Thermal failure mechanism of fiber ropes when bent over sheaves

Abstract

Thermal damage is an important failure mechanism that affects the bending failure of fiber ropes. This is relevant because synthetic fibers often have a relatively low melting point and low thermal conductivity. In cyclic bending over sheave (CBOS), the heat generated by friction and deformation is not conducted rapidly to the external environment, and the temperature of the rope core increases quickly. This higher temperature greatly reduces the mechanical properties of the fiber, thus accelerating the final rope failure. In this paper, evidence of thermal damage in the bending process of a braided synthetic fiber rope is given. The test conditions inducing thermal damage are discussed, including stress level, bending frequency and diameter ratio. The reasons for the heat generation and the dynamic process of heat accumulation inside the rope during CBOS are also discussed. This study aims to provide theoretical and experimental guidance for the design and use of fiber rope.

Keywords

rope;bend over sheaves thermal damage bending fatigue

Fiber rope is a textile structure coupling the soft with the rigid. In the radial direction, this type of rope shows excellent flexibility, which can be easily carried and stored. The flexibility of fiber rope also allows knotting. In the axial direction, the rope is rigid, which makes it an important tool in industry and our daily lives for lifting, bundling, mooring, etc.¹ Steel wire rope is also widely used in production for its high strength and modulus, which make it the main choice for dragging and hoisting. However, in long-distance lifting, the large self-weight of steel wire rope reduces its load capacity. In sea conditions, steel wire rope corrodes, which imposes a further limitation on its use. Over the past decade, with the development of high-performance fibers, high-performance ropes have been widely developed and used, especially high-strength high-modulus polyethylene (HMPE) fiber ropes and aramid ropes. For the same diameter, the strength of these high-performance ropes is more or less equal to that of steel wire rope, but with as little as one-seventh the weight. The higher strength-to-weight ratio allows fiber ropes to replace steel wire ropes in many fields and they have become significant in many engineering applications.² They play a unique role in the ultra-deep sea, aerospace and other areas with lightweight requirements.³

For fiber ropes, initial studies focused on strength. The influence of structures and materials on the strength of rope are determined mainly through experiment,^4,5 theory^6,7 and simulation analysis.^8–12 At present, fiber ropes have been able to satisfy the requirements of many of the application conditions. However, most of the rope in use is required to roll over sheaves, fairleads, etc., so the bending properties have necessarily become the focus of research.

The study of the bending properties of fiber ropes begins with the bending performance of the yarn. Based on the ideal bending yarn model, Backer¹³ first deduced the relationship between the fiber stress, strain and curvature within the rope structure. Costello and Butson¹⁴ built the simplified bending model of wire ropes based on ideal wire rope structures. Based on this model, a series of relationships are discussed, including the stress, strain and relative motion when bending over sheaves. Popper,¹⁵ taking the fiber assembly as a research object, discussed the behavior of strands and fiber in rope when bending over sheaves based on the analysis of friction between fiber and strands. Cornelissen and Akkerman¹⁶ analyzed the nonlinear bending behavior of multi-stranded bundles, and obtained the relationship between the bending moment and bending curvature. Based on the model, a numerical simulation model is given to evaluate the effect of shear stiffness. Burgoyne et al.¹⁷ introduced the results of fatigue testing of parallel aramid bundles under different pre-tensions. Hobbs and Nabijou¹⁸ also conducted cyclic bending over sheave (CBOS) testing of aramid ropes and compared the results with those of HMPE fiber ropes. Ridge et al.^19,20 designed a CBOS testing device for wire ropes and discussed the bending properties of wire rope under tension and bending; the relevant parameters and the failure mechanisms were discussed based on a series of experiments. Nabijou and Hobbs^21–23 discussed the main problems in rope bending over sheaves, including relative motion, the friction in the rope and other parameters that affect the bending fatigue properties of wire ropes. Sloan et al.,²⁴ referring to wire rope research, discussed the effect of fiber material, coating and rope structures on the rope bending fatigue life. In their research, the authors emphasized the importance of thermal effects on bending failure and gave an improved rope bending solution. Based on experiments and theoretical work, Do Vu et al.²⁵ revealed the changes of stress and strain when fiber rope is bent over sheaves using finite element analysis. Michael et al.²⁶ reported that fiber ropes would be an option to replace steel wire ropes in many technical applications and showed the mechanisms to correctly analyze the application-related mechanical properties of fiber ropes and the testing machinery.

