Reliability Estimation of Non-rotating ropes Based on ...

International Journal of Performability Engineering, Vol. 12, No. 5, September 2016, pp. 471-480 © Totem Publisher, Inc., 4625 Stargazer Dr., Plano, Texas 75024, U.S.A

Reliability Estimation of Non-rotating ropes Based on Cumulative Damage HOUDA MOURADI1, *, ABDELLAH EL BARKANY1 and AHMED EL BIYAALI1 1 Mechanical Engineering Laboratory, Faculty of Science and Technology, University Sidi Mohamed Ben Abdellah, Fez, Morocco.

(Received on June 22, 2016, Revised on August 14, 2016) Abstract: Steel wire ropes consist of several steel wires twisted together to make complex structure with huge mechanical properties combining axial strength and stiffness with bending flexibility. The use of these ropes has known tremendous growth and has had significant effect on several industrial applications. Particularly, non-rotating wire ropes, which prevent the rotation of the suspended load under important hoisting heights. Industrial experience shows that the sudden breaking of a large part of steel wire ropes in service is usually due to the cumulative damage of their components, which could lead to serious accidents. Therefore, wire ropes must be monitored during their operation so as to be able of changing them timely. Thus, this work aims to interpret the non-rotating rope’s residual strength in the quantitative sense, using an analytical model which enables us to track and to estimate the rope’s reliability in terms of its strands’ cumulative damage. A numerical application is given to show the performance of the model. Keywords: Non-rotating wire rope, failure criterion, reliability, cumulative damage, monitoring.

1.

Introduction

A wire rope (Fig. 1) is a mechanical complex system generally made of several strands laid helically and symmetrically in one or multiple layers around a straight central core strand. The strand itself consists of several wires regularly arranged around a central wire in one or multiple layers. A special wire rope structure called non-rotating wire rope made of several layers of strands twisted in opposite direction from one layer to the other. This rope structure prevents the rotation of the suspended load under important hoisting heights. The geometric composition of non-rotating ropes is chosen so that the turning torque of the steel core and the outer strands cancel each other in a wide load range. It avoids in this way the kinking of the ropes. The basic material of wire ropes is initially steel with an elevated strength thanks to its high carbon content and its very fine grain structure. The division of the load bearing capacity between parallel wires assures the combination of high axial strength and stiffness with bending flexibility. Wire ropes are generally identified by several parameters including size, grade of steel used, minimum breaking load, metric mass, the number of strands and the number of wires in each strand.

_________________________________ *Corresponding author’s email: [email protected]

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Houda Mouradi, Abdellah El Barkany and Ahmed El Biyaali

Fig. 1: Schema illustrating the different components of steel wire rope

Non-rotating ropes, during use, are dynamically complex systems composed of numerous moving parts. Due to their mechanical properties (bending flexibility and torsional stiffness) and their ability to relatively resist large axial loads, non-rotating ropes have been widely used in fulfilling difficult tasks such as lifting loads, suspended bridges, elevators, boat caching… On the other hand, in the great majority of applications, a rope is subjected to repeated bending, fluctuating loads, internal contact forces, the wear due to these forces and high contact stresses with sheaves or drums whose direct consequences are the significant changes of geometrical and mechanical characteristics of the wire rope’s components. This results in a reduction of the rope’s reliability and makes the development of a completely analytical approach to the latter extremely difficult. Industrial experience shows that the sudden breaking of a large part of steel wire ropes in service is usually due to the cumulative damage of their components, which could lead to serious accidents. Thus, companies have interest in eliminating or controlling the risk of accidents that are linked to the wire rope use so as to guarantee a smooth running of their activities but most importantly, ensure the staff security. To keep abreast of degradation, wire ropes must be periodically inspected. More dependable than visual inspections, non-destructive inspection methods enable monitoring the rope in service. Two main features of deterioration are usually registered by a magnetic flux detector: distributed losses of the metallic area and localized faults, such as wire breaks [1]. But these data are not sufficient to indicate the rope’s strength in the quantitative sense. Therefore, it must be interpreted using an appropriate mechanical model to track the residual strength of the rope in service. For this, this paper aims to apply an analytical

Reliability Estimation of Non-rotating ropes Based on Cumulative Damage

473

model which enables us to monitor and to estimate the non rotating rope reliability during its operation, at different levels of its strands’ damage. Thus, this model will help us plan and organize actions of preventive maintenance so as to be able of changing it at the appropriate time. 2.

