PHILOSOPHICAL SOCIETY OF WASHINGTON 
BULLETIN VOL. XIII, pp. 87-102, PI. 6 
STEEL CYLINDERS FOR GUN CONSTRUCTION 
STRESSES DUE TO INTERIOR COOLING 
BY 
ROGERS BIRNIE 
Read befoke the Philosophical Society of Washington 
May 11, 1895 
WASHINGTON 
PUBLISHED BY THE SOCIETY 
June, 1895 BULL. PHIL. SOC. WASH. 
1895, VOL. 13, PLATE 6. 
Measured Stresses in hollow forging treated by interior cooling 
And computeij elastic resistance to interior pressure 
Before Annealing 
Breech Section 
Breech Middle- SEbrioN 
Muzzle Middle Section 
Muzzle Section 
FORGjNG REDUCED SCALE 
W6T. 690 LBS. 
Curves 
dhow the measured stresses in the /?vyj«y 
" corj-es/iondtny stresses deduced by i/ieo-y. 
*' stresses /or the stcde o/'ccct/on (/&, meaccmum) 
j /S/torr etroun areas /or the sta/e o/~rest. 
1 .. .# ” " •• •• •'action 
| Common to both strain areas. 
BreegH Section 
Areas 
After Annealing 
Muzzle Section 
STRESSES IN GUN FORGINGS. STEEL CYLINDERS FOR GUN CONSTRUCTION- 
STRESSES DUE TO INTERIOR COOLING. 
BY 
Rogers Birnie. 
[Read before the Society May 11,1895.] 
This experiment is the first important step taken by the 
Ordnance Department of the United States Army to inves- 
tigate the merits of making cannon from a single steel forg- 
ing with initial tension produced by interior cooling. The 
experiments of similar import with reference to built-up or 
hooped steel guns which were made by this department in 
1884-’6 established on a ver}*- firm basis the manufacture in 
this country of that description of gun, and it is not im- 
probable that the present investigation may lead to equally 
important results for the single forging construction. 
The inception of this work is due to Captain Frank Hobbs, 
of the Ordnance Department, and the experimental forging 
was furnished, through his instrumentality, by the Bethle- 
hem Iron Co. The subject has received attention in other 
places, particularly in the interesting work of General 
Nicholas Kalakoutsky,* of the Russian artillery, but up to 
this time guns have not been made upon the plan proposed. 
At Le Creusot, France, interior cooling has, however, been 
used to improve the condition of cylinders used in built-up 
guns. As to the treatment of the forging, it may suffice to 
say that its preparation was similar to that of a forging in- 
tended for a hooped gun, by the usual methods of casting, 
*Kalakoutsky (General Nicholas). Investigations into the internal 
stresses of cast iron and steel. London, George Reveirs, 1888. 
13-Bull Phil. Soc., Wash., Vol. 13. 
87 88 
BIRNIE. 
forging, annealing, oil-tempering and annealing. In the 
final annealing, while still in the annealing furnace and 
uniformly heated to redness, water was passed into and 
through the bore of the forging until it was cool enough to 
handle. 
The several circular sections shown in the drawings, each 
about 0.5 of an inch thick, were then cut from different parts 
of the forging to ascertain the strains in concentric element- 
ary cylinders of which the section may be conceived to be 
composed. Each section was marked to be divided into a 
number of circular rings about 0.15 of an inch in radial 
thickness. Before cutting out these rings datum points were 
marked on the face of each to measure two diameters at 
right angles. The rings were removed consecutively and 
measurements of the diameters made at each stage of the 
operation. The change in diameter of a ring on being re- 
leased from the section is taken as a measure of the circum- 
ferential strain or stress to which it was subjected in the 
forging. A ring which expands on being released was evi- 
dently under circumferential compression in the forging, and 
one which contracts was under tension. The datum points 
for the curves of initial tension shown in the figures are de- 
rived from the difference in the original diameters of the 
rings and their diameters after release. As seen in these 
curves, the compression is greatest at or near the surface of 
the bore, whence it gradually decreases to zero at the neutral 
point. At this point the strains of tension begin and in- 
crease gradually toward the exterior of the cylinder. The 
strains of compression and extension are in equilibrium. 
