Finite element analysis of hoisting rope and fretting wear ...

Engineering Failure Analysis 27 (2013) 173–193

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Engineering Failure Analysis

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Finite element analysis of hoisting rope and fretting wear evolutionand fatigue life estimation of steel wires

Dagang Wang a,c, Dekun Zhang b,⇑, Songquan Wang b, Shirong Ge a

a School of Mechatronic Engineering, China University of Mining and Technology, Xuzhou 221116, Chinab School of Materials Science and Engineering, China University of Mining and Technology, Xuzhou 221116, Chinac Technical Mechanics, University of Wuppertal, Wuppertal 42119, Germany

a r t i c l e i n f o a b s t r a c t

Article history:Received 2 March 2012Received in revised form 21 August 2012Accepted 23 August 2012Available online 13 September 2012

Keywords:Finite element analysisHoisting ropeSteel wireFretting wear evolutionFatigue life prediction

1350-6307/$ - see front matter � 2012 Elsevier Ltdhttp://dx.doi.org/10.1016/j.engfailanal.2012.08.014

⇑ Corresponding author. Address: School of MaterChina. Tel.: +86 516 83591918; fax: +86 516 83591

E-mail address: [email protected] (D. Zhang

This paper is concerned with the finite element analysis of hoisting rope and three-layeredstrand for the exploration of fretting fatigue parameters and stress distributions on thecross-section. Also, the Archard’s wear law based evolution of fretting wear depth of wirescrossed at different angles and implications to fatigue life estimations of fretted wires werepresented. The results show that different wires in the rope or strand and distinct materialmodels in the analyses both induce different stress distributions and fretting fatigueparameters. The predicted fretting wear depths of wires show good agreement with exper-imental results.

� 2012 Elsevier Ltd. All rights reserved.

1. Introduction

Ropes, due to their capacity to support large axial loads with comparatively small bending or torsional stiffness, havebeen widely employed in various engineering applications such as cranes, lifts, aerial ropeways, ski lifts and mine hoists[1,2]. During mine hoisting, the ropes are subjected to axial cyclic stretching load and cyclic bending load, which resultsin the microscopic motion of individual wires against their neighboring wires in the rope, i.e. fretting wear between contact-ing wires. The interaction of fretting wear and cyclic load causes the crack initiation, propagation and final fracture, i.e. fret-ting–fatigue, which accelerates the failure of the rope [3–5]. In order to prolong the rope’s service life, the rope is lubricatedto reduce the coefficient of friction between friction wires. However, the lubricant in the rope is not capable to withstandhigh operating pressures and then to flow back after a large number of load cycles, which results in the contact areas exposedto local high pressure and thereby the induced accelerated fretting damage [6]. The contacting wires in the rope thus expe-rience approximately the frictionless elastic state at the beginning and the frictional elastic–plastic state during the remain-ing service life. Therefore, in order to explore the failure mechanisms of the rope in service, it is significant to investigate thestate of stress and inter-wire displacement of the rope during different stages of the service life, establish mathematicalmodels of fretting wear evolution and to discuss the fatigue life estimation of fretted wires.

Due to the increasing demand in predicting the rope behavior, several theoretical models of strands or ropes have beenbuilt in which approximations and assumptions were made [7–10]. With the development of the finite element method andthe associated computing capacity, progress has been made in the analysis of rope or strand behavior which takes into ac-

. All rights reserved.

ials Science and Engineering, China University of Mining and Technology, Daxue Road, Xuzhou 221116,916.).

http://dx.doi.org/10.1016/j.engfailanal.2012.08.014

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http://dx.doi.org/10.1016/j.engfailanal.2012.08.014

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Nomenclature

CW, IW, OW central core wire, single helical wires in inner or outer layers in the straight strandOCW, OIW, OOW single helical core wire, double helical wires in inner or outer layers in the helical strandXi, Yi, Zi coordinate values along the centerline of the single helical wire in the ith layerri, ai helix radius or helical angle of the single helical wire in the ith layerXoj, Yoj, Zoj coordinate values along the centerline of the double helical wire in the jth layerrOj, aOj helix radius or helical angle of the double helical wire in the jth layerU1, U2, U3 translations in R, h or Z directions in the cylindrical coordinate systemUR1, UR2, UR3 rotations about R, h or Z directions in the cylindrical coordinate systeme strand axial strainF normal contact load between contacting wiresCPt sum of contact pressure of all contact nodes on the slave surfaceAs total contact area of the slave surfacen number of contact nodes on the slave surfaceW total wear volumeS total accumulated displacementDx relative displacementN fretting wear cyclesk Archard’s wear coefficientu crossing angle between contacting wiresp average contact pressurea variable radius of the contact area during fretting weara0 initial contact radius of the contacting wireR1, R2 radius of the upper or lower contacting wire in the case of u = 90�, R1 = R2

D1, D2 diameter of the upper or lower contacting wire, D1 = D2

CF contact shape dependent constanth1, h2 wear depth of the contacting wire in cases of u = 90�or u – 90�R3, R4 radius of the upper or lower contacting wires in the case of u – 90�, R3 = R4

