OVERCHARGED AND UNDERCHARGED MINES.

7. For overcharged and undercharged mines in which the L. R. R. and crater radius differ materially in length the results deduced from the preceding equations are not applicable. For such mines the following equations, due to Gumpertz and Lebrun, are in common use, viz.:

For an overcharged mine,

C = C´(11/6)[l + (7/8)(r - l)]3. (6)

For an undercharged mine,

C = C´(11/6)[l + (7/8)(l - r)]3. (7)

In which C = charge of explosive in pounds, l = L. L. R. in yards, r = crater radius in yards, C´ = amount of explosive in pounds necessary to throw out one cubic yard of earth in a common mine in the same soil.

These formulæ are deduced as follows, viz.:

It was found by experiments made independently by Belidor and Marescot that 3660 lbs. of powder in a mine with L. L. R. equal to 4 yards gave a crater with a radius of 12 yards in earth requiring for a common mine 1½ lbs. of powder per cubic yard. The charge for a common mine in the same soil with L. L. R. equal 4 yards is

(11/6)(4 yds.)3 × (1½) = 176 lbs.

Representing by l the L. L. R. for a common mine requiring a charge of 3660 lbs., since the charges of common mines are proportional to the cubes of their lines of least resistance, we have

176 : 3660 :: 43 : l3 = 1330.8,

whence

l = 11y; 113 = 1331.

To find from these data the relations between charges for overcharged mines, construct Figs. 2 and 2a, ([Pl. XI])

Fig. (2) gives mines with crater radii of 4y and 12y and a common L. L. R. of 4y.

Divide the distance between A and B into four equal parts, and assume the points of division as the extremities of the crater radii of overcharged mines, each of which exceeds the one next smaller by (¼)AB, and all corresponding to a L. L. R. of 4y.

Fig. (2a) gives common mines with lines of least resistance of 4y and 11y. Divide the distance A´B´ also into four equal parts, and assume the points of division as the extremities of the crater radii of common mines each of which exceeds the one next smaller by (¼)A´B´.

Since the charges for the common mines whose lines of least resistance are respectively 4y and 11y are identical with those of the overcharged mines whose crater radii are 4y and 12y respectively, it is assumed that the charges for the intermediate common mines are the same as would be required to produce the corresponding intermediate overcharged mines.

The increment of the crater radius and line of least resistance of any one of these common mines is equal to 7/8 the increment of the crater radius of the corresponding overcharged mine; consequently the charge which gives an overcharged mine whose L. L. R. and crater radius are l´ and r´, respectively, will produce a common mine whose L. L. R. l will be given by the equation

l = l´ + (7/8)(r´ - l´). (a)

Since the charge for a common mine is obtained from equation (4), C = C1(11/6)l3, the charge for the overcharged mine will be

C = C1(11/6)[l´ + (7/8)(r´ - l´]3,

as above.

For ordinary earth and gunpowder, when L. L. R. is measured in feet, eqs. (6) and (7) become, respectively:

For an overcharged mine,

C = (1/10)[l + (7/8)(r - l)]3 (6´)

For an undercharged mine,

C = (1/10)[l - (7/8)(l - r)]3 (7´)

8. Giving to l the same value in equations (4), (6), and (7), we have

C´ = C((7/8)[r/l] + (1/8))3, (8)

In which C = charge for common mine with L. L. R. and crater radius = l. C´ = charge for over or undercharged mine with L. L. R. = l and crater radius r. Equations (6), (7), and (8) having been deduced from the relations existing between C, l, and r for mines varying from common mines up to those in which r = 3l may safely be used for overcharged mines up to this limit.[9] In their applications to undercharged mines they become uncertain when r = (½)l; and when r = (⅜)l the computed charge generally produces a camouflet.

These computed charges are:

for r = (½)l, C´ = 0.1779C; for r = (⅜)l, C´ = O.1636C.

A rule of the French engineers states that a charge which will produce a common mine with L. L. R. = l will produce a camouflet if the L. L. R. is increased to 7/4l. At this depth C´ = 0.187C, and the formula gives a crater radius of 25/49.

As a safe “rule of thumb,” we may assume that a charge which will give a common mine with L. L. R. = l will give a camouflet with L. L. R. = 2l (r´ from formula = (3/7)l). Conversely, a camouflet will be produced by ⅛ of the charge which will produce a common mine.

9. Radius of Rupture.—The determination of the radius of rupture is an important consideration in underground warfare, since, when it is known, miners may so place their chambers as to break in the galleries of the enemy without injuring their own.

As different mining galleries, however, differ from each other so much in strength to resist crushing, and as the cost of an exhaustive series of experiments to determine their relative strength would be so great both in time and money, but little well-established data exist upon which to found a rule for determining the radius of rupture.

10. The rule deduced by Gumpertz and Lebrun, however, from the material available at their time corresponds very nearly with the results of later experiments and observations, and is generally admitted as sufficiently near correct for practical use.

This rule is based upon the theory that the surface of rupture is an oblate spheroid, ([Pl. XI], Fig. 3), with its axis of revolution vertical and its centre at the centre of the charge; the intersection with the surface of the ground AD coinciding with the edge of the crater. The ratio between the semi-transverse axis CF and the semi-conjugate axis CH of the generating ellipse of this assumed spheroid is the same as that between the radius of explosion CD and L. L. R., CK. The rule is, that the radius of rupture in any direction is equal the corresponding radius of this spheroid.

From the conditions assumed the following values of the semi-transverse and semi-conjugate axes h and v (which are the horizontal and vertical radii of rupture) are obtained, viz.:[10]

h = l√(1 + 2(r/l)2);
v = l√[(1 + 2(r/l)2)/(1 + (r/l)2)].

For common mines these formulas give:

h = 1.732l = (7/4)l = (7/4)r;
v = 1.225l = (5/4)l = (5/4)r.

For six-line craters,

h = 4.358l = (35/8)l = (3/2)r;
v = 1.378l = (11/8)l = (½)r.

11. The English authorities adopt the value of 7/4l´ for the horizontal and l´ √(2) = 1.41421 l´ = (7/5)l´ for the vertical radius of rupture of all classes of mines. In which l´ = L. L. R. of an equivalent common mine = l + (7/8)(r-l), etc.

Some later experiments at Chatham have given

v = (5/3)l for a 4-lined crater;
v = 2l for a 5-lined crater;

and

v = (5/2)l for a 7½-lined crater.

12. There are other good reasons for believing that Lebrun’s value for the vertical radius is too small; but as its use leads to increasing the charges designed to produce crushing effects, the error, if it exists, is in the right direction, and justifies the use of the formula until more exact data are available.