Duplicate sections from the breech and muzzle ends of the 
forging are illustrated, the originals being taken directl}' 
after the treatment by interior cooling and tbe duplicates 
when the parts of the forging to which they belonged had 
been subjected to partial annealing. The object of this 
treatment was to show how the strains originally produced 
could be controlled and ameliorated, if necessary, by anneal- 
ing a forging after interior cooling. STRESSES IN GUN FORGINGS. 
89 
The accuracy of the results is of course dependent upon, 
the measurements of diameters. Of this, however, there is 
every indication that proper care and skill were exercised. 
The measurements were made with a micrometer scale and 
read to the fractional part of TFJ7(r °f au inch. The sensi- 
tiveness of the results is such that an error of towo' °f an 
inch in the reading of an average diameter (five inches) cor- 
responds to 600 pounds per square inch in the expressed 
stress of tension or compression. 
Deductions from the Application of the Formulas for Gun 
Construction.—These and other similar experiments show 
the favorable condition of strains produced in a hollow forg- 
ing by interior cooling. The strains are analogous to those 
produced by shrinkage in the built-up construction and serve 
the same purpose. The present experiments are particularly 
instructive in that they deal with a hollow forging of vary- 
ing thickness of wall, with sectional dimensions correspond- 
ing to the service field gun. 
The strains directly produced by the treatment are found 
to be more intense than is necessary and show the desira- 
bility of an amelioration of that treatment in future opera- 
tions ; but the strains left after annealing the treated forging 
are moderate and satisfactory. The elastic resistance of the 
sections, both before and after annealing, is shown to be 
superior to that of corresponding sections of the built-up 
field gun. This, however, is in part attributable to higher 
qualities of metal. 
The physical qualities of the forging, determined from 
tensile-test specimens taken from it after treatment, are as 
follows: 
Breech end. 
Muzzle end. 
Elastic limit, pounds per square inch 
68,090 
75,500 
Tensile strength, “ 
u u u 
126,500 
128,400 
Ultimate extension, 
per cent 
9.50 
11.625 
Reduction of area. 
U U 
12.14 
16.35 
For present purposes the elastic limit for extension will be 
taken at a reduced value, 6 = 60,000, and the elastic limit 
for compression will be defined in each section by the actual 90 
BIKNIE. 
measurements made, the highest being p = 78,280 at the 
bore of the breech middle section. It will be understood 
that the stated measured stresses correspond to the measured 
strains per inch for a modulus of elasticity, E = 30,000,000 
pounds. For example, the value of p just stated is derived 
as follows: 
f = 80.000.00° = 78,280 (stress). (1) 
The sections taken for examination are the breech, breech 
middle, muzzle middle, and muzzle before annealing, and 
the breech and muzzle after annealing. In the dimensions 
of these sections we have nearly the counterpart of four 
principal cross sections of the 3.2-inch field gun. 
The principal objects of discussion will be— 
1. To compare the measured stresses in the forging after 
treatment with those anticipated by theory and required to 
make the resistance to interior pressure a maximum. This 
will show the degree of uniformity in the actual stresses and 
how nearly they conform to the requirements of the law for 
maximum resistance. 
2. Taking the actual stresses as measured in each section, 
to determine the elastic resistance of the section to interior 
pressure. This, while admitting every irregularity of the 
stresses or strains induced by the treatment, will give a final 
measure of its efficacy. 
The formulas* to be applied, which are fundamentally 
the same as those for the built-up construction, relate to a 
gun or cylinder made of a single piece, with initial tension 
produced by interior cooling. 
= * TST? = “ p (2) 
(t)3-1 
p _ 3 (i?12 -R02) P /q\ 
U (4+ 2 JR*) - 3 (jy - Itf) a 
* For these formulas see Gun Making, Appendix B, Military Service 
Institution Monograph, 1888. STRESSES IN GUN FORGINGS. 
91 
In these equations Pu is the interior pressure per square 
inch which, if applied, would produce an uniform stress, 
9U, in the whole thickness of the wall of the cylinder, de- 
pending only upon the dimensions of the cylinder and the 
initial compression p. 
