On the Elastic and Plastic Elongation of Hoisting Ropes
Abstract Generated abstract
This paper analyzes the elastic and plastic elongation of steel hoisting ropes used in mine hoists, emphasizing that unstretched ropes undergo plastic elongation at stresses below the elastic limit of their wires because of rope structure and wire displacement. It replaces the rope tensile diagram with a practical polygonal approximation and derives formulas for initial plastic elongation, elastic deformation during landing operations, operational plastic elongation over time, and additional deformation under emergency loading or preliminary stretching. The work relates these deformations to rope characteristics, suspended length, hoisting height, load conditions, motor starting, and service time, and provides numerical examples for typical skip hoists. It argues that total relative deformation can serve as an indicator of rope stress and serviceability, provided operational elongation and emergency elongation are carefully recorded.
Full Text
ON THE ELASTIC AND PLASTIC ELONGATION OF HOISTING ROPES
N. I. KLYKOV
(Presented by the scientific seminar of the electromechanical departments of the mining faculty)
Steel wire ropes have become widespread in mine hoists. The material of the rope wires possesses elastic-plastic properties. In addition, under the action of tensile loads, a certain mutual displacement of the wires occurs in the rope; the rope receives an additional plastic elongation—“stretching”—as a result of which, over time, it becomes necessary to adjust the length of the rope. The correct solution of the problem of regulating the rope length is connected with the need to determine the numerical values of the elastic and plastic deformation of the rope. The values of these quantities should also be taken into account in solving other problems: establishing a criterion of rope strength under operating conditions, more accurately determining the length of the suspended rope, etc.
In this work an analysis is made of the principal factors affecting the magnitude of the plastic and elastic deformation of the rope.
The general form of the tensile diagram of the individual steel wires of the rope is shown in Fig. 1. Up to point \(A\) on the diagram, corresponding to the elastic limit, the material obeys Hooke’s law. The segment \(OAA_1\) is the zone of elastic deformations. To the right of the line \(AA_1\) lies the zone of elastic-plastic deformations of the material.
Fig. 1. Tensile diagram of the rope material.
As observations show, an unstretched hoisting rope, along with elastic elongation, receives plastic elongation at stresses considerably lower than the elastic limit of the wire material. This is explained by the structural features of the hoisting rope. Consequently, with sufficient accuracy it may be assumed that an unstretched rope operates in the zone of elastic-plastic deformations.
Conventionally, the actual tensile diagram of the rope may, with accuracy sufficient for practical purposes, be replaced by a polygonal one,
consisting of two inclined sections (Fig. 2). In view of the fact that an unstretched rope works in the region of elastic-plastic deformations, we transfer the origin of coordinates to the point \(O_1\). Along the vertical axis we plot the reduced stresses, and along the horizontal axis—the values of the relative deformations of the rope. Let the angles of inclination of the lines to the horizontal axis be denoted on the diagram by \(\alpha\) and \(\beta\), respectively for the elastic and elastic-plastic regions. We shall call the quantity \(E'_k=\operatorname{tg}\beta\) the modulus of total deformation of the rope, in contrast to the modulus of elasticity of the rope \(E_k=\operatorname{tg}\alpha\) [1, pp. 308, 312].
Fig. 2. Straightened tensile diagram of a rope.
In mine hoisting installations, new, as yet unstretched ropes are most often installed. The initial plastic elongation of the rope corresponds to the period of installation and initial adjustment of the rope length. This period is insignificant in comparison with the service life of the rope; therefore the time factor is not taken into account in the calculation.
In accordance with Fig. 2, the magnitude of the relative initial plastic deformation is
\[ \varepsilon_{\text{pl.n}} = \frac{S}{\sum F} \left( \frac{1}{E'_k} - \frac{1}{E_k} \right), \tag{1} \]
where \(S\) is the greatest tensile force, kg;
\(\sum F\) is the cross-sectional area of all the wires of the rope, cm\(^2\).
The tensile force, taking into account the end load, the rope’s own weight, shaft resistances, possible overloading of the conveyances, and dynamic loads, is
\[ S = Q_0 + p\frac{H_0}{2} + \frac{k'Q_0}{n+1} + \frac{k_nQ_0}{n+1} + \left( Q_0+p\frac{H_0}{2} \right) \frac{j_c}{g} \cdot \frac{j_m}{j_c}, \tag{2} \]
where \(Q_0\) is the end load, kg;
\(p\) is the weight of 1 m length of rope, kg/m;
\(H_0\) is the greatest suspended length of the rope, m;
\(k'\) is the coefficient of shaft resistance for one branch;
\(n\) is the ratio of the hoisting vessel’s own weight to the weight of the useful load;
\(k_n\) is the coefficient of overloading of the vessel;
\(j_c\) is the average acceleration, m/sec\(^2\);
\(j_m\) is the maximum acceleration, depending on the nature of the motor starting, m/sec\(^2\);
\(g\) is the acceleration due to gravity, m/sec\(^2\).