Today, we have a more in-depth understanding of fatigue and damage of fiber ropes when bent over sheaves. Reference to wire rope behavior has proved relevant. However, differences in the mechanical and thermal properties between wire and fiber were, to some extent, ignored. This paper focuses on the heat generated by rope bending over sheaves and discusses thermal effects on rope bending fatigue.

Experiments

Experimental device

The bending fatigue tests were conducted on the bending test machine shown in Figure 1, which consists of the drive pulley, testing sheave, hydraulic tensioning system, crank, driveshaft, swing wheel, connectors, etc.

Figure 1.

Illustration of the bending test machine.

As shown in Figure 1, the test rope is wound over the drive pulley and test sheave. The drive pulley has a larger diameter compared to the test sheave. This arrangement would ensure that rope failure occurs at the test section rather than the drive section. The drive pulley is connected to the variable frequency motor via a driveshaft and crank. This makes the drive pulley rotate backwards and forwards. The variable frequency motor controls the test speed, that is, the bending frequency or period. The test sheave connects to the hydraulic tension system, which provides tension on the test ropes.

Sample treatments

The test rope is fitted with two purpose designed resin socket connectors, as shown in Figure 2. The rope is wrapped around the drive and test pulleys and the connectors are bolted together, as shown in Figure 1. For CBOS testing, the rope can be divided into two bending areas. One is the DBZ (double bending zone) and the other is the SBZ (single bending zone). For our bending tests, the thermocouples were placed in the DBZ.

Figure 2.

Fixing method and different bending zones. SBZ: single bending zone; DBZ: double bending zone.

In order to detect the heat generation and distribution during bending over sheave, two thermocouples were placed; one lay on the rope surface where the rope could not contact with the sheave, and the other was in the center of the rope core, as shown in Figure 3. Both of the thermocouples were connected to the temperature tester and linked to the computer to record the temperature changes. They were tightly secured in place by a thin fixed bandage. In order to avoid the influence of the bandage on the thermal conductivity, a thin net-like bandage was employed. The bandage is elastic and porous and would not obviously affect the relative movements and heat diffusion.

Figure 3.

Distribution of the thermal sensors.

Testing parameter setting

Although different from steel wire, synthetic fibers are generally similar to each other in thermal conductivity and heat generation. Consequently, this paper does not discuss the effects of different fiber materials on rope bending properties. All the test samples were made of polyethylene terephthalate (PET) fiber with a breaking stress of 8.0 cN/dtex, breaking elongation of 12.5–14% and a modulus of 121 cN/dtex.

The test ropes were a traditional double braided structure with the diameter of 10 ± 0.01 mm. The sheath was produced with arrangement 1 full 1 empty with floating length of one²⁷ 32 braided structure with a thickness of 1 mm, and the braiding angle was 25°. The rope core is also a braided structure with arrangement 2 full 2 empty with floating length of one and eight strands. The braiding angle is 20°. The groove profile of the test sheave was a “U”. During testing, the diameter ratio (D/d) varied with five diameter ratios: 8:1, 10:1, 15:1, 20:1 and 25:1. The corresponding sheave diameters were 80, 100, 150, 200 and 250 mm.

The bending frequencies were 0.1, 0.2, 0.5, 1.0 and 2.0 Hz, giving bending periods of 10, 5, 2, 1 and 0.5 s, respectively.

During the bending test, there was no added coolant or lubrication.

Before CBOS testing, the breaking strengths of ropes were tested by a universal tester. The pre-tension was applied by the hydraulic tension servo system (see Figure 1) and measured by the tension sensor that lay between the hydraulic tension servo system and the testing sheave. The stress level is the percentage of pre-tension divided by the breaking strength. In this test, six stress levels were selected: 60%, 65%, 70%, 75%, 80% and 85%. The breaking stress and the pre-tension of different stress levels are given in Table 1.

Table 1.

Breaking stress and pre-tension of different stress levels

Stress levels	Breaking	85%	80%	75%	70%	65%	60%
Stress (Pa)	3.16E + 05	2.69E + 05	2.15E + 05	1.61E + 05	1.13E + 05	7.33E + 04	4.40E + 04
Pre-tension (N)	2.48E + 04	2.11E + 04	1.69E + 04	1.27E + 04	8.86E + 03	5.76E + 03	3.45E + 03

All these tests are conducted under an air-conditioned indoor environment, where the temperature is 24 ± 2℃, and the relative humidity is 65 ± 5%.