0 — p _l 4 R? P02 ~1 p 
l3 (Px2 — Rf) 3r2 (Px2 — P02) J l j 
In this 9 is the increase of stress caused at radius r by the 
application of any interior pressure, P, within the limit of 
elastichy of the cylinder. 
0 P -^Q2 % I 4 Rx2 R02 p . „ /r\ 
p ~ [_3 (Pt2 — P02) + 3r2 (Pf2 — P02) J1 “ + " ( } 
In this p is the relief of stress at radius r, with given values 
of Pu and 0U, under the assumption that the pressure Pu 
has been applied and is withdrawn. 
Replacing 9u in (5) by its value as expressed in (2) and 
designating by px a stress at radius ru the ratio of the stresses 
at two different points in the wall may be expressed by 
r 1 P02 2 R*R02 -| 
(&A f _ Xi t Xo (A2 —^o2) r2 
p 1 P02 2 R'Rf Pl [) 
| _ P,*- P02 (P^-P,,2)r,2 
From this the stress p at a given radius, r, may be deter- 
mined when px for the radius rx is given. The symbols p 
and p1 in this equation may express either compression or 
tension. 
The radius of the circle on which should be found the 
neutral point of every curve of stress incident to the system 
at rest is found from the equation 
/l©!z1 92 
BIRNIE, 
Additional formulas deduced from those given in Ap- 
pendix B, “ Gun Making,” can be given to determine, first, 
the changes in the stresses at given radii due to reaming 
out or enlarging the bore of a cylinder under initial ten- 
sion ; second, the changes due to turning off or reducing the 
exterior of the cylinder. Such formulas would be useful in 
the practical working of this method, but it will be sufficient 
to state here that any reduction of the thickness of wall 
should cause a lowering of the initial strains or stresses. 
This result has not been uniformly shown in the present 
experiments. Exceptions may be noted in the earlier stages 
of dismantling the breech and the breech middle sections. 
In these cases it would appear that the metal near the sur- 
face of the bore was overcompressed by the treatment, and 
local strains were produced which became manifest when a 
part of the metal was removed. 
To explain the application of the formulas we may take, 
for example, the breech section before annealing. 
Stress Curves for the State of Rest.—Taking the measured 
compression, 71,260 pounds, on the diameter, 3.81 inches, as 
a basis for constructing the “deduced” curve of stresses, we 
first find the corresponding compression at the surface of 
the bore from equation (6), in which 
r=R0 — 1.8, rx = 1.905, R1 = 4.86, Pl = 71260. 
Then : 
1.0647 — 0.15898 — 2.0695. _ 1.41228 w 71 oen _oc „ 
Po 1.0647 — 0.15898 — 1.5446 Pl 1.16378 X 71260 86477 lbs- 
Next, applying equations (3) and (2) with p0 given, we 
find: 
p, 3 X 20.38 61.14 Q('Air7 qia 
P- = 100.96 — 43.41 p-= X 86477 = 91870 Pounds' 
9a = aPa = 0.70997 X 91870 = 65225 pounds. 
These latter values express the theoretical condition, 
assumed only for auxiliary purposes, that, having a com- STRESSES IN GUN FORGINGS. 
93 
pression of 71,260 pounds at the intermediate radius, 1.905 
inches, an applied interior pressure of 91,870 pounds would 
produce throughout the whole wall of the cylinder an uni- 
form tension of 65,225 pounds per square inch. 
The remaining points of the deduced curve of stresses for 
the state of rest are now derived by equation (5). Having 
0U = 65,225 and Pu = 91,870, we find : 
459970 ' 
P == 55488 — -—5— 
r 
in which, by substituting the several values of r, there re- 
sults : 
D0= 3.6, R0 = 1.8: P = 55488 —141965 = — 86477 
d = 3.81, r = 1.905: P = 55488 — 126750 = — 71260 
d = 4.41, r = 55488— 94604 = —39116 
(Proof.) 
&c., &c., 
as given in table A and shown on the accompanying 
plate 6. 