The rope calculation is performed by the formula [2, p. 48]
\[ p = \frac{Q_0}{ \dfrac{\sigma_{\text{pc}}\cdot 10^6}{\gamma_0 m} - H_0 }, \tag{3} \]
where \(\sigma_{\text{ult}}\) is the ultimate strength of the rope, kg/mm\(^2\);
\(m\) is the safety factor of the rope;
\(\gamma_0\) is the fictitious weight of \(1\ \text{m}^3\) of rope, kg/m\(^3\).
From (3), the design end load will be
\[ Q_0=p\left(\frac{\sigma_{\text{ult}}\cdot 10^6}{\gamma_0 m}-H_0\right). \tag{4} \]
Substituting the value of the design end load (4) into expression (2), we obtain
\[ S=p\left\{ \frac{l_m+g}{g} \left( \frac{\sigma_{\text{ult}}\cdot 10^6}{\gamma_0 m} -\frac{H_n}{2} \right) + \frac{k'+k_n}{n+1} \left( \frac{\sigma_{\text{ult}}\cdot 10^6}{\gamma_0 m} -H_0 \right) \right\}. \tag{5} \]
The modulus of elasticity of the rope [3, p. 16]
\[ E_k=0.86\cdot 10^6\left(\frac{\sum F}{p}\right)^4. \tag{6} \]
The value of the modulus of total deformation, determined experimentally, is on average
\[ E'_k \simeq 0.6\,E_k . \tag{7} \]
Substituting the values of the greatest tensile force (5), the modulus of elasticity of the rope (6), and the reduced modulus of deformation (7) into expression (1), we obtain
\[ \varepsilon_{\text{pl.n}} = 0.8\cdot 10^{-6} \left(\frac{p}{\sum F}\right)^5 \left[ \frac{l_m}{g} \left( \frac{\sigma_{\text{ult}}\cdot 10^6}{\gamma_0 m} -\frac{H_n}{2} \right) + \frac{k'+k_n}{n+1} \left( \frac{\sigma_{\text{ult}}\cdot 10^6}{\gamma_0 m} -H_0 \right) \right]. \tag{8} \]
Thus, the magnitude of the relative initial plastic elongation of the rope depends mainly on the characteristics of the rope, the hoisting height, and the nature of the starting of the hoisting motor.
The value of the relative initial plastic deformation of the rope for skip hoists with contact starting of an asynchronous motor and an average hoisting height is (Fig. 3)
\[ \varepsilon_{\text{pl.n}}\simeq 0.002. \tag{9} \]
During operation of a hoisting rope, every change in the end load corresponds to a definite value of elastic deformation (lines \(AB\) in Fig. 2).
When solving problems connected with regulating the length of the rope, the magnitude of the elastic deformation of the rope before the hoisting vessel is set under load is of substantial importance. The magnitude of the relative elastic deformation of the rope in this case will be
\[ \varepsilon_y=\frac{kQ_0}{\sum F\cdot E_k}, \tag{10} \]
where \(k\) is the end-load coefficient.
When lowering a load, the end-load coefficient is equal to unity.
When landing an empty vessel,
\[ k=\frac{n+a}{n+1}, \tag{11} \]
where \(a\) is the coefficient of loading of the vessel in motion (taken into account only when automatically loaded skips are used).
Fig. 3. Initial plastic elongation of the rope.
Substituting into expression (10) the value of the calculated end load (4) and of the modulus of elasticity of the rope (6), we obtain
\[ \varepsilon_y = 1.16\cdot 10^{-6}\, \frac{n+a}{n+1} \left(\frac{P}{\sum F}\right)^{\chi} \left( \frac{\sigma_{\mathrm{vr}}\cdot 10^{5}}{\gamma_0 m} - H_0 \right), \tag{12} \]
Thus, the magnitude of the relative elastic deformation of the rope before the hoisting vessel is landed on the lower receiving platform depends on the characteristics of the rope and on the hoisting height. The influence of the rope’s own weight was not taken into account in the calculation, since under operating conditions the rope is not relieved of the action of its own weight.
