Methods and Formulas for Calculating Concentrated Charges
Abstract Generated abstract
This article reviews the historical development of principles, methods, and formulas for calculating concentrated explosive charges in mining blasting, with attention to normal throw, reduced throw, loosening, and compression charges. It compares early charge formulas based on crater geometry and proportionality to the cube of the line of least resistance, then examines later correction functions for the blast action index and approaches incorporating charge depth and the physical and mechanical properties of the blasted medium. The review groups the principal formulas by their underlying assumptions and dependencies, showing how successive methods attempted to account for observed variation in explosive consumption with charge conditions. It concludes that existing formulas represent stages or special cases in an unresolved theoretical problem requiring further experiments and observations.
Full Text
METHODS AND FORMULAS FOR CALCULATING CONCENTRATED CHARGES
S. D. OSNOVIN
(Presented by the scientific conference of the Mining Operations Faculty).
At conferences on drilling-and-blasting operations in 1939 and then in 1952 in Moscow, it was noted that the problems of calculating charges still do not have the necessary theoretical basis, and that the level of charge calculation lags behind the pace of development of drilling-and-blasting practice; this lag is becoming more and more appreciable. In the resolution adopted at the latter conference, the scientific elaboration of this problematic question was set as one of the principal and priority tasks.
The purpose of the present article is to give a general review of the development of ideas, principles, methods, and formulas for calculating concentrated charges.
1. Charges of Normal Throw
The following elements of an explosion crater are distinguished (Fig. 1):
\(W\) — the shortest distance from the center of the charge to the free surface, called the line of least resistance and denoted by LLR;
\(r\) — the radius of the explosion crater;
Fig. 1. Explosion crater.
\(R\) — the radius of action of the explosion, determining the distance from the apex of the overturned cone to any point of its edge;
\(n\) — the ratio of the radius of the explosion crater to the line of least resistance \(\left(n=\dfrac{r}{W}\right)\), called the index of the action of the explosion (the throw index).
A charge that forms on the surface of a medium (rock) a crater whose radius is equal to the LNR \((r = W)\), and whose blast action index is equal to unity
\[ \left(n = \frac{r}{W} = 1\right) \]
is called a charge of normal throw; the crater formed by the explosion of such a charge is a crater of normal throw.
The destructive action of the explosion of a charge is in direct dependence on the resistance of the medium. From this it was assumed that the magnitude of the charge must be proportional to the resistance that it has to overcome. This principle was expressed as early as 1628 and up to the present time has not been changed. The prevailing resistance that the charge must overcome was taken to be the resistance of the force of gravity of the medium being torn away, and, as a consequence, it was considered that the weight of the charge should be proportional to the weight of the medium being torn away or, in other words, to the volume of the crater of normal throw. Therefore, the proposed formulas for calculating the magnitude of a charge of normal throw are based on the theoretical calculation of the volume of the crater formed by the explosion and on the experimental determination of the quantity of explosive required to throw out one unit of its volume.
A number of formulas were proposed for calculating charges of normal throw. The principal ones are given in Table 1.
Table 1
Formulas for calculating charges of normal throw
| Sequential No. of formula | Year | Formula | Adopted crater form | Crater volume |
|---|---|---|---|---|
| 1 | 1812 | \(L = 1.83 W^{3} P\)¹ | Inverted truncated cone | \(1.83 W^{3}\) |
| 2 | 1871 | \(L = c W^{3}\) | Cone of revolution | \(W^{3}\) |
| 3 | 1838–1905 | \(L = 1.896 W^{3} z\) | Paraboloid of revolution | \(1.896 W^{3}\) |
The meaning of the quantities entering into the formulas:
\[ L \text{ — weight of the charge in kg;} \]
\[ W \text{ — LNR in m.} \]
The quantities \(P\), \(c\), and \(z\), the consumption of explosive per \(1\ \text{m}^{3}\) of rock torn away, are called the specific consumption of explosive.
Formula 1. The formula is based on a crater of normal throw in the form of an inverted truncated cone, the radius of whose lower base is equal to one-half the radius of the upper base. The volume of such a cone is
\[ V = \frac{\pi}{3}\cdot W \left(r^{2} + r \cdot \frac{r}{2} + \frac{r^{2}}{4}\right) = \frac{\pi}{3}\cdot W \cdot 1.75 r^{2}. \tag{1} \]
Since for a crater of normal throw \(r = W\), then
\[ V = \frac{\pi}{3}\cdot W \cdot 1.75 W^{2} = 1.83 W^{3}. \tag{2} \]
¹ For powder charges.
Hence:
$$ L = 1.83 \cdot W^3 \cdot P. \tag{3} $$
Formula 2. The ejection crater was taken in the form of a straight inverted cone of revolution with its apex at the center of the charge.
The principal property of a crater of normal ejection is \(r = W\).
The volume of the crater, expressed in terms of the LLR \(= W\), will be:
$$ V = \frac{1}{3}\pi \cdot r^2 \cdot W = \frac{1}{3}\pi \cdot W^3 \cong W^3. \tag{4} $$
Hence, for calculating charges the formula obtained was:
$$ L = c \cdot W^3. \tag{5} $$
Formula 3. It is based on an ejection crater in the form of a paraboloid of revolution with its focus at the center of the charge.
The volume of a paraboloid of revolution is
$$ V = \frac{1}{2}\pi \cdot r^2 \cdot h, \tag{6} $$
where \(r\) is the radius of the base;
\(h\) is the height.
Then, using the properties of the parabola and expressing \(h\) and \(r\) through the LLR \(= W\), the volume of the crater is determined:
$$ h = \frac{R + W}{2}; $$
$$ r = W;\quad R = \sqrt{W^2 + W^2} = W\sqrt{2}; $$
$$ V = \frac{1}{2}\pi \cdot W^2 \cdot \frac{W \cdot \sqrt{2} + W}{2} = 1.896\,W^3 \tag{7} $$
and the following formula was proposed for calculating the magnitude of the charge:
$$ L = 1.896\,W^3 \cdot z \tag{8} $$
Formulas 1, 2, and 3 are constructed on the basis of one and the same principle.
For a comparative evaluation of these three formulas it is necessary to remember that the consumption of explosive per unit volume of the crater is determined exclusively experimentally, proceeding from the magnitude of the charge for which the radius of the base of the explosion crater is equal to the LLR.
