Predicting the Reliability of Steel Ropes at the Design Stage

Introduction. Steel ropes are critically important load-bearing elements of lifting machines that determine their safety and operational efficiency. According to Rostechnadzor and global industry research, up to 20% of lifting accidents are related to critical rope defects. Rope failure can lead to economic losses due to downtime and disruptions of logistics cycles, as well as man-made consequences. Statistics show that over 5,000 incidents occur annually due to broken traction and load-bearing elements, with approximately 30% having consequences for human life and health [1].

A steel rope is a complex mechanical and technical system that distributes the load between interconnected elements that operate in different conditions and are subject to aging, corrosion, wear, and fatigue damage. This makes it difficult to assess the reliability of the entire system. Existing design methods compensate for uncertainty through significant safety factors. However, practice shows that this approach does not provide the required reliability for modern high-power lifting machines with high work intensity —it does not exclude sudden failures and is economically inefficient. In these conditions, in order to reduce man-made risks and improve operational efficiency, it is necessary to move from the use of stock coefficients and visual control to predictive design and analytics that provide an estimated level of reliability based on predictive failure models.

Research on steel ropes reliability has been conducted for decades and covers the stages of design, production and operation. In 1963, with the support of OITAF and RILEM organizations, the international organization for the study of rope fatigue, OIPEEC, was established.

Modern development of artificial intelligence and digital vision has significantly advanced the issues of predictive analytics of steel ropes. The developed methods and automated digital control systems described in the works of M.N. Khalfin [2][3], A.A. Korotkov [4][5], A.V. Panfilov [6][7] and A.A. Kulchitskiy [8] are being actively implemented in operational practice.

An important stage for the development of predictive design is the updating of ISO 16625, which involves determining the margin coefficient and evaluating fatigue life, taking into account a variety of factors, which marks the transition from simplified calculations to deep modeling of real-world operating conditions.

The complexity of distribution of mechanical properties and loads between the elements is a determining factor in the reliability of a steel rope. The multilayer structure of the rope is hierarchical in nature: the internal elements serve as a support for the external ones. The violation of these supporting links leads to degradation of the rope structure and changes in the working conditions of its elements. Wahid A. [9][10] designates this phenomenon as the effect of “systemic wear” that occurs when core stability is lost.

The basis for the implementation of predictive design is the consideration of the rope as a system. Mouradi H. [11] proposed a method for predicting durability using majority logic, where the key aspect is the mathematical relationship between the probability of trouble-free operation and the degree of accumulated damage. Bassir Y. [12] notes that the analysis of the hierarchical structure makes it possible to transform the failure statistics of the basic elements into an accurate forecast of the reliability of the entire rope. Xia Y. [13] suggests conducting finite element analysis at three hierarchical levels: at the micro level — wire contact, at the meso level — the interaction of strands, and at the macro level — the behavior of the entire rope. This approach takes into account local friction and intermittent sliding during bending, described in the Han Y. model [14], as well as the loss of cross-sectional area from wear over time, considered by Salleh S. [15]. Studies by Peng Y. [16] and Xu C. [17] focus on the processes of internal friction and inter-wire wear, confirming that the degradation rate directly depends on lay parameters and lubrication rate. V.P. Golovin [18] demonstrates the effectiveness of synthetic thickeners of rope lubricants, and Peng H. [19] emphasizes the need to take into account the degradation of lubricant properties as a key factor in rope durability. V.Yu. Volokhovsky [20] examines the effect of thermal cycles on the ropes of metallurgical cranes and suggests a transition from deterministic calculations to risk assessment as the probability of a random event in which the diagnostic indicator of the rope exceeds the established rejection level.

The analysis of modern research shows that, despite the in-depth study of certain aspects of rope operation, the issue of assessing their reliability as machine elements remains insufficiently studied. The gap between theoretical degradation models and practical design methods prevents the full realization of the potential of the predictive approach. As a result, reliability rationing becomes an urgent task, requiring the establishment of quantitative normative values and the selection of adequate evaluation criteria. There is an objective need to create comprehensive reliability forecasting models that take into account the design features of the rope, the expected operating conditions and the requirements of regulatory and technical documentation.

The aim of this research is to develop a model for predicting the reliability of a steel rope, taking into account the multicomponent structure, operating conditions and requirements of regulatory and technical documentation (using the example of a two lay rope GOST 7668 as part of gantry crane mechanisms).

Research objectives:

• perform the analysis of the requirements of regulatory and technical documentation and determine the limits of the steel rope's operability;
• determine boundary values of indicators corresponding to the transition of the system to the limiting state, taking into account the dominant mechanisms of destruction;
• integrate regulatory criteria into the reliability forecasting model;
• develop a comprehensive mathematical model for reliability assessment.

