CALCULATION OF HOISTING ROPES BY LOAD FACTORS AND END LOAD

I. A. BALASHEV

At the present time, the calculation of ropes for hoisting installations is carried out according to the static load. “The calculated static load of a hoisting rope consists of the weight of the cage or skip with attached devices, plus the weight of the maximum working load, plus the weight of the rope of length from the point where it leaves the sheave to the point of attachment to the cage or skip located at the lower receiving (loading) platform of the lowest level” [7].

The calculation is based on the strength equation

\[ Q_0+pH_0=\frac{\sigma_{\mathrm{vr}}}{m}S. \tag{1} \]

On the left-hand side we have the external forces; on the right, the elastic forces of the rope. Hence, by transformations, the calculation formula for the weight of one running meter of rope is obtained:

\[ p=\frac{Q_0}{L_0-H_0}([[unclear: expression]]-m), \tag{2} \]

where \(Q_0\) is the end load, \(L_0\) is the strength length of the rope, and \(H_0\) is the length of the part of the rope hanging from the sheave.

After selecting a rope according to OST, the verification of the strength reserve of the selected rope, as well as of a rope in service, is carried out by the formula

\[ m=\frac{Q_{\mathrm{usl}}}{Q_0+pH_0}. \tag{3} \]

Formula (2) is known in the literature under the name of the calculation formula of Academician M. M. Fedorov [5].

The action on the rope of other forces, besides static ones, is not introduced into the calculation, and such forces are taken into account by limiting strength reserves. Bending forces are limited by the limiting ratio of the diameter of the sheave and drum to the diameter of the rope, and rope wear by the limiting angle of rope deviation. The calculated and rejection values of the strength reserves are given in Table 1.

Table 1. Strength reserves

For all types of surface installations: a) the ratio of the diameter of the drum and sheave to the diameter of the rope must be not less than 80; b) the ratio of the diameter of sheaves and drums to the diameter of the rope wires must be not less than 1200; c) the deviation angle (deflection) of the rope strand must not exceed \(1^\circ 30'\).

The restrictions indicated in items a, b, and c guarantee that the actual strength margin of the rope remains within permissible limits and are justified by many years of operating experience.

At the same time, however, the calculation method presented contains a number of shortcomings. Let us indicate the principal ones:

1. The strength margin remains constant, irrespective of the hoisting height, which is in contradiction with well-founded theoretical and experimental studies of recent years.

2. In the standards for strength margins (Table 1), the type of vessel and the method of controlling the machine drive are not taken into account, as a result of which the strength margins obtained are excessive in some cases and insufficient in others (with contactor control of an asynchronous drive), which brings the loaded elements of the machine into an emergency condition and causes rapid wear of the ropes, and in some cases their rupture.

3. In the calculation, the actual strength margin remains unknown; moreover, this margin with respect to the fatigue limit is assumed to be the same for all types of installations, irrespective of the method of control (of the character of the accelerating torque).

The indicated shortcomings are so substantial that they make it necessary to seek new methods of rope calculation which would, to the fullest possible extent, reflect the principal factors determining the load on the rope.

One of the possible methods of calculating a rope, in which the actual loads are more fully taken into account, is the method of calculation by load factors and end load.

The essence of the proposed method is as follows:

a) selection of the rope from the catalogue is made according to the breaking force of the rope, \(Q_{\mathrm{razr}}\), which is determined from the coefficient of the general strength margin, \(m_{\mathrm{obsh}}\), and the end load, \(Q_0\), by the formula

\[ Q_{\mathrm{razr}} = m_{\mathrm{obsh}} Q_0, \tag{4} \]

b) checking the strength-margin coefficient of a new rope, or of one already in service, is determined by the formula

\[ m'_{\mathrm{obsh}} = \frac{Q'_{\mathrm{razr}}}{Q_0}, \tag{5} \]

where \(Q'_{\mathrm{razr}}\) is the conditional breaking force of the rope, and \(m_{\mathrm{obsh}}\) is the general strength-margin coefficient.