If, however, the quantities \(P\), \(c\), and \(z\) are calculated on the basis of one and the same experiment, on the basis of the same experimental data, then one or another form of the calculation formula has no essential differences, and according to each of these three formulas the same calculated charge magnitudes should be obtained as a result of the calculation.
Therefore these formulas, as having no essential difference, may be reduced to one general formula:
$$ L = q \cdot W^3, \tag{9} $$
\(q\) is the specific consumption of explosive when the LLR \(= 1\) m.
This formula of general form is also used at present for calculating concentrated charges of normal ejection.
A characteristic feature of this formula is that it assumes:
1) proportionality of the magnitude of the charge to the volume of the ejection crater or to the cube of the LLR;
2) the constancy of the specific consumption of explosives with variation of the LLR, all other conditions being equal.
Later studies and practical observations showed that in carrying out blasting operations under ordinary mining conditions, when only fragmentation of the rock is required without any considerable throw, or only its loosening, the specific consumption of explosives does not remain constant, and as the LLR increases an ever smaller specific consumption of explosives may be used. In this connection new formulas were proposed for calculating charges, which came to be called formulas for calculating charges of reduced throw and loosening charges.
2. Charges of reduced throw and loosening charges
For a charge of reduced throw, the index of blasting action is
\[ n=\frac{r}{W}<1 \]
and a crater is formed with a base radius smaller than the LLR, i.e. \(r<W\).
Practice has shown that when the index of blasting action is \(n \le 0.75\), the rock is not thrown out, and the external action of the charge is expressed only in the loosening and bulging of the blasted rock. Such charges are called loosening charges.
If, however, the value is \(n<0.5\), then no bulging of the rock at the surface occurs; in this case there occurs only the formation of a spherical chamber around the location of the charge. These charges are called compression charges and are used in the shooting of boreholes and drill holes to form pot-shaped enlargements.
All formulas for calculating a charge of reduced throw and loosening charges differ in outward form from the general formula for charges of normal throw (9) only in that a correction coefficient \(f(n)\) is introduced into them as a multiplier.
In general form, the formulas for calculating charges of reduced throw may be represented as:
\[ Q=f(n)\cdot q\cdot W^3=f(n)\cdot L, \tag{10} \]
where \(Q\) is the weight of a charge of reduced throw or a loosening charge, in kg;
\(f(n)\) is the function of the action of the explosion (the throw index);
\(q\) is the specific consumption of explosives, in kg/m\(^3\);
\(W\) is the LLR, in m;
\(L\) is the magnitude of a charge of normal throw with LLR \(=W\).
The task of all authors consisted mainly in deriving a mathematical expression for \(f(n)\).
Quite a number of formulas for expressing \(f(n)\) have been proposed. A considerable part of these formulas has not found practical application and at present is only of historical interest. We shall dwell only on those formulas which, to one degree or another, have retained their significance up to the present time.
Formulas for calculating charges of reduced throw may be combined into three groups according to the features most characteristic of them. Among these should be included the basic principle of constructing the formula, and on which quantities entering into the formula the dependence is established.
Formulas of the first group
To this group we have assigned the formulas, except for the formula of A. V. Kovazhenkov (1945), that are the earliest in time (Table 2).
Characteristic features of these formulas:
1) the basic principle adopted for charges of normal throw—that the magnitude of the charge is proportional to the weight of the rock torn away or to the volume of the throw crater—was not changed;
2) the value \(f(n)\) was made dependent only on the blast action index \(n\);
3) in deriving the value \(f(n)\), a powder charge or a charge of brisant explosive was taken into account.
In deriving the formulas for \(f(n)\) of this group, the authors, applying various methods of processing empirical data, proceeded from the following general propositions:
1) throw craters with a blast action index \(n\) not equal to 1, at a constant LNS, can be obtained by increasing or decreasing the charge;
2) with an increase or decrease in the magnitude of the charge, if the LNS remains constant, the blast action index \(n\) will change its value; consequently, the magnitude of the charge may be regarded as a function of the blast action index \(n\)—\(f(n)\).
Formula 1 (1812) [3 and 12]. Of the formulas that have found application, this formula is the earliest.
Fig. 2. Study of \(f(n)\).
To derive the mathematical expression for \(f(n)\), experimental data were used (Fig. 2). The magnitude of the powder charge for a given LNS \(= W\), forming the normal throw crater \(mOn\), was established, as well as the magnitude of the charge forming, at the same LNS, the crater of intensified throw \(MON\) with the greatest value of the blast action index for powder charges, \(n = 3\). Having then calculated at what LNS \(= W_n\) the same charge of intensified throw in the given medium would form the normal throw crater \(MLN\), the increment \((r_1 - r)\) was determined from the experimental data, and, on the other hand, the increment of the calculated LNS \(= W_n\) relative to the actual LNS \(W\), equal to \((W_n - W)\), was determined; it was accepted that the ratio
\[ \frac{W_n - W}{r_1 - r} \]
for all values \(n = \mathrm{const} = 0.91\).
Proceeding further from the proposition that “the magnitudes of two normal throw charges of one and the same explosive, acting in one and the same medium, other conditions being equal but with different LNS, are related to one another as the cubes of their LNS,” and then carrying out the corresponding simple mathematical transformations, the following formula was proposed:
\[ f(n)=\left(\frac{W_n}{W}\right)^3=(0.09+0.91\,n)^3. \tag{11} \]
Formula 2 [3 and 12]. The values \(f(n)\), calculated by the originally proposed formula (11), for values of the index of explosive action \(n\) tending to zero, turn out to be underestimated; this was noted by the author of the formula himself, and at the same time he proposed a new formula for calculating charges of reduced throw, in the derivation of which exclusively experimental data were used.
A series of experimental explosions established that a charge of normal throw with LNS \(= W\) at LNS \(7/4\,W\) becomes a charge of maximum compression, and at LNS \(= 7/5\,W\)—a loosening charge.
Similarly, a charge of maximum compression with LNS \(= W\) at LNS \(= 4/7\,W\) becomes a charge of normal throw, and a loosening charge with LNS \(= W\) at LNS \(= 5/7\,W\)—also a charge of normal throw.
Taking into account the data from experimental explosions, a formula was constructed that would satisfy the above conditions.
This formula proved to be
\[ f(n)=\left(\frac{4+3n}{7}\right)^3 . \tag{12} \]
Formula 3—Frolov (1868) [3 and 12]. Frolov proposed a simple method for deriving a formula for determining \(f(n)\).