Materials and Methods. The research was based on the proposed hierarchical decomposition of steel rope reliability by degradation levels and algorithmization of the “weak link” principle for sequential systems according to the principles of calculating the probability of trouble-free operation of elements of lifting cranes RTM 24.090.25–76. The object of the simulation was a two lay steel rope with a diameter of 27 mm. 6×36(1+7+7/7+14)+1 WS FC according to GOST 7668–80 as part of lifting mechanism of Kirovets gantry crane 16/20 (Fig. 1, Table 1).

Reliability was accepted as an indicator of the failure-free operation of a steel rope in accordance with GOST R 27.102–2021¹. The choice of the indicator was due to the unmaintainability of the rope as a separate element of the lifting machine and the continuous nature of the processes of corrosion of wires and aging (decomposition) of the core, which could occur regardless of the intensity of operation.

To establish the limits of the working capacity of a steel rope, an analysis of RD ROSEC 012–97² rejection standards was performed, which took into account defects caused by natural wear and aging of the rope material (Table 2) and the acceptable number of wire breaks, taking into account the wear rate and the classification group (operating mode) of the mechanism (Table 3). The analyzed defects were systematized by the nature of degradation: A — wire breaks, B — wire wear, C — fiber core degradation. Taking into account the discrete nature of the damage accumulation process, the calculated values of the number of breaks were rounded upward. Critical defects that occurred instantly, such as creases, kinks, electric arc damage, lightning, fire, etc., were excluded from consideration.

To synthesize the forecasting model, we decomposed the steel rope's reliability by degradation levels and determined generalized limiting states for groups A, B, and C (Table 4). We implemented a hierarchical relationship between degradation levels through a system of dynamically dependent parameters. In this system, the predicted values of wear and deformation at the current time step acted as variable boundary conditions for evaluating the subsequent states of the system. The method of calculating losses of metal cross-section was based on a combined consideration of mechanical wear of wires and atmospheric corrosion. We introduced the parameters of medium aggressiveness into the model as an additive degradation factor that determined the rate of decrease in wire diameter in the outer layer of the rope.

The methodology for substantiating the generalized limit state for group B was implemented through the calculation of the total loss of metal section area as a function of surface wear of wires, taking into account the dynamic breakage threshold N_lim, which determined the point of joint achievement of the limit state according to criteria B1 (wear) and B2 (loss of cross-sectional area) (Fig. 2, 3).

The dynamically changing threshold for the acceptable number of N_lim breaks was determined based on the approximation of discrete dependencies presented in Table 3 (Fig. 4, 5).

To verify the results, a comparative analysis of the predicted reliability curves with the calculated value of the median service life for M6 operating mode according to ISO 16625 was applied. Mathematical data processing was performed using MS Excel 14.0.4760.1000 and Mathcad 14.0.0.163. The dependencies were approximated by a polynomial function of 3–4 orders of magnitude; the coefficient of determination was in the range 0.9425–0.9998.

Results. During the study, we obtained the dependencies of the total loss of cross-sectional area of the rope metal part on the amount of surface wear of wires in the outer layer (Fig. 2, 3). Based on the curves obtained, we found that, considering the contribution of the dynamic number of wire breaks N_lim and formal compliance with regulatory requirements for wear (Table 2), a critical threshold of 17.5% (defect B2) was achieved with surface wear values less than 40% (defect B1).

As a result of hierarchical decomposition of rope reliability by degradation levels, a generalization of regulatory defects was performed (Tables 2, 3) and a selection of mathematical models for predicting reliability was made. The formulated generalized criteria for limit states and the corresponding calculation apparatus were systematized and described in Table 4.

Based on the data in Table 3, analytical dependencies of the acceptable number of N_lim wire breaks on the degree of surface wear and corrosion of the outer layer of wires (expressed as a percentage of the nominal diameter of wires) were obtained, determining the dynamically changing limits of rope operability in the reliability model (Fig. 4, 5).

To determine the acceptable number of wire breaks N_lim(х), we obtained the following expressions for ropes:

– long lay in sections 6d in M1-M4 mode and 30d in M5-M8 mode, as well as cross lay in section 6d in M5–M8 mode:

(1)

– cross lay in sections 6d in M1-M4 mode and 30d in M5-M8 mode, as well as long lay in in section 30d in M1-M4 mod:

(2)

– cross lay in section 30d in M1–M4 mode:

(3)

– long lay in section 6d in M5–M8 mode:

(4)

Based on reasonable criteria for limiting conditions (Table 4), an algorithm for predictive reliability modeling has been developed, presented as a flowchart in Figure 6.

Description of the model. The following is an analytical description of the reliability assessment models according to criteria groups C, B, and A according to Table 4 and the flowchart in Figure 6.