The general strength-margin coefficient in the calculation is determined as

\[ m_{\mathrm{obsh}} = K_{\mathrm{nag}} K_{\mathrm{izn}} K_{\mathrm{zap}} . \tag{6} \]

Here \(K_{\mathrm{izn}}\) is the rope-wear coefficient,

\(K_{\mathrm{zap}}\) is the strength margin of the rope wires, determined as

\[ K_{\mathrm{zap}} = \frac{\sigma_{\mathrm{vr}}}{\sigma_{\mathrm{dop}}}, \tag{7} \]

\(K_{\mathrm{nag}}\) is the load factor.

The load factor, in turn, consists of the sum of five partial factors:

\[ K_{\mathrm{nag}} = K_{\mathrm{kond}} + K_{\mathrm{kan}} + K_{\mathrm{izg}} + K_{\mathrm{kol}} + K_{\mathrm{upr}} . \tag{8} \]

In the right-hand side, the partial load factors express, in fractions of the end load, the equivalent forces from the end load, from the weight of the rope, from bending on the sheave and drum, from oscillations of the rope during starting, and from the method of controlling the motor.

In expanded form, the design formula (4) has the form:

\[ Q_{\text{des}} = K_{\text{res}}\, K_{\text{wear}}(K_{\text{end}} + K_{\text{rope}} + K_{\text{bend}} + K_{\text{osc}} + K_{\text{ctrl}})Q_0 . \tag{9} \]

Partial load factors

End-load factor

The end load \(Q_0\) consists of the nominal payload, \(Q_{\text{p}}\), and the dead weight \(Q_{\text{d}}\) of the vessel and mine cars. The weight of the payload is not a strictly constant quantity either for coal or for ore deposits. The influence of moisture, lump size, and contamination by included rock causes a change in the specific weight of the mineral and, consequently, in the weight of the load in the cars and skips.

Individual cases of overloading of the payload in cars and skips indicate that deviations from the nominal weight in skips in the direction of increase reach up to 110%.

In calculating the rope it may be taken that:

a) for skip hoists,

\[ K_{\text{load}} = 1.1 \]

b) for cage hoists,

\[ K_{\text{load}} = 1.05 . \]

Factor for the weight of the rope

The rope’s own weight depends on the hoisting height and can be quite significant. Its introduction into the design formula in the form of a constant factor, \(K_{\text{rope}}\), is justified by the following considerations.

By definition, the load factor due to the weight of the rope is equal to

\[ K_{\text{rope}} = \frac{pH_0}{Q_0} = \frac{\gamma_0 H_0}{\dfrac{Q_0}{S}} . \tag{10} \]

It can be shown that the ratio \(\dfrac{Q_0}{S}\) varies within narrow limits, 1500–1600, if \(Q_0\) is expressed in kg and the rope section \(S\) in \(\text{cm}^2\).

The conventional specific weight \(\gamma_0\) also varies only slightly for different rope constructions and fluctuates within limits of about \(9500 \,\text{kg}/\text{m}^3\).

Taking the average values of these quantities, we obtain

\[ K_{\text{rope}} = \frac{9500H_0}{1600 \cdot 10^4} = 6 \cdot 10^{-4} \cdot H_0 . \tag{11} \]

On the basis of this formula we compile Table 2.

Table 2

From the table it is evident that the load due to the weight of the rope increases and, at a shaft depth of 1000 m, reaches 60% of the end load.

Using Table 2, one may introduce into the calculation a load factor due to the weight of the rope as a function of the hoisting height. Theoretical and experimental work by Bogan, Nekoz, A. S. Il’ichev, and others [8] has shown that the magnitude of the stress wave caused by the dynamic force during abrupt changes in velocity increases up to a certain “critical” rope length, amounting by weight to \(0.35Q_0\), while with further increase in the rope length this stress either remains constant or even decreases.

In practice this circumstance is taken into account by differentiating the scale of safety factors according to the hoisting height, as shown in Table 3.

