A New Method for Calculating Elongated Charges of Reduced Throw and Loosening Charges

S. D. OSNOVIN

(Presented to the scientific seminar of the departments of development of bedded deposits, development of ore deposits, mine construction, safety engineering, and mine ventilation)

In the resolution of the Twentieth Congress of the Communist Party of the Soviet Union on the report of the Central Committee of the CPSU it is stated: “The decisive condition for the further growth of all industrial production is its continuous technical progress.”

For the correct and effective performance of blasting operations, an adequate theoretical basis and a sufficient level of scientific development in the calculation of charges are necessary. Meanwhile, questions of charge calculation still do not have the necessary theoretical basis, and the scientific development of this problematic question has been set as one of the principal and most urgent tasks.

1. Specific consumption of explosive¹⁾ and its variation depending on the magnitude of the LLR²⁾

To perform useful work, an elongated charge during an explosion, according to the fundamental proposition advanced by us, like a concentrated charge, must:

1) break the bond of the rock being detached with the massif;
2) overcome the force of gravity of the rock being detached;
3) overcome resistance from the pressure of the air column;
4) fracture and displace the entire volume of the detached rock.

The surface along which the detached rock is separated from the general massif we shall call the surface of detachment.

The number of units of the surface of detachment per unit volume of the rock detached from the massif we shall call the specific surface of detachment.

Let us initially assume that the detachment of rock from the general massif occurs in the form of a triangular prism, for which the conventional designations are (Fig. 1):

\[ W \text{ — LLR;} \]

\[ r \text{ — radius of the prism;} \]

\[ R \text{ — radius of action of the explosion;} \]

¹⁾ Explosive substance.
²⁾ LLR — line of least resistance.

\(\alpha\) — angle of the prism half-opening;

\(l\) — length of the charge or prism;

\(S\) — fracture surface of the prism;

\(V\) — volume of the prism;

\(F\) — area of the base of the prism.

The fracture surface for such a prism will be:

\[ S = 2Rl + 2rW = \frac{2Wl}{\cos \alpha} + \]

\[ + 2W^2 \tg \alpha, \tag{1} \]

Fig. 1. Shape of the body being fractured—a triangular prism

Introduce the following notation:

\[ m = \frac{l}{W}; \qquad l = mW. \tag{2} \]

Substituting the value of \(l\) into the preceding expression, we then obtain:

\[ S = \frac{2W^2 m}{\cos \alpha} + 2W^2 \tg \alpha = \frac{2W^2 m}{\cos \alpha} + \frac{2W^2 \sin \alpha}{\cos \alpha} = \]

\[ = \frac{2W^2}{\cos \alpha}\left(m+\sin \alpha\right). \tag{3} \]

The volume of the prism:

\[ V = Fl = rWl = W^2 \tg \alpha \, l = mW^3 \tg \alpha. \tag{4} \]

The specific fracture surface (per unit volume) will be:

\[ \frac{S}{V} = \frac{2W^2}{\cos \alpha}\left(m+\sin \alpha\right) : mW^3 \tg \alpha = \]

\[ = \frac{2W^2}{\cos \alpha}\left(m+\sin \alpha\right) : mW^3 \frac{\sin \alpha}{\cos \alpha} = \frac{2}{W}\left(\frac{1}{m}+\frac{1}{\sin \alpha}\right). \]

\[ \tag{5} \]

It follows from expression (5) that for constant \(m\) and angle \(\alpha\), i.e., for similar prisms, the magnitude of the specific fracture surface under frac-

rock in the form of a triangular prism will vary inversely proportionally to the L.L.R.

Let us now consider how the value of the specific fracture surface will vary with a change in the L.L.R., when the detachment of rock from the massif occurs not in the form of a triangular prism, but as a body of arbitrary shape.

The fracture surface of a triangular prism (3)

\[ S=\frac{2W^2}{\cos\alpha}\left(m+\sin\alpha\right). \]

The volume of the prism (4)

\[ V=mW^3\operatorname{tg}\alpha. \]

If the detached body is of a shape other than a triangular prism, then, for the same L.L.R., its fracture surface and volume will be different.

The fracture surface of any body \(S_1\) may be expressed through the fracture surface of the prism—\(S\), for which the latter must be changed by some, unknown to us, \(n\)-fold number of times. In exactly the same way, the volume of any body \(V_1\) may be expressed through the volume of the prism \(V\), changing it by some, unknown to us, \(n_1\)-fold number of times.