The “deduced” stress curves for the state of rest in the 
remaining sections considered are derived in a similar 
manner. In each case the measured stress which is taken 
as a basis and so forms a common point on both the meas- 
ured and deduced curves of stress is designated (see table 
and plate 6) by figures in parentheses, as, for example 
(75,000) in the breech middle section, (59,910) in the muzzle 
middle section, and so on. 
Resistance to Interior Pressure.—The limit of elastic resist- 
ance of the metal under extension will be taken as before 
stated, 6 = 60,000. The value of P0 will then depend upon 
the condition that this limit shall not be exceeded at any 
point. By a comparison of the measured and deduced 
stress curves for the state of rest, or, if need be, by a pre- 
liminary computation, the most dangerous measured stress— 
that is to say, the one which, under the action of an interior 
pressure, would be the first to reach the limit, 0 = 60,000, 
can readily be selected; Consequently the points which 
must be taken upon which to base the value of P0 for the 
several sections are selected and designated (see table) by 94 
BIRNIE 
underlining the critical measured stress; for example, 42920 
on the diameter 9.54 inches in the breech section, and so on. 
In each case it is seen from the table that the corresponding 
stress in action is 60,000 pounds, while the stresses on other 
diameters are less than this; hence the condition is fulfilled. 
Taking again the breech section (second stage) before an- 
nealing as an example of the method of computation, the 
elastic resistance of the section and the stresses on given 
diameters for the state of action are derived as follows: 
The measured stress which in this section will first reach 
the limit, 60,000, is 0 = 42,920 on the diameter, 9.54 inches. 
The increase of stress allowable on this diameter in passing 
from the state of rest to action is therefore: 
0 = 60000 — 42920 = 17080 pounds. 
The corresponding value of P0 is then found from (4), 
with r — 4.77. 
0 = 17080 == 0.32602 P0 . • . P0 = 52390 pounds. 
The interior pressure being thus determined, equation (4) 
is further applied to determine the increase of stress at other 
given radii. For this purpose it is convenient to reduce it 
to the form by substituting known values: 
0 = 5553 -f 262300 
r 
in which, hy substituting the several values of r, we obtain 
the increase of stress for that radius, and, taking the alge- 
braic sum of this result and the measured stress at the same 
point (at rest), we have finally the stress pertaining to the 
applied interior pressure, P0 = 52,390 pounds. Thus: 
Increase. Measured. Q (action). 
d —3.81, r = 1.905: 6 = 5553 + 72280 = 77833 — 71260 = + 6573 pounds, 
d —4.41, r — 2.205: 0 — 5553 + 53949 = 59502 — 56800 = + 2702 “ 
* * * * * * * 
d = 9.54, r = 4.77: 6 = 5553 -f 11528 = 17080 + 42920 = + 60000 “ 
(Proof.) 
as given in table A and shown on the accompanying 
plate 6. 95 
STRESSES IN GUN FORGINGS. 
The radius of the neutral circle of stress for each section 
has been computed by (7) and is noted in the table. 
Table A. 
Measured and Deduced Stresses for the State of Rest and Action. 
BEFORE ANNEALING. 
f 7} — q 34 inches 
Breech section, second stage. Original diameters, | — inches' 
State of rest—stresses. 
State of action. 
Diameters. 
Measured. 
Deduced. 
Pressure. 
Stresses. 
Pounds 
Pounds 
Pounds 
Pounds 
Inches. 
persq. inch 
per sq. inch 
per sq. inch. 
per sq.inch. 
— 86,477 
—(71,260) 
+ 34 
+ 6,573 
3.81 
— 71,260 
calibers, 0.85. 
4.41 
— 56,800 
— 39,116 
+ 2,702 
5.05 
— 30,000 
— 16,658 
o 
+ 1,694 
5.70 
— 2,370 
— 1,142 
a 
■2 
+ 35,477 
6.30 
+ 14,290 
+ 9,132 
O 33 
+ 46.278 
6.95 
+ 23,960 
+ 17,396 
+ 51,235 
7 57 
+ 32,900 
+ 23,382 
10 o 
+ 56,762 
8.20 
+ 35,400 
+ 28,125 
‘-5 
+ 56,557 
8.82 
+ 36,700 
+ 31,837 
§ 
+ 55,740 
9.54 
+ 42,900 
+ 35,272 
+ 60,000 
Elastic limit. 