The absolute elastic deformation of the rope in this case is
\[ \Delta l_y=\varepsilon_y H_0. \tag{13} \]
Example. Determine the magnitude of the relative and absolute elastic deformation of the rope before landing the vessel, if the hoisting installation has: hoisting height \(390\ \mathrm{m}\); rope diameter \(39\ \mathrm{mm}\); six-ton skips with a sector gate.
Solution. We take the length of the rope suspension to be \(400\ \mathrm{m}\); the coefficient of loading of the skip in motion, \(0.1\); the ratio of the weight of the vessel to the pay-
to the useful weight of the load, 0.805 [2, p. 32]; weight of 1 m of rope length, 5.312 kg/m [2, p. 52]; cross-sectional area of all wires, 5.6562 cm²; ultimate strength of the rope, 180 kg/mm²; fictitious weight of the rope, 9200 kg/m³; design safety factor of the rope, 6.5.
The relative elastic deformation of the rope is
\[ \varepsilon_y =1.16\cdot 10^{-6}\, \frac{0.805+0.1}{0.805+1} \left(\frac{5.312}{5.6562}\right) \left(\frac{180\cdot 10^6}{9200\cdot 6.5}-460\right) =0.0011 . \]
The absolute magnitude of the elastic deformation is \(\Delta l_y=0.0011\cdot 400=0.44\) m.
The magnitude of the relative plastic deformation of the rope during operation depends mainly on two factors: the greatest tensile force and the time of operation,
\[ \varepsilon_{\text{pl}\cdot \text{e}} = \frac{0.9S\,t^{0.163}}{\sum F\cdot E_k'}, \tag{14} \]
where \(t\) is the time of operation of the rope from the moment of suspension, months.
The values of the coefficient 0.9 and of the exponent 0.163 were determined experimentally, by measuring the plastic elongation of the rope at known intervals of time. The tensile force was calculated from expression (5).
Substituting into expression (14) the value of the tensile force (5) and of the reduced modulus of deformation (7), we obtain
\[ \varepsilon_{\text{pl}\cdot \text{e}} = 1.75\cdot 10^{-6} \left(\frac{p}{\sum F}\right)^{5} t^{0.163} \left[ \frac{lm+g}{g} \left( \frac{\sigma_{\text{pr}}\cdot 10^6}{\gamma_0 m} -\frac{H_0}{2} \right) + \frac{k'-k_n}{n+1} \left( \frac{\sigma_{\text{pr}}\cdot 10^6}{\gamma_0 m} -H_0 \right) \right]. \tag{15} \]
In practical calculations, the value of the service relative plastic deformation of the rope for skip hoisting installations in mines of medium depth, depending on the time of operation of the rope, may with sufficient accuracy be taken as (Fig. 4)
\[ \varepsilon_{\text{pl}\cdot \text{e}} = 5.6\cdot 10^{-3}\cdot t^{0.163}. \tag{16} \]
Example. Determine the total plastic elongation of two ropes after four months, if before this the first rope had worked for two months, and the other rope for eight months. The greatest suspended length of the rope is 400 m; the length of the rope strand is 50 m.
Solution. The absolute plastic elongation of one rope is
\[ \Delta l_{\text{pl}\cdot 1} = 5.6\cdot 10^{-3}(400+50) \left[(2+4)^{0.163}-2^{0.163}\right] = 0.55\ \text{m}. \]
The absolute plastic elongation of the other rope is
\[ \Delta l_{\text{pl}\cdot 2} = 5.6\cdot 10^{-3}(400+50) \left[(8+4)^{0.163}\right] - 8^{0.163} = 0.25\ \text{m}. \]
The total plastic elongation of the ropes is
\[ \Delta l_{\text{pl}} = 0.55+0.25 = 0.8\ \text{m}. \]
In the calculations given, an unstretched rope was assumed. In the case of preliminary stretching of the rope by a known force \(S_v\), the magnitudes of the na-
…initial and operational plastic deformation must be determined with allowance for this force. If the force of preliminary stretching of the rope is less than the greatest operational tensile force, then the magnitude of the relative initial plastic deformation is
\[ \varepsilon_{\text{pl.n}}=\frac{S-S_{\mathrm{v}}}{\sum F \cdot E'_k}, \tag{17} \]
Fig. 4. Operational plastic elongation of the rope.
and the magnitude of the relative operational deformation is determined from expressions (14), (15), or (16).