Results and discussion

Thermal effects on rope morphology

A reflective microscope was employed to observe the morphology of the failed rope strands and fibers, as shown in Figure 4. From Figure 4(a), it can be seen that there was a thin net linking adjacent fiber bundles; the fibers in this area connected two strands. It is not easy to recognize which strand the fibers belong to. When separating these strands, see Figure 4(b), it appears that the net has tightly connected the adjacent bundles, as the net fibers and the bundle fibers have merged with each other. The net has to be broken to separate them. This kind of connection is not a sort of interlacing and the boundary of the net fiber and the bundle fiber is not clear.

Figure 4.

Cross-linking caused by heating (stress level: 70%; testing frequency: 1 Hz, D/d: 20): (a) the condition after the bending test, when a kind of net was formed among different strands; (b) when strands were separated, the adhesion between the fibers can be clearly observed.

The thin net was formed during the bending process. When bending over sheave, there were tiny relevant movements among these strands. These kinds of movements loosen the touched fiber of strands and the loosened strands from adjacent strands merged with each other. At the same time, the high-speed movements heat the fibers and make them sticky. The sticky fibers connected with each other to form the net-like structures. When separating adjacent strands, the net would be observed. This phenomenon shows us that during the CBOS test, the generated heat may partially melt the fibers to form this kind of net.

In order to confirm this, a scanning electron microscopy (SEM) examination was conducted, as shown in Figure 5. Figure 5(a) shows the image of the fiber bundle before CBOS, which illustrates a clear fiber boundary, and that the fiber bundles are loose. After CBOS, it is not easy to observe single fibers; all the fibers are tightly bonded together to form a thick fiber rod with a smooth surface. Based on this comparison, it can be deduced that during CBOS testing, the temperature in the rope was high enough to partially melt the fiber to bond adjacent fibers together to form a fiber rod, which was hard and not easy to bend.

Figure 5.

Morphological changes in fiber bundles caused by thermal damage (stress level: 70%; testing frequency: 1Hz, D/d: 20): (a) before bend testing, the fiber bundles were loose and easily separated; (b) the tested rope before bending fatigue; (c) after bend testing, fiber bundles were changed to a fiber rod with a smooth surface that was hard to bend; (d) the failed rope after bending fatigue.

From the results of macro and micro morphology of the fiber rope before and after CBOS, it can be observed that the boundaries of adjacent fibers are not clear, and the melting phenomenon is obvious. This means that significant heat was generated and accumulated in the inner rope.

Effects of stress level on rope temperature

Figure 6 shows that the rope temperature changes with the stress level, including the surface temperature (T_s), the core temperature (T_c) and the difference between these two temperatures (T_d). All of these three sets of data show a good linear relationship with the stress level. The correlation coefficients of T_c and T_d are more than 0.99, while that of T_c is 0.9795, which is a little smaller. The reason may be that the temperature is also affected by the environment and not only depends on the rope. For the T_s and T_c, the difference is the rate of temperature increase. For the surface temperature, the slope of the fitted line is 2.8, which is smaller than that of the core temperature (7.8). This means that the core is more sensitive to the stress level. For T_d, the slope of the fitted line is 4.9, and the correlation coefficient of T_d is higher, which means that this is steadier and mostly depends on the material and the structure of the braid.

Figure 6.

The temperature of the different parts of the rope varies with the stress level (bending period = 2 s).

At the lowest stress level (60%, 4.40E + 04 Pa), the temperature difference between the core and sheave is small, about 5–10℃. With increasing stress levels, the difference becomes larger; when the stress level approaches 85% (2.69E + 05 Pa), the temperature difference is almost 40℃ and the core temperature increases to 80℃. This may greatly impair the mechanical properties of some fiber ropes.