Exterior, 9.72 
+ 36,014 
Breech middle section, second stage. Original diam-/ A —2.80 inches. 
eters, 1A == 8.68 inches. 
* Neutral point, d = 5.76. 
Bore, 
3.20 
78,280 
90,721 
+ 
21,115 
Thickness of section in 
3.38 
— 
75,000 
— 
75,000) 
+ 
14,789 
calibers, 0.81. 
4.02 
— 
42,910 
— 
35,109 
o 
+ 
22,595 
4.04 
— 
10,070 
— 
11,209 
a 
cS 
+ 
40,140 
5 20 
+ 
7,410 
+ 
4,791 
o 
JO 
+ 
48,450 
5.90 
+ 
18,500 
+ 
10,304 
D 
+ 
52,566 
0.50 
+ 
28,380 
+ 
24,159 
o 
+ 
57,587 
7.08 
+ 
34,320 
+ 
29,934 
+ 
60,000 
Elastic limit. 
7.05 
+ 
35,490 
+ 
34,378 
Js 
+ 
58,454 
8.24 
+ 
30,760 
+ 
38,041 
+ 
51,485 
Exterior, 8.38 
+ 
31,330 
+ 
38,799 
+ 
51,593 
Muzzle middle section, second stage. Original diam- f A = 2.80 inches. 
eters, \ A = 6.44 inches. 
* Neutral point, d = 5.04. 
Bore, 
3.20 
_ 
65,300 
_ 
77,348 
, 
+ 
670 
Thickness of section in 
3.38 
— 
59,910 
— 
59,910) 
CD 
+ 
56 
calibers, 0.45. 
4.02 
— 
13,430 
— 
15,737 
+ 
31,328 
* 
i-> u 
o 
4.64 
+ 
19,070 
+ 
10,838 
d 
+ 
54,679 
5.26 
+ 
30,500 
+ 
28,583 
00 
CD 
o3 
+ 
60,000 
Elastic limit. 
5.90 
+ 
22,630 
+ 
41,352 
c3 
+ 
47,734 
Exterior 
6.06 
+ 
24,510 
+ 
43,930 
+ 
48,726 
♦ Neutral point, d= 4.354. 
14-Bull. Phil. Soc., Wash., Vol. 13. 96 
BIRNIE, 
Table A—Continued. 
BEFORE ANNEALING. 
Muzzle section, third stage. Original diameters, 
D0 = 2.80 inches. 
1)1 = 5.39 inches. 
Diameters. 
State of rest—stresses. 
State of action. 
Measured. 
Deduced. 
Pressure. 
Stresses. 
Inches. 
Bore, 3.2 
3.35 
* 
3 90 
4.46 
Exterior, 4.60 
Pounds 
per sq. inch. 
Pounds 
per sq. inch. 
— 36,748 
—(26,000) 
+ 3,378 
+ 22,836 
+ 26,808 
Pounds 
per sq. inch 
« s 
©.S3 
CO —' GO 
W<D 
Pounds 
per sq inch. 
+ 14,784 
+ 21,900 
+ 36,049 
+ 60,000 
Thickness of section in 
calibers, 0.22. 
Elastic limit. 
— 26,000 
— 1,925 
+ 28,600 
* Neutral point, d = 3.82. 
Breech section, third stage. Original diameters, 
AFTER ANNEALING. 
7)0 = 3.34 inches. 
D1 = 9.77 inches. 
Bore, 3.6 
— 29,170 
— 30,905 
+ 60,000 
Elastic limit. 
3.81 
— 25,20 ■ 
—(25,200) 
+ 55,119 
4.41 
— 31,290 
— 13,164 
o 
+ 30,330 
Thickness of section in 
5.05 
— 18,120 
— 4,752 
+ 30,443 
calibers, 0.75. 
5.70 
— 790 
+ 1,058 
° ? 