C, m material constants in Paris equationNf fatigue cyclesDK stress intensity factor rangeh crack depth during crack propagationRs stress ratio during the fatigue testKc critical stress intensity factorKmax, Kmin maximum or minimum values of stress intensity factor in one fatigue cycleDKth crack growth thresholdhf crack depth at the fracture failure of steel wireY(c) normalized stress intensity factorc ratio of crack depth to the wire diameterr axial stress in one fatigue cyclermax, rmin maximum or minimum axial stresses in one fatigue cycle

174 D. Wang et al. / Engineering Failure Analysis 27 (2013) 173–193

count the contact deformation, friction, plasticity and residual stress. Nawrocki and Labrosse [11] dealt with axially loadedtwo-layered straight strand and found inter-wire pivoting governs the cable response. Jiang et al. [12,13] proposed concisefinite element models for 1 � 7 wire strand under axial extension and pure bending load, respectively, and studied the con-tact stress between wires, wire radial displacement and global response of the strand. Páczelt and Beleznai [14] developedthe p-extension concept in elastic finite element method and analyzed the two-layered strand subjected to tension, torsionand bending loading, respectively. Jiang et al. [15] and Shibu et al. [16] investigated the global behavior, i.e. axial force andreaction torque, of the three-layered strand under axial extension. Erdonmez and Imrak [17] modeled 6 � 7 stranded ropeand presented the global responses of the rope and axial forces of its internal wires. However, from the literature study men-tioned above, there are few studies on finite element analyses of the 6 � 19 wire rope or three-layered strand under axialextension from the viewpoint of determination of fretting parameters (normal contact load, relative displacement and axialforce).

Considering fretting wear of steel wires, Zhang et al. [18,19] conducted fretting wear tests of steel wires under dry fric-tion, lubrication and different corrosive mediums, respectively, and addressed the effect of various fretting parameters onthe fretting wear behavior of steel wires. Cruzado et al. [20] found that wear volume and wear rate of steel wires increasedwith increasing contact load and displacement amplitude. Argatov et al. [21] and Urchegui et al. [22] examined the weardegradation of the rope subjected to cyclic bending over a sheave and discussed the relationship between wear severity

D. Wang et al. / Engineering Failure Analysis 27 (2013) 173–193 175

and the sheave diameter. Giglio and Manes [23] reported several analytical formulations to estimate the state of stress in theinternal and external wires of a rope. It’s found that previous efforts have been focused on fretting wear tests of steel wires invarious conditions and theoretical relationship between wear degradation of the rope and the sheave diameter. It is to bedeplored that few studies on the evolution model of fretting wear depth of steel wires crossed at different angles have beenreported in the literature.

Over the last years, several advances have been made in the understanding of fatigue fracture behaviors of steel wires.Dieng et al. [24] revealed the impact of lubrication and zinc coating on the fretting–fatigue behavior of high strength steelwires. Périera et al. [25,26] pointed out the impact of water and sodium chloride on the fretting–fatigue behavior of bridgecable wires. Llorca and Sanchez–Galvez [27] and Beretta and Matteazzi [28] both stated that the fatigue limit and fatigue lifeof steel wires were highly dependent upon their surface condition and on the microcrack growth rate. Mahmoud and Fisher[29] explored the effect of stress corrosion and hydrogen embrittlement on the degradation of bridge cables. Li et al. [30]studied fatigue crack growth of suspension bridge wires utilizing multiscaling and mesoscopic fracture mechanics. To date,however, fatigue life estimation of fretted wires has not been reported.

The objective of this study is to explore the state of stress and fretting fatigue parameters of steel wires employing finiteelement analyses of 6 � 19 + IWS rope or three-layered strand under axial extension. Also, the evolution model of frettingwear depth of steel wires crossed at different angles during fretting wear as well as implications to the fatigue life estimationof fretted wires will be discussed. The remainder of this paper is organized as follows. Section 2 presents the finite elementmodels for 6 � 19 + IWS rope and (1 + 6 + 12) straight stand. The global behavior, i.e. axial force and reaction torque, and dis-tributions of von Mises equivalent stress, axial stress and wire displacement on the cross-section of the rope are shown bythe elastic analysis of the rope. Also, the wire axial force and relative displacement between neighboring wires in the ropeare obtained. Then the global behavior and distributions of various stresses on the cross-section of three-layered strand arepresented using both elastic and elastic–plastic material models. In the latter case, local contact, friction, plastic deformationand the exact boundary conditions have all been considered. Meanwhile, the sub-model with fine mesh near to contact re-gions is utilized to investigate the accurate results. Furthermore, the variations of wire axial force and contact load betweenneighboring wires with the strand axial strain in both finite element analyses are given. In Section 3, based on the Archard’swear law and theory of contact mechanics, the evolution models of fretting wear depth of steel wires crossed at differentangles have been developed, which correlates the fretting wear depth of contacting wire with fretting parameters. Fatiguelife estimations of fretted wires based on the crack growth equations and linear elastic fracture mechanics are discussed inSection 4. In this section, we introduce the one-regime model of normalized stress intensity factor for crack growth of steelwires in tension.