Having analyzed the results of experimental explosions, Frolov showed that, for equal LNS, the magnitude of the charge increases when the radius of the base of the crater increases; if the radii are equal, then the magnitudes of the charges increase with increasing LNS. These same experiments showed that for charges of intensified throw \((n>1)\) the value \(f(n)\) lies within the limits:
\[ 1<f(n)<n^3, \tag{13} \]
and for charges of reduced throw \((n<1)\):
\[ 1>f(n)>n^3 . \tag{14} \]
On the other hand, at \(n=1\) the value \(f(n)\) is also \(=1\).
The three indicated conditions can be satisfied by only one formula of the general form:
\[ f(n)=A+B n^3, \tag{15} \]
where
\[ A+B=1. \tag{16} \]
Applying this method, but using data from earlier and widely known explosions, Frolov determined the values of the coefficients \(A=0.5\) and \(B=0.5\) and proposed, for determining \(f(n)\), the formula:
\[ f(n)=(0.5+0.5\,n^3). \tag{17} \]
Frolov was the first to point out the major role of the connectedness of different media and the need to take this connectedness fully into account.
Formula 4—Boreskov (1871) [3 and 12]. Developing the method proposed by Frolov, but refining the experimental data, Boreskov proposed the following expression for determining \(f(n)\):
\[ f(n)=(0.4+0.6\,n^3). \tag{18} \]
Formula 5 [3 and 12]. Using the very same method, the values of the coefficients \(A=0.2\) and \(B=0.8\) were calculated from experimental data, and the formula was proposed:
\[ f(n)=(0.2+0.8\,n^3). \tag{19} \]
Formula 6 (1871) [3 and 12]. The main purpose pursued in deriving this formula was to simplify the calculation of charges by allowing for a certain overestimation of them in blasting operations in military engineering, with the sole aim of destruction.
It was assumed that \(W_n = n \cdot W = r\), and for \(f(n)\) the following expression was proposed:
\[ f(n)=\left(\frac{W_n}{W}\right)^3=\left(\frac{n\cdot W}{W}\right)^3=n^3. \tag{20} \]
Formula 7 (1873) [3 and 12]. Through numerous experimental blastings of charges with increased and reduced throw, it was established that for them the radius of explosion \(R\) (Fig. 3) can be determined through \(W_n\) by the following equality:
\[ R = W_n + 0.41\,W. \tag{21} \]
Fig. 3. Investigation of \(f(n)\).
On the other hand,
\[ R=\sqrt{W^2+r^2}. \tag{22} \]
It was further assumed that
\[ f(n)=\left(\frac{W_n}{W}\right)^3. \tag{23} \]
After replacing \(r\) by its equivalent \(n\cdot W\), and solving the three equations given above, (21), (22), and (23), the formula obtained was:
\[ f(n)=\left(\sqrt{1+n^2}-0.41\right)^3. \tag{24} \]
Formula 8 — A. V. Kovazhenkov (1945) [6]. The author, considering that the index of the action of an explosion \(n\) (the throw index) does not fully encompass all the features of the action of a charge, in order to supplement the concept of charge action introduces a new index, the “throw coefficient” \(k\), characterizing the ratio of the volume of the throw crater to the volume of the sphere of destruction. Having then proved that the “throw coefficient” depends only on the index of the action of the explosion, and solving the equations obtained, after transformations he proposes the following formula:
\[ f(n)=0.356\,(1+n^2)^{\frac{3}{2}}. \tag{25} \]
Table 2
Values of \(f(n)\) according to formulas of the first group
| No. | Formula designation | Year | Formula for \(f(n)\) | \(n=0\) | \(n=0.25\) | \(n=0.5\) | \(n=0.75\) | \(n=0.8\) | \(n=0.9\) | \(n=1.0\) | \(n=1.25\) | \(n=1.5\) | \(n=1.75\) | \(n=2.0\) | \(n=2.25\) | \(n=2.50\) | \(n=2.75\) | \(n=3.0\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1 | \(1^{1)}\) | 1812 | \(f(n)=(0.09+0.91n)^3\) | — | — | — | — | — | — | 1.0 | 0.86 | 3.08 | 4.70 | 6.97 | 9.78 | 13.93 | 17.36 | 22.43 |
| 2 | \(2^{2)}\) | — | \(f(n)=\left(\dfrac{4+3n}{7}\right)^3\) | 0.187 | 0.31 | 0.49 | 0.73 | 0.76 | 0.88 | 1.0 | — | — | — | — | — | — | — | — |
| 3 | 3—Frohde | 1868 | \(f(n)=(0.5+0.5n^3)\) | 0.50 | 0.51 | 0.56 | 0.71 | 0.75 | 0.86 | 1.0 | 1.48 | 2.19 | 3.18 | 4.50 | 6.25 | 8.31 | 10.90 | 14.0 |
| 4 | 4—Boreskov | 1871 | \(f(n)=(0.4+0.6n^3)\) | 0.40 | 0.41 | 0.48 | 0.65 | 0.70 | 0.84 | 1.0 | 1.57 | 2.43 | 3.62 | 5.20 | 7.30 | 9.78 | 12.88 | 16.6 |
| 5 | \(5^{3)}\) | — | \(f(n)=(0.2+0.8n^3)\) | 0.20 | 0.21 | 0.30 | 0.54 | 0.60 | 0.78 | 1.0 | 1.76 | 2.90 | 4.48 | 6.60 | 9.40 | 12.71 | 16.80 | 21.8 |
| 6 | \(6^{4)}\) | 1871 | \(f(n)=n^3\) | 0 | 0.016 | 0.125 | 0.42 | 0.51 | 0.73 | 1.0 | 1.95 | 3.38 | 5.36 | 8.0 | 11.39 | 15.63 | 20.8 | 27.0 |
| 7 | 7 | 1873 | \(f(n)=\bigl(\sqrt{1+n^2}-0.41\bigr)^3\) | 0.20 | 0.25 | 0.35 | 0.59 | 0.66 | 0.82 | 1.0 | 1.68 | 2.69 | 4.14 | 6.08 | 8.64 | 11.86 | 15.93 | 20.8 |
| 8 | 8—A. V. Kovazhenkov | 1945 | \(f(n)=0.356(1+n^2)^{3/2}\) | 0.356 | 0.37 | 0.50 | 0.70 | 0.75 | 0.87 | 1.0 | 1.46 | 2.06 | 2.91 | 4.00 | 5.30 | 6.98 | 8.92 | 11.8 |
1) Only for powder charges of enhanced throw.