Then the fracture surface of any body will be:

\[ S_1=n\frac{2W^2}{\cos\alpha}\left(m+\sin\alpha\right), \tag{6} \]

its volume:

\[ V_1=n_1mW^3\operatorname{tg}\alpha, \tag{7} \]

and the specific fracture surface:

\[ \frac{S_1}{V_1}=\frac{n}{n_1}\frac{2}{W}\left(\frac{1}{m}+\frac{1}{\sin\alpha}\right). \tag{8} \]

If the detached body preserves its similarity with a change in the L.L.R., then the ratio \(\frac{n}{n_1}\), \(m\), and the angle \(\alpha\) will remain constant values, and, consequently, the magnitude of the specific fracture surface for any shape of the detached body, with a change in the depth of emplacement of the charge, will likewise vary inversely proportionally to the L.L.R.

Thus, for elongated charges (whether the detachment of rock from the general massif occurs in the form of a triangular prism, or with the base of the prism in the form of an ellipse, or the detachment of rock occurs in the plane along the line of boreholes and in the plane of the bench toe

ledge, as occurs in series blasting, etc.) the specific breakaway surface, while preserving the similarity of the body being broken away, with a change in the L.L.R. of the charge will always change inversely proportionally to the L.L.R.

But this still does not exhaust the question, since the resistance of the rock to the action of the explosion over the elementary areas of the breakaway surface may not be equivalent. Depending on the position of these elementary areas relative to the system of forces developing during the explosion, there may occur rupture—shearing, or shearing—rupture, or only shearing.

The breakaway surface of any body may be determined from expression (6).

To express the breakaway surface in units equivalent in resistance, it is necessary to introduce into expression (6) some correction coefficient, as yet unknown to us, \(n_2\).

Then the breakaway surface \(S_1\), in units equivalent in resistance, will be:

\[ S_2 = n_2 n \frac{2 W^2}{\cos \alpha}\left(m+\sin \alpha\right). \tag{9} \]

The volume of the body will remain unchanged (7). Hence the specific breakaway surface will be expressed as:

\[ \frac{S_2}{V_1} = \frac{n_2 n}{n_1 W}\frac{2}{W} \left(\frac{1}{m}+\frac{1}{\sin \alpha}\right). \tag{10} \]

Since, for constant \(m\) and angle \(\alpha\), with a change in the L.L.R. the bodies being broken away will be similar, all quantities on the right-hand side of the equation, except the L.L.R., will remain unchanged. From this follows a second important conclusion: the magnitude of the specific breakaway surface, expressed in units equivalent in resistance, for similarity of the bodies being broken away, regardless of their shape, with a change in the L.L.R. will always change inversely proportionally to the L.L.R.

This means that the deeper the charge is placed, the smaller the influence on the magnitude of the specific consumption of explosives will be exerted by the breakaway surface, and the greater the influence exerted by the volume or weight of the rock being broken away.

The number of units of area upon which the air column exerts pressure, per unit volume of the rock being broken away, we shall call the specific pressure area of the air column.

Let us suppose that the body being broken away is in the form of a triangular prism (Fig. 1).

The air column exerts pressure on the area \(ABVG\). The area \(ABVG\) will be:

\[ S^1 = 2 r l = 2 W l \operatorname{tg}\alpha, \tag{11} \]

or

\[ S^1 = 2 W m W \operatorname{tg}\alpha = 2 W^2 m \operatorname{tg}\alpha. \tag{12} \]

The magnitude of the specific pressure area of the air column:

\[ \frac{S^{1}}{V}=\frac{2 W^{2} m \operatorname{tg}\alpha}{m W^{3}\operatorname{tg}\alpha}=\frac{2}{W}, \tag{13} \]

i.e., the value of the specific area of pressure of the air column, as the depth of emplacement of the charge changes, varies inversely proportionally to the l.n.s. This will hold for any shape of the body being detached.