2. Finite element analyses of 6 � 19 + IWS rope and (1 + 6 + 12) staight strand

2.1. Geometry

6 � 19 + IWS wire rope consists of a straight strand surrounded in a symmetrical way by six outer helical strands. Thestraight strand consists of a central core wire (CW) surrounded in a symmetrical way by six identical single helical wiresin the inner layer (IW) and by six pairs of single helical wires in the outer layer (OW1, OW2). The outer helical strand consistsof a central single helical core wire (OCW) surrounded by six double helical wires in the inner layer (OIW1–6) and by twelvedouble helical wires in the outer layer (OOW1–12). The structure and geometry parameters of the rope and strands areshown in Figs. 1 and 2, and Table 1, respectively.

It is convenient to describe the structure of the rope using a Cartesian coordinate system (X, Y, Z), with the Cartesian frame{eX, eY, eZ} where eZ is the rope axis. The location along the centerline of the single helical wire in the ith layer is given by [31]

Xi ¼ ri cosðhiSÞYi ¼ ri sinðhiSÞZi ¼ hiSri tanðhiÞ

ð1Þ

Fig. 1. Schematic of geometic parameters of the rope.

Fig. 2. 1/6 of pitch length of the rope structure.

Table 1Geometry parameters of the rope.

Strand Strandangle (�)

Pitch length(mm)

Diameter(mm)

Wire layer Wire diameter(mm)

Layangle (�)

Helix radius(mm)

Pitch length(mm)

6 � 19 + IWS Rope

Straight – – 5 Core wire 1 – – –Inner layer 1 6.64 1 54Outer layer 1 13.1 2 54

Helical 16.2 108 5 Core wire 1 – – –Inner layer 1 6.64 1 54Outer layer 1 13.1 2 54


where ri represents the helix radius of the wire in the ith layer, ai denotes the helical angle of the wire in the ith layer and hiS

equals the sum of hiS0 and hS. The parameter of S in the subscript represents the wire number of wires in the ith wire layer.hiS0 is the single helix phase angle and hS defines the rotational angle of the centerline of the wire around Z axis (rope axis)relative to X axis.

The location along the centerline of the single helical core wire in the outer helical strand can be obtained from Eq. (1).The location along the centerline of the double helical wire in the jth layer is written as [17]

XOj ¼ XCW þ rOj cosðhOjSÞ cosðhCWÞ � rOj sinðhOjSÞ sinðhCWÞ sinðhOjÞYOj ¼ YCW þ rOj cosðhOjSÞ sinðhCWÞ þ rOj sinðhOjSÞ cosðhCWÞ sinðhOjÞZOj ¼ ZCW � rOj sinðhOjSÞ cosðhCWÞ

ð2Þ

where XCW, YCW and ZCW are coordinate values along the centerline of the single helical core wire in the outer helical strand,rOj is the helix radius of the wire in the jth layer of the outer helical strand, hCW corresponding to the single helical core wire isa similar angle with hiS in Eq. (1), aOj is the helical angle of the wire in the jth layer and hOjS is given by hOjS = mhCW + hOjS0. Theparameter m is estimated by m = hs/(hw cosaOj), where hs and hw are the pitch lengths of the outer helical strand and the dou-ble helical wire in the jth layer, respectively, and hOjS0 is phase angle of the double helical wire.

2.2. Finite element model

2.2.1. Generation of the rope and strandThe core wire and the single helical wires in inner and outer layers in the straight strand are easily generated using extru-

sion and extrusion with twist, respectively, in Abaqus/CAE. Due to the complex wire geometry of the outer helical strand,Pro/EngineerWildfire 5.0 is employed for its construction. The centerlines of single and double helical wires of an outer heli-cal strand are generated by Eqs. (1) and (2). The solid wires are constructed by sweeping the wire cross-sections along cor-responding centerlines to compose the outer helical strand. Then all wires in the outer helical strand are imported intoAbaqus/CAE and assembled with three wires in the straight strand. To complete the construction of 6 � 19 + IWS rope, singlehelical wires in the straight strand and all wires in the outer helical strand are radially patterned along the rope axis, respec-tively. Taking advantage of the symmetric feature of the rope (straight strand), only 1/6 of pitch length of the rope (straightstrand) is chosen to build the finite element model. The coordinate system chosen is such that Z-axis is parallel to but doesnot coincide with the rope symmetric axis, and the origin is located on the front cross-section plane of the rope (straightstrand) model (fixed end, Z = 0). The opposite back cross-section plane of the rope (straight strand) model is defined asthe tensile end (Z = 1/6 of pitch length of the rope or straight strand). In order to precisely explore the stress states nearthe local contact region, sub-modeling technique is employed because more accurate results can be obtained with very small


computation cost. Fig. 3b shows the sub-model of the straight strand of which the length is 1 mm. It is obviously observedthat the meshes are fine near to the local contact regions between neighboring wires.