The stress level plays a role in two ways: friction and heat due to deformation. Under a relatively low stress level, the stress has little influence on the deformation heat; instead, it mainly affects the core temperature due to friction. The sheath, being exposed to the environment, easily exchanges the heat with the environment, so when the stress level is small, the temperature of the sheath mainly depends on the temperature of the environment. For the core, due to friction among the strands and fiber, some heat is generated, so the core temperature is a little higher than that of the sheath. With rising stress levels, the pressure between the intertwined strands increases, with the same displacement or relative motion. This leads to an increase of friction and heat, and thus the temperature rises. At the same time, the increase of stress means an increase of strain, and the hysteresis loop becomes larger. This means that during one cycle, more heat is generated by strand deformation. This deformation heat also increases the temperature. Both internal friction and deformation lead to an increase in temperature rise. So, under higher stress levels, the core temperature increased greatly. When the core temperature rises, some heat transfers to the sheath, causing a slower temperature rise.

Effects of bending period on rope temperature

During the bending of the rope around the pulley, the change of temperature is a dynamic equilibrium process between heat generation, accumulation and dissipation. All of these processes are time-dependent. Figure 7 shows core and sheath temperatures varying with bending period.

Figure 7.

The temperature of the different parts of the rope versus bending period.

From the figure, it can be seen that the temperature of the sheath rises slightly and settles between 30℃ and 40℃, whilst the temperature of the rope core clearly increases with shorter bending periods, from 40℃ to 70℃. The temperature difference between the core and the sheath is shown in the histogram. It rises from 10℃ at 10 Hz to 30℃ at 0.5 Hz and shows a good linear relationship with the bending period. These results are consistent with the classical temperature rise formula, shown in Equation (1)

\overset{\cdot}{Δ T} = \frac{π f σ^{2} J ″ (f, T)}{ρ C_{p}}

(1)

where f is the bending frequency, σ is the rope stress, ρ is the material density, C_p is the heat capacity of the material and J is a function of frequency (f) and ambient temperature (T).

During the bending process, there are two ways to generate heat: frictional heat and strain heat. Frictional heat is closely dependent on bending frequency. Assuming that the heat generated during each frictional process is constant, the increase in frequency will increase the power of heat generation, that is to say, the speed of heat generation is increased, while the heat diffusion remains the same. This leads to the rise of temperature of the rope core. For the rope sheath, the frictional heat is limited and the temperature rise is mainly due to the temperature rise of the rope core.

Effects of D/d on rope temperature

The diameter ratio is directly related to the relative displacement between the strands during the bending process. The smaller the diameter ratio, the greater the relative displacement between strands, and vice versa. The greater the relative displacement among the strands during the bending fatigue, the more work done by friction. There is more severe frictional damage and frictional heat. Figure 8 shows the changes of temperature at the rope core and sheath with the diameter ratio.

Figure 8.

The variation of the temperature of different parts of the rope with the D/d.

It can be clearly seen that as the diameter ratio increases, the core temperature decreases, while the temperature of the rope sheath changes little. When the diameter ratio is 8:1, the temperature of the rope core reaches 60℃. At this temperature, the properties of some fiber ropes would be severely impaired. With increasing diameter ratio, the temperature of the rope core decreased. At a diameter ratio of 20:1, the temperature of the rope core is steady around 40℃. The temperature of the rope sheath rises slightly with the diameter ratio. This is mainly due to the fact that the temperature of the rope sheath depends mostly on the temperature of the rope core and less on friction.

The diameter ratio influences the relative slip among the strands when bending over sheaves. Keeping other conditions unchanged, the greater the slip, the greater the work done by friction; consequently, more heat is generated, resulting in a higher temperature. When the diameter ratio increases, the relative slip decreases and, correspondingly, the generated heat decreases with a lower core temperature rise. At 20:1 the relative slip is small and the generated heat per cycle is also small and has little effect on the core temperature.

Heat generation and the accumulation process

Heat generation

When a rope bends over sheaves, friction plays two roles. Firstly, it leads to wear in the rope by abrasion and, secondly, the friction generates heat, leading to a temperature rise in the rope.

During the bending process, the rope is subjected to a large external force, and has different strain profiles in different parts of the cross-section. Above the neutral axis, the rope is stretched severely and, when it leaves the sheave, the degree of this stretching decreases. Below the neutral axis, the rope is compressed, and when it leaves the sheave, this part is stretched again. When these two changes happen repeatedly, due to the viscoelastic properties of fiber, there is an energy loss in the process of stretching and compression, most of which transforms into heat and causes an increase of temperature.¹⁰

Due to the low thermal conductivity and the existence of a sheath, the generated heat quickly accumulates in the core of the rope and leads to the temperature rise. Meanwhile, many fibers have a low glass transition temperature and melting point. Both the low thermal conductivity and low glass transition temperature make the fiber vulnerable at higher temperatures and can lead to final failure. As Table 2 shows, compared to steel wire, these common synthetic fibers have a lower glass transition temperature and lower thermal conductivity.