The foregoing shows that, for elongated charges, the specific surface of detachment and the specific area of pressure of the air column, with changes in the value of the l.n.s. and in the similarity of the bodies being detached, will vary in exactly the same way as the corresponding quantities for concentrated charges [4; 5]. And since this is so, the formula for determining the value of the specific consumption of explosive as a function of the absolute value of the l.n.s. for elongated charges will be identical with the formula for concentrated charges [4; 5]:

\[ q=f(k)\,q_{\text{norm}}\ \text{kg}; \tag{14} \]

\[ f(k)=\frac{1}{k+1}\left(\frac{k}{W}+1\right), \tag{15} \]

\(f(k)\) is the correction to the normal specific consumption of explosive \(q_{\text{norm}}\) when the l.n.s. \(\ne 1\) m.

Theoretically, the value of \(f(k)\) can vary within the limits from

\[ 1\ (k=0)\ \text{to}\ \frac{1}{W}\ (k=\infty). \]

For rocks these limits will be narrower.

The principal properties of \(f(k)\) for elongated charges are as follows:

1) when the l.n.s. \(=1\) m, for all values of the coefficient \(k\), \(f(k)=1\);

2) for an l.n.s. \(\ne 1\) m, the change in the numerical values of \(f(k)\) with a change in the l.n.s. will occur not according to one, but according to different laws, and the more sharply the greater the value of the coefficient \(k\), i.e., the stronger the rock;

3) a sliding scale of change in the numerical values of \(f(k)\) with a change in the l.n.s., depending on the value of the coefficient \(k\).

From the properties of \(f(k)\) it follows that:

1) with an increase in the l.n.s., for all values of the coefficient \(k\ne 0\), the value of the specific consumption will decrease;

2) the decrease in the value of the specific consumption of explosive will occur not according to one, but according to different laws, and the more sharply the greater the value of the coefficient \(k\), i.e., the stronger the rock.

The reasons for the possible reduction in the specific consumption of explosive with an increase in the depth of emplacement of the charge had not previously been clarified. The theoretical propositions set forth above concerning the action of charges in a medium explain the physical causes of this phenomenon.

When calculating the specific consumption of explosive by formula (14), the normal specific consumption of explosive must be determined for the same position of the cha—

row in the object, the order of blasting, and with the same ratio \(\frac{l}{W}=m\)

or \(\frac{H}{W}=m_1\), at which blasting operations are subsequently to be carried out. In the process of blasting operations, however, it is often necessary to use charges at different values of \(m\) and \(m_1\). In this case, use of the derived formula would make it necessary, when determining the normal specific consumption of explosives, to determine it empirically for all values of \(m\) and \(m_1\) that will occur in the performance of blasting operations. This method is the most accurate, but may not always prove acceptable. Then another method may be used—the method of recalculating the normal specific consumption of explosives, approximate though it is, but sufficiently meeting the requirements that may be imposed in the performance of blasting operations. For recalculating the normal specific consumption of explosives in this case, Table 1 may serve; it gives data on the relative change in the magnitude of the normal specific consumption of explosives as a function of the value \(\frac{l}{W}=m\) or \(\frac{H}{W}=m\), compiled on the basis of theoretical investigation and the corresponding calculations.

Table 1

Relative change in the magnitude of the normal specific consumption of explosives as a function of the value \(\frac{l}{W}=m\) or \(\frac{H}{W}=m_1\)

For calculating loosening charges, the normal specific consumption of explosives must be determined for the loosening charge. For charges of normal throw, the normal specific consumption of explosives is taken for charges of normal throw, and the numerical value \(f(k)\) for all l.n.s. is taken equal to unity. If, however, for one or another l.n.s. it is necessary to regulate the throw of rock between charges of normal throw and a loosening charge, this can be achieved by assigning intermediate values to the specific consumption of explosives for one or another l.n.s. between the charge of normal throw and the loosening charge.

For determining the specific consumption of explosives, we gave formula (14).

At an infinitely large value of l.n.s.,

\[ q=\frac{k}{(k+1)\cdot \infty}\,q_{\text{norm}}+\frac{1}{k+1}\,q_{\text{norm}}= \]

\[ =0+\frac{1}{k+1}\,q_{\text{norm}}=\frac{1}{k+1}\,q_{\text{norm}}. \tag{16} \]

The specific explosive consumption—$q$—for all numerical values of the coefficient $k$ will tend toward these limit-minimum values with an increase in the line of least resistance, i.e.:

1) the limit toward which the specific explosive consumption—$q$—will tend as the line of least resistance increases will depend on the value of the normal specific explosive consumption $q_{\text{norm}}$ and on the value of the coefficient $k$;

2) with a line of least resistance $=30\ \text{m}$, the specific explosive consumption—$q$—for all values of the coefficient $k$ will be very close to its limit-minimum value.