2.2.2. Material properties and structural discretizationA commercial finite element analysis program (ABAQUS) was employed throughout. All wires were made of a homoge-

neous and isotropic material. 6 � 19 + IWS rope was analyzed using the elastic material model due to the complex structureand CPU limitations, while the global model and sub-model of the straight strand were computed employing both elastic andelastic–plastic material models. The Young’s modulus is 203 GPa and Poisson’s ratio is 0.3. The actual stress–plastic straindata for wire material after yielding is shown in Table 2, which was obtained by the uniaxial tension test. Three-dimensionalsolid linear brick elements were used for the structural discretization. The finite element meshes of the rope and straightstrand are shown in Fig. 3. The rope is meshed using 51456 elements and 71422 nodes; the global model and sub-modelof the straight strand are meshed using 10944 and 38830 elements, respectively, and with 15067 and 49720 nodes, respec-tively. Each node has three degrees of freedom, i.e. translations in X, Y and Z directions (U1, U2, U3).

2.2.3. Interaction propertiesThe contact surface interactions were defined between contacting wires via the contact pair approach in ABAQUS, which

used the master-slave algorithm and finite sliding slip mode to enforce the contact constraints. Only normal contact prop-erties were defined during elastic analyses of the rope and straight strand, while the elastic–plastic analysis of the straightstrand employed the penalty algorithm to deal with the local contacts between all possible contact surfaces (Penalty, frictionof coefficient = 0.15 [32]).

2.2.4. Constraints and boundary conditionsReference points (3) and 2 (4), located at the rope (straight strand) centerline, were established at a distance away from

the front and back cross-section planes of the rope (straight strand), respectively, to couple the nodes on the corresponding

(a) 6×19+IWS wire rope

(1) Global model (2) Sub–model

(b) (1+6+12) straight strand

Fig. 3. Finite element mesh.

Table 2Data for wire material after yielding.

Actual stress (MPa) Plastic strain

1450 01498.61 0.0001071570.51 0.0007391634.2 0.001581657.56 0.002281680.94 0.002901690.94 0.003591696.52 0.00429


cross-section. The kinematic (distributing) coupling mode was employed for the rope (strand) analysis in order to achievethe same displacement between the reference point and corresponding coupled nodes. All degrees of freedom of referencepoint 1 (3) are constrained, while reference point 2 (4) is restrained not to rotate about U3 direction, i.e. UR3 = 0, and hastranslational degree of freedom along U3 direction, i.e. U3 = e, to apply the axial extension strain. The continuum distributingcoupling used is softer than the kinematic distributing coupling and allows the relative deformation between coupled sur-face nodes.

In the straight strand analysis, the length of the strand is assumed sufficiently long for the clamping conditions to be neg-ligible. To ensure the symmetric feature of the strand, precise boundary conditions are maintained by using the constraintequations which relate the displacements of the corresponding nodes on front and back cross-sections of the straight strandmodel. To ensure the ‘‘matching nodes’’ to still remain as ‘‘matching nodes’’ after deformation, when a pure extension is ap-plied at the two ends of the strand, the deformation relationship between any pair of corresponding nodes n (U1, U2, 0) andn0(U1, U2, U3) on the opposite cross-sections can be expressed as

Fig

DU01 ¼ DU1

DU02 ¼ DU2

U03 ¼ U3 þ eUP

ð3Þ

where DU1, DU2, U3 and DU01, DU02, U03 are the displacement components of node n at the front cross-section and its corre-sponding node n0 at the back cross-section in the coordinate system, respectively, UP is 1/6 of the pitch length of the strandand e is the axial strain. The induced axial force and torque, which keeps the rope unrotated, can be derived by the summa-tion of the reaction nodal forces on any one of the two cross-sections of the rope or strand finite element model. In the sub-model, displacements of nodes of two opposite cross-sections obtained in the global model are utilized as boundaryconditions.

2.3. Numerical results and discussion

2.3.1. 6 � 19 + IWS ropeIt can be seen from Fig. 4 that the finite element analysis predicts very closely rope responses in the linear region as given

by Costello’s theory [7]. The 6 � 19 + IWS rope has a tensile strength of 1570 MPa and a minimum breaking force of 133 kN.The maximum static tension permitted for the rope is 17733 N assuming the safety factor of 7.5 according to safety regula-tions in coal mine in China. The rope axial tension is 17522 N with applied axial strain of 0.001 as shown in Fig. 4a. Therefore,the response of the rope with the axial strain of 0.001, a typical loading level for engineering applications, is mainly discussedas follows.

Figs. 5a and b show the symmetric distributions of Von Mises equivalent stress and axial stress on the rope cross-section.The average stress level decreases in order of the core wire, helical wires in inner and outer layers in the straight strand,which is the case for both stresses. The axial forces of the core wire, helical wires in inner and outer layers are 160.1 N,152.5 N and 132.8 N, respectively, as shown in Fig. 6a, which coincides with the result in Fig. 5b. The symmetric axis of stressdistribution on the helical strand cross-section is the radial normal rope axial line, which is the case for both stresses. Theaverage Von Mises equivalent stress and average axial stress of steel wire both decrease gradually with increasing radial dis-tance from center of the rope. The axial force of the core wire is 130.1 N as compared to axial force ranges of 111–148.4 N and81.9–155.1 N for wires in the inner and outer layers in the helical strand as shown in Figs. 6b and c, which coincides with theresult in Fig. 5b.

