Much has been written on the subject of elastic strength of guns in the endeavor to make the subject perfectly clear, with the result that many books on the subject are filled with formulas. Lately the subject has been illustrated with geometrical diagrams quite as complicated as the formulas themselves, so that experts who examine them will probably find themselves wondering if, after all, they know anything about the question. It is thought that the subject may be presented in an elementary way without the use of many formulas, illustrating by numerical examples.
The basis of the modern theory is that within the elastic limit, that is, the limit at which if the load is removed a metal will assume its original dimensions, the strain (stretch or shortening per unit length) is proportional to the stress (or load per unit area). This is probably only approximately true for some metals. For example, some mild steel stretches less just within the elastic limit for a positive increment of stress than it does lower down. It seems to save up, so to speak, for a greatly disproportionate stretch just above the elastic limit ; after this excessive stretch, the amount of strain for an increment of stress becomes nearly as small as below the elastic limit and then increases gradually till the ultimate load is obtained, the specimen breaking at a load somewhat lower than the highest point reached.
The highest stress obtained is the tensile strength and the stress that strains a metal to its elastic limit is the elastic strength of the metal (for extension or compression, as the case may be). A distinction should be carefully observed between the elastic strength of a metal and the elastic strength of a gun built of the metal, the last being the pressure per square inch which, when applied to the gun internally, will permanently deform it by straining some portion of the metal beyond its elastic limit.
Following the ordinary custom, we will consider the strain within the elastic limit as proportional to the stress in the same direction, when this is the only stress applied. We will, moreover, only consider metals while acting inside their elastic limits, the various fundamental rules given applying only with this condition.
Within the elastic limit, then, if a thousand pound pull is given to a specimen of metal, and then a thousand more is added, the stretch for the second will be the same as for the first. This is equivalent to saying that the strain due to any stress is independent of prior stress. A second rule is that when a stress of tension (or compression) is exerted upon a metal, the strains in directions at right angles are l of that in the line of the stress and are contractions (or extensions). The coefficient ¼ is sometimes used in place of 1/3, but this latter is used almost exclusively by the Army and Navy ordnance bureau officers. If two or more stresses at right angles act upon a specimen, the total strains in each direction will be the algebraic sum of the strains in that direction, each component strain being independent of prior strain and each being readily calculated by the rule for the strains produced by a single stress.
The ratio of the stress to the strain caused by it in the same direction is called the modulus of elasticity of the metal and will be denoted by E.
It is necessary to use calculus only once in the subject of elastic strength of guns, namely, to get the fundamental formulas for tension and radial pressure.
The stresses acting upon a point in the thickness of a gun may be resolved in the three directions of the length, the thickness, and perpendicular to a radius of the gun and the length. Any point at rest is held by equal and opposite forces or otherwise it would move in the direction of the greater force. So that we may represent the radial pressure on any particle by two opposite and equal arrows pointed towards each other along a radius, and the circumferential tension at any point by two equal and opposite arrows pointed tangentially and away from each other. The longitudinal stress would likewise be represented by equal and opposite arrows in the direction of the length.
We will denote the radial stress by p, the circumferential or tangential stress by t and the longitudinal stress by q. Similarly, strains in the same directions will be denoted by ['p1], [t] and [q.]
If we denote the pressure inside of a closed cylinder by P0, outside by P1, the internal radius by R0 and the outside by R1, and suppose a plane surface passed through its axis, removing one half of the cylinder, it will be evident that the total tension of the cylinder at the points of junction with the plane must be 2(P0R0-P1R1) multiplied by the length of the cylinder, since this will be the total pressure on the flat surface tending to separate it from the half cylinder. But the presence of the flat surface will in nowise alter the forces causing stress in the cylindrical part, and this is, evidently therefore the total tension throughout the two thicknesses of the original cylinder.
By dividing this quantity by the area of the section of the metal, we find the mean tension per unit area. A similar method of finding tension will apply to any cylindrical element, so that we readily find a differential expression and then, by integration, the law of radial pressures and tensions throughout the thickness. This is the occasion in the study of elastic strength of guns where we are compelled to use calculus. Such calculus as is necessary is however very simple, but we will not go into it here, and will consider the results only.
The three conditions necessary to the deduction of these equations (which are absolutely independent otherwise of the metal) are uniform elasticity (or constant modulus), uniform longitudinal stress, and uniform longitudinal strain throughout the thickness. In them, c2 and c1 are constants. The longitudinal tension q is, following Clavarino, equal to c1, and according to Birnie, zero. Birnie's assumption will give lower elastic strength generally and therefore being safer, is taken here. It seems moreover to agree more nearly with the facts.
From these it will follow that a wire-wound gun will be weaker than another built-up gun of larger dimensions, but the same modulus, in which the same compression before firing of the same interior tube is accomplished. The greatest elastic strength possible with a gun of the same metal and same inside and outside dimensions as the one we have been using, calculated in this way, would be over 80,000 lbs.