As an example, it may be noted that for the coefficient $k=10$ the limit-minimum value of the specific explosive consumption is $0.091\,q_{\text{norm}}$, for $k=1$ it is $0.50\,q_{\text{norm}}$, and for $k=0.1$ it is $0.9\,q_{\text{norm}}$.

At a line of least resistance $=\infty$, the specific explosive consumption will, in essence, be expended only to overcome the weight of the rock being detached.

For a line of least resistance $<1\ \text{m}$, the specific explosive consumption—$q$—will, on the contrary, increase, tending toward its maximum limit.

The degree of rock fragmentation, the relative coarseness of the rock, and the degree of its throw, as functions of the magnitude of the line of least resistance for an elongated charge, will vary in exactly the same way as for a concentrated charge.

Formulas for Calculating Charges

Formula (14), for determining the magnitude of the specific explosive consumption $q$ as a function of the depth at which the charge is placed, is applicable for any shape of the body of rock detached from the massif.

To calculate the magnitude of the charge, the detached body of rock may likewise be taken to be of any shape. It is important only that the shape of the detached body and its similarity be preserved both when determining the normal specific explosive consumption (line of least resistance $=1\ \text{m}$) and in the subsequent calculations, i.e., when the line of least resistance $\ne 1\ \text{m}$.

The general form of the formula for calculating charges will be:

\[ Q=f(k)\,q_{\text{norm}}\,V\ \text{kg}, \tag{17} \]

where $Q$ is the weight of the charge in kg;
$V$ is the volume of the detached rock in $\text{m}^3$.

In differentiated form, the formulas for calculating elongated charges, for different positions of them in the object being blasted and under different blasting conditions, may be presented as follows.

1. For calculating single elongated charges parallel to a free surface, with an unconfined action:

\[ Q=\frac{1}{k+1}\left(\frac{k}{W}+1\right)q_{\text{norm}}\,W^2\,l\ \text{kg}, \tag{18} \]

or

\[ Q=f(k)\,q_{\text{norm}}\,W^2\,l\ \text{kg}, \tag{19} \]

where $W$ is the line of least resistance, in m;
$q_{\text{norm}}$ is the normal specific consumption (line of least resistance $=1\ \text{m}$), in kg;
$l$ is the length of the charge, in m.

1. For calculating single charges in benches (above the edge), when there is one free surface, one may use formula (19), and also the formula:

\[ Q=\frac{1}{k+1}\left(\frac{k}{W}+1\right) q_{\text{norm}} W^2 H \ \text{kg}, \tag{20} \]

or

\[ Q=f(k) q_{\text{norm}} W^2 H \ \text{kg}, \tag{21} \]

where \(W\) is the L.N.S. or the design line of resistance (at the toe of the bench, if the bench is not vertical), in m;
\(H\) is the height, in m.

Formulas (19) and (21), when calculating charges, must give identical results; the difference lies only in the method of determining the normal specific explosive consumption \(q_{\text{norm}}\). According to formula (19), this consumption is determined on the basis of the volume \(W^2 l\), and according to formula (21), \(W^2 H\).

1. For calculating charges in benches (above the berm) when they are arranged in rows and blasted simultaneously:

\[ Q=\frac{1}{k+1}\left(\frac{k}{W}+1\right) q_{\text{norm}} a m W^2 \ \text{kg}, \tag{22} \]

or

\[ Q=f(k) q_{\text{norm}} a m W^2 \ \text{kg}, \tag{23} \]

or

\[ Q=f(k) q_{\text{norm}} a W H \ \text{kg}, \tag{24} \]

where \(a\) is the distance between charges in a row, in m;
\(W\) is the L.N.S. or the design line of resistance, if the bench is not vertical, in m;
\(m\) is the ratio \(\dfrac{l}{W}\).

When choosing the distance between charges in a row, the magnitude of the charge in the second and subsequent rows, the ratios \(\dfrac{l}{W}=m\) or \(\dfrac{H}{W}=m_1\), the depth of the berm and other elements, one should use the formulas, parameters, other elements, and established provisions designated for this purpose.