I. Reliability assessment in accordance with the limiting condition of group C:

(5)

where T_В — fiber core characteristic resource, h; β_В — parameter that determines the intensity of core aging;

where k_дег. — constant of the core degradation process, h–1; E_В — core elasticity modulus, MPa; q — radial pressure of the strands, MPa.

Deformation of the core under load at time t according to the Kelvin-Voigt model:

where η_В — dynamic viscosity of the rope lubricant, MPa⸱h; t — estimated time, h.

To account for changes in the structural stability of the core when assessing reliability according to the limiting state of group B, it is proposed to determine the coefficient of wear intensification in case of structural instability of the core:

where ε_lim — maximum acceptable deformation.

II. Reliability assessment in accordance with the limiting condition of group B:

(6)

where k_A — coefficient of acceptable loss of metal section of the rope; А₀ — nominal cross-sectional area of the metal part of the rope, mm²; ΔА_∑ — cumulative loss of the cross-sectional area of wires of outer and inner layers, mm²; E[ΔА_∑] and D[ΔА_∑] — mathematical expectation and variance of a random value of cumulative loss of cross-sectional area of the metal part of the rope.

The mathematical expectation of the cumulative loss of cross-sectional area of the metal part of the rope is determined by the expression:

,

where ΔА_внеш.(t) and ΔА_внут.(t) — cumulative loss of cross-sectional area of outer and inner wires at time t.

and ,

where Z_внеш. and Z_внут. — number of wires in outer and inner layers; f_ΔA(h(t)) — function that determines the area loss depending on the amount of wear h(t).

Mathematical expectations of the amount of wear on outer and inner wires are determined by the Archard model of wear kinetics with a corrosion additive:

and ,

where K_ω — coefficient of wear rate (depends on the conditions of friction and lubrication); p — average contact pressure in the “wire — block groove” pair, MPa; υ — average relative sliding speed of the rope in the groove, mm/h; υ_кор. — average corrosion rate for a specific category of medium, mm/h; Н — hardness of the wire material, MPa; К_f — coefficient of fretting wear of wires (determined from reference data); К_стр. — coefficient of wear intensification with structural instability of the core; σ_кон. — contact stress between the wires inside the strand, MPa; δ — amplitude of wire slippage during bending, mm; h_внеш.(t) — amount of wear on the outer wires, mm; h_внут.(t) — amount of wear on the inner wires, mm; t — estimated time, h.

The average corrosion rate for a specific category of medium is proposed to be determined according to GOST ISO 9226³.

The dispersion of the cumulative cross-sectional area losses of the metal part of the rope is determined by the expression:

Accordingly, the expression includes variances of the random value of the loss of the cross-sectional area of the outer and inner wires:

and ,

where ν_внеш. — coefficient of variation of loss of the cross-sectional area of outer wires; ν_внут. — coefficient of variation of loss of the cross-sectional area of inner wires.

As well as the covariance matrix:

where ρ — correlation coefficient.

III. Reliability assessment in accordance with the limiting condition of group A. The model assumes the use of Poisson distribution, where the intensity of defects is modeled by Weibull's law and increases with wear:

(7)

where N_lim — safety threshold for the number of breaks (decreasing with wear); k — accumulated number of breaks over time; Λ(t) — mathematical expectation of the number of breaks at time t; n_пр — number of strands; t — estimated time, h.

To take into account the predicted wear when estimating the probability of trouble-free operation under limiting conditions of group B, you should determine A(t) using function f_ΔA(h(t)) (see part II), and the maximum number of breaks N_lim using dependencies 1 – 4, assuming х = 100 (А₀ – A(t)) / А₀.

The frequency of breaks, considering the accumulation of fatigue damage:

where β_A — shape parameter that determines the wear rate; η(A(t)) — scale parameter that determines the resource, h.

The scale parameter defines the following dependency:

where m — indicator of sensitivity of the characteristic resource to overstress (indicator of the angle of inclination of the fatigue curve); η₀ — characteristic resource of the new rope with nominal cross-section А₀, h; А₀ — nominal cross-sectional area of the steel rope wire without wear, mm²; A(t) — cross-sectional area of the steel rope wire at time t, mm².

The scale parameter assumes that the equivalent stress in the wire section increases as the wire cross-sectional area decreases:

,

where S — equivalent load on the wire, N.

It should be noted that this model uses the dependence of the resource on the cross-sectional area, which corresponds to the Weller model:

.

Since stresses σ(t) are inversely proportional to cross-sectional area of wire A(t), the following expression can be derived:

In this case, indicator of the inclination angle of fatigue curve m in this model is equivalent to the indicator of the slope of the fatigue curve of the wire material.

where N_циклов — number of cycles before cracks appear, ω — frequency of operation — the number of work cycles per hour, h⁻¹; n_б — number of blocks; k_п — multiplicity of work per cycle.