0.000 0.002 0.004 0.006 0.008 0.0100.0

5.0x104

1.0x105

1.5x105

2.0x105

Axi

al fo

rce

(N)

Rope axial strain

Elastic finite element (Frictionless) Theory of elasticity (Costello)

0.000 0.002 0.004 0.006 0.008 0.0100

1x105

2x105

3x105

Torq

ue (N

mm

)

Rope axial strain

Elastic finite element (Frictionless) Theory of elasticity (Costello)

(a) (b). 4. Global rope responses: (a) variation in rope axial force with rope axial strain, (b) variation of rope reaction torque with rope axial strain.

(a) Von Mises equivalent stress (b) Axial stress (S33)

(c) Axial displacement (U3)

Fig. 5. Stress and displacement distributions of the rope section (Z = 9 mm) under the axial strain of 0.001.

0

60

120

180

Axi

al fr

oce

(N)

Straight strand

IHGFEDCB

(b)(a)

Steel wire number

Helical strand

(c)

Helical strand

A

Fig. 6. Axial forces of steel wires in the rope (A–CW; B–IW; C–OW; D–OCW; E–OIW1; F–OIW3; G–OOW1; H–OOW4;I–OOW7).


It is observed from Fig. 5c that the difference in axial displacement of steel wires in the helical strand is relatively large ascompared to that in the straight strand. The largest difference corresponds to the neighboring wires between adjacent helicalstrands. The difference in the axial displacement of contacting wires on the rope cross-section can be used to demonstratethe relative movement between contacting wires. The maximum relative movements between the core wire and helical wirein the inner layer, between adjacent wires in inner and outer layers, between adjacent helical wires in the inner layer, be-tween adjacent helical wires in the outer layer in the straight strand are 1.6 lm, 4.3 lm, 1.4 lm and 3.9 lm, respectively, asdisplayed in Fig. 7a, while they are 5.6 lm, 5.6 lm, 4.4 lm and 6.6 lm, respectively, in the helical strand as shown in Fig. 7b.The maximum relative movements between neighboring wires in adjacent helical strands and between neighboring wires instraight and helical strands are 14.4 lm and 6 lm, respectively, as exhibited in Fig. 7c. The hanging rope during the lifting

0

10

20 CWIW

IWOW

0 5 10 15 200

10

20 IW1IW2

Dis

plac

emen

t (μm

)D

ispl

acem

ent (

μm)

Dis

plac

emen

t (μm

)

0 5 10 15 20

OW1OW2

Distance along rope axis (mm)



(a) Neighboring wires in the straight strand

0

10

20 OCWOIW3

OIW3OOW5

0 5 10 15 200

10

20 OIW4OIW5

0 5 10 15 20

OOW8OOW9

(b) Neighboring wires in the helical strand

0 5 10 15 200

5

10

15

20 Helical strandHelical strand

Rel

ativ

e di

spla

cem

ent

0 5 10 15 20

Straight strandHelicals trand

(c) Neighboring wires in adjacent strands

Fig. 7. Displacements of steel wires along the rope axis with the rope axial stain of 0.001.


process has hundreds of pitch lengths, which will result in the relative movement between contacting wires ranging fromthe micron scale to the millimeter scale [33].

2.3.2. (1 + 6 + 12) Straight strandIn order to simulate the actual operation condition of the rope, the frictional elastic–plastic finite element analysis of the

rope under different axial extensions should be discussed. That’s because the material in the local contact area yields attrib-uted to the failure of the lubricant and the high local contact pressure between contacting wires after a larger number of loadcycles during operation. Meanwhile, the periodic overload of steel wires occurs due to the decrease in endurance strength ofhoisting rope before the frequent fall accident of the cage induced by the rope sudden breaking during lifting. However, due


to the complex structure of the rope, we mainly investigate the three-layered straight strand subjected to various axialextension strains (0.001–0.01) using both elastic and elastic–plastic material models.

Fig. 8 shows that the finite element analysis with the elastic material model predicts very closely strand responses in thelinear region as given by Costello’s theory [7]. A comparison of the results by Costello’s theory and the elastic–plastic analysisof the straight strand in Fig. 8 reveals that the strand responses are in good agreement when the strand axial strain is lowerthan 0.008, while plastic behavior of steel wires occurs when the axial strain reaches 0.008 in the elastic–plastic analysis.