Features of the derived formula:

1) it is a formula of general form;
2) the change in the weight of a charge calculated by this formula, with a change in the depth of its placement depending on the value of the coefficient \(k\) (rock strength), will occur not according to one law, but according to different laws.

Determination of the Values of the Coefficient \(k\)

The general form of the formula for determining \(f(k)\):

\[ f(k)=\frac{1}{k+1}\left(\frac{k}{W}+1\right). \]

The value of \(f(k)\), depending on the value of the coefficient \(k\), theoret—

can vary practically within the limits from 1 to \(\dfrac{1}{W}\). The value \(f(k)=\dfrac{1}{W}\) will correspond to the case when the charge overcomes only the resistance along the surface of separation and the resistance due to the pressure of the air column. Such a case cannot occur in blasting rock.

According to Hauser’s formula, the value \(f(n)=n^3\). For charges of maximum compression and brisant explosives, the value of the exponent of blast action \(n\) in this formula is conventionally taken as \(0.5\), for which

\[ f(n)=0.5^3=0.125=1/8. \]

For brisant explosives, loosening charges are taken within the limits from

\[ \frac{1}{3}\ \text{to}\ \frac{1}{8} \]

of demolition charges.

Analyzing Lares’s formula, it is not difficult to establish that, with an increase in the least resistance line above \(30\ m\), the limiting value of \(f(n)\) is also \(0.125=1/8\).

According to V. K. Karchevsky’s formula, the value of \(f(n)\) at l.r.l. \(=30\ m\) is equal to

\[ 0.13=\frac{1}{7.7}. \]

Although charges with the value \(f(n)=0.125\), by certain indications, belong to charges of maximum compression, practical experience confirms that under mining conditions sufficiently good results are obtained when such charges are used.

At l.r.l. \(=30\ m\) and the value of the coefficient \(k=9.5\), the numerical value \(f(k)=0.125\). For \(k=10\), the numerical value of \(f(k)\) at l.r.l. \(=30\ m\) will be \(0.121\), and for \(k=15\) it is \(0.094\).

The value of the coefficient \(k\) increases as the strength of the rock increases. Therefore the value of the coefficient \(k=9.5\) may be taken as its limiting—maximum—value for rocks.

According to Lares’s classification of rocks by blastability, the rocks of greatest strength are, to the highest degree, tough and dense quartzites; consequently, for quartzites it is necessary to adopt the maximum limiting value of the coefficient \(k=9.5\), at which, for l.r.l. \(=30\ m\), the numerical value \(f(k)=0.125\).

According to the indicated classification, for highly tough and dense quartzites the normal value of the specific consumption of explosive with a working capacity of \(480\ \mathrm{cm}^3\), when blasting a homogeneous rock onto a side dump, with one free surface and l.r.l. \(=1\ m\), with the charge-density coefficient and the tamping density equal to 1, is \(q_1=1.5\ \mathrm{kg}\). The volumetric weight of highly dense and tough quartzites is \(3.0\ \mathrm{t/m^3}\).

According to expression (16), the specific consumption of explosive for overcoming the force of gravity of the detached rock, taking into account the expenditure of its energy for its crushing, is

\[ q_v=\frac{q_{\mathrm{norm}}}{k+1}. \]

For quartzites, with the above-indicated value of the normal specific consumption of explosive,

\[ q_v=\frac{1.5}{9.5+1}=0.143\ \text{kg}=143\ \text{g} \tag{25} \]

or per \(1\ t\) of their weight:

\[ a=\frac{q_v}{3}=\frac{143}{3}\simeq 48\ \text{g}/t. \tag{26} \]

The magnitude of the charge expended in overcoming the force of gravity for one weight unit of the rock being detached will be influenced by the following factors:

1. The capacity for work of the explosive.
2. The direction of the impact force relative to the direction of the forces of gravity, i.e., whether the rock is to be lifted, torn off downward, cast sideways, etc.
3. The number of free surfaces or the number of lines of impact.
4. The physico-mechanical properties of the rock being blasted.

The energy of the charge in an ultradynamic impact during an explosion is expended not only on useful work, i.e., on overcoming the force of gravity of the rock being detached, but also on internal deformations: compression of the rock, its crushing, elastic and plastic deformations. Since the character of these internal deformations will differ depending on the physico-mechanical properties of the rock being blasted, their influence on the magnitude of the charge will also differ.

