Page:EB1911 - Volume 20.djvu/244

ii. _𝑛S_𝑛+1 denotes the shrinkage between the 𝑛^𝑡ℎ and (𝑛 + 1)^𝑡ℎ hoops.

𝑛S𝑛+1	＝2𝑟𝑛/M{ (𝑡𝑛 − 𝑡′𝑛)}	(8)
	＝2𝑟𝑛/M [ (𝑡𝑛 − 𝑡𝑛−1) + 𝑟𝑛2 − 𝑟𝑛2−1/𝑟𝑛2 + 𝑟𝑛2−1 {(𝑡𝑛−1) + (𝑝𝑛)} ]	(9).

Here M can be taken as 12,500 tons per square inch for gun steel. In the example already calculated the shrinkage between the jacket and barrel is 0·009 in.

S＝2 × 4·1/12,500 [ 6·43 + 11·25 + (4·1)² − (2·5)²/(4·1)² + (2·5)²(−11·25 + 3·4) ]＝0·009 in.

In that portion of the gun in which wire is used in the construction, exactly the same principles are involved. It may be assumed that the tube on which the wire is wound is so large, in comparison to the thickness of the wire, that the compression of the concave surface of the wire and the extension of its convex surface may be neglected without Wire guns. sensible error.

The greatest advantage is obtained from the wire coils when in the Firing Stress the tension T is uniform throughout the thickness of the wiring. The Firing Stress T in the wire may be as low as 25 tons per square inch and as high as 50 tons, but as the yielding strength of the wire is never less than 80 tons per square inch nor its breaking strength less than 90 tons, there is still an ample margin especially when it is remembered that the factor of safety is included in the calculation.

If the wire is wound direct on to the barrel and is covered by a jacket, 𝑟₀, 𝑟₁ being the radii in inches of the barrel, 𝑟₁, 𝑟₂ the radii of the internal and external layers of wire, and 𝑟₂, 𝑟₃ the radii of the jacket; then for the Firing Stress in the wire

T(𝑟2−𝑟)＝P𝑟−P2𝑟2

(9),

or

T(𝑟−𝑟1)＝P1𝑟1−P𝑟

(10).

By combining these the gunmakers’ formula for the wire is obtained

P1＝𝑟2 − 𝑟1/𝑟1 (T + P2) + P2

(10a).

As T is to be uniform, when the gun is fired, the Initial Tensions of the wire are arranged accordingly, and the tensions at which the wire must be wound on to the guns have now to be determined.

Let θ	＝the winding tension at radius 𝑟 in.
(𝑡)	＝the initial tension at radius 𝑟 in.
(𝑝)	＝the radial pressure between any two layers of wire at radius 𝑟 in.

It is assumed that M is uniform for the gun steel and wire.

Then

θ＝(𝑡) + (𝑝)𝑟2 + 𝑟0 2/𝑟2 − 𝑟02

(11),

where

(𝑡)＝T − P0𝑟02/𝑟2 𝑟32 + 𝑟2/𝑟32 − 𝑟02

(12),

and

(𝑝)＝P − P0𝑟02/𝑟2 𝑟32 − 𝑟2/𝑟32 − 𝑟02

(13).

By means of these two equations and (9) the expression (11) becomes

θ＝E/𝑟 + F/𝑟 − 𝑟0 + G/𝑟 + 𝑟0

(14),

where

E＝−(T+P₂)𝑟₂

F＝(T + P₂)𝑟₂ − (T + P₀)𝑟₀

G＝(T + P₂)𝑟₂ + (T + P₀)𝑟₀

To compare with the previous example, the stress for a 4·7-in. Q.F. wire gun will be calculated. This consists of a barrel, intermediate layer of wire and jacket.

Here 𝑟₀＝2·5; 𝑟₁＝3·75; 𝑟₂＝5·5; 𝑟₃＝7·5 inches; the firing tension T₁ to T₂ of the wire＝25 tons per square inch, suppose.

Take P₀ = 21 tons per square inch and consider that the jacket fits tightly over the wire, but has no shrinkage. Then for the Firing Stress, from (2), P₂＝2·25 tons,
and from (9) and (10), T₁(𝑟₂-𝑟₁) = P₁𝑟₁-P₂𝑟₂

P₁＝14·97, say 15 tons;

from (4) we can obtain T₀ and T₂ since P₀, P₁ and P₂ are known; from (3) T₀ = 0·6 tons. T₂ = 7·5 tons.

T′₂＝−5·4 tons (a compression),

and

T₃＝5·25 tons.