Figs. 9 and 10 present various symmetric stress distributions on the strand cross-section obtained from the global modeland sub-model, respectively. It is clearly seen that overall trends of stress distributions from the global model agree wellwith those from the sub-model with fine mesh near the contact region which gives accurate results. Meanwhile, those re-sults are in good agreement with the results in Fig. 5. It is observed from Figs. 9a and 10a that the von Mises equivalent stresspeaks are all located near to the contact surfaces between neighboring wires in the same wire layer and between adjacentwire layers. The largest stress value corresponds to the contact between the helical wires in the outer layer. The descendingorder of overall equivalent stress levels of steel wires in different layers is the core wire, wires in the inner and outer wirelayers. The stress of the core wire is uniform across the section; the stress at the inner side of the inner layer wire is higherthan that at the outer contact edge; the stress of the outer layer wire presents a very similar manner to that in the inner layerwire (ignoring the contact region). It can also be seen that the equivalent stress is quite uniformly distributed along the wiresdue to the unwinding and slight straightening of these wires [15]. By comparison with the elastic analysis, the elastic–plasticanalysis presents slightly larger von Mises equivalent stress level in the vicinities of core wire-inner layer wire contact re-gions and inner layer wire-outer layer wire contact regions except the smaller peak stress value as a consequence of theresultant contact deformations (Fig. 11a), while the elastic–plastic analysis exhibits smaller equivalent stress near to thecontact surfaces of neighboring wires in the outer wire layer attributed to the local friction effect (Figs. 9a and 10a) [15].More detailed equivalent stress distributions along a radial normal axial line from the strand center through the trellis con-tact point (Y-axis in Fig. 3b. 2) with varying axial strain are given as shown in Fig. 11a. It can be clearly seen in Fig. 11a thatthe equivalent stress always presents an overall downward trend along the radial normal axial line and sudden changes nearthe contact line with each strand axial extension strain due to the local contact deformation. As the strand axial strain isincreased, the stress in each wire increases linearly in the elastic analysis, whereas the stress increases linearly until plasticdeformation begins to occur in the elastic–plastic analysis. When the strand axial extension strain reaches approximately0.008 in the elastic–plastic analysis, the material in the local contact area between adjacent wires begins to yield resultingin the decrease of overall von Mises equivalent stress level as compared to the elastic analysis with the same axial strain.However, the difference in von Mises equivalent stress along the radial line is not obvious in the linear region between bothfinite element analyses with the same axial strain except in the vicinities of contact regions as illustrated above.

It is clearly seen in Figs. 9b and 10b that the axial stress distribution pattern in the bulk of the wires (ignoring the contactregions), is similar to that for the equivalent stress distribution in Figs. 9a and 10a, but the values are somewhat lower, indi-cating that the axial stress is the predominant stress. In the local vicinity of contact regions between neighboring wires theaxial stress becomes smaller as a result of contact deformations [12,15]. Fig. 11b shows the more detailed axial stress dis-tributions along the radial normal axial line from the strand center. It is obviously observed that the axial stress presents anoverall downward trend along a radial line and sudden decreases near the contact region between neighboring wires in adja-cent layers as illustrated above in both analyses with each axial strain. An increase of axial strain induces the linear increasein the axial stress in each wire in the elastic analysis, while in the elastic–plastic analysis the axial stress increases linearly inthe elastic region until plastic deformation begins to occur at the axial strain of approximately 0.008. The magnitude of de-crease in the axial stress in the contact region increases with increasing axial strain, which is the case in both analyses. How-

0.0020.000 0.004 0.006 0.008 0.0100.0

5.0x103

1.0x104

1.5x104

2.0x104

2.5x104

3.0x104

Axi

al fo

rce

(N)

Strand axial strain

Elastic finite element (Fricitionless) Elastic-plastic finite element (Fricitional) Elastic theory (Costello)

0.000 0.002 0.004 0.006 0.008 0.0100.0

2.0x103

4.0x103

6.0x103

8.0x103

1.0x104

Torq

ue (N

mm

)

Strand axial strain

Elastic finite element (Frictionless) Elastic-plastic finite element (Frictional) Elastic theory (Costello)

(a) (b) Fig. 8. Global strand responses (a) Variation in strand axial force with strand axial strain (b) Variation of strand reaction torque with strand axial strain.

(a) Von Mises equivalent stress

(1) (2)

(b) Axial stress

(c) Max. principal stress

(1) (2)

(1) (2)

Fig. 9. Stress distributions of the strand cross-section (Z = 9 mm) with the strand axial strain of 0.001 (Global model).


ever, the magnitude of decrease in the axial stress in the contact region is larger in the elastic–plastic analysis than that inthe elastic analysis with the same strand axial strain.

It is observed from Figs. 9c and 10c that the descending order of maximum principal stress levels of wires in three layersis the core wire, spiral wires in the inner and outer layers, which indicates the core wire is subjected to the largest stressconcentration and thereby its premature failure is more easily to occur. Meanwhile, compared to the frictionless elastic anal-ysis, the frictional elastic–plastic analysis presents smaller maximum principal stresses across the core wire cross-section,contact regions between neighboring wires in adjacent wire layers and the contact region between spiral wires in the innerwire layer, which reveals that frictionless elastic state is more likely to induce the stress concentration and thus easily resultsin the premature failure of the strand [6,34]. Therefore, the coefficient of friction can be reduced by means of rope lubricationbut should not drop to zero, otherwise the service life of the rope will be reduced [6]. Fig. 11c presents the radial displace-ment distribution along a radial normal axial line with varying strand axial strain. It can be seen that there are slight smoothchanges near the contact region due to the local contact deformation [12].When the strand axial extension strain reachesapproximately 0.008 in the elastic–plastic analysis, the radial displacement of each node begins to be larger than that inthe elastic analysis resulted from the plastic deformations due to higher contact stress between neighboring wires.