2) Only for powder charges of reduced throw.
3) For powder charges under any conditions and for explosive charges when \(n<1\).
4) For explosive charges when \(n \geq 1\) and for powder charges in rock, masonry, and concrete.
In a response (as part of a discussion) to the article by A. V. Kovazhenkov, I. M. Naumov [8] considers that, in deriving the formula for \(f(n)\):
1) the collapse zone was incorrectly assumed to be spherical, since only a cone of loosening, and not a sphere, is formed around the crater of ejection;
2) the derived formula for \(f(n)\) is incorrect.
Table 2 gives numerical values of \(f(n)\) according to formulas of the first group, in which the value of \(f(n)\) was made dependent only on the blast-action index \(n\). The numerical values of \(f(n)\) according to the formulas of the first group differ substantially from one another, depending on what empirical data and what method of processing them the authors used in deriving these formulas. The closest numerical values of \(f(n)\) are those given by formulas (19) and (24). The mean numerical values of \(f(n)\) are given by formula (24), which has become most widespread in Western countries.
All the investigators who proposed formulas for determining \(f(n)\) that we have assigned to the first group sought only to introduce a correction into the basic formula (9), adapting it to practical data. In general, none of these formulas reflected the physical law of the action of a charge in a medium. In addition, the choice of the blast-action index \(n\) was not determined by objective factors. As a result, the calculated charge magnitudes obtained diverged from reality, and none of these formulas received general recognition in charge calculations.
Formulas of the Second Group
When formulas of the first group were used, practice showed that, in choosing the magnitude of the blast-action index \(n\), and consequently \(f(n)\), it is necessary to take into account the absolute value of the least resistance line (LLR). This circumstance was taken into account, and formulas were proposed that we have assigned to the second group (Table 3), in whose derivation:
1) the basic principle remained the same;
2) the value of \(f(n)\) was made dependent not on the blast-action index \(n\), but on the absolute magnitude of the LLR;
3) the formulas were intended for brisant explosives.
Formula 9 (1934) [3]. The formula is intended for calculating charges of reduced ejection in mining operations in those cases where only fragmentation of the rock is required without significant ejection of it.
The author analyzed in detail the factors affecting the magnitude of the specific consumption of explosives and gave tables of coefficients accounting for the degree of this influence.
When a charge is exploded, the presence of two cones is established (Fig. 4). Part of the rock in the volume of the cone \(AOB\) will be ejected, forming an ejection crater \(AOB\) with base radius \(r\). Beyond the limits of the ejection crater, the weakened wave of blast action, producing only loosening of the rock, forms the loosening cone \(COD\) with base radius of the crater \(r_1\), called the crater of charge action.
In accordance with the establishment of two cones, two characteristics are introduced:
\[ \frac{r}{W} = n \]
—the ejection index, and
\[ \frac{r_1}{W} = n^1 \]
—the blast-action index.
On the basis of practical experience, the following basic assumptions were adopted as initial ones:
1) for any charge of reduced ejection, the blast-action index
\[ \frac{r_1}{W} = n^1 = 1 \]
and consequently the cone of blast action is a right-angled cone;
2) good fragmentation is obtained only if, between the throw index \(n\) and the LNS (in meters), there exists a dependence expressed by the equality:
\[ n^{2}\cdot W=1. \tag{26} \]
On the basis of these propositions, the following relation between the elements of the throw funnel was established:
\[ r=n\cdot W = W\sqrt{\frac{1}{W}}=\sqrt{W}, \tag{27} \]
i.e., the radius of the throw sphere, which in practice cannot be measured, is equal to the square root of the LNS.
In addition to the listed supplementary definitions, the author of the formula also introduced the value of the arrow \(k\)—the excess of the throw radius \(R\) (Fig. 4) over the LNS \(W\).
Fig. 4. Throw funnel and funnel of charge action.
Using these basic dependences by means of algebraic transformations, and taking earlier investigations as a basis, the author obtained the following formula for \(f(n)\), expressed as a function of the LNS value:
\[ f(n)=\left(\sqrt{\frac{W+1}{W}}-k\right)^{3}; \tag{28} \]
\[ k=W\left(\sqrt{\frac{W+1}{W}}-1\right), \tag{29} \]
where \(W\) is the LNS in m;
\(k\) is the arrow, or correction to the LNS, in m.