Table 3. Standards for rope safety factors as a function of hoisting height (under static load), R. Pil’—Handbook of Mining

From this table it is evident that the design safety factor for large hoisting heights is half that for small heights. A similar, but more substantiated, scale of safety factors was proposed by Corresponding Member of the Academy of Sciences of the USSR A. S. Il’ichev [6].

In this connection it appears possible to take the load factor due to the weight of the rope as constant and equal to

\[ K_{\text{rope}} = 0.3. \tag{12} \]

In this case, in short ropes we shall have an increased reserve, and in long ropes a reduced one, which will be compensated by an increase in the forces from free oscillations in the first case and by their decrease in the second.

Coefficient due to rope bending

When the rope passes over the guide sheave and is wound onto the drum, bending stresses arise. The magnitude of the bending force is determined by the formula

\[ Q_{\text{bend}} = \frac{E_k \cdot \delta \cdot S}{D}. \tag{13} \]

The load factor due to bending forces will be

\[ K_{\text{bend}} = \frac{Q_{\text{bend}}}{Q_0} = \frac{E_k}{\dfrac{D}{\delta} \cdot \dfrac{Q_0}{S}} = \frac{0.8 \cdot 10^6}{1000 \cdot 1600} = 0.5. \tag{14} \]

The Young’s modulus for ropes of accepted constructions is taken from experimental data of MakNII [8]; the ratio \(\dfrac{D}{\delta}\) is taken as the minimum permissible, and \(\dfrac{Q_0}{S}\) according to the established constancy of this ratio.

Fig. 1. Vibrograms of rope oscillations:
a) excavator operation; b) excavator control; c) hoisting machine.

a) Excavator operation

Empty cage moving upward.

Start.

Steam engine.

Engine operation at a speed of 2.5 m/s.

Deceleration.

Rope 0.87; rope KP-6×19-1, \(d = 21\) mm; [[unclear: remaining technical note]]. Pavlovich mine.

b) Automatic control

Cage moving upward with an 8 t load.

Empty skip moving downward; [[unclear: load/no-load note]].

Start.

Engine operation at a speed of 5.2 m/s.

Deceleration.

Loading: 10 s.

[[unclear: machine note]]
\(P = 640\) kg, \(Q = 8\) t, \(V = 5.2\) m/s; drum [[unclear: dimensions]]; rope KP 6×37-1, \(d = 47\) mm; main shaft [[unclear]], VIII. 1947.

c) Automatic control — [[unclear: speed/acceleration diagram]]

Cage moving upward with a 7 t load.

Empty skip moving downward from the zero level.

Acceleration.

Rope oscillation.

\(g =\) [[unclear]]; [[unclear: technical note]]
Hoisting machine: \(H = 225\) m, \(P = 800\) kg, \(Q = 7\) t; rope KP 6×37-1, \(d = 45\) mm, left-hand lay.

Raising the skip.

Deceleration.

Stopping from balanced rope.

Rope unwinding.

Landing the loaded rope [[unclear]].

Locking K2.

[[unclear: “K” marks at several points on the traces]]

Coefficient from Starting Oscillations

Owing to elastic properties, forced oscillations arise in the rope under the action of driving forces. The nature of the oscillatory process is described by the wave equation. These oscillations cause additional stresses in the rope, moving along its length. Especially large stresses arise at the beginning of the hoisting operation in those cases when the rope had slack or looseness. The wave equation, likening the rope to an elastic thread, describes well the phenomenon itself of the oscillatory process, but does not provide sufficient material for calculating the rope. Experimental data of the Committee on Safety in the Mining Industry in England, obtained at 100 mines, show that the maximum rope tension from oscillations during starting in most cases amounts to \((1.4—1.8)Q_{\text{st}}\) [8].

In unfavorable cases the rope tension during starting may reach \(2.5 Q_{\text{st}}\). This tension includes the inertia forces arising during accelerated motion. As a rule, the maximum tension occurs when the end load starts from rest; then, as the speed increases, they decrease to a value of about \(1.6 Q_{\text{st}}\).