Fig. 12 shows that both finite element analyses present the decrease of wire axial force in order of the core wire, the heli-cal wires in inner and outer layers, which coincides with the results in Fig. 6. The ranges of wire axial force in elastic andelastic–plastic analyses are 154–1764 N and 154–1398.2 N, respectively, when the strand axial extension strain increasesfrom 0.001 to 0.01.

(a) Von Mises equivalent stress

(b) Axial stress

(1) (2)

(1) (2)

(1) (2)

(c) Max. principal stress

Fig. 10. Stress distributions of the strand cross-section with the strand axial strain of 0.001 (submodel).


The normal contact load between contacting wire cannot be directly derived by the total force due to contact pressure(CFN) attributed to the three dimensional contact, whereas it can be calculated from the contact pressure of the nodes inthe slave surface [34]

F ¼ CPtAs

nð4Þ

where F is the contact load between contacting wires, CPt is the sum of contact pressure of all contact nodes on the slavesurface, As is the total contact area of the slave surface and n is the number of contact nodes on the slave surface.

0.0 0.5 1.0 1.5 2.0 2.5 0.0 0.5 1.0 1.5 2.0 2.5 0.0 0.5 1.0 1.5 2.0 2.5 0.0 0.5 1.0 1.5 2.0 2.5

0.0 0.5 1.0 1.5 2.0 2.50.0 0.5 1.0 1.5 2.0 2.5

0

500

1000

1500

2000

2500

OWIW

ε=0.001 ε=0.004ε=0.006 ε=0.008ε=0.010

Von

Mis

es e

quiv

alen

t str

ess

(MPa

)

Radial distance from center of strand (mm)

CW

(1) (2)

ε=0.001 ε=0.004ε=0.006 ε=0.008ε=0.010

IWCW OW 0

500

1000

1500

2000

2500

OWIW

ε=0.001 ε=0.004ε=0.006 ε=0.008ε=0.010

S33

(MPa

)


CW

(1) (2)

ε=0.001 ε=0.004ε=0.006 ε=0.008ε=0.010

IWCW OW

(a) Von Mises equivalent stress (b) Axial stress

-0.010

-0.008

-0.006

-0.004

-0.002

0.000

0.002

OWIW

ε=0.002 ε=0.004ε=0.006 ε=0.008ε=0.010

Rad

ial d

ispl

acem

ent (

mm

)


CW

(1) (2)

εε=0.002 ε=0.004ε=0.006 ε=0.008ε=0.010

IWCW OW

(c) Radial displacement of nodes with different radial distance from center of strand

Fig. 11. Distributions of stresses and radial displacement along the radial normal axial line from the strand center through the trellis contact point: (1)frictionless elastic case, (2) frictional elastic–plastic case.

0.000 0.004 0.008 0.0120

500

1000

1500

2000

Axi

al fr

oce

of s

teel

wire

(N)

CW IW OW

(a)0.000 0.004 0.008 0.01

(b)

CW IW OW

Strand axial strain

Fig. 12. Variation of axial forces of wires in three layers with the strand axial strain.


Fig. 13a presents the variations of contact load between contacting wires with the strand axial strain in elastic and elas-tic–plastic analyses. The contact load decreases in order of between helical wires in the outer layer, between core wire andthe inner layer wire, between inner layer wires and between neighboring wires in inner and outer layers in the elastic case,

0

200

400

600

800

1000C

onta

ct lo

ad (N

)

Strand axial strain

CW-IW IW-IW IW-OW OW-OW

(1)0.000 0.004 0.008 0.012 0.000 0.004 0.008 0.012

(2)

CW-IW IW-IW IW-OW OW-OW

CW-IWIW-IWIW-OWOW-OW0

200

400

600

800

1000

Con

tact

load

(N)

CW-IWIW-IWIW-OWOW-OWWires in contact

(1) (2)

(a) (b) Fig. 13. Contact load with strand axial strain: (a) variation of contact load between contacting wires with the strand axial strain; (b) ranges of contact loadbetween contacting wires with the strand axial strain ranging from 0.001 to 0.01: (1) frictionless elastic case; (2) frictional elastic–plastic case (CW–IWdenotes core wire and wires in contact in the inner layer; IW–IW represents adjacent wires in the inner layer; IW–OW defines neighboring wires in theinner and outer wire layers; OW–OW means adjacent wires in the outer layer).


while the contact load between inner layer wires is the smallest in the elastic–plastic case. As the strand axial extensionstrain increases from 0.001 to 0.01, the overall range of contact load between contacting wires is 3.5–904.9 N in the elasticanalysis, whereas the overall range is 20.2–551.3 N in the elastic–plastic analysis. By comparison with the elastic case, theelastic–plastic analysis presents larger contact load between contacting wires with the same axial extension strain exceptthe contact load between outer layer wires as shown in Fig. 13b, which reveals that the contact load between outer layerwires is greatly affected by the friction effect and plastic behavior of the wire material.