Table 2.

Comparison of thermal parameters of different materials²

Materials	T_m (℃)	T_g (℃)	Thermal conductivity W/(M·K)
Steel	1300–1400	–	132
PET	255–260	67–81	0.26
Nylon	215–220	45–50	0.24
PP	200–210	25–35	0.12

PET: polyethylene terephthalate; PP: polypropylene.

Dynamic process of heat accumulation in the rope

The change of the internal temperature of the rope is a dynamic process. The heat generation includes frictional heat and deformation heat, and the heat generated will diffuse into the environment through the rope structure.

The temperature change during rope bending over sheaves is shown in Figure 9. It can be observed that temperature rises for the rope core and the sheath are different. The whole temperature curve can be divided into four zones. The first one is the initial zone. In this zone, the bending process is just beginning; the generated heat and thermal effects slowly accumulate, and the temperature of the rope core and the sheath are nearly the same. As the bending continues, the heat generated by friction and deformation continue to accumulate, leading to a steady rise of the core temperature. As the bending continues further to a certain degree, the core temperature rises to a plateau and remains steady. During this process, the temperature of the rope sheath also rises but with a relatively lower amplitude. Generally, there is an obvious lag in the temperature change of the rope core and the sheath. The lag mainly depends on the experimental conditions and rope structures.

Figure 9.

Changing curve of temperature of rope components with time (stress level: 70%; testing frequency: 1 Hz, D/d: 20).

Conclusion

This paper studies the fatigue process of fiber rope in bending over sheaves and discusses the thermal damage on the rope together with fiber morphology and the effects of stress level, bending frequency and diameter ratio on the temperature. Based on the above discussion, the following can be concluded.

During the bending process, in addition to mechanical failure caused by friction among strands, friction and deformation lead to an energy loss, and thus heat accumulation and a temperature rise. Due to the lower thermal conductivity and lower glass transition and melting temperatures, the accumulated heat can greatly decrease the properties of fiber rope and accelerate failure. Thus, the thermal failure mechanism is an important consideration for fiber rope fatigue failure.

Stress levels determine the degree of friction and deformation during the bending process. The higher the stress level, the greater the frictional effect and deformation heat, leading to higher temperatures.

The bending frequency plays a key role in heat accumulation. When the bending frequency is low, the generated heat does not accumulate in the rope core, so there will be little obvious rise in temperature. In this case, the thermal failure mechanism would not play an important role for rope failure. Conversely, when the bending frequency is high, even when the stress level is not high and the diameter ratio is not excessive, the generated heat is also accumulated in the rope core due to the low thermal conductivity of synthetic fiber.

The diameter ratio determines the degree of bending and has a direct relationship with relative displacement. A smaller diameter ratio would lead to a greater relative displacement between strands and more heat generated by friction.

The thermal failure mechanism for fiber rope failure is a time-dependent mechanism closely dependent on bending frequency. At lower frequencies, the generated heat has enough time to dissipate and keep the temperature of the rope steady. Abrasion plays the main role for rope fatigue failure. When the bending frequency rises, the heat does not have enough time to dissipate and accumulates in the core. This leads to a rise in temperature and, in this case, the thermal failure mechanism must not be ignored as it may play the main role in the final failure.

Footnotes

Declaration of conflicting interests

The authors declared no potential conflicts of interest with respect to the research, authorship and/or publication of this article.

Funding

The authors received no financial support for the research, authorship and/or publication of this article.

References

McKenna

Hearle

JWS

O’Hear

. Handbook of fibre rope technology, 1st ed. Cambridge: Woodhead Publishing Limited, 2004, pp. 432–432.

Foster

. Advantages of fiber rope over wire rope. J Ind Text 2002; 32: 67–75.

Davies

Forest

Lacotte

. Large synthetic fiber ropes for marine applications - evaluating twisting and bending effects for deep-ocean mooring and handling. Sea Technol 2007; 48: 10–13.