Formula (28) became widely used in the USSR. Later, however, a number of critical remarks concerning the method for determining \(f(n)\) and the numerical values of the latter were made by N. V. Melnikov, P. Ya. Taranov, V. I. Terent’ev, Sh. I. Ibraev, A. V. Kovazhenkov, and others [7; 14; 15; 4 and 6]. In summary form these remarks are as follows:
1) the formula is based on the incorrect principle of constancy of the radius of explosion (throw) \(R\) for different LNS values;
2) the dependence \(n^{2}\cdot W=1\) was obtained by mathematical transformations, and not on the basis of experimental explosions;
Table 3
Values of \(f(n)\) according to formulas of the second group
| No. | Formula designation | Year | Formula for \(f(n)\) | \(n=-3.16\) LNS \(=-0.1\) |
\(n=-1.41\) LNS \(=-0.5\) |
\(n=-1.00\) LNS \(=-1.00\) |
\(n=-0.82\) LNS \(=-1.5\) |
\(n=-0.71\) LNS \(=-2.0\) |
\(n=-0.63\) LNS \(=-2.5\) |
\(n=-0.58\) LNS \(=-3.0\) |
\(n=-0.50\) LNS \(=-4.0\) |
\(n=-0.45\) LNS \(=-5.0\) |
\(n=-0.41\) LNS \(=-6.0\) |
\(n=-0.38\) LNS \(=-7.0\) |
\(n=-0.35\) LNS \(=-8.0\) |
\(n=-0.33\) LNS \(=-9.0\) |
\(n=-0.32\) LNS \(=-10.00\) |
\(n=-0.26\) LNS \(=-15.00\) |
\(n=-0.22\) LNS \(=-20.00\) |
\(n=-0.18\) LNS \(=-30.00\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1 | 9 | 1934 | \(f(n)=\left(\sqrt{\dfrac{W+1}{W}}-k\right)^3\) | 30.0 | 2.6 | 1.00 | 0.62 | 0.49 | 0.38 | 0.33 | 0.28 | 0.24 | 0.22 | 0.21 | 0.19 | 0.18 | 0.17 | 0.16 | 0.15 | 0.15 |
| 2 | 10—N. V. Melnikov | 1944 | \(f(n)=\dfrac{\left((W+1-k)\sqrt{\dfrac{W+1}{W}}\right)^3}{2\sqrt{2W^3}}\) | 13.6 | 2.55 | 1.00 | 0.76 | 0.68 | 0.58 | 0.55 | 0.53 | 0.47 | 0.44 | 0.43 | 0.42 | 0.41 | 0.40 | 0.39 | 0.38 | 0.37 |
| 3 | 11—V. E. Finindova | 1944 | \(f(n)=\left(\dfrac{W+1}{2W}\right)^{3/2}\) | 12.90 | 1.82 | 1.00 | 0.76 | 0.65 | 0.58 | 0.54 | 0.50 | 0.46 | 0.45 | 0.44 | 0.42 | 0.41 | 0.40 | 0.39 | 0.38 | 0.37 |
Table 4
Values of \(f(n)\) according to formulas of the first group
(converted to LLR)
| No. | Formula designation | Year | Formula for \(f(n)\) | Indicator of explosive action \(=3.16\) LLR value \(=0.1\) |
Indicator of explosive action \(=1.41\) LLR value \(=0.5\) |
Indicator of explosive action \(=1.00\) LLR value \(=1.0\) |
Indicator of explosive action \(=0.82\) LLR value \(=1.5\) |
Indicator of explosive action \(=0.71\) LLR value \(=2.0\) |
Indicator of explosive action \(=0.68\) LLR value \(=2.5\) |
Indicator of explosive action \(=0.58\) LLR value \(=3.0\) |
Indicator of explosive action \(=0.50\) LLR value \(=4.0\) |
Indicator of explosive action \(=0.45\) LLR value \(=5.0\) |
Indicator of explosive action \(=0.41\) LLR value \(=6.0\) |
Indicator of explosive action \(=0.38\) LLR value \(=7.0\) |
Indicator of explosive action \(=0.35\) LLR value \(=8.0\) |
Indicator of explosive action \(=0.33\) LLR value \(=9.0\) |
Indicator of explosive action \(=0.32\) LLR value \(=10.0\) |
Indicator of explosive action \(=0.26\) LLR value \(=15.0\) |
Indicator of explosive action \(=0.22\) LLR value \(=20.0\) |
Indicator of explosive action \(=0.18\) LLR value \(=30.0\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1 | \(1^{1)}\) | 1812 | \(f(n)=(0.09+0.91\,n)^3\) | 25.9 | 2.57 | 1.00 | 0.59 | 0.41 | 0.29 | 0.24 | 0.17 | 0.125 | 0.097 | 0.085 | 0.068 | 0.059 | 0.055 | 0.039 | 0.024 | 0.016 |
| 2 | \(2^{2)}\) | — | \(f(n)=\left(\dfrac{4+3n}{7}\right)^3\) | — | — | 1.00 | 0.80 | 0.68 | 0.59 | 0.55 | 0.49 | 0.45 | 0.42 | 0.40 | 0.37 | 0.36 | 0.36 | 0.32 | 0.30 | 0.28 |
| 3 | 3—Frolov | 1868 | \(f(n)=(0.5+0.5n^3)\) | 16.3 | 1.90 | 1.00 | 0.77 | 0.68 | 0.63 | 0.60 | 0.56 | 0.55 | 0.53 | 0.53 | 0.52 | 0.52 | 0.52 | 0.51 | 0.51 | 0.50 |
| 4 | 4—Boreskov | 1871 | \(f(n)=(0.4+0.6n^3)\) | 19.3 | 2.08 | 1.00 | 0.73 | 0.62 | 0.55 | 0.52 | 0.48 | 0.45 | 0.44 | 0.43 | 0.43 | 0.42 | 0.42 | 0.41 | 0.41 | 0.40 |
| 5 | \(5^{3)}\) | — | \(f(n)=(0.2+0.8n^3)\) | 25.2 | 2.42 | 1.00 | 0.64 | 0.50 | 0.40 | 0.36 | 0.30 | 0.27 | 0.26 | 0.24 | 0.23 | 0.23 | 0.23 | 0.21 | 0.21 | 0,[[unclear: value cut off]] |
| 6 | \(6^{4)}\) | 1871 | \(f(n)=n^3\) | 31.5 | 2.80 | 1.00 | 0.55 | 0.38 | 0.25 | 0.20 | 0.125 | 0.091 | 0.069 | 0.065 | 0.041 | 0.036 | 0.033 | 0.018 | 0.011 | 0.058 |
| 7 | 7 | 1873 | \(f(n)=\left(\sqrt{1+n^2}-0.41\right)^3\) | 24.1 | 2.30 | 1.00 | 0.68 | 0.53 | 0.46 | 0.39 | 0.35 | 0.33 | 0.30 | 0.29 | 0.27 | 0.26 | 0.26 | 0.25 | 0.23 | 0.22 |
| 8 | 8—A. V. Kovazhenkov | 1945 | \(f(n)=0.356(1+n^2)^{\frac{3}{2}}\) | 12.9 | 1.84 | 1.00 | 0.77 | 0.66 | 0.59 | 0.57 | 0.50 | 0.47 | 0.45 | 0.43 | 0.42 | 0.41 | 0.40 | 0.38 | 0.37 | 0.36 |
1) Only for powder charges with enhanced ejection.
2) Only for powder charges with reduced ejection.
3) For powder charges at any \(n\) and for explosive charges when \(n<1\).
4) For brisant explosives when \(n\geq 1\) and for powder charges in rock, masonry, and concrete.
