being placed in position the hoop may prematurely grip on the
gun and may consequently have to be sacrificed by cutting it off
and shrinking on another.
The dilatation must be so adjusted that the required temperature to obtain it is not higher than that used for annealing the forging, otherwise the effect of this annealing will be modified. There is, therefore, for this reason, considerable risk in shrinking up long hoops of small diameter.
Before heating hoops of large diameter two or three narrow reference bands are turned on the exterior and their diameter measured; special gauges are prepared to measure these plus the dilatation required. After heating the hoop but before shrinking it, the diameter of the reference bands when tested by these gauges should not be in excess of them. The temperature can then be easily ascertained by dividing the dilatation by the coefficient of expansion of steel per degree F. or C., taking of course the diameter into account.
For small hoops this method is not convenient, as the hoop cools too quickly; the dilatation must then be obtained by ascertaining the temperature, and this is best done by the use of some form of pyrometer, such as a Siemens water pyrometer, before the hoop is withdrawn from the furnace.
It may also be desired to obtain a given striking energy or velocity at some definite range — then, the weight of the projectile being decided upon, the muzzle velocity is found from the formulas (see Ballistics) given in Exterior Ballistics. From this and the length of the gun allowable the designer has, with the aid of former experience and the formulas given in Internal Ballistics, to decide on the weight and nature of the powder charge necessary and the internal dimensions of the powder chamber and bore. These data are used to plot what is termed a “gun makers' curve,” i.e. the curve of pressures along the bore which the powder charge decided upon will give. The factor of safety and the maximum allowable stress of the steel forgings or steel wire also being known, the necessary strength of each section of the gun can be easily found and it remains to so proportion each part as to conform to these conditions and to meet certain others, such as facilities for manufacture, which experience only can determine.
When the second course consists of a single long tube into which a tapered barrel is driven, as in the system adopted by the English government, the two tubes are treated as a single tube equal in thickness to the two together; but when the second course consists of several tubes shrunk on to the barrel the additional strength, obtained by the initial tension of the shrunk tubes, is sometimes taken account of in the calculation, or the two may be treated as one thick tube.
The gun makers' formulas for the strength of the gun are obtained from considering the strength of a thick cylinder exposed to unequal internal and external pressures. Supposing a transverse section of the gun to cut through n tubes, the internal radius of the barrel is r0 in., the external radius ri in., the external radius of the second course is ra and so on; and the external radius of the jacket is r„. Then if T = a circumferential stress (tension) in tons per square inch, Tn = a circumferential stress at radius r„ in., P = a radial stress (pressure) in tons per square inch, and P„ = a radial stress at radius rn in., the formulas used in the calculation of the strength of built-up guns are as follows:—
T＝Pn−1-ir„ i^-P„r„', r„, V„= P„, -P„
P＝r- r„—r^, '-?-„"-r„, -^
where r is any intermediate radius in the thickness of a tube
| Tn−Pn＝T−P | (3) |
in the same tube; also the pressure between the (n−1)th and nth
hoops is
| Pn−1＝rn2 − rn−12rn2 + rn−12 (Tn−1 + Pn) + Pn | (4). |
Equation (4) is usually known as the Gunmakers’ formula and from it, when Pn and Tn−1, Tn−2 . . . are known the other pressures can be found. The proof tension of the material is kept well below the yielding stress. For ordinary carbon gun steel it is usual to consider that the proof tension of the barrel should not exceed 15 Ions and of the outer hoops 18 tons per square inch; with nickel gun steel these become 20 tons and 24 tons respectively. If the h"" hoop is the exterior tube then P„ = o; neglecting the atmospheric pressure.
In all gun calculations for strength three cases must be considered:
(a) When the built-up gun is fired, the stress is called the Firing Stress and is obtained by the repeated use of equation (4);
(b) When the gun, supposed to be a solid homogeneous block of metal is fired, the stress is termed the Powder Stress and is obtained from the equations (1) and (2);
(c) When the built-up gun is in repose, the stress is then called the Initial Stress or Stress of Repose.
Between these three cases the following relations hold:—
Initial Stress + Powder Stress＝Firing Stress (5).
It is best to use different symbols to distinguish each kind of stress. We will use for the Firing Stress P, T; for Powder Stress p, t; and for the Initial Stress (p), (t).
The method of working will be illustrated by a practical example. Take, for instance, a section across the chamber of a 4·7-in. Q.F. gun, for which the diameter of the chamber is 5 in., that of the barrel 8·2 in., and the external diameter of the jacket 15 in.
Here
r0＝2·5; r,＝4·1; r2＝7·5
T0＝15; T1＝18; P2＝0.
From (4) for the Firing Stress
P1＝(7·5)2 − (4·1)2(7·5)2 + (4·1)2×18＝9·72 tons per square inch.
P0＝(4·1)2 − (2·5)2(4·1)2 + (2·5)2×(15+9·72)+9·72＝21 tons per square inch.
From (3) the tension T′n of the outer fibres of the hoops is obtained; thus
T′2 = P2+T1−P1＝18−9·72 = 8·28 tons per square inch.
T′1 = P1+T0−P0＝9·72+15−21 =3·72 tons per square inch.
For any intermediate radius r the stress can be found by using equations (l) and (2) or (l) or (2) and (3).
For the Powder Stress equations (1) and (2) are used by putting n＝1, and then p1＝0 (also remembering that, as there are two hoops, the outer radius must be written r2); the formulas become
t＝...
(6)
(7).
When r = r0 = 2·5, t =t0, p0 = P0 already found and:
t0＝(7·5)2 + (2·5)2(7·5)2 − (2·5)2×21＝26·25 tons.
For the tension of the fibres at the outer circumference
t2＝26·25 − 21＝5·25 tons,
from (3) and for a radius r2 = 7·5 inches.
The stress for any intermediate radius r can be obtained from (6) and (7) or, from (6) or {7) and (3).
Subtracting the Powder Stress from the Firing Stress the Initial Stress is obtained, and the various results can be tabulated as follows:—
| At Radius. | Tensions. | Pressures. | |||||
| Firing Stress. |
Powder Stress. |
Initial Stress. |
Firing Stress. |
Powder Stress. |
Initial Stress. | ||
| Barrel | r0＝2·5 | 15·0 | 26·25 | −11·25 | 21·0 | 21·0 | 0 |
| r0＝4·1 | 3·72 | 11·57 | −7·85 | 9·72 | 6·32 | 3·4 | |
| Jacket | r0＝4·1 | 18·0 | 11·57 | 6·43 | 9·72 | 6·32 | 3·4 |
| r0＝7·5 | 8·28 | 5·25 | 3·03 | 0 | 0 | 0 | |
It is generally stipulated that the initial compression of the material at the interior surface of the barrel shall not exceed 26 tons per square inch, i.e. (t0) =−26 tons; in the example above (t0) =−11·25 tons only, but in wire-wound guns special attention to this condition is necessary.
It now remains for the designer so to dimension the several hoops that they shall, when shrunk together, give the stresses found by calculation. To do this the exterior diameter of the barrel must be a little larger than the interior diameter of the covering hoop; after this hoop is shrunk on to the barrel its exterior diameter is turned in a lathe so that it is slightly larger than the interior of the next course hoop and so on. It will be seen that the fibres of the barrel must be compressed while the fibres of the superimposed hoop are extended, and thus produce the Initial Stress. The shrinkage S may be defined as the excess of the external diameter of the tube over the internal diameter of the hoop, when separate and both are in the cold state. Then