Literature review

In pursuance of exploiting the steel wire rope in exemplary conditions, a fine analysis of its security must be associated to inspection procedures defined accordingly. The safety analysis consists in linking the residual strength estimation of the rope to inspection data (when they are available). This link is difficult to establish especially if we assume that the concept of the rope safety factor is lightly insufficient to define its current state. In this context, Vorontsov et al. [1] proposed a strength assessment model for predicting the operating time of steel wire ropes that have deteriorated using magnetic NDT data. This model was based on the measurement of the distributed losses of the sectional metallic area and local wire breaks by a magnetic flux detector. These measures are used as input data for the strength evaluation. Jomdecha and Prateepasen [2] presented the construction of main-flux equipment for wire rope inspection which can be adjusted high electromagnetic field strength to produce leakage filed from flaws of various largediameter ropes. Moreover, this equipment is sufficiently sensitive to detect a smallest surface flaw of 1×2 mm at 5 mm equipment lift off. The inspection signals produced shown locations, levels and deterioration quantities. However, Kresak et al [3] deduced that the widely used magnetic method gives unreliable results in the most critical point of steel wire ropes, the anchor. Therefore, they introduced an application of the acoustic and thermographic method in the defectoscopic testing of immobile steel wire ropes at this critical point. The application of these two methods enabled the increase of safety measures of steel wire ropes in suspend bridges, towers and in hoisting equipments. However, the inspection data are not sufficient to estimate the rope’s strength in the quantitative sense. Therefore, it must be interpreted using an appropriate mechanical model to track the residual strength of the rope in service. For there, Statistical methods of strength and reliability estimation of damaged parallel wires have been discussed by Camo [4]. Giglio and Manes [5] presented a comparison between several analytical formulations used to estimate the state of stress in internal and external wires of steel wire ropes in order to predict the reliability fatigue life for the components. Blokus [6] applied the results of the reliability functions of parallel-series composed systems of dependent components and their limit forms to the reliability evaluation of the shipyard rope transportation parallel-series system. Chouairi et.al [7] have applied a practical method in order to analyze the steel wire ropes reliability. This method consists in developing an analytical approach of availability and maintainability using reliability-based optimization. Liu and chen [8] have established a finite element model to analyze the reliability of prestressed steel cable in regards to temperature and expansion coefficients action. On the other hand, in the context of multi-scale models suggested for predicting the reliability of steel wire ropes, Kolowrocki [9] developed a multi-scale modeling allowing the estimation of a wire rope lifespan, where he distinguished the strand scale, the scale of the layer of strands and that of the wire rope. He considers the rope as a mixed system (series-parallel). The choice of this system is justified by the fact that the external layer of

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the rope is made of strands with diameters superior than those of the internal layer. Moreover, the failure of one of these strands engenders that of the wire rope (series system). The external layer of strands is in parallel with the internal layer (parallel system). In addition, Cremona [10] has suggested model based on probabilistic approach in order to evaluate the residual strength of a wire rope. In his study, he has chosen to proceed according to two objectives: the first one consisting in evaluating the effects of factors influencing the long term performance of the rope. This has obliged him to develop a model of the rope’s strength at different levels of damage of its components and to relate the response of the wire with that of a group of wires considered collectively (strands) and finally to that of a group of strands (wire rope). The second objective was to develop a model of the wire rope residual lifespan in given conditions of use given that the physical and mechanical characteristics of a rope evolve over time because of the influence of the environment and conditions of use. According to the multi-scale approach developed by Elachachi et al. [11], a wire rope can be considered as a system consisting of group of strands arranged in parallel. A strand is itself consisting of stubs of strands arranged in series. Each stub is made of several wires arranged in parallel. The choice of these three scales is indicated knowing that, firstly wires are twisted and wrapped one with the other, a broken wire has the capacity to recover its share of applied load on a given length from the break, called recovery length. Its value, according to Raoof [12] is ranging from 1 to 2, 5 times the lay lengths. Secondly, a strand’s behavior is deeply related to that of the weakest stub (series system). Thirdly, strands’ being arranged in parallel, the rope’s strength depends on their individual strength and on the distribution of solicitation. Another multi-scale suggestion has been developed by Meksem et al. [13] in order to predict the reliability of hoisting ropes by taking into consideration the failure criteria which are based on the unacceptable number of broken wires in the rope. In their model, they consider the rope as a system made of several strands arranged in series. Each strand consists of several wires arranged of a majority logic system. The choice of these scales is justified as follows: one broken wire does not lead to the failure of the wire rope. However, starting from a certain number of broken wires, the wire rope can be declared as being failed (majority logic system). On the other hand, the strands are arranged in series the failure of one of these strands engenders that of the wire rope (series system). 3.