+ 38,749 
6.30 
+ 9,048 
-j- 4,906 
of u 
+ 42,613 
6.95 
+ 16,620 
+ 8,002 
10 o 
+ 45,378 
7 57 
+ 19,220 
+ 10,243 
"co 
+ 44,497 
8.20 
+ 20,600 
+ 12,020 
+ 43,119 
8.82 
+ 21,500 
+ 13,410 
+ 41,860 
Exterior, 9.00 
+ 18,500 
+ 13,761 
+ 38,315 
* Neutral point, d = 5 562. 
D0 — 2.80 inches. 
Dy — h.38 inches. 
— 20,254 
—(14,330) 
+ 59,710 
+ 60,00U 
3.35 
— 14,330 
<D 
©.2 o 
Elastic limit. 
* 
3.90 
4.46 
Exterior, 4.00 
— 3,460 
+ 10,100 
+ 1,861 
+ 12,586 
+ 14,676 
<M P 
C5 CO C$ 
<N m-n 
•Z, OG 
<D 
+ 55,467 
+ 58,825 
Thickness of section in 
calibers, 0.22. 
Muzzle section, third stage. Original diameters, 
The Possible Maximum Resistance of the Sections.—Suppose 
the iuitial tension curve to be such as is required by theory 
to give a maximum resistance, several propositions may be 
stated : 
1st. The point of critical strain may always be taken at 
the surface of the bore where the compression at rest or the 
* Neutral point, d — 3.82. STRESSES IN GUN FORGINGS. 
97 
extension in action cannot exceed the elastic limit of the 
metal. 
2d. To make the resistance to interior pressure a maxi- 
mum in any cylinder, the state of initial tension should be 
such that when the pressure acts from within the whole 
thickness of metal in the wall should be, as nearly as prac- 
ticable, uniformly strained to the elastic limit of the metal. 
3d. If p and $ be taken equal and the compression of bore 
carried to the limit p, there is but one thickness of cylinder 
(0.65 caliber, nearly)* for which a condition of uniform 
strain in action equal to the elastic limit of the metal can 
be attained. 
4th. For cylinders of greater thickness than 0.65 caliber 
a state of uniform strain in the wall will be reached in 
action and passed before the elastic limit of the metal is 
attained, and with increasing pressure this limit will be 
fully reached only at the surface of the bore, thus determin- 
ing the limit of pressure. For such cylinders the best con- 
ditions of resistance will be obtained by utilizing the full 
limit of compression of the metal in the initial tension. 
5th. But for cylinders of less thickness than 0.65 caliber 
a state of uniform strain in action equal to the elastic limit 
of the metal can be attained with a compression of bore less 
than the limit p. The thinner the cylinder the less should 
be the initial compression imposed. It follows that the 
possible maximum resistance of such cylinders will be ob- 
tained by adjusting the initial compression within limits. 
If the full limit of initial compression were given, the elastic 
limit of the metal would be reached in action at the exterior 
of the cylinder sooner than at the bore. 
6th. As a consequence, also, of the preceding, the resist- 
ance of cylinders of less thickness than 0.65 caliber, treated 
by interior cooling, should be directly proportional to the 
thickness. This treatment gives the means of imparting the 
greatest resistance so far known to such cylinders. 
*See Appendix B, “Gun Making” and “ Modern Gun Construction 
and Breech Mechanism,” Congress of Engineers, 1893. 98 
BIRNIE. 
7tli. When one point of the initial tension curve is given 
(either an assigned or measured value of p) the curve will 
be fixed, as has been illustrated in the examples worked out. 
It is important to observe that this curve as defined and 
laid down is, barring the presence of local strains in the 
metal, that which should be naturally formed under the 
conditions of equilibrium between the positive and negative 
strains in the wall of the cylinder at rest. This equilibrium 
must exist, and the curve as defined fulfills this condition, 
since it is dependent upon it. The straight line, represent- 
ing the state of uniform strain in action, used in every case 
as the datum line for the initial tension curve, is dependent 
upon the condition of equilibrium between the pressure and 
the elastic strains in the metal. That line, however, is the 
initial tension curve itself in a particular position. On the 
withdrawal (supposed) of the force P the right line falls to 
the position of the initial tension curve, having lost nothing 
of its property to express the equilibrium of the forces which 
cause it to exist. This being, then, the only curve which 
can be formed under the circumstances, and since the value 
of p cannot exceed the given elastic limit, it may be seen 
why in cylinders thicker than 0.65 caliber the conditions of 
maximum pressure and uniform strain to the elastic limit 
of the metal in action cannot exist together. An initial ten- 
sion curve, starting with the limit p at the bore and having 
higher strains than the natural curve toward the exterior, 
might' be laid down which would become a straight line with 
P increased sufficiently to stretch the bore to the elastic 
limit, but such an initial tension curve is not attainable in 
practice. 
