On the Methodology for Calculating and Selecting a Mine Hoisting Rope
Abstract Generated abstract
This paper addresses the calculation and selection of steel wire ropes for mine hoisting installations, arguing that prevailing practice emphasizes safety requirements while giving insufficient attention to economic efficiency. It proposes a design method that begins with the justified load capacity and winding dimensions of the hoisting machine, then derives rope diameter and the required wire strength limit, with further evaluation based on rope construction and operating conditions such as abrasion, corrosion, and fatigue. A worked comparison suggests that the method can lead to smaller hoisting-machine dimensions than the conventional procedure, while analysis of 104 Kuzbass cases indicates that a wire strength limit of about 160 kg/mm2 is often sufficient under contemporary regional conditions. The study concludes that frequent use of higher strength ropes is not generally necessary and may increase costs without corresponding technical benefit.
Full Text
On the Methodology for Calculating and Selecting a Mine Hoisting Rope
V. F. Kutsenalenko
(Presented by Prof., Doctor of Technical Sciences I. A. Balashev)
Mine hoisting installations are major consumers of steel wire ropes. At these facilities in Kuzbass alone, about 150 thousand meters of rope, with a total weight on the order of 700 tons and a cost exceeding 1.5 million rubles, are used annually. Being expensive, the rope at the same time affects the dimensions and cost of all hoisting equipment, and therefore its economy.
At present, calculation of a hoisting rope is carried out with allowance for the sole requirement of the Safety Rules, namely that its operational reliability be ensured. Insufficient attention is paid to the economic aspect of the calculation.
Of great economic importance is the characteristic of the strength of the rope wires under simple uniaxial tension \((\sigma_{pu})\). A suitably chosen value of \(\sigma_{pu}\) increases the service endurance of the rope and reduces capital and operating costs for the hoisting installation.
In the existing literature, the question of developing methods for a justified selection of strength limits is not reflected. In practice this characteristic is adopted at the discretion of the mine-hoist designer, within the limits of its standard values \((130—200\ \text{kg}/\text{mm}^2)\). Apparently, in connection with this situation, ropes operating in mines in a large number of cases cannot be justified with respect to the adopted values of \(\sigma_{pu}\). For example, in Kuzbass, at depths of most mines up to \(250\ \text{m}\) and terminal rope loads up to \(15\ \text{t}\), \(\sigma_{pu}\) is taken within the range \(130—190\ \text{kg}/\text{mm}^2\), with about 60% of all ropes having \(\sigma_{pu}=150—160\ \text{kg}/\text{mm}^2\) and about 35% having \(170—180\ \text{kg}/\text{mm}^2\).
Recommendations on this question have been proposed by individual specialists. Thus, Prof. D. G. Zhitkov, studying the endurance of ropes on testing machines, obtained the best results with a wire strength limit of \(180\ \text{kg}/\text{mm}^2\). He recommends this value for ropes of mine hoists, especially in cases of severe operating conditions [1]. V. A. Tsvetkov considers a strength limit of \(150—160\ \text{kg}/\text{mm}^2\) expedient for mine depths up to \(400\ \text{m}\) [2].
The recommendations cited are isolated from specific rope designs and specific production conditions, and therefo-
Practice does not always justify high strength limits. On the main cage hoist of the Temir-Tau mine (Kuznetsk Metallurgical Combine), ropes with $\sigma_{nu} = 180\text{–}190\ \mathrm{kg/mm^2}$ have been operating for many years; however, their service life is only 130–150 days. Examining the hoisting installation, we found that the reduction in the endurance of the ropes here is caused by premature fatigue failure of the wires due to abrasive damage by ore and rock dust. Under such conditions, ropes made of wires with lower values of $\sigma_{nu}$ will operate significantly better.
In studying the question of selecting the strength limit, we considered its connection with other factors that determine the technical and economic rationality of the hoisting installation. Such factors include: the dimensions of the rope and of the hoisting machine, operating conditions, and the rope construction.
As a result of the study, a method was developed for designing the hoisting machine and the rope, and also for selecting an expedient value of $\sigma_{nu}$.
The design method currently in use is based on the relationships:
$$ d = f(\sigma_{nu}, \gamma_0), \tag{1} $$
$$ D_b, B = f(d), \tag{2} $$
where $d$ is the rope diameter,
$\gamma_0$ is the conventional volumetric weight of the rope,
$D_b, B$ are the diameter and width of the hoisting-machine drum.
The value of $\gamma_0$ for standard hoisting ropes is 8900–10000 $\mathrm{kg/m^3}$ and, in calculating the rope for strength, is taken as any value lying within the indicated limits.
The dimensions of the drums of the hoisting machine, obtained according to the relationships given above, have the most substantial significance for the hoisting installation, since they largely determine its economy. But, as is evident, these dimensions are determined through the quantities $\sigma_{nu}$ and $\gamma_0$, which are less important and, moreover, are selected with a certain arbitrariness. Consequently, the requirements of economic expediency are not properly reflected here.