It is quite obvious that, if there is slack or looseness of the rope—and this will always be the case in cage hoists with catches—the force will be significant; while in skip hoists, where the rope is always taut, these forces will be smaller.

Experiments carried out by the author on skip and cage installations of the Kuzbass show that oscillations of the conveyance occur not only during the starting period, but also in all positions of the moving conveyance. The experimental procedure consisted in placing a vibrograph in the conveyance, while a normal cycle of the hoisting operation was performed.

Processing of the experimental vibrograms leads to the following conclusions:

1. The greatest amplitudes of oscillations occur in skip installations controlled by contactors.

2. In skip installations with a liquid rheostat and controlled according to the Leonard system, the amplitudes of oscillations are considerably smaller.

3. In cage installations controlled by a liquid rheostat, the amplitudes of oscillations during starting are insignificant, since they are damped by the action of the parachute spring.

4. Oscillations occur throughout the entire hoisting cycle, increasing in frequency and amplitude during the starting and stopping periods.

As an illustration, Fig. 1 gives vibrograms of typical installations. The magnitude of the stresses caused by oscillations, for the given specific case for the loaded branch moving upward, as a rule, increases as it approaches the unloading bunker. The numerical value of the stresses and additional forces caused by oscillations is determined as follows.

The stress in the rope is

\[ \sigma = iE_k, \tag{15′} \]

where \(i\) is the relative elongation and \(E_k\) is the modulus of elasticity of the rope; moreover

\[ i=\frac{\lambda}{H_0}, \tag{16} \]

where \(H_0\) is the length of the suspended rope. We have

\[ \sigma=\frac{\lambda}{H_0}\cdot E_k. \tag{15″} \]

or

\[ \sigma H_0 = \lambda E_k . \tag{15} \]

In one of the vibration records taken experimentally, with a threefold increase, we have the amplitude of oscillations \(21\ \mathrm{mm}\). Consequently, the absolute elongation caused by the oscillations will be

\[ \lambda = 21 : 3 = 7\ \mathrm{mm}. \]

Taking for the rope \(E_k = 0.85 \cdot 10^6\ \mathrm{kg}/\mathrm{cm}^2\), we obtain the equation of a hyperbola whose asymptotes are the coordinate axes

\[ \sigma H_0 = 7 \cdot 0.85 \cdot 10^6 . \]

If \(H_0\) is expressed in meters, then \(\sigma H_0 \cong 600\).

As \(H_0\) varies, we shall have the value of the stress given in Table 4.

Table 4

The stresses in the rope increase as the skip approaches the unloading bunker.

For the given hoisting height of \(5\ \mathrm{m}\), rope cross-section of \(7\ \mathrm{cm}^2\), and end load of \(12000\ \mathrm{kg}\), we shall have the greatest value of the rope load factor due to oscillations

\[ K_{\mathrm{osc}} = \frac{7 \cdot 1000}{12000} = 0.59 . \tag{17} \]

Comparing the experimental data obtained with the above-mentioned results of experiments and theoretical investigations, one may propose for calculations the following values for the coefficient characterizing the additional stresses in the rope due to oscillations:

a) for cage hoists with cams on both landings

\[ K'_{\mathrm{osc}} = 0.8 \div 1.1 \cong 0.9, \tag{18} \]

b) for skip hoists and cage hoists with swinging platforms on one of the levels

\[ K''_{\mathrm{osc}} = 0.4 \div 0.6 \cong 0.5 . \tag{19} \]

There is reason to believe that this type of additional load is specific and differs not only for different installations, but also for a given installation in different periods of operation, and also depends on the condition of the surfaces of the drums and sheaves.

Coefficient Depending on the Method of Control

Additional stresses in the rope also arise from the character of the excess (accelerating) torque. In time of occurrence, this type of oscillation coincides with the preceding one, i.e., it takes place during accelerated motion. However, the causes of their occurrence are entirely different: whereas starting oscillations depend on the different velocities at the ends of the rope, i.e., whether the rope is taut or has a large slack and the upper end acquires…

some velocity, while the lower one has zero velocity, then the oscillations arising from the method of control occur independently of the condition of the rope.