The Powder Stress is obtained in the same way as in the previous example, so also is the Initial Stress; therefore we may tabulate as follows:—

At Radius.			Tensions.			Pressures.
			Firing Stress.	Powder Stress.	Initial Stress.	Firing Stress.	Powder Stress.	Initial Stress.
Barrel	${\Big \{}$	𝑟0＝2·5	0·6	26·25	−26·25	21·0	21·0	0
		𝑟1＝3·75	−5·4	13·125	−18·125	15·0	7·875	7·125
Wire	${\Big \{}$	𝑟0＝3·75	25·0	13·125	11·875	15·0	7·875	7·125
		𝑟2＝5·5	25·0	7·125	17·5	2·25	2·25	0
Jacket	${\Big \{}$	𝑟2＝5·5	7·5	7·5	0	2·25	2·25	0
		𝑟3＝7·5	5·25	5·25	0	0	0	0

As the wire is wound on, the pressure of the external layers will compress those on the interior, thus producing an extension in the wire which is equivalent to a reduction in the winding tension θ of the particular layer at radius 𝑟 considered. If τ represents this reduction then

θ＝(𝑡) − τ,

where

τ＝−𝑟_𝑛² + 𝑟₀²/𝑟² − 𝑟₀²(𝑝)

At the interior layer of wire τ is the initial stress on the exterior of the barrel and the winding tension must commence at

θ＝ 11·875 + 18·525＝30·4 tons per square inch.

As the jacket is supposed to have no shrinkage T＝0 and consequently

θ＝(𝑡)＝17·5 tons per square inch.

These winding tensions can be found directly from formula (14) and then

E＝−149·875; F＝34·875; G＝264·875.

Sir G. Greenhill has put these formulas, both for the built-up and wire-wound guns, into an extremely neat and practical geometrical form, which can be used instead of the arithmetical processes; for these see Text-Book of Gunnery, Treatise of Service Ordnance, 1893, and Journal of the United States Artillery, vol. iv.

The longitudinal strength of the gun is very important especially at the breech end; along the forward portion of the gun the thickness of the barrel and the interlocking of the covering hoops provide ample strength, but at the breech special provision must be made. It is usual to provide for this by means of a strong breech piece or jacket in small guns orLongitudinal stress. by both combined in large ones. Its amount is easily calculated on the hypothesis that the stress is uniformly distributed throughout the thickness of the breech piece, or jacket, or of both. If 𝑟₀ is the largest radius of the gun chamber, 𝑟₀₁ the radius of the obturator seating, 𝑟₁ the external radius of the barrel, and P₀ the maximum powder pressure, then, with the usual form of chamber adopted with guns fitted with obturation other than cartridge cases, there will be a longitudinal stress on the barrel at the breech end of the chamber due to the action of the pressure P₀ on the rear slope of the chamber, of

π/4(𝑟₀² − 𝑟₀₁²)P₀ tons

this is resisted by the barrel of section π/4 (𝑟₁²−𝑟₀²) so that the resistance

R＝𝑟₀²−𝑟₀₁²/𝑟₁²−𝑟₀² P₀ tons.

This portion of the longitudinal stress is not of great importance as the breech end of the barrel is supported in all modern designs by the breech bush. In Q.F. guns, i.e. those firing cartridge cases, the breech end of the chamber has the largest diameter, and 𝑟₀−𝑟₀₁ so that there is no longitudinal stress on the chamber part of the barrel.

For the breech piece or outer tube of radii 𝑟₁ and 𝑟₂, the resistance

R＝𝑟₀₁²/𝑟₂²−𝑟₁² P₀ tons for B.L. guns
 ＝𝑟₀²/𝑟₂²−𝑟₁² P₀ tons for Q.F. guns

If the longitudinal stress is taken by a jacket only, the resistance is found in the same way.

Generally for ordinary gun steel, the longitudinal stress on the material is always kept below 10 tons per square inch or 13 tons for nickel steel; but even with these low figures there is also included a factor of safety of 1·5 to 2. In large guns it is best to consider the jacket as an auxiliary aid only to longitudinal resistance, as, owing to the necessary connexions between it and the breech bush and its distance from the centre of pressure, there is a possibility that it may not be taking its proportionate share of the stress.

The thread of the breech screw and of the breech bush (or opening) must be so proportioned as to sustain the full pressure on the maximum obturator area; V or buttress shaped threads are always used as they are stronger than other forms, but V threads have the great advantage of centring the breech screw when under pressure.

In most modern B.L. guns fitted with de Bange obturation the