To predict the number of cycles before the appearance of cracks N_циклов, it is proposed to use the empirical formula of Professor K. Feirer:

where D/d — ratio of the diameter of the carriage rollers to the diameter of the rope. S — rope tension, N; R₀ — marking group of wire strength, N/mm²; b₀, b₁, b₂, b₃ — empirical constants that take into account lay density and the shape of wires (b₀ = 2.634; b₁ = 4.375; b₂ = –1.72; b₃ = –0.4).

According to the proposed model, the predicted rope failure is a consequence of the parallel development of several degradation mechanisms. Despite the fact that the processes of wear, corrosion and fatigue occur simultaneously in the rope, the principle of sequential connection of the system elements is embedded in the structure of the model. The model assumes that the system's performance stops when any of the three established failure criteria is reached — the “weak link model”. The mutual influence of degradation processes in the proposed model is realized through dependent parameters and coefficients. Therefore, the overall reliability of a steel rope is determined by the probability of trouble-free operation of a sequential system with dynamically dependent parameters:

(8)

where P_В(t) — probability of failure free operation within a given time interval t according to the criteria of group C; P_Б(t|В) — probability of failure free operation within a given time interval t according to the criteria of group B, taking into account the coefficient obtained based on the forecast of the model of group C; P_А(t|Б) — probability of failure free operation within a specified time interval t according to the criteria of group A, taking into account the forecast of the dependent parameters of models of Group B.

Example of calculation and verification of the model. As an example, the calculation of the probability of trouble-free operation of a steel rope was performed during operation as part of the lifting mechanism of the Kirovets gantry crane 16/20 (operating mode group M6). The calculated parameters of the rope and operating modes are presented in Table 5. Based on the calculation results, a graph of the dependence of the probability of trouble-free operation on time is shown in Figure 7.

For verification, the graph shows the calculated value of the median service life of a steel rope according to ISO 16625 — T_М6 = 3200 hours for a given operating mode M6.

Discussion. The analysis of the results shows that the proposed set of models consistently bridges the gap between the theoretical assessment of reliability and operational documentation regulations. A key feature of the proposed interpretation of failures is the consideration of the synergetic interaction of several degradation mechanisms that form an integral picture of wear and damage. When modeling rope survivability, the central methodological issue is the contradiction between the physical nature of the processes and the mathematical scheme used. Within the framework of the proposed concept, the predicted rope failure is interpreted as the result of the parallel development of several degradation mechanisms — wear, corrosion and fatigue — despite the fact that the mathematical model purposefully includes the principle of sequential connection of elements (“weak link model”). Such a statement is justified by the fact that achieving any of the limiting criteria leads to a loss of performance of the system as a whole.

An essential feature of the model is the realization of the mutual influence of degradation processes through a system of dependent parameters and coefficients. Rheological degradation of the core and wear kinetics change the stress-strain state of wires, thereby modifying the rate of accumulation of fatigue damage and the redistribution of local loads. As a result, the overall reliability of the rope is determined by the probability of failure-free operation of a sequential system with dynamically dependent parameters, which provides higher forecast accuracy compared to additive approaches that ignore interprocess communications.

Unlike common studies, where degradation factors are treated as independent variables, this paper implements the concept of dynamic parameter dependence, reflecting the actual connectivity of mechanisms. The applicability of the developed apparatus as an analytical tool for design calculations of rope reliability in order to prevent critical defects and optimize design solutions is shown. At the same time, the heterogeneity of the apparatus used complicates the assessment of the total error by standard methods, which necessitates the development of a special reliability criterion that takes into account the particular errors of each component of the model and their possible correlation.

Conclusion. In the course of the research, we developed a comprehensive predictive mathematical model for steel rope reliability, which describes the combined effects of wire breaks, wear, atmospheric corrosion, and core rheological degradation. The hierarchical decomposition of rope reliability allows us to justify the calculation scheme with interdependent parameters and take into account synergistic effects of degradation, reducing the risk of sudden failure. The proposed calculation apparatus incorporates regulatory requirements into the forecasting process. Verification confirmed the consistency of predicted reliability curves with calculated values of median resource according to ISO 16625. At the same time, estimated median resource turned out to be 37% more conservative compared to traditional methods. The scope of applicability is limited to the assessment of the reliability of double lay ropes with an organic core according to GOST 7668–80 at the design stage and assumes the availability of statistical parameters of degradation models, as well as data on the strength and load of rope elements. Further research will focus on the development and experimental verification of models for ropes of different design groups, taking into account the specifics of operation, and their application in engineering practice for the reasonable selection of parameters and optimization of maintenance regulations.