Chapter I: Nomenclature and Theory

=1.= =Military Mining= includes all the operations necessary for placing charges of explosive underground and exploding them at the time desired, for the purpose of destroying the men, materials, or works in their vicinity, or for breaking up the surface of the ground either to advance or retard the operations of a siege.

The excavation for receiving the _charge_ is called the _chamber_. The approaches leading to the chamber when horizontal or somewhat inclined are called _galleries_, and when vertical are known as _shafts_. When very steep they are sometimes called _slopes_. The charge, chamber, and approaches taken together constitute a _mine_.

The pit formed by the explosion is called the _crater_.

When the ground is homogeneous and its surface horizontal, the intersection of its surface by the crater is approximately a circle, the radius of which is called the _crater radius_, _AB_, Pl. XI, Fig. 1.

The right line joining the centre of the charge with the nearest point of the surface toward which the explosion will take place, generally the surface of the ground, is called the _line of least resistance_ (written generally L. L. R.), _C B_, Pl. XI, Fig. 1.

A right line from the centre of the charge to the edge of the crater is called the _radius of explosion_, _C D_, Pl. XI, Fig. 1.

The distance from the centre of the charge at which an ordinary mining gallery will be broken in by the explosion is called the _radius of rupture_, _C L_, Pl. XI, Fig. 3. The radius of rupture varies in length with its inclination to the horizontal.

Craters whose _diameters_ are once, twice, etc., their lines of least resistance are called _one-lined_, _two-lined_, etc., craters.

Mines in which the L. L. R. is equal to the _crater radius_ are called _common mines_. (Their craters are _two-lined_.) Those in which the crater radius exceeds the L. L. R. are called _overcharged mines_ or _globes of compression_; when it is less, they are _undercharged mines_; and when the charge is so small that no exterior crater is formed, they are known as _camouflets_.

=2.= In the explosion of military mines on land it may safely be assumed that the circumstances of combustion of the charge when fired are such that the energy developed is directly proportional to the charge. A portion of this energy is generally lost by the escape of the compressed gases into the air, by the heat given up to the surrounding media, and by the transmission of earth-waves or shock; the remainder and greater part, however, is expended in rupturing the case containing the charge, compressing the soil in its immediate vicinity, separating that lifted up from that forming the sides of the crater, breaking up the portion thrown out into large or small fragments, projecting them to a greater or less distance, and disintegrating the soil around the crater to a distance which varies with the soil and with the quantity and character of the explosive used.

As the proportional part of the energy expended in each of the effects above named cannot be determined in any particular case, and as each case differs in some respect from every other, it is manifestly impossible to express in any mathematical formula a rule for determining the exact amount of explosive required for any particular mine.

From the results of long experience, however, engineers have concluded that computations sufficiently exact for practical purposes can be made upon the hypothesis that _for common mines and those approximating closely to them in form_, the volumes of the craters are directly proportional to the charges used.

=3.= In order to apply this rule in practice the volumes of craters formed by known charges must be measured; but since the soil in the immediate vicinity of the crater is more or less disintegrated, and the crater itself is partly filled up by the material which falls back into it, the outlines of the original crater cannot usually be recognized or its exact geometrical figure be determined. Besides, the craters formed under circumstances seemingly identical differ more or less among themselves.

For convenience in computation, however, several simple geometrical figures have been assumed as giving with sufficient accuracy the form of the crater of a common mine. See Pl. XI, Fig. 1. Among these Vauban assumed a cone, _ACD_, with its vertex at the centre of the charge; Valière a paraboloid of revolution, _AHD_, with its focus at the centre of the charge; Müller truncated this paraboloid by a horizontal plane through its focus; while Gumpertz and Lebrun adopted the form in common use at their time, and which has been generally accepted since, viz., a frustum of a cone, _AEFD_, the smaller base of which passes through the centre of the charge and has a radius, _EC_, equal to one-half the crater radius, _AB_ (or one-half L. L. R., _CB_).

The volumes of these figures are as follows:

Vauban’s cone 1.05 (L. L. R.)^3,
Valière’s paraboloid 1.90 (L. L. R.)^3,
Müller’s truncated paraboloid 1.84 (L. L. R.)^3,
The frustum of a cone 1.83 (L. L. R.)^3 = nearly (11/6)(L. L R.)^3.

The cone of Vauban (lately assumed also by Höfer) was abandoned as unsatisfactory, because it did not conform to the craters produced, and, as treated by Höfer, because the charges computed by its use were found to be too small (an error in the wrong direction). The paraboloid of Valière or Müller would seem to conform more nearly to the actual shape assumed by the crater; but it will be observed that the volume of the latter is sensibly the same as that of the truncated cone, and as the volume of earth thrown out is the quantity to be considered, the truncated cone will be assumed as the measure for it.