Apropos of wire drawing it may be said that if a specimen of steel is placed in the testing machine and permanently strained by say 90,000 lbs. pull per square inch, and is then taken out of the machine and retested as a metal, it will now show over 90,000 lbs. elastic strength, and from its smaller original area in this last case, greater ultimate strength than before. Does this cold drawing increase its elastic strength for compression, or even keep it the same, and is the metal any better for gun purposes than it was before? If so, would it not be well to fire heavy proof charges, as in the old converted guns, and then finish bore and rifle the gun?
The effect of successive shrinkages is to continually compress the metal next the bore.
Now, in calculating the strength of the gun we found the pressures at contact surfaces when the maximum firing pressure (the elastic strength of the gun) acted on the inside, and o on the outside of the gun. If we take a simple tube of the same dimensions and modulus, and suppose the same firing pressure to act, the stresses and strains at the same radii will represent the changes in the stresses and strains of the compound tube, and by subtracting these from the final stresses and strains of the built-up gun, we obtain those in the gun when no pressure is acting. That is, in (1) we place p equal to the elastic strength of the gun, and r equal to the inner radius. Next we place p= 0 and r equal to the outer radius; we thus have two equations and two unknown quantities, c1 and ce. Solve for these, and plot the curve shown by (1). The ordinates are the changes in radial pressure in the compound tube due to firing with a pressure equal to the elastic strength of the gun, and from these and the calculated pressures at the contact surfaces in the gun when in action, we find the pressures at rest by subtraction.
The pressure at rest on the outside of the inner tube is what causes the compression of the bore. In (1) we place p equal to this pressure at rest, and r equal to the outside radius of the inner tube. Next we place p = 0 and r equal to the radius of the bore. From these two equations we find C1 and c2 as before. We substitute these in (4), making r equal to the radius of the bore, and the result is the strain (per unit) of the circumference, which is the same as the strain (per unit) of the radius, and by multiplying by the diameter (and taking the negative) we get its total shortening, or the total compression of the bore. The compression per unit should not exceed the elastic limit of compression. In the gun that we have arbitrarily assumed the increase in radial pressure at the contact surface of 2" radius, caused by 61,036 lbs. inside and o pressure outside the gun, is 12,207 lbs. (c1 = 4069 and c2= 65,104 for simple gun), and as the pressure during firing at this surface is 39,313 lbs., the pressure when the gun is at rest is 39,373 — 12,207 = 27,106 lbs. Can the inside tube stand this pressure when the inside pressure is o .? We find under the given conditions that c2 = c1 , that 27,106= — ¾c , and on the inside that [t] = —.0024, which is greater numerically than —.002, the elastic limit of compression. The tube would therefore be permanently deformed, and we cannot build a gun of these dimensions and this metal so that m firing the interior of the second tube will be strained to its elastic limit. The shrinkage on the inner tube may be determined, however, to meet the condition that this is not deformed when the gun is at rest.
It should be noted that the pressure at rest outside the inner tube is due partly to the successive outside shrinkages. If, now, the compression per unit of the diameter of the bore is less than the elastic limit of compression, we may shrink on another hoop if desirable with such shrinkage as will bring the compression of the bore to this limit. Thus, in (4) we place [t] equal to the strain (negative) required, and r equal to the inner radius. In (1) we place p = 0 and r equal to the inner radius of the gun. From these two equations we find c1 and c2, and substitute them in (1), making r the original outside radius of the gun (before putting on the hoop). The result is the pressure required between the original gun and the hoop.
We have now the pressure at the surface of contact, and we require the shrinkage.
In (1), we substitute this pressure for p and the radius of contact for r. Next we place p = o, and r equal to the radius of the bore. Solve for c1 and c2 (they have already been found, however), and substitute in (4), making r the radius of contact. The result is the strain, and taken with a negative sign is the compression (per unit) of the outer diameter of the inside portion. In (1) we substitute the pressure at the surface of contact for p, and the radius of contact for r. Next o for p and the external radius for r. Solve for c1 and c2. Substitute in (4), taking r the radius of the surface of contact, and the result is the extension per unit of the inner diameter of the hoop. Add this extension to the preceding compression and multiply by the diameter, and the result is the required shrinkage. Various other problems might be solved in the same way but will not be gone into here. Our endeavor has been to avoid the usual nomenclature of books on the elastic strength of guns, which is something appalling, by the use of equations that will keep what we are doing clearly before us. Certain minor operations are duplicated in the description, which would not really be necessary with actual numerical cases, and it is thought that this adds to the clearness. It may be remarked that the pressures at the surfaces of contact of a built-up gun when the gun is at rest may be used to find the shrinkages, in exactly the same way as shown in this note using the maximum firing pressure. In fact, the corresponding pressures at these surfaces under any given conditions may be used.
If two adjacent cylinders in a built-up gun have the same elastic limit, the most advantageous intermediate radius will be a mean proportional between the other two radii. The proof of this will not be given here as it would unnecessarily complicate what is intended only as a brief summary of the subject. Other considerations frequently interfere with this. For example, the jacket of a modern gun is made of sufficient thickness to withstand the longitudinal pull between trunnions and breechblock. If P0 and R0 are the elastic strength of the gun and the radius of the exposed nose of the breech block respectively, the elastic strength of the metal and R1 and R2 the interior and exterior radii of the jacket, so that no unit area may be exposed to a lengthwise pull greater than the elastic strength of the metal.