304 



NA TURE 



[July 24, 1890 



THE SCIENTIFIC PRINCIPLES INVOLVED 

 IN MAKING BIG GUNS. 



I. 



CTEAMSHIPS are now called boats, and the largest cannon 

 '-' are called guns, according to a process in language which 

 philologists have explained ; but while steamships have increased 

 in size and complication, the gun, however big, satisfies the 

 Hibernian definition of a cylindrical hole with metal placed 

 round it ; and the most difficult problem of the gun-maker 

 is to dispose the metal in the most efficient manner, hampered 

 as he is by the limitations of the metallurgical art. 



The difficulties increase with the size of the gun, according to 

 the well-known law of Mechanical Similitude. 



Geometrical Similitude is independent of scale ; a geometrical 

 theorem is true, however large the figure may be drawn ; but 

 the laws of Mechanical Similitude are complicated, when we 

 notice the differences between a simple girder and the Forth 

 Bridge, or between the anatomy oflarge and small animals. 



As an example of mechanical similitude, consider what sort of 

 a steamship would be required to reduce the voyage to America 

 from six to five days. The present steamers crossing in six days 

 have a speed of 20 knots, and displacement of io.ock) tons, and 

 the indicated horse-power is close on 20,000. To cross in five 

 days the speed would have to be increased 20 per cent., to 24 

 knots ; and now if we apply Froude's law that, at corresponding 

 speeds as the sixth root of the displacements, the resistances are 

 as the displacements, we shall find that the steamer would have 

 to be of 30,000 tons, and of 65,000 horse-power, thus exceeding 

 even the Great Eastern^s dimensions. 



With given material, say steel, the strongest with which we 

 are familiar, a limit of size is soon reached at which the structure 

 falls to pieces almost of its own weight ; and recent experience 

 with the heaviest artillery seems to show that we are nearing 

 this limit. 



The larger the gun or structure, then, the greater the necessity 

 for careful and scientific design and proportion. It is proposed 

 to give here a sketch of the fundamental principles which guide 

 the gun-maker, and which he applies to secure the safety of the 

 gun under the greatest pressure it can ever be called upon to 

 sustain. 



While reaping almost all the glory of success, the gun-maker 

 cannot risk the disgrace of a failure ; on the other hand, the 

 carriage-maker can work with a small margin of safety, as ample 

 warning would be given of any failure, and breakage is easily 

 repaired ; but the failure of a gun may be so disastrous that it 

 must be avoided at all cost, so that the gun-maker never allows 

 himself to work very close to the limits which his theory allows. 



At the present time the design and employment at sea or in 

 forts of such monsters as our no-ton or Krupp's 135-ton guns is 

 severely criticized and condemned in certain circles ; but it is a 

 maxim in artillery that one big gun is worth much more than its 

 equivalent weight in smaller guns ; and for naval engagements 

 a few line-of-battle ships armed with the heaviest artillery are 

 invincible, if properly flanked and protected by the light cavalry 

 of frigates. 



So, too, with steamships ; the largest and fastest always fill 

 with passengers, and by making rapid passages, and therefore more 

 in a given time, are found to be more profitable in spite of 

 their great initial cost and expense of working. 



The size of the gun is settled by the thickness of armour it is 

 required to attack ; the calibre increasing practically as the 

 thickness to be pierced, but the weight of the gun mounting up 

 as the cube of the calibre. Thus if an 8-inch gun weighing 

 13 tons can pierce 12 inches of armour, a 16-inch gun is required 

 to pierce 24 inches, and the 16- inch gun will weigh 104 tons. 



Part I. — The Stresses in a Gun. 



(i) The theory of gun-making begins with the investigation of 

 the stresses set up in a thick metal cylinder, due to steady 

 pressures, applied either at the interior, or exterior, or at both 

 cylindrical surfaces. 



So far, the dynamical phenomena which arise from the propa- 

 gation and reflexion of radial vibrations are beyond our powers 

 of useful analysis ; so that we restrict ourselves to the investigation 

 of the elastical problem of the thick cylinder of elastic material, 

 subject to given internal and external pressures, applied steadily, 

 as in the case of a tube tested under hydraulic pressure. 