Table 5
Values of \(f(n)\) and \(f(k)\) according to formulas of the third group
| No. | Formula designation | Year | Formulas for \(f(n)\) and \(f(k)\) | \(=0.1\) | \(=0.5\) | \(=1.00\) | \(=1.5\) | \(=2.0\) | \(=2.5\) | \(=3.0\) | \(=4.0\) | \(=5.0\) | \(=6.0\) | \(=7.0\) | \(=8.0\) | \(=9.0\) | \(=10.0\) | \(=15.0\) | \(=20.0\) | \(=30.0\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1 | 12—S. D. Osnovin | 1939 | \(\displaystyle f(n)=\left(\sqrt{\frac{\sin\alpha(3\cos\alpha+\sin\alpha)}{4\cos^{2}\alpha}}\right)^{3}\) | 10.6 | 1.81 | 1.00 | 0.66 | 0.52 | 0.44 | 0.37 | 0.29 | 0.24 | 0.22 | 0.21 | 0.16 | 0.14 | 0.13 | 0.10 | 0.08 | 0.06 |
| 2 | 13—V. K. Karchevskii | 1945 | \(\displaystyle f(n)=\left(\frac{0.9}{W}+1\right)\) | 9.1 | 1.90 | 1.00 | 0.77 | 0.55 | 0.46 | 0.40 | 0.32 | 0.28 | 0.25 | 0.23 | 0.21 | 0.20 | 0.19 | 0.16 | 0.14 | 0.13 |
| 3 | 14—S. D. Osnovin | 1953 | \(\displaystyle f(k)=\frac{1}{k+1}\left(\frac{k}{W}+1\right)\) | \(K=1^{*}\) | 1.00 | 0.84 | 0.75 | 0.70 | 0.67 | 0.63 | 0.60 | 0.58 | 0.57 | 0.56 | 0.56 | 0.55 | 0.54 | 0.53 | 0.52 | |
| 3 | 14—S. D. Osnovin | 1953 | \(\displaystyle f(k)=\frac{1}{k+1}\left(\frac{k}{W}+1\right)\) | \(K=9.5^{*}\) | 1.00 | 0.70 | 0.55 | 0.44 | 0.40 | 0.32 | 0.28 | 0.25 | 0.23 | 0.21 | 0.20 | 0.19 | 0.16 | 0.140 | 0.125 |
*) \(K=1\) and \(K=9.5\) are the extreme limits for rocks; between them, intermediate values of \(f(k)\) are obtained on a sliding scale.
3) mathematical errors were made in deriving the formula;
4) the value of the arrow \(k\) was used incorrectly;
5) the results of calculating charges at large LLR values are underestimated.
Formula 10—N. V. Melnikov (1944) [7]. Mining engineer N. V. Melnikov notes that the author of formula (28), although he did take into account the valuable study of the change in the specific consumption of explosives with increasing LLR, nevertheless made errors in deriving the formula.
Correcting the errors which, in N. V. Melnikov’s opinion, were made in deriving formula (28), he gives the following expression for determining \(f(n)\):
\[ f(n)= \frac{\left(W+1-k\sqrt{\frac{W+1}{W}}\right)^3} {2\sqrt{2W^3}}, \tag{30} \]
where \(W\) is the LLR in m;
\(k\) is the arrow or correction \(k\) of the LLR in m.
Formula 11—V. E. Filippovich (1944) [16]. Using the data of the article by mining engineer N. V. Melnikov, V. E. Filippovich proposes another method of solution for deriving the value \(f(n)\), considering the method proposed by him to be simpler and more rational.
The formula derived by V. E. Filippovich has the following form:
\[ f(n)=\left(\frac{W+1}{2W}\right)^{\frac{3}{2}}. \tag{31} \]
As can be seen from Table 3, the formulas of N. V. Melnikov and V. E. Filippovich give higher numerical values of \(f(n)\) at large LLR values than are obtained from formula (28).
For comparison of the numerical values of \(f(n)\) according to the formulas of the second and first groups, Table 4 gives the numerical values of \(f(n)\) according to the formulas of the first group, recalculated for the same LLR values that were adopted for the indicators of blasting action \(n^1\) in formula (28) (Table 3).
All the researchers who proposed formulas for determining \(f(n)\), which we have assigned to the second group, likewise sought to introduce only corrections into the basic formula (9), adapting it to practical data. Nevertheless, this was a step forward, since instead of the blasting-action index \(n\), which is not conditioned by any objective factors, \(f(n)\) was made dependent on the quantity being determined—the LLR. However, these formulas too, when applied in practice to calculate charges for reduced throw in rocks of different kinds, i.e., in rocks with different physicomechanical properties, give contradictory, unreliable results.
All this compelled the search for new ways and new methods for calculating charges of reduced throw.
Third group of formulas
In the third group (Table 5) we have included formulas based on principles and methods different from those adopted in the formulas of the first and second groups.
The works of this trend include the works of G. I. Pokrovsky, A. F. Sukhanov, S. D. Osnovin, M. P. Brodsky, M. N. Kosachev, D. D. Bizyukin, V. K. Karchevsky, and other authors.
Formula 12—S. D. Osnovin (1939) [9]. S. D. Osnovin assumed that it is not the volume of the medium being blasted, but the system of forces acting—
…passing along the lateral surface of the breakout funnel, and the system of resistance forces along this surface determines the action of any charge. Having determined how the resistances change along the lateral surface of a breakout funnel with \( \mathrm{LLR}=W \) and with an index of explosive action \( n \) not equal to 1, when the same charge gives a normal breakout funnel with \( \mathrm{LLR}=W_{\text{norm.}} \), a relationship is thereby established between \( W_{\text{norm.}} \) and \( W \). Assuming further that
\[ f(n)=\frac{W_{\text{norm.}}^{3}}{W^{3}}, \tag{32} \]
the following general expression was obtained for determining \( f(n) \):
\[ f(n)=\left(\sqrt{ \frac{\tg \alpha \left(\tg \varphi \cdot \cos \alpha+\sin \alpha\right)} {\sqrt{2}\cdot \cos \alpha \left(\tg \varphi \cdot \cos 45^\circ+\sin 45^\circ\right)} }\right)^{3}. \tag{33} \]
For \( \tg \varphi=3 \):
\[ f(n)=\left(\sqrt{ \frac{\sin \alpha \left(3\cos \alpha+\sin \alpha\right)} {4\cos^{2}\alpha} }\right)^{3}, \tag{34} \]
where \( \alpha \) is the angle between the axis of the funnel and the generatrix;
\( \varphi \) is the angle of internal friction, or resistance, of the rock.
Formula 13—V. K. Karchevskii (1945) [5]. According to V. K. Karchevskii’s basic proposition, during an explosion the charge must:
1) detach a block from the mass;
2) break up the block detached from the mass into pieces;
3) lift and displace the entire volume of the blasted mass.
The work of the charge, consequently, must be proportional to the magnitude of the breakout surface, the planes of crushing, and the volume of the mass displaced by the explosion.