If an asynchronous electric motor is controlled by means of a liquid rheostat, then in this case the excess (accelerating) torque has an almost constant, smoothly increasing value. In the same case, when we have a metallic rheostat, the excess torque will have a sawtooth character during the transition from one stage of the rheostat to another. To the upper end of the rope, in addition to a force equal to the static one, an accelerating force is imparted in impulses, which will cause oscillations in the rope throughout the entire period of accelerated motion. It is obvious that in the first case of control the oscillations will be insignificant, while in the second case (with contactor control) we shall have favorable conditions for the creation of considerable amplitudes of oscillation.

Analysis of vibrograms, observations, direct study and analysis of accidents that have occurred, as well as statistical materials on the service life and working capacity of ropes on hoisting installations with liquid rheostats and contactor control, lead to the following conclusions [1, 2]:

a) the average service life of ropes, in time, with contactor control, according to data on 60 ropes, is 30% lower than in installations with a liquid rheostat;

b) the working capacity of the rope in the same cases is 20% lower;

c) the number of broken wires with contactor control is greater, which also serves as a reason for replacing the rope;

d) check calculations of accident cases indicate that, with contactor control, the additional forces depending on the control factor amount to 40–50% of the static force.

The influence of the control factor on the rope and the machine elements, so far as is known, has not been studied and is posed here for the first time.

As a first approximation, on the basis of the data indicated, it is proposed to adopt:

1) for installations with an asynchronous drive and a liquid rheostat, and also for the Leonard system,

\[ K'_{\mathrm{ctrl}} = 0.1, \tag{20} \]

2) for installations with a metallic rheostat of contactor control, or with a drum controller,

\[ K''_{\mathrm{ctrl}} = 0.5. \tag{21} \]

General load factor

On the basis of the foregoing, for various methods of control, and also depending on the type of vessels, we can compile a table of the proposed values of the load factor. These data are given in Table 5.

Additional dynamic forces

In addition to the indicated principal forces acting on the rope during the normal cycle of the hoisting operation, there may be factors that cause additional stresses in the rope.

These include resonance phenomena in the rope, dynamic loads during service and safety braking, oscillations during loading of the skip or of the cage suspended on the rope, and impacts when jamming occurs in the unloading curves or in the shaft.

Table 5. Values of the load factor

Resonance phenomena in the rope may arise when the period of the rope’s natural oscillations coincides for a prolonged time with the period of a force external with respect to the rope.

It can be shown that the period of the rope’s natural oscillations during a cycle of hoisting operation, varying according to a parabolic law, for ordinary depths lies within the limits from 3 sec. to 0.3 sec. Indeed, the period of the rope’s natural oscillations is

\[ T = 2\pi \sqrt{\frac{Q_0 H}{g E_k \cdot S}}, \tag{22} \]

or

\[ T = 2\pi \sqrt{H_{(\mathrm{m})}\frac{Q_0}{S}\cdot \frac{1}{gE_k}} . \tag{22'} \]

If we substitute the values \(\frac{Q_0}{S}=1600\), \(g=981\), and \(E_k=0.8\cdot 10^6\), then we obtain the equation of the parabola

\[ T = 2\pi \sqrt{\frac{H_{(\mathrm{m})}1600\cdot 100}{981\cdot 0.8\cdot 10^6}} = \frac{\sqrt{H_{(\mathrm{m})}}}{11}. \tag{22''} \]

For \(H=1000\) m and 10 m we obtain, respectively—\(T_{1000}=2.88\) sec. and \(T_{10}=0.288\) sec.

It follows from this that, in order to create conditions of resonance, the external cause must vary according to the same parabolic law at least over part of the section. Eccentricities on sheaves and cylindrical drums vary according to a linear law and cannot cause resonance phenomena. Resonance phenomena can arise only in special cases of conical and bicylindrical drums, over part of the winding member, when the linear speed of the rope may likewise vary according to a parabolic law [4].