We consider it useful to change the design principle by relating the quantities of interest to us by a dependence of the form
$$ \sigma_{nu} = f(D_b, B). \tag{3} $$
Having selected justified dimensions of the winding organs, one can find the necessary rational value of the strength limit.
In designing the hoisting machine, one should proceed from the maximum utilization of its load capacity and winding surface. This circumstance is not at all emphasized in present-day practice and literature, and is insufficiently taken into account.
The following design procedure is proposed.
- The hoisting machine is selected according to its load capacity, in accordance with the inequality
$$ F_{\max} > Q_0, \tag{4} $$
where $F_{\max}$ is the maximum static rope tension in kg permitted by the design of the machine,
$Q_0$ is the end load of the rope in kg.
- The value of the winding surface of the drum is determined
$$ \frac{\pi D_b B \cdot 10^3}{d + \varepsilon} = H_d + l_z + l_{vt}, \tag{5} $$
where \(\pi = 3.14\),
\(\varepsilon\) is the clearance between two adjacent turns of the rope on the drum, in mm,
\(H_0\) is the maximum length of the working part of the rope permitted for winding, in m,
\(l_z\) is the length of the reserve part of the rope, in m,
\(l_{fr}\) is the length of the friction turns, in m.
In calculations it is usually taken that: \(\varepsilon = 2\text{--}3\) mm, \(l_z = 30\text{--}35\) m, \(l_{fr} = \pi D_b n_{fr}\), where \(n_{fr} = 3\) is the number of obligatory friction turns.
From equation (5) the rope diameter is determined as
\[ d = \frac{\pi D_b B \cdot 10^3}{H_0 + l_z + l_{fr}} - \varepsilon . \tag{6} \]
Dividing both sides of equation (6) by \(D_b \cdot 10^3\), we obtain
\[ \frac{d}{D_b \cdot 10^3} = \frac{\pi B}{H_0 + l_z + l_{fr}} - \frac{\varepsilon}{D_b \cdot 10^3}. \tag{7} \]
The quantity
\[ \frac{\varepsilon}{D_b \cdot 10^3} \ll 0.001 \]
and may be neglected.
In accordance with the requirements of the Safety Rules, and depending on the magnitude of the end load and the hoisting height, the most suitable ratio
\[ \frac{D_b \cdot 10^3}{d} \]
is taken within the limits
\[ \frac{D_b \cdot 10^3}{d} = 80\text{--}100. \]
From equation (7) the value \(H_0\) is determined:
\[ H_0 = (80\text{--}100)\pi B - (l_z + l_{fr}). \tag{8} \]
- The hoisting machine is checked with respect to the winding surface, according to the condition
\[ H_0 \geq H, \tag{9} \]
where \(H\) is the calculated hoisting height.
- From equation (6) the value \(d\) is determined. From the catalog, a rope of standard diameter nearest to the calculated one, in the direction of increase or decrease, is selected, and the values \(d\), \(p\), \(s\) are entered, where \(p\) is the weight of 1 m of rope length, in kg,
\(s\) is the cross-sectional area of the rope metal, in mm\(^2\).
- The required value of the ultimate strength is determined from the equation
\[ \sigma_{pc} \geq \frac{(Q_0 + pH_0)m}{s}\ \frac{\mathrm{kg}}{\mathrm{mm}^2}, \tag{10} \]
where \(m\) is the static safety factor of the rope, adopted according to the standards of the Safety Rules,
\(H_0\) is the calculated length of the rope suspension.
According to the obtained $\sigma_{nch}$, a standard value is adopted. The selected rope is evaluated from the standpoint of endurance, determined by its internal properties and by the conditions of the hoisting installation on which it will operate. In doing so, the following are taken into account:
a) Rope construction.
Ropes of the LK type, with linear contact of wires, are very durable in operation and will justify the use of higher strength limits. The ordinary construction of ropes, especially the construction $6 \times 37 + I_{oc}$ at diameters up to 39 mm, does not ensure the use of the metal before the onset of fatigue, and therefore here it is inexpedient to adopt a high-strength wire material;
b) Operating conditions of the rope.
Attention is given to the intensity of operation of the hoisting installation (frequency of hoisting operations), the presence and possible influence of factors reducing the fatigue strength of the metal (corrosion, abrasiveness, conditions of mechanical wear), the condition of the mine shaft, the method of loading the conveyances, the type of hoisting motor, and the method of controlling it.
The endurance of the rope must be combined with the severity factor of its operating conditions.
- A check is carried out:
a) for the load capacity of the machine
\[ F_{\max} \geq Q_0 + pH_0, \tag{11} \]
b) for the safety factor of the rope
\[ m = \frac{\sigma_{nch}s}{Q_0 + pH_0}. \tag{12} \]
For round standard ropes of domestic manufacture the relation holds
\[ s = 0.37d^2. \tag{13} \]
Substituting into equation (10) the value of $s$ from (13) and $d$ from (6), we obtain:
\[ \sigma_{nch} \geq \frac{(Q_0 + pH_0)m} {0.37\left( \frac{\pi D_b B \cdot 10^3}{H_0 + l_z + l_{vm}} - \varepsilon \right)^2}. \tag{14} \]
Equation (14) shows that the strength limit is a function of many quantities, including the dimensions of the hoisting machine, and therefore it corresponds to the form of relation (3) that is taken as the basis of the calculation method.
