( 1 - ----- ) (4)
R + r0 \ r_x / \ r_x /
In order to represent more clearly the distribution of stresses and
pressures in the metal of a homogeneous ideally perfect hollow cylinder,
let us take, as an example, the barrel of a 6 in. gun--153 mm. Let us
suppose T = 3,000 atmospheres; therefore, under the most favorable
conditions, P0 = 1.41 T, or 4,230 atmospheres. From Equation (1) we
determine R = 184.36 mm. With these data were calculated the internal
stresses and the pressures from which the curve represented in Fig. 1 is
constructed. The stresses developed under fire with a pressure in the
bore of 4,230 atmospheres are represented by a line parallel to the axis
of the abscissæ, since their value is the same throughout all the layers
of metal and equal to the elastic limit, 3,000 atmospheres. If, previous
to firing, the metal of the tube were free from any internal stresses,
then the resistance of the tube would be
R² - r²_0
P0 = U ----------- ,
R² + r²_0
or 2,115 atmospheres--that is, one-half that in the ideally perfect
cylinder. From this we perceive the great advantage of developing useful
initial stresses in the metal and of regulating the conditions of
manufacture accordingly. Unless due attention be paid to such
precautions, and injurious stresses be permitted to develop themselves
in the metal, then the resistance of the cylinder will always be less
than 2,115 atmospheres; besides which, when the initial stresses exceed
a certain intensity, the elastic limit will be exceeded, even without
the action of external pressures, so that the bore of the gun will not
be in a condition to withstand any pressure because the tensile stress
due to such pressure, and which acts tangentially to the circumference,
will increase the stress, already excessive, in the layers of the
cylinder; and this will occur, notwithstanding the circumstance that the
metal, according to the indications of test pieces taken from the bore,
possessed the high elastic limit of 3,000 atmospheres.
[Illustration: Fig. 1]
In order to understand more thoroughly the difference of the law of
distribution of useful internal stresses as applied to homogeneous or to
built-up cylinders, let us imagine the latter having the external and
internal radii of the same length as in the first case, but as being
composed of two layers--that is to say, made up of a tube with one hoop
shrunk on under the most favorable conditions--when the internal radius
of the hoop = sqrt(R v0) or 118.7 mm., Fig. 2, has been traced,
after calculating, by means of the usual well known formulæ, the amount
of pressure exerted by the hoop on the tube, as well as the stresses and
pressures inside the tube and the hoop, before and after firing. A
comparison of these curves with those on Fig. 1 will show the difference
between the internal stresses in a homogeneous and in a built-up
cylinder. In the case of the hooped gun, the stresses in the layers
before firing, both in the tube and in the hoop, diminish in intensity
from the inside of the bore outward; but this decrease is comparatively
small. In the first place, the layer in which the stresses are = 0 when
the gun is in a state of rest does not exist. Secondly, under the
pressure produced by the discharge, all the layers do not acquire
simultaneously a strain equal to the elastic limit. Only two of them,
situated on the internal radii of the tube and hoop, reach such a
stress; whence it follows that a cylinder so constructed possesses less
resistance than one which is homogeneous and at the same time endowed
with ideally perfect useful initial stresses. The work done by the
forces acting on a homogeneous cylinder is represented by the area _a b
c d_, and in a built-up cylinder by the two areas _a' b' c' d'_ and _a"
b" c" d"_. Calculation shows also that the resistance of the built-up
cylinder is only 3,262 atmospheres, or 72 per cent. of the resistance of
a homogeneous cylinder. By increasing the number of layers or rows of
hoops shrunk on, while the total thickness of metal and the caliber of
the gun remains the same, we also increase the number of layers
participating equally in the total resistance to the pressure in the
bore, and taking up strains which are not only equal throughout, but are
also the greatest possible. We see an endeavor to realize this idea in
the systems advocated by Longridge, Schultz, and others, either by
enveloping the inner tubes in numerous coils of wire, or, as in the
later imitations of this system, by constructing guns with a greater
number of thin hoops shrunk on in the customary manner. But in wire
guns, as well as in those with a larger number of hoops--from four to
six rows and more--the increase in strength anticipated is acknowledged
to be obtained in spite of a departure from one of the fundamental
principles of the theory of hooping, since in the majority of guns of
this type the initial compression of the metal at the surface of the
bore exceeds its elastic limit.[3] We have these examples of departure
from first principles, coupled with the assumption that initial stresses
do not exist in any form in the metal of the inner tube previous to the
hoops having been shrunk on; but if the tube happen to be under the
influence of the most advantageous initial stresses, and we proceed
either to hoop it or to envelope it with wire, according to the
principles at present in vogue, then, without doubt, we shall injure the
metal of the tube; its powers of resistance will be diminished instead
of increased, because the metal at the surface of the bore would be
compressed to an amount exceeding twice its elastic limit. An example of
injury inflicted in this way is to be found in the method adopted for
hooping cast iron tubes cast by Rodman's process. If we take into
consideration the undoubted fact of the existence to a considerable
extent of useful initial stresses in these tubes, then the hoops should
be put on them either with very little shrinkage or none at all, whereas
ordnance authorities everywhere have applied to this case methods which
are only correct for tubes which are free from initial stresses.