Thus, resonance oscillations in ropes can arise only in special cases and are not a general phenomenon; consequently, there is no basis for introducing them into the general calculation formula.

The stresses arising in the rope during service and safety braking are examined in detail in the report by A. S. Il’ichev [8].

On the basis of tests carried out by the author on a large number of brakes of hoisting installations, the results of which have been partially published [2], it has been established that the braking torque of safety pneumatic and oil brakes rises smoothly to its nominal value over 0.6–0.8 sec., and only in simple load brakes is this time the smallest—about 0.3 sec.; moreover, in the latter case the greatest fluctuations in the braking-torque characteristic are observed.

Thus, the upper part of the rope during safety braking is not stopped instantaneously, but over a fairly considerable period of time.

Experimental data establish the magnitude of the stress during braking as up to 2.28 times greater than the stress from the static force [8].

As can be seen, the magnitude of these stresses is considerably less than the stresses during starting and therefore should not be introduced into the calculation, since their simultaneous action is excluded. The stresses arising when the skip suspended on the rope is loaded by mineral falling from the measuring bin, or when a mine car is placed in the cage, appear in the same form.

As for impact applications of force to the rope when the vessel jams in the shaft or in the unloading curves, such cases, being of an emergency character, cannot be introduced into the calculation.

Rope wear factor

Rope wear is closely connected with its service life. It has been established [10] that the aggregate strength of a rope is proportional to its wear. Wear of the surface wires by more than half their diameter leads to their rapid breakage, and this degree of wear must be regarded as limiting. The number of surface wires is, for a rope of construction \(6 \times 19 + 1\), 63%, and for a construction \(6 \times 37 + 1\), 49% of the total number of wires. Thus, in terms of the wear factor due to abrasion of the wires, one may take, for the first construction,

\[ K_{\text{wear}} = 1.3, \]

and for the second—1.25.

In the current standards for the calculated and limiting factors of safety, the coefficients of total wear adopted are:

a) for cage installations—1.25;
b) for skip installations—1.30.

These values must be adopted for calculation.

It should only be noted that the degree of rope wear is closely connected with the optimum service life of the rope and depends on many factors of operating conditions.

Factor of safety

By definition,

\[ K_{\text{sf}} = \frac{\sigma_{\text{vr}}}{\sigma_{\text{allow}}}, \tag{23} \]

where \(\sigma_{\text{vr}}\) is the temporary resistance of the rope material, and \(\sigma_{\text{allow}}\) is the allowable stress.

The temporary resistance of rope wires is established by the standards and, for hoisting ropes of our plants, amounts to \(130\text{–}160\ \mathrm{kg/mm^2}\) [9]. In the coming years this standard will evidently be raised to \(180\ \mathrm{kg/mm^2}\). While there is sufficient certainty with respect to the temporary tensile resistance of the wires, the same cannot be said with respect to the allowable stress.

As a consequence of the fact that the rope is subjected to the action of forces having the character of periodic, repeatedly short-duration loads, the permissible stress must be limited by the fatigue (endurance) stress of the metal. There are no direct experiments to determine the fatigue characteristics of wires. Wernle’s experiments [3] indicate that the rope has a critical load, the exceeding of which, even within the limits of the rope strength, considerably shortens its service life. Apparently this must be understood to mean that the limit of fatigue stresses lies below the limit of elastic deformations.

At one time, Academician M. M. Fedorov, on the basis of literature data, established for the safety factor with respect to rope endurance a value of 2.12–1.26 [8].

Analyzing the above Table 5, it may be concluded that the greatest fluctuations of the loading force, depending on the load factors \(K_{\text{col}}\) and \(K_{\text{upr}}\), occur in cage hoists controlled by a metallic rheostat with an asynchronous drive, and the smallest—in a liquid rheostat and a skip (freight) hoist. It must be assumed that the permissible stresses for fatigue should stand in the same relation.

By virtue of the adopted calculation method, it appears correct to consider the safety factor the same both for a new and for an old rope.