=4.= The principle that “the volumes of the craters are proportional to the charges used” is the general statement of the _miner’s rule_. Assume _C_ and _C´_ to represent the charges of two mines whose volumes are _V_ and _V´_, lines of least resistance _l_ and _l´_, and crater radii _r_ and _r´_. Assume also that the craters are frustums of cones, the radii of whose larger bases are twice those of the smaller. Then

_C_ : _C´_ :: _V_ : _V´_ :: (11/6)(_lr_^2) : (11/6)(_l´r_´^2),

or

_C´_ = _C_ (_V´_/_V_) = _C_[(11/6)(_l´r_´^2)/(11/6)(_lr_^2)] = _C_[(_l´r´_^2)/(_lr_^2)] ... (1)

Equation (1) is applicable to _mines in which r does not differ materially from l or r´ from l´_.

From an experimental mine giving a crater of this general type the relations between _C_, _l_, and _r_ may be determined, and assuming any two of the quantities _C´_, _l´_, and _r´_ for a mine with a crater nearly similar in form, the other may be found from eq. (1).

When _l_ = _r_ and _l´_ = _r´_, we have

_C_ : _C´_ : :(11/6)_l_ : (11/6)_l_´^3,

and

_C´_ = _C_[(11/6)(_l_´^3)/(11/6)(_l_^3)] = _C_[(_l_´^3)/(_l_^3)] (2)

Equation (2) is applicable to common mines, and shows that _in common mines the charge varies as the cube of the line of least resistance_.

Assuming _C__{´}_ as the charge which will produce a crater with a volume of unity, equations (1) and (2) become, by omitting the primes from _l_ and _r_,

_C_ = _C__{´}_(11/6)_lr_^2, (3)

and

_C_ = _C__{´}_(11/6)_l_^3 (4)

Equation (4) gives the rule for determining the charge for common mines whose L. L. R. is given, viz.: _Multiply 11/6, the cube of the line of least resistance in yards, by the quantity of explosive required to throw out one cubic yard_.

The latter quantity is determined by experiment. A similar rule may be written out from eq. (3) for mines differing but little from common mines.

=5.= The quantity of gunpowder required to throw out a cubic yard of material has been calculated from a great number of mines fired in different kinds of soil. The following table gives the quantities required according to Lebrun and Macaulay, respectively the French and English authorities on the subject:[8]

TABLE A.

Number. Description of Earth, Rock, or Weight per Charge, Charge, Proportional
Masonry. cubic foot. Gumpertz Macaulay. value
and Lebrun. of charge.

lbs. lb. oz. lb. oz.
1 Light sandy earth (_common earth, 85 1.8 1.13 1.12
Lebrun_)

2 Hard sand 111 1.10¾ 2.0 1.25

3 Fat earth mixed with sand and gravel
(_common earth, Macaulay_) 116 1.5⅓ 1.10 1.00

4 Wet sand 118 1.12 2.2 1.30

5 Earth mixed with stones 118 1.14 2.4 1.40

6 Clay mixed with tufa 124 2.1 2.8 1.55

7 Fat earth mixed with pebbles 143 2.4 2.12 1.69

8 Rock 143 3.0 3.10 2.25

9 New or old moist brickwork or masonry 2.2 1.30

10 Inferior brickwork or masonry 2.11 1.66

11 Good, new ditto 3.10 2.25

12 Good, old ditto 4.1 2.50

13 Roman ditto, or other equally good
in warm climates 4.11 2.90

=6.= For _common mines_ in _ordinary earth_ a convenient rule, very generally used, and which gives results nearly the same as those deduced from the table, is:

_The charge of gunpowder in pounds is equal to one tenth the cube of the line of least resistance in feet_, or

_C_ lbs. = (1/10)_l_^3 ft. (5)

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 :: 4^3 : _l_^3 = 1330.8,

whence

_l_ = 11^_y_; 11^3 = 1331.

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

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

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 4^_y_.

Fig. (2_a_) gives common mines with lines of least resistance of 4^_y_ and 11^_y_. 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 4^_y_ and 11^_y_ are identical with those of the overcharged mines whose crater radii are 4^_y_ and 12^_y_ 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_ = _C__{1}(11/6)_l_^3, the charge for the overcharged mine will be

_C_ = _C_{1}(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_ = 3_l_ 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_´ = 0.1779_C_; for _r_ = (⅜)_l_, _C_´ = O.1636_C_.

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 (7/4)_l_. At this depth _C_´ = 0.187_C_, 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. = 2_l_ (_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_ = 1.732_l_ = (7/4)_l_ = (7/4)_r_;

_v_ = 1.225_l_ = (5/4)_l_ = (5/4)_r_.