With an increase in the LLR, the volume increases rapidly (as the cube) and the area more slowly (as the square). The specific consumptions of explosive for overcoming resistance along the breakout surface, for forming cracks and crushing \( q_{1} \), and for displacing the blasted mass \( q_{2} \), will be different. Moreover, the specific consumption of explosive \( q_{1} \), with a change in the depth of charge placement, will vary proportionally to \( W^{2} \), while the specific consumption of explosive \( q_{2} \) will vary proportionally to \( W^{3} \). The total charge weight will be the sum of the explosive expenditures for these two kinds of work.
Assuming, on the basis of experimental and calculated data, that
\[ \frac{q_{1}}{q_{2}}=\mathrm{const}=9, \tag{35} \]
V. K. Karchevskii proposed the following formula:
\[ q=0.9\,q\cdot W^{2}+0.1\,q\cdot W^{3}, \tag{36} \]
or, in simplified form,
\[ Q=q\cdot W^{2}\left(0.9+0.1\,W\right). \tag{37} \]
If this formula is reduced to the form of the general formula for calculating charges of reduced throw, then the value of \( f(n) \) will be
\[ f(n)=\left(\frac{0.9}{W}+0.1\right). \tag{38} \]
Formula 14—S. D. Osnovin (1953) [10; 11]. To perform useful work, a charge, according to the proposition put forward by S. D. Osnovin, must simultaneously overcome:
1) the resistance to separation of the rock from the mass;
2) the force of gravity and inertia of the rock being detached;
Table 6
Value of \(f(k)\) for various values of the coefficient \(\kappa\)
| LNS \(m\) | \(\kappa=0\) | 0,1 | 0,5 | 1,0 | 1,2 | 1,4 | 1,6 | 1,8 | 2,0 | 3,0 | 4,0 | 5,0 | 6,0 | 7,0 | 8,0 | 9,0 | 9,5 | 10,0 | 10,5 | 15,0 | \(\infty\) |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 0,1 | 1 | 1,82 | 3,33 | 5,50 | 5,91 | 6,25 | 6,54 | 6,79 | 7,00 | 7,75 | 8,20 | 8,50 | 8,71 | 8,88 | 9,00 | 9,10 | 9,14 | 9,18 | 9,22 | 9,38 | 10,00 |
| 0,5 | 1 | 1,09 | 1,33 | 1,50 | 1,55 | 1,58 | 1,62 | 1,64 | 1,67 | 1,75 | 1,80 | 1,83 | 1,86 | 1,88 | 1,89 | 1,90 | 1,90 | 1,91 | 1,91 | 1,94 | 2,00 |
| 1,0 | 1 | 1,00 | 1,00 | 1,00 | 1,00 | 1,00 | 1,00 | 1,00 | 1,00 | 1,00 | 1,00 | 1,00 | 1,00 | 1,00 | 1,00 | 1,00 | 1,00 | 1,00 | 1,00 | 1,00 | 1,00 |
| 2,0 | 1 | 0,95 | 0,83 | 0,75 | 0,73 | 0,71 | 0,69 | 0,68 | 0,66 | 0,63 | 0,60 | 0,58 | 0,57 | 0,56 | 0,56 | 0,55 | 0,55 | 0,54 | 0,54 | 0,53 | 0,50 |
| 3,0 | 1 | 0,94 | 0,78 | 0,67 | 0,64 | 0,61 | 0,59 | 0,57 | 0,55 | 0,50 | 0,47 | 0,44 | 0,43 | 0,42 | 0,41 | 0,40 | 0,40 | 0,39 | 0,39 | 0,38 | 0,38 |
| 4,0 | 1 | 0,93 | 0,75 | 0,63 | 0,59 | 0,56 | 0,54 | 0,52 | 0,50 | 0,44 | 0,40 | 0,37 | 0,36 | 0,34 | 0,33 | 0,32 | 0,32 | 0,31 | 0,31 | 0,30 | 0,25 |
| 5,0 | 1 | 0,93 | 0,73 | 0,60 | 0,56 | 0,53 | 0,51 | 0,49 | 0,47 | 0,40 | 0,36 | 0,33 | 0,31 | 0,30 | 0,29 | 0,28 | 0,28 | 0,27 | 0,27 | 0,25 | 0,20 |
| 6,0 | 1 | 0,92 | 0,72 | 0,58 | 0,54 | 0,51 | 0,49 | 0,46 | 0,44 | 0,37 | 0,33 | 0,31 | 0,29 | 0,27 | 0,26 | 0,25 | 0,25 | 0,24 | 0,24 | 0,22 | 0,17 |
| 7,0 | 1 | 0,92 | 0,71 | 0,57 | 0,53 | 0,50 | 0,47 | 0,45 | 0,43 | 0,36 | 0,31 | 0,29 | 0,27 | 0,25 | 0,24 | 0,23 | 0,23 | 0,22 | 0,22 | 0,20 | 0,14 |
| 8,0 | 1 | 0,92 | 0,71 | 0,56 | 0,52 | 0,49 | 0,46 | 0,44 | 0,42 | 0,34 | 0,30 | 0,27 | 0,25 | 0,24 | 0,22 | 0,21 | 0,21 | 0,20 | 0,20 | 0,18 | 0,12 |
| 9,0 | 1 | 0,92 | 0,71 | 0,56 | 0,51 | 0,48 | 0,45 | 0,43 | 0,41 | 0,33 | 0,29 | 0,26 | 0,24 | 0,22 | 0,20 | 0,20 | 0,20 | 0,19 | 0,19 | 0,17 | 0,11 |
| 10,0 | 1 | 0,92 | 0,70 | 0,55 | 0,51 | 0,47 | 0,45 | 0,42 | 0,40 | 0,32 | 0,28 | 0,25 | 0,23 | 0,21 | 0,20 | 0,19 | 0,19 | 0,18 | 0,18 | 0,16 | 0,10 |
| 15,0 | 1 | 0,91 | 0,70 | 0,54 | 0,50 | 0,46 | 0,43 | 0,40 | 0,38 | 0,30 | 0,25 | 0,22 | 0,20 | 0,18 | 0,17 | 0,16 | 0,16 | 0,15 | 0,15 | 0,13 | 0,07 |
| 20,0 | 1 | 0,91 | 0,69 | 0,53 | 0,48 | 0,45 | 0,42 | 0,39 | 0,37 | 0,29 | 0,24 | 0,21 | 0,19 | 0,17 | 0,16 | 0,145 | 0,140 | 0,136 | 0,133 | 0,11 | 0,05 |
| 30,0 | 1 | 0,90 | 0,68 | 0,52 | 0,47 | 0,44 | 0,40 | 0,38 | 0,36 | 0,28 | 0,23 | 0,19 | 0,17 | 0,15 | 0,14 | 0,130 | 0,125 | 0,121 | 0,118 | 0,094 | 0,038 |
3) resistance to the crushing of the rock and to the displacement of the entire volume of detached rock;
4) air resistance.