On the basis of the very incomplete data on this question, one can speak only of approximate values of this factor, classifying it according to the magnitude of the dynamic component of the loads on the rope; namely, according to Table 5, take:

1) for vessel type a) Table 5 — \(K_{\text{zap}} = 2.3\)
2) ” ” ” b) ” ” ” 2.1
3) ” ” ” c) ” ” ” 2.2
4) ” ” ” d) ” ” ” 2.0

General safety factor

The calculated general safety factor for different types of vessels and methods of their control is given in Table 6.

Table 6. General safety factor

The last two columns are adopted—the first as the calculated rope safety factors, and the second as the limiting rejection factors, upon reaching which the rope must be replaced as having no reserve against wear.

To evaluate the obtained safety factors, we shall compare them with the accepted strength reserves under static load.

Comparison of Safety Factors

In order to compare the rope safety factors, it is necessary to exclude from the load the weight of the rope, which is estimated by the constant load factor \(K_{\mathrm{rope}} = 0.3\). Then we obtain Table 7.

Table 7. Comparison of calculated and rejection conditional strength reserves of the rope under end load with those adopted by the PTE for static force

It follows from the table that in installations with metallic rheostats the rope has an insufficient reserve of 10–15%, whereas in the case of installations with liquid rheostats there is an excessive reserve of 7–12%.

The explanation of the comparison results given in Table 7 may be twofold: a) the adopted partial coefficients in the first case are large, and in the second are small; b) the reserves adopted according to the PTE for cage and skip installations with metallic rheostats are small.

As for the adopted values of the partial coefficients, and in particular those characterizing the dynamic forces in the ropes—from oscillations during starting, depending on the control—as well as the strength reserve due to permissible fatigue stresses, they undoubtedly require theoretical, experimental, and operational refinement. We are inclined to think that the refinement of these coefficients will proceed in the direction of reducing the numerical value relative to those adopted. However, these refinements are unlikely to change the position of the four accepted types of installations.

The second explanation, that the adopted reserves for the first two types of installations are indeed insufficient, is convincingly confirmed by operating data. Under operating conditions, an insufficient reserve will cause increased wear of the rope and a reduction in its service life.

A convincing fact in this respect is provided by two installations in the Kuzbass with eight-ton skips, operating in the same shaft, having identical drum diameters of 4 m and the same rope construction—\(6 \times 37 + 1\) and diameters of 47 mm—but controlled: the southern one by a liquid rheostat, and the northern one with contactor control. The operating results of these installations over the last five years are given in Table 8.

Consideration of the data given in Table 8 shows that the significant increase in all operating indicators for the hoisting installation with a liquid rheostat (southern hoist) is due to the lower resulting stress in the rope.

For the southern hoisting installation, a rope with a diameter not of 47 mm but of 43 mm is sufficient, whereas for the northern one (with contactor control) there are no grounds for reducing the rope diameter.

Table 8

Example of a comparative calculation by the proposed method and the existing one

For a skip installation having \(Q_n=6\ t\) and \(H_o=600\ m\), it is necessary to calculate the rope for two cases: variant a) with an asynchronous electric drive with a liquid rheostat and b) with contactor control. The dead weight of the skips is \(6\ t\), \(D_b=D_{\text{shk}}=4\ m\).

Variant a). \(Q_{\text{break}}=6.5\cdot12000=78000\ kg\).

A suitable rope of construction \(6\times19+1\), diameter \(37\ mm\), having \(Q_{\text{break}}=82500\ kg\) (with \(\sigma_{\mathrm{vr}}=160\ kg/mm^2\)) and \(p=4.6\ kg/m\).

The design safety factor will be

\[ m_{\text{calc}}=\frac{82500}{12000}=6.88. \]

Variant b). \(Q_{\text{break}}=8\cdot12000=96000\ kg\).

A suitable rope of construction \(6\times19+1\), diameter \(40\ mm\), \(p=5.4\ kg/m\) and \(Q_{\text{break}}=96800\) (with \(\sigma_{\mathrm{vr}}=160\ kg/mm^2\)).

