Issue 1/2011


02/02/11

The service lives of running wire ropes under the influences of size effect

Prof. Dr.-Ing. Klaus Feyrer

The service lives of running wire ropes are dependent above all on rope tensile force S/d2 and the ratio of the diameters of the sheave and the rope D/d. Additional significant factors are rope diameter d and the length of the section of the rope that is actually subjected to bending l. The method used for calculating rope drives in the past [1] did not take sufficient account of these influences. In the present paper the factors associated with flexure – the bending cycle factors – are modified to take into account to the greatest possible extent the influences of the rope diameter and of the section exposed to bending. In addition to the sources cited, a prime source of information was data drawn from the archives of the Institute of Mechanical Handling and Logistics (IFT) at the University of Stuttgart, where Professor K.-H. Wehking is Head of the institute.
Category: Issue 1/2011
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Calculation used in the past

The numbers of bending cycles to failure NE or to retirement age NAE as determined for the most common classes of rope designs are referenced to a uniform section of rope with diameter of dE = 16 mm and bending length – i.e. the section of the rope subjected to fl exure – of lE = 60dE [1]. The mean numbers of bending cycles here are N – E and N – AE. At any arbitrary rope diameter d and at any arbitrary rope bending length l, the mean numbers of bending cycles are
The numbers of bending cycles N – E and N – AE continue to apply without change. To be determined for the actual form are the bending cycle factors for rope diameter fd and for rope bending length fl.
The bending cycle factor for rope diameter can be traced back primarily to D. G. Shitkow and I. T. Pospechow who, in their book entitled Drahtseile [2], published a chart in which the relationship of the number of bending cycles can readily be seen.
This equation has been integrated into the calculation methods for traction drives [1]. R. Ciuffi and G. Rocatti [3], making reference to a chart by Hugo Müller [4], have determined that rope diameter probably has a greater influence on rope service life than what is reflected in equation (2).
The above-mentioned chart by Müller [4] is reproduced in Figure 1. It is possible to derive from the four different numbers of bending cycles for the four rope tension levels (from 20 to 60 kg/mm2) a mean exponential of almost b = –1. But the chart also shows the very severe jumps in services lives from one rope to the next, even though the diameters differ only minimally. In spite of the very high average decline in the number of bending cycles to failure as rope diameter increases, the 20 mm rope, for instance, achieves a service life which is longer than the 16 mm rope by a factor of 2.5.
A study conducted by Virsik [5], carried out under my supervision, initially confirmed equation [2]. In the meantime, however, doubts have arisen as to whether this equation takes sufficient account of the influence of the rope diameter.
In just the same way as for the rope diameter, it is necessary to examine whether the bending cycles factor for rope bending length used in the past, to wit,
takes sufficient account of this influence. This factor is based on standard deviation lgs0 = 0.05 for a uniform bending length of lE = 60d.
The influence of rope diameter
The bending cycle factor fd is used to take account of the influence of rope diameter on rope service life. This factor is derived from the number of bending cycles through to failure. It is assumed that the bending cycle factor can also be used for rope service life through to the rope retirement point.
The bending cycle factor expresses the relationship between the number of bending cycles to failure (or to retirement point) for a section of rope (with diameter d and bending length l = 60d) and the uniform rope section exhibiting bending length of lE = 60 dE and diameter dE = 16 mm. Here, with the exception of the rope diameter, all the data for the ropes to be compared and the test environments have to be identical. Taking the number of bending cycles Ni from bending test i using a rope with diameter d and bending length l = 60d, we find that the bending cycle factor is
The bending cycle factor for all those ropes tested with rope diameter d is the geometric mean of the bending cycle factors fdi determined in accordance with equation (4)
The advantage of this method is found in the fact that the results of many different ropes and test environments can be merged and that a common bending cycle factor fd can be found for the rope diameter d being considered – this instead of a scatter plot. Shown in Figure 2 is the result derived from the evaluation of many bending trials.
With a single exception, the ropes of the rope classes are used to determine the bending cycle factor fd, for which the constants are given in [1] and with which the mean service life N – E of the uniform rope piece can be calculated for use as the denominator in equation (4). Inserted as the numerator in equation (4) is the number of bending cycles Ni derived from tests with ropes of the particular diameter, taken from the IFT archives. Observed here are the numbers of simple bending cycles (straight – curved – straight) through to rope failure, drawn from the practical loading range.
The exception previously mentioned concerns the three ropes with 2 mm diameter in the simpler 6 x 7-FC-sZ design. No constants are available to calculate of the mean number of bending cycles to failure at 16 mm diameter. In this exceptional case, the mean number of bending cycles is determined on the basis of fourteen trials using four 16 mm ropes. The bending cycle factor, referenced to this mean number of bending cycles, comes to fd (d=2) = 5.70.