1. The technique of charging and blasting, i.e., the correctness of charging, the quality of stemming, and the blasting conditions: sequential or simultaneous.

Investigation and study of this question have shown that the weight of the charge for overcoming the force of gravity of one weight unit of detached rock \(a\), for the classes of rocks adopted in Lares’ classification, with an explosive capacity for work of \(480\ \text{cm}^3\) and the same blasting conditions, taking into account the possible accuracy of calculating charges in such an inconstant medium as rock, may practically be considered a constant value equal to \(48\ \text{g}/t\).

For explosives not having a capacity for work of \(480\ \text{cm}^3\), and also taking into account the charging density and the quality of stemming, the weight of the charge for overcoming the force of gravity of one weight unit of detached rock will be:

\[ a_1=\frac{ed}{\Delta}a\ \text{g}/t, \tag{27} \]

where \(e\) is the coefficient characterizing the capacity for work of the explosive used relative to an explosive with a capacity for work of \(480\ \text{cm}^3\);

\(d\) is the coefficient characterizing the quality of stemming;

\(\Delta\) is the coefficient characterizing the charging density.

The specific consumption of explosive for overcoming the force of gravity of the rock being detached is expressed as:

\[ q_v=a_1\gamma\ \text{g}/\text{m}^3, \tag{28} \]

where \(\gamma\) is the unit weight of the rock, \(t/\text{m}^3\).

In general form, the normal specific consumption of explosive for a single

the charge with one free surface and blasting the rock into a side dump can be determined by the formula:

\[ q_{\text{norm}}=q=q_1\,s\,v\,\frac{e}{\Delta}\,d . \tag{29} \]

The specific consumption of explosives for overcoming the resistance along the breakaway surface and from the pressure of the air column will be:

\[ q_w=q_{\text{norm}}-a_1\gamma\ \text{g}/\text{m}^3 . \tag{30} \]

Hence the value of the coefficient \(k\):

\[ k=\frac{q_w}{q_v}=\frac{q_{\text{norm}}-a_1\gamma}{a_1\gamma} =\frac{q_{\text{norm}}}{a_1\gamma}-1 . \tag{31} \]

After substituting into formula (31) the values for \(q_{\text{norm}}\) and \(a_1\) for a single charge with one free surface and blasting the rock into a side dump, we obtain:

\[ k= \frac{q_1\,s\,v\,\dfrac{e}{\Delta}\,d} {\dfrac{e}{\Delta}\,d\,a\gamma} -1 = \frac{q_1\,s\,v}{a\gamma} -1, \tag{32} \]

where \(s\) is the coefficient of the rock structure;
\(v\) is the confinement coefficient of the blasted rock.

Formula (32) establishes the dependence of the value of the coefficient \(k\) on five quantities. The influence of each of them on the coefficient \(k\) is clear from the formula.

If the normal specific consumption of explosives \(q_{\text{norm}}\) is determined empirically, then formula (31) should be used to calculate the coefficient \(k\). In this case the value of \(q_{\text{norm}}\) will be equal to the actual normal specific consumption of explosives.

If the rock is being lifted, then the formula for determining the value of the coefficient \(k\) for a single charge with one free surface will be:

\[ k=\frac{q_w}{q_v}=\frac{q^1_{\text{norm}}}{a_2\gamma}-1. \tag{33} \]

\[ q^1_{\text{norm}}=q_{\text{norm}}\,v_1, \tag{34} \]

\[ a_2=a_1+\frac{q^1_{\text{norm}}-q_{\text{norm}}}{\gamma}\ \text{g}/\text{m}, \tag{35} \]

where \(a_1\), \(q_{\text{norm}}\), and \(\gamma\) have their previous meanings;
\(a_2\) is the magnitude of the charge expended on overcoming the force of gravity of one weight unit when lifting rock in \(\text{g}/\text{m}\); \(v_1\) is the confinement coefficient of the blasted rock depending on the direction of the impact force relative to the force of gravity.