To compute the possible maximum resistance of the pres- 
ent sections of cylinders we will take, as before, the con- 
servative limits, & = 60,000 = p. The breech (after anneal- 
ing) and the breech middle sections are respectively 0.75 and 
0.81 caliber in thickness. Their maximum resistance will 
therefore depend upon the assumption that the bore is ini- 
tially compressed to the limit p at rest and extended to the STRESSES IN GUN FORGINGS. 
99 
limit 0 in action, and its value will be found from equation 
(1) (Appendix B, “ Gun Making ”); whence 
p= 4**+£$('’ + v=imx 120000- 
70000 pounds per square inch. 
44 94 
Breech middle: P= = X 120000 = 71652 pounds per 
square inch. 
The muzzle middle and muzzle sections are less than 0.65 
caliber thickness, being respectively 0.45 -and 0.22 caliber. 
The desired initial compression, or that value of p, in terms 
of the limit 0, which will cause the wall to be uniformly 
strained to 60,000 pounds per square inch when the limit of 
interior pressure is reached, is found by combining equa- 
tions (1) and (2) of the work cited; whence 
r4R* + 2R* "I 
p~ Ls(iv-io J (8) 
From which we find: 
Muzzle middle: p = 0.6769 X 60000 = 40614 
Muzzle: p = 0.3174 X 60000 = 19046 
These reduced values of p being given, the maximum re- 
sistance will be found byr applying equation (1) as for the 
other sections. Then 
Muzzle middle: P = jjq (40614 -f- 60000) = 47650 pounds 
per square inch. 
3 19 
Muzzle : P = (19046 + 60000) = 24634 pounds per 
square inch. 
The value 24,634 for the muzzle section is less than 24,920, 
which was computed, from the most dangerous one of the 
actually measured stresses, to be the resistance after anneal- 
ing. This slight discrepancy is due to the irregularities of 100 
BIRNIE. 
the curve of measured stresses and that there is no measured 
stress for the exterior surface, where, as every indication 
points, the critical strain should actually be located. It 
will be observed that the curve of deduced stress at rest in- 
dicates a stress of 14,676 pounds at the exterior surface. 
Taking the latter to govern the resistance, we find, from 
equation (4), P0 — 24,167. It may be said, therefore, that 
the probable resistance, based upon an extension limit of 
60,000 pounds as taken, lies between 24,167 and 24,634 (theo- 
retical maximum) instead of being 24,920. 
Similar conditions are also present in the breech and 
muzzle sections before annealing, and the values given for 
them in table A and drawing should probably be reduced 
in the same proportion, or about 3 per cent. This reduction 
is made in the following table, which gives a comparison of 
the resistance of these sections and those of corresponding 
dimensions in the built-up field gun as now made. It must 
be observed, however, that in the field gun the tube is not 
hooped in the two forward sections. 
Comparative Resistance of Initial Tension Cylinders and the 3.2-inch Built-up 
Meld Gun. 
Resistance, estimated. 
Breech sec- 
tion. 
Breech mid- 
section. 
Muzzle mid- 
section. 
Muzzle sec- 
tion. 
T ... , , . (On measured IBefore annealing 
Initial tension 1 stresses> } After annealing... 
Pounds 
per sq. 
inch. 
50,818 
52,015 
70,000 
Pounds 
per sq. 
inch. 
59,350 
Pounds 
per sq. 
inch. 
31,316 
Pounds 
per sq. 
inch. 
15,578 
24,167 
24.634 
cylinder. { Theoretical maximum .7. 