Ridge IML, Wang P, Grabandt O and O'Hear N. Appraisal of Ropes for LNG Moorings. In: Ridge IML (ed.), OIPEEC Conference 2015. Stuttgart: Springer International Publishing, 24-26 March 2015, pp. 216–227.

Davies P, Lacotte N, Kibsgard G, et al. Durability of fibre ropes for deep sea handling operations. In: W.Yeung R, (ed.), 32nd International Conference on Ocean, Offshore and Arctic Engineering. San Francisco: Springer International Publishing, June 8-13,2013, p. 121–132.

Ghoreishi

Cartrauda

Davies

et al.

Analytical modeling of synthetic fiber ropes subjected to axial loads. Part I a new continuum model for multilayered fibrousstructures. Int J Solid Struct 2007; 44: 2924–2942.

Ghoreishi

Davies

Cartraud

et al.

Analytical modeling of synthetic fiber ropes. Part II: a linear elastic model for 1 + 6 fibrous structures. Int J Solid Struct 2007; 44: 2943–2960.

Ning

. Computer-aided modeling of braided structures overbraiding non-cylindrical prisms based on surface transformation. Adv Eng Software 2016; 98: 69–78.

Ning

Potluri

et al.

Geometrical modeling of tubular braided structures using generalized rose curve. Text Res J 2017; 87: 474–486.

10.

Overington

Leech

. Modeling heat build-up in large polyester ropes. Mech Based Des Struc 1996; 54: 56–69.

11.

Seo

Backer

et al.

Structural modeling of double-braided synthetic-fiber ropes. Text Res J 1995; 65: 619–631.

12.

Banfield

Flory

. Computer modelling of large, high-performance fiber rope properties. Compos Struct 1989; 20: 1563–1573.

13.

Backer

. The mechanics of bent yarns. Text Res J 1952; 22: 668–681.

14.

Costello

Butson

. Simplified bending theory for wire rope. J Eng Mech Divis 1982; 108: 219–227.

15.

Popper

. Mechanics of bending of fiber assembies, Manchester City, MA: Department of Mechanical Engineering, Massachusetts Institute of Technology, 1966, pp. 215–219.

16.

Cornelissen

Akkerman

. Analysis of yarn bending behaviour. Mech Based Des Struc 2009; 35: 9–16.

17.

Burgoyne

Hobbs

Strzemiecki

. Tension-bending and sheave bending fatigue of parallel lay aramid ropes. Text Res J 1989; 45: 34–46.

18.

Hobbs

Nabijou

. changes in wire curvature as a wire rope is bent over a sheave. J Strain Anal Eng 1995; 30: 271–281.

19.

Ridge IML. Bending-tension Fatigue of Wire Rope. Reading, UK: University of Reading, 1992, pp. 87–92.

20.

Ridge

IML

Zheng

Chaplin

. Measurement of cyclic bending strains in steel wire rope. J Strain Anal Eng 2000; 35: 545–558.

21.

Nabijou

Hobbs

. Frictional performance of wire and fiber ropes bent over sheaves. J Strain Anal Eng 1995; 30: 45–57.

22.

Nabijou

Hobbs

. Relative movements within wire ropes bent over sheaves. J Strain Anal Eng 1995; 30: 155–165.

23.

Nabijou

Hobbs

. Fatigue of wire ropes bent over small sheaves. Int J Fatigue 1994; 16: 453–460.

24.

Sloan F, Nye R and Liggett T. Improving bend-over-sheave fatigue in fiber ropes. In: Kosters M (ed.), OCEANS 2003 Proceedings. London: Springer, 2-5 May 2003, pp. 1054–1057.

25.

Do Vu T, Durville D and Davies P. Bend-Over-Sheave Of Synthetic Braided Ropes: Approach To Internal Mechanisms Through Finite Element Simulation. In: Cartrauda P (ed.), 11th World Congress on Computational Mechanics. Barcelona, Spain: woodheaded publisher, 12-14 August 2014, pp. 1123–1124.

26.

Michael M, Mammitzsch J and Heinze T. MechaJulynical behavior of high performance fiber ropes in technical applications. In: Gibson G (ed.), 20th International Conference on Composites Materials. Copenhagen Elsevier,19-24 July 2014, pp. 26–38.

27.

Kyosev

. Generalized geometric modeling of tubular and flat braided structures with arbitrary floating length and multiple filaments. Text Res J 2015; 86: 1270–1279.