When there are two or more free surfaces, the same formulas may be used to determine the value of the coefficient \(k\), but in this case the value of \(a_1\) must be determined from the expression:

\[ a_1=\frac{ved}{\Delta}\ \alpha z/m, \tag{36} \]

where \(v\) is the clamping coefficient of the rock being blasted, depending on the number of free surfaces.

The formulas derived for determining the coefficient \(k\) should also be used in serial blasting of charges.

Examples of the Method for Calculating Charges

The calculation of charges in Examples 1, 2, and 3 is carried out on the basis of the previously adopted Lares classification of rocks by blastability.

Example 1. Calculate an elongated charge with one free surface and blasting of the rock to a side spoil bank. Data: the rock to be blasted is shale with strength coefficient \(f=6\), \(\gamma=2.5\ t/m^3\), \(s=1\), \(v=1\), \(\Delta=0.8\), \(d=1\), bench height \(H=5.5\ m\), burden line \(=5\ m\), explosive—ammonite No. 10, \(e=1.72\).

The normal specific consumption of explosive according to formula (29) is

\[ q_{\text{norm}}=q_1 sv\frac{e}{\Delta}d=\frac{0.3\cdot1.72}{0.8}=0.645\ kg. \]

According to formula (32):

\[ k=\frac{q_1 sv}{a\gamma}-1=\frac{300}{48\cdot2.5}=1.5. \]

According to formula (15), the numerical value is

\[ f(k)=\frac{1}{k+1}\left(\frac{k}{W}+1\right)=\frac{1}{1.5+1}\left(\frac{1.5}{5}+1\right)=0.52. \]

The weight of the charge for the main blast according to formula (21) is:

\[ Q=f(k)q_{\text{norm}}W^2H=0.52\cdot0.645\cdot5^2\cdot5.5=46.8\ kg. \]

The specific consumption of explosive at burden line \(=5\ m\) is

\[ q=f(k)q_{\text{norm}}=0.52\cdot645=335\ g/m^3. \]

Example 2. Calculate an elongated charge (overdrilling) with one free surface and blasting of the rock to a side spoil bank. Data: coal is being blasted, its strength coefficient is \(f=2\), \(\gamma=1.35\ t/m^3\), \(s=0.8\), \(v=1\), explosive ammonite No. 10, \(e=1.72\), \(\Delta=0.8\), \(d=1.3\), bench height \(H=11\ m\), burden line \(=7\ m\).

Normal specific consumption of explosives according to formula (29):

\[ q_{\text{norm}}=\frac{0.1\cdot 0.8\cdot 1.72\cdot 1.3}{0.8}=0.223\ \text{kg}. \]

According to formula (32):

\[ k=\frac{100\cdot 0.8}{48\cdot 1.35}-1=0.23. \]

According to formula (15), the numerical value is

\[ f(k)=\frac{1}{0.23+1}\left(\frac{0.23}{7}+1\right)=0.84. \]

Weight of the charge at the base of the blast

\[ Q=f(k)\,q_{\text{norm}}\,W^2H=0.84\cdot 0.223\cdot 7^2\cdot 11=100.7\ \text{kg}. \]

Specific consumption of explosives at L.R. = 7 m

\[ q=f(k)\,q_{\text{norm}}=0.84\cdot 223=187\ \text{g}/\text{m}^3. \]

Example 3. Calculate an elongated charge with one free surface and blasting onto the side dump (over the edge). Data: the rock to be blasted is sandstone, for which it has been experimentally determined that \(q_{\text{norm}}=1\ \text{kg}\), \(\gamma=2.5\ \text{t}/\text{m}^3\); the explosive is ammonite No. 10, \(e=1.72\), \(\Delta=0.8\), \(d=1\), bench height \(H=10\ \text{m}\), L.R. = \(6.6\ \text{m}\).