[Footnote 3: In certain cases this, of course, may be an
advantage, as, for instance, when the inner tube is under
injurious initial stresses; but then, in order to be able to
apply the necessary shrinkage, we must know the magnitude of
these stresses.]
[Illustration: Fig. 2]
During the process of hooping guns it is very important to know how to
take into account the value and mode of distribution of the prejudicial
stresses in the inner tube, should such exist. Knowing these stresses,
it is possible, by regulating the tension of the hoops, to reduce the
compression of the metal at the surface of the bore to the proper
extent, thus doing away with the previously existing tension, and by
that means removing a source of weakness in the tube. In precisely the
same way in the shrinkage of gun hoops attention must be paid to the
character and value of the stresses which arise in the course of their
manufacture; otherwise it will be impossible to hoop the barrel
throughout in a proper manner. If prejudicial stresses exist in the
metal of a hoop before it is put in its place, then, when the gun is
fired, if it had been shrunk on with the degree of tension usually
allowed, the layer situated in the internal radius will be extended
beyond admissible limits, thereby causing the resistance of the gun to
be less than that prescribed.[4]
[Footnote 4: When the inner tube is strengthened by means of
wire, the initial or natural stresses in the latter may be
neglected on account of its thinness; but when the thickness of
the hoops is reduced, and the number of layers thereby increased,
then the value of the initial stresses in these hoops is a very
important factor with respect to the decrease or increase Of the
powers of resistance of the gun.]
It is evident, from what has been said, that in order to determine
precisely the resistance of hollow cylinders to internal pressures, and
to make the correct calculations for hooping tubes, it is absolutely
necessary to know whether internal initial stresses exist in the tube
and in the hoops, and to ascertain what their nature and intensity may
be--that is to say, whether they are useful or detrimental; yet it is
incontestable that in the construction of modern ordnance no attention
has been paid to the investigations indicated. If it be possible to
ignore these considerations in the manufacture of guns of small caliber,
and where the thickness of metal is not sufficiently great to admit of
strongly developed internal stresses, such is by no means the case with
the colossal and costly weapons of the present day. In these the
thickness of metal in the tube and hoops is very great; hence the
extreme probability of very considerable internal stresses developing
themselves. That the strength of large guns is often far below that
anticipated is demonstrated, year by year, by the repeated cases of
failure. Consciousness as to the want of strength in such guns is made
evident by the precautionary measures as to their use everywhere
adopted. The heavy artillery produced in the gun factories of Europe is
constructed with all the skill, science, and experience which engineers
and artillerists can command, and therefore it would seem that instances
of defective strength should not arise. Such cases, however, do occur
everywhere, and irresistibly give rise to the suspicion that not only is
the system of construction of guns of large caliber faulty, but also
that the conditions of their manufacture must be considered as
defective. Bearing in mind the enormous sums of money expended by every
nation in order to secure an armament of completely trustworthy guns,
this question demands speedy and searching investigation. The first step
in this direction is the study of the internal stresses inherent in the
metal; because, if such exist, and are capable of attaining, under
certain conditions, considerable magnitudes, then it is absolutely
necessary to take advantage of them in order to increase the resistance
of the metal, instead of allowing them to act to its detriment.
The study of natural internal stresses is of importance, not only with
reference to gun making, but also in respect of other structures where
great resistance is required. All have heard of the sudden failure of
crank shafts and piston rods, of the bursting of boiler shells and
tubes, of the breaking of tires, etc. In the majority of cases the
investigations into the causes of such sudden failures have not led to
any definite results. It has usually been found that the metal possessed
a satisfactory elastic resistance, and satisfied all the conditions set
down in the specifications. Had attention been paid during these
investigations to the state of the internal stresses in the metal, the
cause of unlooked-for accidents might have been explained, and steps
would consequently have been taken to avoid them in future.
We are also familiar with the development of considerable internal
stresses in various kinds of steel articles which are subjected to
hardening and tempering; for example, as dies, tools of various
description, sword blades, and thin plates rolled at a low temperature
or subjected to cold hammering. In the foundry the appearance of
internal stresses is of still more frequent occurrence. The neglect of
certain practical rules in casting, and during the subsequent cooling,
leads to the spontaneous breakage of castings after a few hours or days,
although taken out of the sand apparently perfectly sound. Projectiles
for penetrating armor plate, and made of cast steel, as well as shells
which have been forged and hardened, and in which the metal possessed an
ultimate resistance of over twelve thousand (12,000) atmospheres, with
an elastic limit of more than six or seven thousand atmospheres, will
crack to a serious extent, and even break up in the lathe, while the
recess for the copper ring is being turned out. In shell of this nature,
as well as in chilled cast iron shell, the heads are apt to fly off
spontaneously either while they are lying in store or during transport.
Such phenomena, it seems to me, demonstrate the existence of internal
stresses of considerable magnitude in the metal of the projectiles, and
it is highly probable that the manufacture of many articles would have
approached nearer to perfection had more attention been bestowed upon
the study of the internal stresses which they were liable to. Having
thus explained the nature and importance of the subject, I will proceed
to describe the experiments which I have made with a view to its
illustration.--_London Engineer._
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