Fig. I is drawn representing the stresses set up in a 



cylinder or tube B, by an internal pressure pi ; we denote by 

 7-i and r,, the inner and outer radii, the suffixes / and denoting 

 inside and outside ; and then r can be used to denote any 

 intermediate radius. 



The stress at any point at a distance r from the axis will 

 consist of a radial pressure, p, and a circumferential tension, /; 

 the radial pressure /decreasing from/, at the inner radius n to 

 zero at the outer radius r^, the atmospheric pressure not being 

 taken into account; while the circumferential tension /at the 

 same time diminishes from ti to to. 



The British units employed in practical measurements with 

 guns are the inch and the ton ; so that r being measured in 

 inches, / and / are measured in tons per square inch. 



(2) To determine the state of stress at any point of the cylinder, 

 we suppose it divided by a diametral plane r,, r,- O r^ r„ ; and 

 the equilibrium of an inch length of either half is considered. 



The stresses / and t being represented graphically by the 

 ordinates of the curves pi pp^, titt„, the equiUbrium of either half 

 of the cylinder requires that the area of the circumferential 

 tension-curve r,7Ar„ and its counterpart should be equal to the 

 area of the rectangle O/.-, and its counterpart, these latter repre- 

 senting the thrust due to the pressure /, on the half cylinder. 



Then, denoting the area ri^,/„r„ by Q, and calling it the 

 resistance of the section r,r„. 



Q =Piri 



(I). 



If we divide the resistance Q by the thickness of the cylinder 

 I'o - f'i, we obtain the average circumferential tension in the 

 material ; and when the cylinder is thin, the maximum circum- 

 ferential tension ti and the average tension Q/(r„ - r,) will not 

 be appreciably different ; so that a knowledge of the average 

 circumferential tension will be sufficient for practical purposes 

 in such cases as, for instance, of the cylindrical shell of a 

 boiler ; and we have thus the elementary formula ordinarily 

 employed in the design of boilers. 



But when, as in a gun or hydraulic press, the thickness has to 

 be made considerable, we must have the means of determining 

 the maximum tension /,-, and of contriving that ti shall not 

 exceed a certain proof limit suitable for the material. 



(3) Now, just as the equilibrum of either half of the cylinder 

 requires that the area r it iter,-, — piri, so the equilibrium of either 

 half of a part of the cylinder bounded internally by the radius 

 ri, and externally by any radius r, requires that the area 

 rititr should equal the rectangle Opt minus the rectangle Op ; 

 or, in the notation of the Integral Calculus — 



tdr = piri - pr 



(2). 



The first attempt at a solution of these equations (i) and (2) 

 is due to Peter Barlow, when called upon to calculate the 

 strength of the cylinder of the Bramah hydraulic press, in a 

 paper read before the Society of Civil Engineers in February 

 1825, and published in the Edinburgh Journal of Science, and 

 in the Trans. I.C.E., vol. i. 1836. 



(4) Barlow assumed that under an internal pressure the metal 

 is compressed radially as much as it is stretched circumferen- 

 tially, so that the cubical compression of the metal is zero, and 

 he is justified therefore in putting / = / in the material of the 

 cylinder. 



Then equation (2) becomes 



/ pdr = piVi - pr ; 



so that, differentiating with respect to r, 



p = - d(pr)ldr, or dpjp + 2drlr = o ; 



and integrating again with respect to r, 



log / -f log r- = constant, 

 or 



pr^ = a, a. constant ; p = t — ar-- . . . . (3) ; 



so that p and t, if equal, vary inversely as the square of the 

 distance from the axis. 



Thus, a cylindrical tube under internal and external pressures 

 which are inversely as the squares of the internal and external 

 radii respectively, will, according to Barlow's law, have at any 

 point a radial pressure and an equal circumferential tension, 

 also inversely as the square of the distance from the axis. 



NO. 1082, VOL. 42] 