Weight of the charge for overcoming the force of gravity of one weight unit of the detached rock, according to formula (27):

\[ a_1=\frac{ed}{\Delta}\,a=\frac{1.72}{0.8}\cdot 48=103\ \text{g}/\text{t}. \]

According to formula (31):

\[ k=\frac{q_{\text{norm}}}{a_1\gamma}-1=\frac{1000}{103\cdot 2.5}-1=2.8. \]

According to formula (15):

\[ f(k)=\frac{1}{2.8+1}\left(\frac{2.8}{6.6}+1\right)=0.40. \]

Weight of the charge for the main blast

\[ Q=f(k)q_{\text{norm}}\, W^2 H=0.40\cdot 1.0\cdot 6.6^2\cdot 10=174.2\ \text{kg}. \]

The specific consumption of explosives at l.l.r. \(=6.6\ \text{m}\):

\[ q=f(k)q_{\text{norm}}=400\ \text{g}/\text{m}^3. \]

The final value of the coefficient \(k\) in the calculation formula, for one or another set of specific conditions, is corrected in accordance with the results of blasts. At present there are several classifications of rocks according to blastability, with sharply divergent data on the normal specific consumption of explosives for rocks with the same physical-mechanical properties. In light of the new theoretical propositions being developed, there is an urgent need to develop a rational unified classification of rocks by blastability and by the coefficient \(k\) for different forms of charges, their position in the object being blasted, and blasting conditions. This work is complex, very labor-intensive, and requires the participation of a large collective of Soviet scientists and production workers.

Conclusion

A number of researchers have been concerned with the calculation of charges, beginning in 1812, when the first formulas were proposed, and continuing up to the present time.

All the proposed formulas, derived by their authors for certain specific conditions, when applied in practice to other specific conditions give unsatisfactory, contradictory results, and not one of them has received general recognition.

The phenomenon of explosion is objective in the physical-mechanical sense and must have a single nature of forces and a single expression.

The formulas we propose for calculating charges for different positions of them in the object being blasted and under different blasting conditions are based on the joint consideration of the factors of overcoming the cohesion forces along the surface of separation, the force of gravity of the rock being separated, and the resistance to crushing of the rock; they have a single nature of forces and, in general form, a single expression.

Many phenomena in blasting, for which no physical explanation had been given—including the causes of the change in the magnitude of the specific consumption of explosives with changes in the depth of charge placement, the dependence of the radius of action of an explosion on the depth of charge placement and on the index of explosion action—have received their physical explanation in light of the theoretical propositions developed. The contradiction among the formulas proposed earlier by researchers has been eliminated.

The overwhelming majority of the formulas proposed earlier are empirical and semi-empirical, constructed on an experimental basis, on the basis of test blasts under specific conditions. On the basis of the theoretical propositions developed, we have generalized these formulas, constructed on an experimental basis; consequently, the proposed formulas are also constructed on an experimental basis, but only with broad use of domestic and world experience.

The sliding scale of numerical values of \(f(k)\) in the formulas opens up great possibilities for controlling the calculation of charges. In this case, the results of calculating charges will depend on how correctly the value of the normal specific consumption of explosives and the coefficient \(k\) have been chosen. We have proposed a method for determining the coefficient \(k\) theoretically. In practical use of the derived formulas, the value of these quantities under the specific conditions of blasting operations can be refined. This is a matter for further work and time. Thus, one need only be able to use these general-form formulas and learn to control the calculation, making proper use of the opportunities that present themselves.

LITERATURE

1. Brodsky M. P. On the action of an explosion in a solid medium. Theory and Practice of Drilling-and-Blasting Operations in Industry, All-Union Engineering and Technical Mining Society, Ugletekhizdat, 1953.

2. Karchevsky V. K. A new basis for calculating charges, Magnitogorsk Mining and Metallurgical Institute, Scientific Transactions, issue 2, Questions of Mining, Metizdat, 1945.

3. Melnikov N. V. Refinement of the formula for calculating charges for blasting operations, Ugol’, No. 3, 1944.

4. Osnovin S. D. The action of concentrated charges and their calculation, Tomsk Polytechnic Institute named after S. M. Kirov, Order of the Red Banner of Labor. Questions in the Development of Thick Seams of the Kuzbass. Moscow, Ugletekhizdat, 1952.

5. Osnovin S. D. The action and calculation of reduced-throw charges and loosening charges. Proceedings of the Tomsk Polytechnic Institute named after S. M. Kirov, Order of the Red Banner of Labor, vol. 79, Tomsk, 1954.

6. Osnovin S. D. A method and formulas for calculating concentrated charges, Proceedings of the Tomsk Polytechnic Institute named after S. M. Kirov, Order of the Red Banner of Labor, vol. 84, Tomsk, 1955.

7. Sukhanov A. F. A unified classification of rocks by blastability and a new method for calculating explosive charges, Gornyi Zhurnal, No. 6, 1955.