71,652 
47,650 
3.2-inch field gun computed resistance 
38,250 
41,030 
21,730 
13,995 
Conclusions.—The graphic representation of the curves of 
stress, &c., on the accompanying plate 6 affords the best 
means of judging the results of the treatment of the forging. 
The close accordance of the measured and deduced curves 
of stress in the forging after treatment is not accidental, 
because, as previously stated, both curves depend upon the STRESSES IN GUN FORGINGS. 
101 
equilibrium of the strains in the forging, and b}r construc- 
tion they have one point in common. Their further coin- 
cidence, therefore, is evidence of the uniformity of results 
obtained by the treatment. The somewhat marked irregu- 
larity of the measured stresses near the bore in the breech 
section, both before and after annealing, has led to the con- 
struction of two deduced curves, of which it will be seen 
that the one marked No. 2 coincides most nearly with the 
measured curve. This No. 2 is based upon the measured 
compression on the second circle from the bore. There is 
apparent evidence in this section that the contractile force 
of the outer layers of metal in cooling was sufficient to over- 
compress the metal near the bore. 
The strains engendered in all the sections by the interior 
cooling were apparently unnecessarily severe, and tended to 
produce too great a strain of tension toward the exterior for 
economy of resistance to interior pressure. Thus, in all of 
the sections, before annealing, it is seen that the curve of 
stress in action departs considerably from a horizontal line, 
and the limit of stress in action is reached first at or near 
the exterior surface.* Of the four, however, the breech 
middle sectiou is exceptionally well disposed. 
It is important to note the general resemblance of the 
measured curves of stress in the four sections as showing 
the regularity of the cooling treatment throughout the length 
of the forging, and that an inspection of the end sections 
would have disclosed the condition of initial tension in the 
whole forging. This forging, as shown on the drawing, had 
marked irregularity of sectional dimensions, yet the degree 
of initial tension in the several sections is in general propor- 
tional to the thickness of the section, and there is no gen- 
eral abnormal distortion of either thin or thick sections. 
* The position of the stress curve for the state of action is necessarily 
influenced by the selection of the value 0 = 60,000. If this value had 
been taken equal to 68,000, as given in the report of physical qualities of 
the metal before quoted, the stress curves in action would be considerably 
more elevated next the bore, and the estimated values of P0 would be 
correspondingly increased. 102 
102 
This result was, however, to be expected, inasmuch as the 
theoretical curve of initial tension depends upon the thick- 
ness of wall in calibers, and the actual curve evidently obeys 
the same law. 
This experiment leaves no room to doubt that initial ten- 
sion strains of as great intensity as are desirable can be pro- 
duced in a hollow forging by interior cooling, and if these 
strains should be more than needed they can be reduced 
by annealing. The effect of the subsequent annealing in 
the present case was beneficial, particularly in the muzzle 
section, which now shows the peculiarly interesting case of 
a resistance to interior pressure which closely approximates 
the possible maximum. The ordinates of the stress curve 
iu action are all nearl}7- equal and differ but little from the 
limit of 60,000 pounds. 
Without disparagement to the built-up, hooped gun, which 
has proved to be excellent, it may be said that the apparent 
superiority of a gun made of a single forging, with initial 
tension produced by interior cooling, rests not only upon 
claims for reduced cost and increased longitudinal stiffness, 
but also for increased tangential strength in every section 
where the actual thickness of wall is insufficient in practice 
for the division of the built-up gun into as many as four 
layers, since this number of is in general required, 
in that construction to enable the bore to be worked through 
the double limit of elastic movement. 
Inasmuch as the walls of built-up, hooped guns below 10 
or perhaps 8 inches caliber cannot be conveniently divided 
into four layers, and this division, moreover, can only be 
applied in the thicker portions (i. e., the reinforce), the con- 
clusion from the theoretical standpoint, at least, is that an 
equality of tangential strength will exist under the two 
modes of construction for the reinforce of guns of 8 or 10 
inches caliber and upwards; but for guns of smaller caliber 
and for the chase portions of all guns the greater tangential 
strength will pertain to the single forging with initial ten- 
sion produced b}7 interior cooling. 
BIRNIE.