If the force of preliminary stretching of the rope is equal to or greater than the operational tensile force, then no initial plastic elongation of the rope occurs, and the relative operational plastic deformation is determined from the expression
\[ \varepsilon_{\text{pl.e}}= \frac{0.9}{\sum F \cdot E'_k} \left[S\left(t^{0.16}+1\right)-S_{\mathrm{v}}\right]. \tag{18} \]
In the operation of a hoisting installation, emergency loads are possible, under which the rope receives an additional elongation
\[ \varepsilon_{\text{pl.em}}= \frac{S_{\text{em}}-S}{\sum F} \left(\frac{1}{E'_k}-\frac{1}{E_k}\right). \tag{19} \]
If the rope has been preliminarily stretched by a force exceeding the magnitude of the operational tensile force, then the magnitude of the relative emergency plastic deformation of the rope will be
\[ \varepsilon_{\text{pl.em}}= \frac{S_{\text{em}}-S_{\mathrm{v}}}{\sum F} \left(\frac{1}{E'_k}-\frac{1}{E_k}\right). \tag{20} \]
To determine the magnitude of the relative emergency deformation of the rope it is obviously necessary to know the magnitude of the emergency force,
which in some cases may be determined by calculation, for example when a safety brake is actuated.
According to the Safety Regulations [4, p. 152, § 828], the magnitude of the emergency elongation of a rope is limited to 0.5% of the length of the rope section that participated in the emergency tensioning. In our opinion, this instruction is not sufficiently justified. In some cases the value 0.5% will be excessive, for example for ropes that have been in service for a long time; in other cases it will be understated, for example for newly suspended ropes.
According to Fig. 2, each value of the total relative deformation of the rope corresponds to a definite value of the reduced stress
\[ \sigma_{\mathrm{pr}}=\varepsilon \cdot E'_k . \tag{21} \]
The value of the total relative deformation of the rope is
\[ \varepsilon=\varepsilon_{\mathrm{pl.n}}+\varepsilon_{\mathrm{pl.e}}+\varepsilon_{\mathrm{pl.exc}}+\varepsilon_{\mathrm{el}} . \tag{22} \]
When determining the maximum value of the reduced stress, the value of the relative elastic deformation of the rope must be taken as the greatest,
\[ \varepsilon_{\mathrm{el}}=\frac{S}{\sum F E_k}. \tag{23} \]
If the concept of the reduced safety factor of the rope is introduced,
\[ n_{\mathrm{pr}}=\frac{\sigma_{\mathrm{tn}}}{\sigma_{\mathrm{pr}}}, \tag{24} \]
and its value for specific conditions is specified in the Safety Regulations, then the strength criterion of the rope may be taken to be its total relative deformation
\[ \varepsilon \ll \frac{\sigma_{\mathrm{tn}}}{n_{\mathrm{pr}}\cdot E'_k}. \tag{25} \]
Conclusions
In determining the elastic and plastic deformation of a rope, it is necessary first of all to take into account the quantities characterizing the structure of the rope, as well as the technical and operating characteristics of the installation.
The total relative deformation of a rope characterizes its stress and serviceability. When introducing strength control of a rope by its total relative deformation, it is necessary to take careful account of the elongation of the rope during operation. For this purpose, the rope-condition log must record the values of the initial plastic elongation, emergency elongation, etc.
The formulas presented may be used to determine the length of the rope to be suspended, in solving questions of regulating the length of the rope, etc.
References
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N. I. Bezukhov. Theory of Elasticity and Plasticity. State Publishing House of Technical-Theoretical Literature, 1953.
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N. K. Pravitskii. Mine Hoisting Installations. Metallurgizdat, 1956.
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F. V. Florinskii. Dynamics of a Mine Hoisting Rope. Ugletekhizdat, 1955.
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Safety Regulations in Coal and Shale Mines. Ugletekhizdat, 1959.
ERRATA NOTED
| Page | Line or formula | Printed | Should read |
|---|---|---|---|
| 5 | formula (8) | $d$ solid |
$g$ hot |
| 80 | 13th line from bottom | solid $\dfrac{100}{\gamma_M}(f_M - f_K)$ |
hot $\dfrac{100}{\gamma_M}(f_M - f_K)$ |
| 85 | formula (1) | $+\dfrac{100}{z_M}(f_M - f_K)$ | $+\dfrac{100}{\gamma_M}(f_M - f_K)$ |