As regards the 2.5 mm diameter 6 x 19-FC-sZ cross lay rope which Müller [4] examined, revealing the results shown in Figure 1, the bending cycle factor is fd (d=2.5) = 5.94. Here the bending cycle numbers are not referenced to the short lived 16 mm rope given in Figure 1 but rather to the mean b E of the uniform section, which can be calculated as per [1]. The ropes in this design class are now used only for secondary purposes.
The bending cycle factor fd is not on a straight line in the double logarithmic chart. Thus it cannot be described in the form of equation (2) with a constant exponent. A simple ratio for the rope diameter’s bending cycle factor is given by
where d is in millimeters. The mediated curve is traced for this equation in Figure 2. The hyperbolic shape can presumably be traced back to the statistical share of the influence exercised by rope size. The influence of the bending cycle factor fd according to equation (5) is far greater than what had been thought valid in the past.
The bending cycle factors listed for the ropes with smaller diameters are, however, valid only as mean values. They are highly dependent on the particular bending cycle number N – E especially for the 2 mm and 2.5 mm ropes. At bending cycle number N – E = 5000 the factor is about fd = 2.5 and at bending cycle number N – E = 50,000 the factor is fd = 20.
The bending cycle factor also grows with the number of bending cycles N – E – even though to a far lesser extent – for the four 4 mm ropes examined. No corresponding behavior was found for ropes with diameters of 6 mm and more. It was found that for these ropes the bending cycle factor is distributed uniformly across the entire loading range. It appears appropriate to limit the application of the calculated bending cycle factor fd to ropes with diameters of d ≥ 6 mm.
At large rope diameters the bending cycle factor fd is documented only through to diameter of d = 44 mm and this by only a very few test results. Even at this diameter – and especially at even greater rope diameters – the bending cycle factor fd as per equation (5) is thus quite uncertain and offers only a rough approximation.
Influence of the rope bending length
As opposed to the rope diameter, the rope service life is influenced only statistically by the rope bending length (statistical size effect). Here it is presumed that the rope being observed was manufactured using uniform materials, at identical machinery settings, and that thus the rope’s service life will vary across the length of the rope only as a matter of probability. The survival probability is PE for the piece of rope with the nominal uniform bending length lE and the corresponding effective uniform bending length LE. Thus, according to the rules of reliability [6, 7 et al.], the survival probability PZ for the rope with the total bending length of L = z · LE, through to the failure of the first piece, is
Here PE is the survival probability for rope sections of effective uniform bending length LE. The number of sections arranged in sequence for bending length L is
Due to the rope’s stiffness in regard to bending the effective bending length L is somewhat shorter than the nominal rope bending length l, determined under the presumption that the rope is completely limp. In fact, and due to the rope’s bending stiffness, a small zone appears at each end of the nominal bending length l, the two together coming to a length of Δl, which is ultimately exposed to less than 90 % of the rising bending load appearing across the bending length [1, 10]. Consequently, thus is practically without influence on the rope’s service life. In the normal loading range this bending length is Δl ≈ 2.5 d. Taking the nominal bending lengths l, the effective ratio of the bending lengths z is
In the statistics books and calculation programs it is not the survival probability P but the failure probability Q that is tabulated for the standard variables. Thus equation (6) is
The rope service life (number of bending cycles to failure or to retirement age) reflects logarithmic Gaussian distribution. At any arbitrary failure probability of QE, the number of bending cycles for sections of a rope with uniform bending length LE/d or with a bending ratio of z = 1 is
As in the past, shown here in curly brackets is the magnitude – the failure probability QE – by way of which the magnitude just in front (here the standard variable u) is determined. As regards the bending length ratio z = 1, the standard deviation can be abbreviated to lgs0 = lgs0.
The number of bending cycles for the uniform bending length at failure probability QE as per equation (8) is at the same time the number of bending cycles for the weakest of the uniform bending lengths situated in sequence and thus is the failure probability Qz for the total effective rope bending length L or for the bending length ratio z
Where the failure probability for the effective rope bending length L is Qz = 0.5, its mean number of bending cycles is
The ratio of the mean number of bending cycles, and with that the bending cycle factor fl for the influence of the bending length, is thus
When processed in the Excel spreadsheet calculation program, the bending cycle factor is
The standard deviation of the number of bending cycles changes with the bending length. The following applies to ropes with bending length ratio z:
When equation (9) is expanded with lgNz and Q it follows that
When processed in the Excel program, using the selected standard variables of u0 = 1, the standard deviation of the number of bending cycles, as a function of the bending length ratio z, is
Earlier testing [10] using sections of a Seale rope with various bending lengths showed good correlation with the theory, even for smaller ratios of z < 1. The bending cycle factor for rope bending length fl is dependent only on the standard deviation lgs0, which is to be examined below.
Standard deviation lgs0