In considering the action of a charge during an explosion, the author of the formula distinguishes the breakaway crater and the throw-out crater. By the breakaway crater is meant the crater bounded by the breakaway surface. By the throw-out crater is meant the crater within which, to one degree or another, the rock is thrown out.
Then, differentiating the specific consumption of explosive, determined for an LNS equal to 1 m, into its component parts in accordance with the principal resistances overcome by the charge, it is proved that, for a breakaway crater of any shape and with its similarity preserved as the LNS changes, the specific consumption of explosive
1) for overcoming the resistance along the breakaway surface will vary inversely proportional to the LNS;
2) for overcoming the resistance of the air column—in the same proportion;
3) for overcoming the force of gravity and the inertia of the rock being detached—will remain a constant quantity.
It is shown how, in this case, the degree of crushing of the rock and the degree of its throw-out will change.
Solving further the system of equations that has been set up, the following formula is proposed for determining \(f(k)\), introduced by the author in place of \(f(n)\) into the general formula for calculating charges of reduced throw-out:
\[ f(k)=\frac{1}{k+1}\left(\frac{k}{W}+1\right), \tag{39} \]
where \(f(k)\) is the correction to the specific consumption of explosive for LNS not equal to 1 m;
\(k\) is a coefficient depending on the physico-mechanical properties of the rock being blasted;
\(W\) is the LNS in m.
For determining the coefficient \(k\), an additional formula of simple form is given.
Numerical values of \(f(n)\) and \(f(k)\) according to the formulas of the third group are given in Table 5. In addition, Table 6 gives, in expanded form, numerical values of \(f(k)\) for different values of the coefficient \(k\). The maximum value of the coefficient \(k\) for rocks is determined as 9.5 (quartzites that are, to the highest degree, strong, dense, and tough). The value of the coefficient \(k\) for homogeneous massive and dense coal with unit weight \(\gamma=1.35\) is taken as equal to 1.
Formula 15—M. P. Brodsky (1953) [2]. On the basis of the general equation given by Frolov, M. P. Brodsky proposed the following formula for determining the optimum ripping charge:
\[ Q_p=0.48H^3\cdot x^3\cdot\gamma+0.07H^2\cdot x^2\cdot\sigma k_2, \tag{40} \]
where \(Q\) is the weight of the ammonite charge in kg;
\(H\) is the depth of charge placement in m;
\(x\) is a coefficient depending on \(H\);
\(\gamma\) is the unit weight of the rock being blasted in t/m\(^3\);
\(\sigma\) is the temporary resistance of the rock to spalling during blasting in t/m\(^2\).
The blast crater is assumed to have the form of a cone of revolution. Tables are given for determining the numerical values of \(x\), \(\gamma\), and \(\sigma\).
If this formula is reduced to the general-form formula for calculating charges (9), we obtain:
\[ Q=\left(0.48\cdot x^3\cdot\gamma+\frac{0.07\cdot x^2\cdot\sigma}{H}\right)H^3. \tag{41} \]
From a comparison of expression (41) with the general formula (9), it follows that the first binomial in parentheses on the right-hand side of equality (41) is essentially the specific consumption of explosives, determined for \( \mathrm{LNS}=1\,m \) and \( \mathrm{LNS}\leq 1\,m \) by calculation, proceeding from the values of the quantities \(x\), \(\gamma\), and \(\sigma\).
The formulas of the third group may be subdivided into two subgroups. To the first subgroup we have assigned the formulas of S. D. Osnovin (1939) and V. K. Karchevskii. To the second subgroup—the formulas of M. P. Brodskii and S. D. Osnovin (1953).
Between these two subgroups of formulas there is an essential difference.
According to the formula of S. D. Osnovin (1939) and the formula of V. K. Karchevskii, \(f(n)\) can have only one order of numerical values as the absolute magnitude of the LNS changes, irrespective of the physico-mechanical properties of the medium being blasted.
According to the formula of M. P. Brodskii, outwardly similar to the formula of V. K. Karchevskii, since in it, as in V. K. Karchevskii’s formula, the charge weight depends on \(H^{2}\) and \(H^{3}\), but it is determined directly from the physico-mechanical properties of the medium being broken, taken into account by the variables \(\gamma\) and \(\sigma\).
According to the formula of S. D. Osnovin (1953), \(f(k)\) can have a different order of numerical values (on a sliding scale) as the absolute magnitude of the LNS changes, depending on the value of the coefficient \(k\), which characterizes the physico-mechanical properties of the medium being broken. The extreme limits of the numerical values of \(f(k)\) are such that, for any LNS, the value of \(f(k)\) for rocks cannot lie outside these limits. Taking the numerical value of \(f(n)\), according to one or another formula for \( \mathrm{LNS}=30\,m \), as equal to the numerical value of \(f(k)\), one can see that the numerical values of \(f(n)\) according to the formulas of N. V. Melnikov and V. E. Filippovich are very close to the numerical values of \(f(k)\) at the coefficient \(k=1.9\); according to the formula of V. K. Karchevskii they coincide completely at the coefficient \(k=9\); according to the formulas of other authors they will be very close at the corresponding values of the coefficient \(k\). Thus, the formulas proposed up to now for determining \(f(n)\) are particular solutions according to this formula. It is also necessary additionally to note a mathematical property of the expression for \(f(k)\), consisting in the fact that, when unity on the right-hand side of the equation is replaced by other numerical values, for each value of the coefficient \(k\) ever new regularities of variation of \(f(k)\) as a function of the absolute magnitude of the LNS will be obtained.
Such, in its main features, is the development of ideas, principles, and methods for calculating concentrated charges of reduced throw and charges for loosening.
A whole series of investigators have dealt with the calculation of charges, each of them contributing his share, depending on the knowledge and methods of obtaining it that existed at the time. Undoubtedly, this work will continue in the future as well and will probably require much more labor, experiments, and observations to resolve this complex and extensive question.
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