For six-line craters,

_h_ = 4.358_l_ = (35/8)_l_ = (3/2)_r_;

_v_ = 1.378_l_ = (11/8)_l_ = (1/2)_r_.

=11.= The English authorities adopt the value of (7/4)_l__{´} 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_ = 2_l_ 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.

EXPLOSIVES.

=13.= No military mining operations of note have been carried on since the introduction of dynamite and other high explosives; consequently our knowledge of their value for work of this kind rests entirely upon the results obtained from experimental mines. Unfortunately but few experiments seem to have been made, and the published results of these are very meagre.

=14.= Two mines fired at Krems in 1873 with L. L. R. of 12 ft. in earth weighing 100 lbs. per cubic foot and charged, one with 173 lbs. gunpowder, the other with 58 lbs. dynamite (kind not stated), gave crater radii, respectively, of 12.75 and 10.25 feet. Lebrun’s formulas applied to these give to gunpowder and dynamite the ratio 1 : 1.688.

Two powder-mines and one dynamite-mine, each of 12 ft. L. L. R., were fired at Willet’s Point in 1878. The powder-mines were each charged with 200 lbs. cannon-powder and the dynamite mine with 82 lbs. dynamite No. 1.

No. 1 powder-mine gave a crater radius of 15½ ft.

No. 2 powder-mine gave a crater radius of 15¼ ft., or a mean of 15⅜ ft.

The dynamite-mine gave a crater radius of 14½ ft.

The relative values of cannon powder and dynamite resulting from the application of the same formulas to these mines is 1 : 1.997.[11]

=15. Choice of Explosive.=--From these experimental mines it may be concluded that for forming craters in ordinary earth dynamite is not quite so efficient as double its weight of good gunpowder. For breaking up hard rock, blowing up strong masonry, and especially in demolitions where tamping is usually defective, this ratio does not hold; but the relative effect of the high explosive increases continually with the lack of tamping and the intensity of the local blow desired, until a point is reached at which the effect of gunpowder is almost imperceptible, while the high explosive does efficient work. This property of the high explosives renders them extremely valuable for use in hasty demolitions, such as blowing up palisades or barriers, destroying guns, etc., etc.

Owing to their varying values in different conditions the choice of explosive to be used in any particular case must evidently depend upon the circumstances attending it.

In underground explosions both gunpowder and high explosives give out noxious gases which penetrate the soil, and which entering a gallery in sufficient quantity would suffocate the miners. Of these gases the carbonic oxide given off by some of the high explosives is probably the most dangerous to human life, and if mixed with the proper proportion of air forms an explosive mixture, resembling in this respect the fire-damp of the coal-mines. Whether in practical mining operations it would ever be retained in the soil in such quantities as to produce this effect remains to be seen.

Some of the high explosives, on the other hand, seem to produce relatively small quantities of noxious gases. The gases produced by gunpowder, while suffocating in their nature, have the advantage of always making their presence known by their odor.

=16.= For use in overcharged mines designed to break in the enemy’s galleries, the high explosives, from the violent character of their explosion and from the phenomena exhibited in submarine mining, promise to give relatively greater radii of rupture than gunpowder; but sufficient data are not available to state this positively.[12]

=17.= Beside the considerations above stated, which refer to the effects produced by the explosive when fired, there are others equally important relating to the safety and facility with which the explosive may be transported, handled, and placed in the mines. The latter will frequently have greater weight than the former in determining the explosive to be used in any particular case which may arise in the practical operations of mining. Of the latter considerations some of the most important in deciding whether to use gunpowder or high explosives are the following, viz.:

Gunpowder is easily obtained, and most enlisted men are more or less familiar with its properties.

It explodes when ignited by fire.

It does not ordinarily explode when struck by a bullet.

It is injured by moisture and destroyed by thorough wetting.

It is not affected by ordinary changes of temperature.

It requires thorough tamping to produce good effects.

Many high explosives are not injured by moisture, and some are unaffected by total immersion in water.

They generally burn without detonation if ignited by flame.

Some of them do not explode when struck by a bullet. The more sensitive ones do.

The properties of some of them are materially changed by freezing.

On account of their greater strength, the same effects may be produced by smaller charges, requiring smaller chambers and cases.

By reason of the violence of their action they produce good results even if imperfectly tamped.

The last two considerations, together with the possibility of using them in wet places without protection against moisture, lessen greatly the time required to excavate, charge, and tamp a mine, and may frequently enable the one using them to anticipate an enemy using gunpowder and thus secure success, when the use of gunpowder would reverse the situation. In mining operations and in expert hands the high explosives, upon the whole, seem to cause fewer accidents than gunpowder.