Reliability model suggested for non-rotating wire ropes

The non-rotating wire rope is a group of interconnected or interdependent elements in a way that makes the state of the rope depends on the states of its constitutive elements [14]. Thus, it is one of the most complicated composed systems. This means that every modeling approach of non-rotating wire ropes will be a multi-scale approach. By relying on different inspection methods of steel wire ropes, which allow the detection of external and internal damage such as broken wires and strands, we propose a multi-scale model allowing the estimation of the non-rotating rope's reliability in terms of its cumulative damage. The suggested model is mainly based on failure criteria which are the unacceptable number of broken strands, in other words, the minimum threshold requirement of functional strands in the rope. This model considers the rope as a system made of several strands arranged of a majority logic system. The strand itself consists of several wires arranged in parallel. The method adopted is a multi-scale approach where we distinguish the scale of the wire, the scale of the strand and that of the wire rope.


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The schema proposed of a non-rotating rope is then a majority logic / parallel system. The choice of these scales is justified by: - One broken strand does not lead to the failure of the non-rotating rope. However, starting from a certain number of broken strands, the rope can be declared as being failed. Therefore, the system is majority logic; - The wires are twisted and wrapped one with the other; a broken wire has the capacity to recover its share of applied load on a given length from the break, called recovery length. This point is pretty important to ensure that the wire rope is tough in the sense that it is tolerant to have local damages, particularly in the case of broken wires. Thus, we can say that the wires are disposed in parallel. According to the suggested model which considers the non-rotating rope as a majority logic/ parallel system, the expression of the reliability is as follow [15]: 𝑅 = 1 − �1 − ∑𝑛𝑘=𝑚 𝐶𝑘𝑛 𝑅𝑖 𝑘 . (1 − 𝑅𝑖 )𝑛−𝑘 �

𝑝

(1)

With p: number of wires; n: number of strands*number of stubs of strand; m: minimal threshold of the number of functional strands. 4.

Application on non-rotating rope 18x7

Knowing the moment in which the damage becomes critical and able to cause a wire rope’s sudden failure, is very important. Reducing the probability of such failure can be made by establishing an ongoing monitoring of the damage state of the components or by analyzing statistical data relating to the history of the wire rope’s operation so as to determine and predict its reliability. But these data are not sufficient to determine the rope’s strength in the quantitative sense. Therefore, it must be processed using mathematical models. The damage being progressive, its variation in terms of the number of cycles is widely influenced by the level of loading [16]. One of the different theories which describe the damage under fatigue solicitation, the unified theory is the mechanical model chosen for presenting the fatigue damage of the wire ropes; it is precisely based on the reduction of the fatigue limit and the loss of the rope’s strength [17]. According to this theory and in the case where the mean stress is null, the expression of the damage based on the fraction of life β is given by the following equation:

With

𝐷=

𝛽

𝛽+(1−𝛽)

𝛾 8 ) 𝛾𝑢 𝛾−1

𝛾−(

β=n/Nf; γ= ∆σ/σ0 and γu = σe/σ0

β: Fraction of life; n: Instant cycle’s number; Nf: Number of accumulated cycles at time of breaking ∆σ: Amplitude of solicitation; σ0: Endurance limit of the virgin material; σe: Endurance instant limit.

(2)


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We put: 𝛼

We obtain:

=

𝛾 8 ) 𝛾𝑢

.

𝐷=

𝛽+𝛼.(1−𝛽)

𝛾−(

𝛾−1

𝛽

.