The design safety factor

\[ m_{\text{calc}}=\frac{96800}{12000}=8.06. \]

When calculating by terminal load, a rope of construction \(6\times37+1\), with diameter \(43\ mm\), \(p=6.2\ kg/m\), having \(Q_{\text{break}}=112000\ kg\), is required.

In this case

\[ m_{\text{calc}}=\frac{112000}{12000+3720}=7.13. \]

The weights of the ropes at a structural length of \(700\ m\) will be as follows:

Variants: a)—\(3220\ kg\), b)—\(3780\ kg\), and in the last case \(4340\ kg\). The ratio of their weights is \(100:117:135\).

The results of the calculation for other hoisting heights and types of vessels are given in Table 9 for skip installations and in Table 10 for cage installations.

Review of the calculation results

For each type of installation and method of control, the calculated safety margins are given both by static force and by terminal load. The obtained safety margins show that, when calculating the rope by

Table 9. Calculation results
A. Skip installation: \(Q_{\mathrm{n}} = 6000,\ Q_{\mathrm{m}} = 6000,\ Q_{0} = 12000\ \text{kg}\)

Table 10. B. Cage installation: \(Q_{\mathrm{n}} = 2000,\ Q_{\mathrm{m}} = 5000,\ Q_{0} = 7000\ \text{g}\)

Note: The weight of the rope with control by liquid rheostat is taken as 100%.

with respect to the end load, the safety factor decreases as the lifting height increases. A similar method of calculation was proposed by Vogan [8], who called the safety factor by end load the coefficient of rope utilization. However, in the method proposed by us, the general safety factor does not remain the same for a given type of hoisting installation, but also depends on the method of machine control, which substantially distinguishes this method from those proposed earlier.

The ratio of rope weights given at the end of each table shows that, when ropes are calculated by the proposed method, a considerable saving of metal is obtained not only for large lifting heights (deep mines), but also for the most common shallow and medium-depth mines.

Conclusions and Proposals

1. The proposed method of calculating ropes by load factors and end load has the following advantages:
   a) the principal factors creating forces in the rope are introduced into the calculation;
   b) a new calculation factor is introduced, dependent on the method of motor control, called the control load factor;
   c) from the computational point of view the method is simple.

2. The use of load factors makes it possible to evaluate more correctly the operation of the rope under the given specific conditions, since the relative weight of each of the load factors depends to a considerable extent on the local operating conditions.

3. Calculation by load factors and end load gives a considerable saving of valuable metal. The significance of this method will be especially important with further improvement in the quality of the metal and in the technology of rope manufacture.

4. The numerical value of the introduced factors is subject to further refinement. The adopted values, as preliminary ones, were taken as the largest of those obtained experimentally or theoretically, and this gives grounds to believe that further refinement should lead to their reduction.

References

1. Balashev I. A.—On rheostats for mine hoisting installations, TPI, Tomsk, 1948.

2. Balashev I. A.—Experimental investigation of the brakes of hoisting machines, journal Ugol’, No. 11, 1940.

1. M. L. Vatna.—Testing of wire ropes, ONTI, M.—L., 1932.
2. A. M. Dinnik.—Resonance in a rope with bicylindrical drums, Mining Journal, No. 12, 1932.
3. A. S. Ilyichev.—Mine hoisting installations, 1933.
4. A. S. Ilyichev.—Reconstruction and further development of mine hoisting, journal Coal, No. 11, 1945.
5. Rules for the technical operation of coal mines, NKUP, Gostoptekhizdat, M.—L., 1941.
6. Proceedings of the conference on mine hoisting ropes, report by A. S. Ilyichev, Institute of Mining, Academy of Sciences of the USSR, M.—L., 1944.
7. Proceedings of the conference on the reconstruction of mine-rope production in the USSR, report by P. P. Nesterov, Institute of Mining, Academy of Sciences of the USSR, 1944.
8. V. A. Tsvetkov.—Wire ropes and their operation in mine hoisting, Proceedings of MakNII, vol. II—ONTI, 1935.