Let us compare, using a specific example, the results of calculations by the existing and proposed methods.
Let the following be given: mine depth—300 m, required terminal rope load—8000 kg.
It is required to select a hoisting machine and a rope for hoisting loads.
After carrying out the necessary operations, we obtain the data presented in the table.
It is clear from the table that the volume of calculation and other operations by both methods is approximately the same. A large difference is obtained with respect to the hoisting machine: with single-layer winding of the rope on the drum, the existing method gives a machine $2 \times 4 \times 1.7$, while the proposed method gives $2 \times 3 \times 1.5$.
Table
| Calculation method | Method of obtaining quantities | $D_b$, m | $B$, m | $H$, m | $H_o$, m | $l_z$, m | $n_{vt}$ | $l_{vt}$, m | $\varepsilon$, mm | $H_d$, m | $\dfrac{D_b}{d}$ | $d$, mm | $p$, kg/m | $s$, mm² | $q_o$, kg/m³ | $\sigma_{mt}$, kg/mm² | $Q_z^*$, kg | $m$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Existing | Accepted | — | — | — | — | 30 | 3 | — | 3 | — | 80 | — | — | — | 9500 | 150 | — | 6.5 |
| Existing | Obtained by calculation | 2.72 | 1.74 | 340 | 350 | — | — | — | — | — | — | — | 3.85 | — | — | — | — | 6.96 |
| Existing | Selected from the catalog for use | 4 | 1.7 | — | — | — | — | — | — | — | — | 34 | 3.8 | — | — | — | 65000 | — |
| Proposed | Accepted | 3 | 1.5 | — | — | 30 | 3 | — | 3 | — | 90 | — | — | — | — | — | — | 6.5 |
| Proposed | Obtained by calculation | — | — | 340 | 350 | — | — | 28.3 | — | 366 | — | 30.5 | — | — | — | 165 | — | 6.7 |
| Proposed | Selected from the catalog for use | 3 | 1.5 | — | — | — | — | — | — | — | — | 31 | 3.1 | 358 | — | 170 | — | — |
*) $Q_z$ denotes the value of the total breaking resistance of all wires of the rope. All other designations were given earlier.
The proposed method has substantial advantages. By its structure it takes into account the load-carrying capacity of the hoisting machine and its reserves, and better satisfies the economic requirements of the calculation. At a number of stages in designing a mine hoist (the selection of the hoisting machine, the adopted ratio
\[ \frac{D_b}{d}, \]
the strength limit, and the evaluation of the adopted rope), the method stimulates the designer’s creative initiative and restricts possible unjustified rule-of-thumb practice.
This method makes it possible to analyze certain important questions. Thus, we analyzed 104 cases of various combinations of rope terminal loads and mine depths occurring in Kuzbass. In each case all the factors were linked through which the influence of the wire strength limit is reflected in the dimensions of the hoisting rope and the machine.
The analysis showed that, for present conditions in Kuzbass, it is most advisable to use a rope-metal strength limit of \(160\ \text{kg}/\text{mm}^2\). This value of the limit has the greatest frequency (47 times), and the greatest number of coincidences with adjacent values of \(\sigma_{nч}\) that, under equal conditions, give the same rope diameter sizes (29 coincidences with \(\sigma_{nч}=170\ \text{kg}/\text{mm}^2\) and 32 with \(\sigma_{nч}=150\ \text{kg}/\text{mm}^2\)). Practical need for strength limits above \(160\ \text{kg}/\text{mm}^2\) occurs only in 28 cases, of which 11 relate to mine depths greater than 300 m, that is, to future Kuzbass conditions. Ropes with wire strength limits from 160 to \(180\ \text{kg}/\text{mm}^2\), at equal terminal loads, do not require changes in the standard sizes of hoisting machines and ensure that loads are transported to the same height. The exceptions are rare cases encountered at mine depths greater than 250 m.
Thus, the use in Kuzbass of a large number of hoisting ropes with wire strength limits above \(160\ \text{kg}/\text{mm}^2\) is not caused by real necessity, and in material terms entails appreciable loss: strength limits of 170–\(180\ \text{kg}/\text{mm}^2\), compared with \(160\ \text{kg}/\text{mm}^2\), make the rope 10–15% more expensive [3].
As is evident from the foregoing, the question of calculating and selecting a hoisting rope is of great economic importance. In solving it, the requirements of economy must necessarily be taken into account. The method proposed for this purpose reflects these requirements more fully than the existing one.
LITERATURE
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Zhitkov L. G., Pospekhov I. T. Steel Ropes for Hoisting-and-Transport Machines. Metallurgizdat, 1953.
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Tsvetkov V. A. Compilation of Standards for the Consumption of Materials in Coal Mines. Report of the Eastern State Scientific-Research Institute of the MUP, 1951.
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Materials and Equipment Used in the Coal Industry. Vol. I, Part I. Ugletekhizdat, 1955.