(3)

From (3) we deduce the expression of the fraction of life β in terms of damage D, which can be written as follows: 𝛽=

𝛼.𝐷

(4)

1−𝐷.(1−𝛼)

In order to establish a link between reliability and damage, we must for starters express reliability, which is an explicit function of time, in terms of the fraction of life β, given by the following relationship [15]: 𝑅(𝛽) = exp(−(𝛽)𝜆 )

Replacing β by

𝛼.𝐷

1−𝐷.(1−𝛼)

(5)

in the equation 5, we get the following

relationship of reliability in terms of damage: 𝑅(𝐷) = 𝑒𝑥 𝑝 �− �

∝.𝐷

1−𝐷(1−∝)

𝜆

� �

(6)

In our study, we will focus on the steel wire rope 18x7 type (18strands 7wires) of non-rotating wire ropes structure illustrated in the figure 2:

Fig. 2: Non-rotating rope 18x7 type

Replacing the reliability Ri in the equation (1) by its expression in terms of the fraction of life β, the reliability equation of the non rotating rope 18x7 according to the suggested model becomes: 𝑝

𝑅(𝛽) = 1 − �1 − ∑𝑛𝑘=𝑚 𝐶𝑛𝑘 (exp(−(𝛽)𝜆 ))𝑘 . (1 − �𝑒𝑥𝑝(−(𝛽) 𝜆 )�)𝑛−𝑘 � .

(7)

477


If we replace Ri by its expression in terms of damage D, we obtain the equation of reliability based on damage of the non rotating rope 18x7 according to the suggested model. 𝑅(𝐷) = 1 − �1 − ∑𝑛𝑘=𝑚 𝐶𝑛𝑘 (𝑒𝑥𝑝 �−(

∝.𝐷

1−𝐷(1−∝)

𝜆

)� )𝑘 . (1 − (𝑒𝑥𝑝 �−(

∝.𝐷

1−𝐷(1−∝)

𝜆

𝑝

)� ))𝑛−𝑘 �

(8)

For α= 0.53 the figure 3 presents the reliability based on damage of the non-rotating rope 18x7 type according to the suggested model taking into account the following failure criteria 10, 7, 4 and 1 broken strand.

Fig. 3: Reliability in terms of damage of the non-rotating rope 18x7 type taking into account the following failure criteria 10, 7, 4 and 1 broken strand

This figure show that reliability increases gradually as we become less demanding in the failure criterion. In other words, we tolerate a greater number of broken strands. Generally, the suggested model is located between a serial system in which the failure criterion is severe (no strand breakage is tolerated) and a parallel system in which no criterion is took into account (the system remains functional till the breakage of the last strand). For a well defined failure criterion, this model tracks the performance of the nonrotating rope in service and allows ascribing to each level of its strands’ damage the corresponding reliability. 5.

Comparative study of reliability models

In order to highlight the failure criterion which is based on the unacceptable number of broken strands or the minimum threshold of functional strands, we suggest making a comparison of the suggested model with those already existing in literature. A model that considers the wire rope as a series-parallel system [9], another one which considers it as a parallel-series-parallel system [11] and a last one which considers it as a series/majority


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logic system [13]. The non-rotating rope 18*7 type will be the subject of this comparison which will be illustrated in the figure 4 for two different failure criterions.

Fig. 4: Reliability in terms of damage according to the four models

According to this figure, we notice that the curves of the two first probabilistic models (series-parallel and parallel-series-parallel) do not change position when changing the failure criterion. Thus, this latter is not taken into consideration in these two models. On the other hand, the series/majority logic model does change position when the failure criterion is changed; therefore this latter is taken into consideration in this multiscale model. However, the representative curves quickly decrease compared to the other curves. In this respect and according to this model, the failure criterion is severe. The suggested model appears to be well suited to the actual situation of using ropes. In fact, it is located between a serial system where the failure criterion is severe and a parallel system where no criterion is taken into account. It is mainly based on a minimum threshold requirement of functional strands in the rope. This model allows the estimation of the non-rotating rope's reliability in terms of its strands’ cumulative damage. 6.

Conclusion

To keep abreast of damage, wire ropes must be monitored. For this, our work is elaborated to apply an analytical model which enables us to track the non rotating rope’s reliability during its operation. The occurrence of unacceptable number of broken strands, In other words, the minimum threshold of functional strands within the rope in service is the action adopted for the rope damage evaluation. The presented model enables us to monitor and to estimate the non rotating rope reliability in terms of its cumulative damage. According


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to the comparative study, this model appears to be well suited to the actual situation of using ropes; it tracks the performance of the non-rotating rope during operation and allows ascribing to each level of its strands’ damage the corresponding reliability. Thus, it helps plan and organize actions of preventive maintenance so as to be able of changing the rope timely. This work has allowed us to shed light on a quiet interesting point. It basically touches on adapting the suggested model for estimating the non-rotating rope’s reliability in terms of the applied stress. This point will be the subject matter of an ulterior work. References [1]