The standard deviation lgs0 is valid for the number of bending cycles for sections of a rope subjected to tensile force and which are bent swelling on a nominal bending length l/d. As above, the standard deviation for the uniform bending length of lE = 60d is abbreviated to lgs0. In the calculations for rope drives [1] lgs0 = 0.05 had been applied in the past. Here it is assumed that lgs0 = 0.05 is valid for the uniform bending length lE = 60d for all rope diameters d. To verify these determinations, all the known standard deviations found in rope bending tests were compiled.
The standard deviations determined on the basis of multiple tests with sections of a rope at uniform loading [8, 9, 10, 11] are calculated in a known fashion [12, 13 et al.] which need not be further discussed at this juncture. The results have been entered in Table 1. Most of the standard deviations show there were determined with bending length l = 45d, commonly used in the past. In the last column equation (12) is used to convert them to the uniform bending length of l = 60d which, due to the standard length of 30d, is given preference for the number of wire breaks that indicate need for retirement.
Of the comparison tests for service life carried out by COMMA1 of the OIPEEC, which Giovannozzi [9] describes, the only tests evaluated are those conducted by Müller as it is only for these that the bending length used is known. Of the nine test series, with five bending tests each at the same loading situation as Müller used, the mean standard deviation was determined to be lgs0 = 0.495, referenced to bending length l = 60d. The standard deviation lgs0 = 0.038 determined by Vogel and Scheunemann [11] for a Seale rope with rope diameter of d = 4 m is, as opposed to the other results for the number of bending cycles, valid through to the rope retirement point.
Bending cycle factor fl – approximation
Basing on the bending length ratio z as per equation (7) and on lgs0 = 0.047, the bending cycle factor as per equation (10a) for rope length l/d is
This equation is not suitable for routine use. Used as an approximation for this is
Since the bending cycle factor fl is referenced to the effective bending length instead of to the nominal bending length, as was the case before, the limit for the bending length for which the bending cycle factor applies can be reduced vis à vis [1]. Nominal bending length of l ≥ 10d appears to be the limit.
 Class standard deviations lgs
The standard deviation lgs is the standard deviation for the number of bending cycles in the ropes within a given rope class. This quite large standard deviation is determinant for the numbers of bending cycles relevant in practice, NA10 and N10, in which a maximum of 10 % of the ropes in this rope class are ready for retirement or have failed. This standard deviation for the rope class is comprised of the standard deviation lgs0, mentioned above, for pieces of one rope in each case as their components [14] and the far larger standard deviation lgsV for the divergences of the mean service lives from one rope in a rope class to the next, exhibiting the relationship
Even for the ropes in the ropes class comprising 18 x 7 low-twist, spiral round-strand ropes with the smallest standard deviation for the tested rope classes (lgs = 0.192) the share of the standard deviation lgs0 = 0.047 accounts for only 3 % of this. Changing this small contribution for other bending lengths modifies the standard class deviation lgs for ropes in a given rope class to only a very limited extent. This can be disregarded when calculating the numbers of bending cycles, N10 and NA10.
Calculating the number of bending cycles
Taking the two bending cycle factors fd and fl, the number of bending cycles that can be achieved, in accordance with equation (1), is
Listed in Table 2 are the constants a for the mean numbers of bending cycles and the numbers of bending cycles that a maximum of 10 % of the ropes in a rope class will not achieve, with 95 % certainty. The constants a are valid in dry rooms for simple bends (straight – bent – straight) for ropes that were very thoroughly lubricated prior to the start of the bending trials but which were not relubricated during bending at steel sheaves with a groove radius of r = 0.53d. The bending cycle factor for the influence of the rope diameter is calculated as per equation (5) while that for the influence of rope bending length fl is calculated as per equation (13).
The numbers of bends to retirement are applicable for the parallel- lay ropes and the spiral round-strand ropes only
  •  if the ropes are monitored by magnetic means or
  •  if testing of the rope in question has demonstrated that readiness for retirement is indicated by wire breaks visible from the outside.
The constants used to calculate rope service life have been changed from bi to ai in Table 2 to make clearer the distinction to the previously valid table in (1). In comparison with b0, the constants a0 are relieved by the constant summand of the bending cycle factors fd and fl previously valid as per equations (2) and (3) but are otherwise unchanged. Also not affected are constants b1, b2 and b4, which are now designated as a1, a2 and a3
Internet
When measuring traction drives there are fi ve limits [1, 15] that have to be observed:
  •  Rope service life
  • Donandt force (yield point)
  •  Break due to force (safety factor)
  •  Indication of retirement age (number of wire breaks at retirement)
  • Economy (optimal rope diameter)
The calculation of these limits is relatively elaborate and time-consuming. A calculation program [Seilleb1. xls in German and Seilleb2.xls in English) is available on the Internet at http://www.uni-stuttgart.de/ift/ forschung/berechnung.html. It calculates these limits and may be used free of charge. The calculation of rope service life in this program is based on equation (15), the constants in Table 2 and equations (5) and (13).
 
Bibliography
[1] Feyrer, K. Drahtseile. 2. ed. Berlin: Springer Verlag, 2000, or Wire
Ropes. Berlin, New York: Springer Verlag, 2007.
[2] Shitkow, D. G. und Pospechow, I. T. Drahtseile. VEB Verlag Technik,
Berlin 1957 (translation from the Russian).
[3] Ciuffi , R. und Roccati, G. “Wire rope and size effect.” OIPEEC
Technical Meeting Stuttgart 1995, Proceedings, pp. 10-1 to 10-
10.
[4] Müller, H. “Drahtseile im Kranbau.” VDI-Bericht No. 98 and dhf
12 (1966), 11, pp. 714-716 and 12, pp. 766-773.
[5] Virsik, K. Einfl uß des Seildurchmessers auf die Seillebensdauer.
Diplomarbeit, Institut für Fördertechnik, Universität Stuttgart
1995.
[6] Bertsche, B. u. Lechner, G. Zuverlässigkeit im Maschinenbau.
Springer-Verlag Berlin, Heidelberg, New York 1990.
[7] O’Connor, P.D.T. Zuverlässigkeitstechnik. VCH Verlagsgesellschaft
mbH, Weinheim 1990.
[8] Feyrer, K. Statistische Auswertung der Ergebnisse von Drahtseil-
Biegeversuchen. Draht 31 (1980), Nr. 6, pp. 404-407 und Nr. 7,
pp. 489-493.
[9] Giovannozzi, R. “Report on the research of the fatigue of wire
rope working party.” COMMA1. OIPEEC-Bulletin 8, Torino 1967;
in particular with the test report by Hugo Müller.
[10] Feyrer, K. Biegewechselzahl von Drahtseilen bei verschiedenen
Biegelängen. Drahtwelt 67 (1981) 4, pp. 86-90.
[11] Vogel, Scheunemann. Lift-Report 35 (2009)6, pp.4ff.
[12] Stange, K. Angewandte Statistik, Vol. 1, “Eindimensionale Probleme”
(1970), Vol. 2, “Mehrdimensionale Probleme” (1971) Springer-
Verlag Berlin, New York.
[13] Rade, L. u. Westergren. Springers mathematische Formeln.
3. ed., Springer, Berlin, Heidelberg 2000.
[14] Pfanzagl, J. Allgemeine Methodenlehre der Statistik, Vol. Il.
Sammlung Göschen. W. de Gruyter Verlag Berlin, New York
1974.
[15] Feyrer, K. “The fi ve dimensioning limits for rope drives.” Int.
Journal of Rope Science and Tech. Bulletin 94 (2007) pp. 5–12.
1/2011

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