[2] [3] [4] [5] [6]

[7]

[8] [9] [10] [11]

[12] [13]

[14] [15]

[16]

Vorontsov, A., V. Volokhovsky, J. Halonen and J. Sunio, Prediction of operating time of steel wire ropes using magnetic NDT data, OIPEEC Conference, Johannesburg, September 2007. Jomdecha, C., and A. Prateepasen. Design of modified electromagnetic main-flux for steel wire rope inspection. NDT & E International, 2009; 42(1): 77-83. Krešák, J., S. Kropuch, and P. Peterka. The anchors of steel wire ropes, testing methods and their results. Metalurgija, 2012; 51 (4): 485-488. Camo, S., Probabilistic strength estimates and reliability of damaged parallel wire cables. Journal of Bridge Engineering, 2003; 8(5): 297–311. Giglio, M., and A. Manes. Life prediction of a wire rope subjected to axial and bending loads. Engineering Failure Analysis, 2005; 12 (4): pp.549-568. Agnieszka Blokus. Reliability analysis of large systems with dependent components. International Journal of Reliability, Quality and Safety Engineering, 2006; 13 (01): 8187. Chouairi, A., M. El Ghorba and A. Benali. Analytical approach of availability and maintainability for structural lifting cables using reliability-based optimization. International Journal of Engineering and Science (RESEARCH INVENTY), 2012; 1 (1): 08-17. Liu,Z. S., and Z. H. Chen. Reliability Analysis of Prestressed Steel Structure in Regards to Temperature Action. Advanced Materials Research. 2010;Vols. 97-101: 4415-4419. Kolowrocki, K., Asymptotic approach to reliability evaluation of rope transportation system. Reliability Engineering & System Safety. 2001; 71 (1): 57-64. Cremona, C., Probabilistic approach for cable residual strength assessment. Engineering Structures. 2003; 25(3): 377–84. Elachachi, S.M., D. Breysse, S. Yotte, C. Crémona. A probabilistic multi scale time dependent model for corroded structural suspension cables. Probabilistic Engineering Mechanics, 2006; 21(3): pp.235-245. Raoof M., and Kraincanic I. Determination of wire recovery length in steel cables and its practical applications. Computers & Structures, 1998; 68: pp. 445–59. Meksem, A., M. El Ghorba, A. Benali and A. El barkany, Optimization by the reliability of the damage by tiredness of a wire rope of lifting. Applied Mechanics and Materials, 2011; 61: pp. 15-24. Song, J., and A. Der Kiureghian, Bounds on system reliability by linear programming. Journal of Engineering Mechanics, ASCE, 2003; 129(6): pp. 627-636. Mouradi, H., A. El barkany and A. El biyaali, Probabilistic approach to reliability evaluation of lifting wire ropes. ARPN Journal of Engineering and Applied Sciences, 2014; 9 (6): pp. 923-928. Castillo, E., A. Fernandez-Canteli, J.R. Tolosa and J.M Sarabia, Stastical Models for Analysis of Fatigue Life of Long Elements. Journal of Engineering Mechanics, ASCE, 1990; 116 (5): pp. 1036-1049.


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[17]

Castillo, E., and A. Fernandez-Canteli, A unified statistical methodology for modeling fatigue damage. Springer Science & Business Media, 2009.

Houda MOURADI is a PhD candidate at the Mechanical Engineering Department in Science and Technology Faculty, Sidi Mohammed Ben Abdellah University, Morocco. She received her BA in design and mechanical analysis, also a master degree in mechanical engineering (2012) from the same university. Her research interests include structural mechanics and more precisely the mechanical behavior of steel wire ropes. Abdellah El BARKANY is a Professor at the Mechanical Engineering Department in Science and Technology Faculty, Sidi Mohammed Ben Abdellah University, Morocco. He obtained his PhD in Mechanical Engineering about “Contribution to the analysis of the mechanical behavior of lifting wire ropes, modeling and experiments” from University Hassan II of Casablanca in 2007. He received his Engineering degree from the same university in 1997. Ahmed El BIYAALI is a Professor at the Mechanical Engineering Department in Science and Technology Faculty, Sidi Mohammed Ben Abdellah University, Morocco. He obtained his PhD about “Atomic and Laser Physics” in 1987, and “Heat of the buildings “from INSA of Lyon and Sciences and Technology Faculty of Tetouan moroccco in 1995.