3. Fretting wear evolution of steel wires crossed at different angles

3.1. Theoretical background

In order to study the wear severity of steel wires during fretting wear, it requires a wear coefficient, kl, expressed as thewear per unit local slip per unit local contact pressure. From experimental results, it is only possible to directly measure thewear coefficient per unit displacement per unit normal load. The latter is the Archard’s wear coefficient, which is defined by[35]

K ¼WSF

ð5Þ

where W is the total wear volume, F is the applied constant contact load, S is the total accumulated displacement equal to2DxN. The component, Dx, is the relative displacement (i.e. a stroke) and N is the total number of fretting cycles. Assumingthat fretting wear is not located in the stick regime and the contact region between neighboring wires presents uniform dis-tribution of contact stress in the steady state, the wear coefficient, kl, is approximately given by

k1 ¼WSF¼ k

að6Þ

where a is a constant ratio of relative slip to relative displacement between contacting wires during the steady state of fret-ting wear [20,32] as shown in Fig. 14. Therefore, this study applies the Archard’s wear coefficient to evaluate the frettingwear depth of wires crossed at different u angles.

3.1.1. u = 90�Let us consider two equal cylindrical wire specimens crossed at right angle during the fretting wear test (Fig. 15), the wear

gap between contact surfaces presents the paraboloid approximation as shown in Fig. 16. The normal contact pressure pat-tern is not much affected by the tangential stresses and displacement amplitude when the contacting bodies have the sameYoung’s modulus [36]. Neglecting the tangential friction between contacting wires, the contact pressure in the steady-stateperiod is written as

p ¼ Fa2 ð7Þ

-150 -75 0 75 150-30

-15

0

15

30

Tang

entia

l for

ce (N

)

Displacement amplitude (μm)

Relative slip

Relative displacement

Fig. 14. The schematic of relative slip and relative displacement between contacting wires in the hysteresis loop of tangential force versus displacementamplitude.

Fig. 15. Two wires in contact with their axes crossed at u angle.

Fig. 16. Scanning electron microscope photo of lateral wear gap of worn wires [18,32].


where p is the average contact pressure and a is the variable radius of the contact area.The evolution of the contact radius during fretting wear is approximately described as follows [37].

a ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffia4

0 þ8kRFDxN

p4

rð8Þ

where a is the contact radius after N cycles, a0 is the initial contact radius and R1 = R2 is the radius of contacting wire.The initial contact radius of contacting wire with the normal contact load F can be calculated from three dimensional

Hertzian contact theory as described in Eqs. (9)–(11) [38].

a0 ¼ CF �ffiffiffiffiffiffiFRE�

3

rð9Þ

1E�¼ ð1� m2

1ÞE1

þ ð1� m22Þ

E2ð10Þ

R ¼ D1D2

D1 þ D2ð11Þ

where D1 and D2 represent diameters, m1 and m2 denote Poisson’s ratios, E1 and E2 are elastic module, and E� is the equivalentelastic modulus. The subscripts, 1 and 2, represent the upper and lower contacting wires, respectively, of which the radii areidentical. CF is contact shape dependent constant which is equal to 0.909 for two identical contacting cylinders in the presentstudy.


The fretting wear depth of contacting wire is given by [21]

Table 3Mechan

Mec

Tens(N/m

1550

h1 ¼a2 � a2

0

2Rð12Þ

3.1.2. u – 90�It is observed from the optical microscope that the wear scar of steel wire after fretting wear is elliptical with contacting

wires crossed at any angle u as shown in Fig. 17. The wear volume at the fretting depth h2 is approximately given by [39]

W ¼ ph22cosec/

ffiffiffiffiffiffiffiffiffiffiR3R4

pð1� h2

8R3� h2

8R4Þ ð13Þ

where h2 is the fretting wear depth of steel wire, R3 = R4 is the radius of two identical contacting wires and u is the crossingangle. The relationship between the wear depth of contacting wire with fretting parameters during fretting wear accordingto Eqs. (5) and (13) is given by

2DxkFN ¼ ph22cosec

ffiffiffiffiffiffiffiffiffiffiR3R4

p1� h2

8R3� h2

8R4

� �ð14Þ

The fretting wear depth can be computed by numerical methods using Matlab (SIMULINK) if all other parameters areknown.

3.2. Theoretical prediction and experimental validation

3.2.1. u = 90�In order to validate the mathematical model of fretting wear, fretting wear tests have been carried out [18,40]. The

mechanical properties and experimental parameters of steel wire are listed in Table 3. Fig. 18 presents the Simulink simu-lation model to determine the wear coefficient k, according to Eqs. (6)–(12). Fig. 19 shows the variations of wear depths withfretting time during fretting wear tests under three contact loads. The average wear coefficients of upper and lower wirespecimens during the test under each contact load are presented in Fig. 20. It’s revealed that the average wear coefficientremains almost constant with respect to normal load variation for the same relative displacement, which is the case for boththe upper and lower specimens. The average value of wear coefficients under three contact loads are chosen for theoreticalprediction and thereby the wear coefficients for upper and lower wire specimens are 2.0 � 10�14 m3/(N m) and2.1 � 10�14 m3/(N m), respectively.

Fretting wear depths of steel wires during fretting tests after 15 h and 20 h under different contact loads are predictedusing the calculated wear coefficients and Eqs. (6)–(12). It is observed from Figs. 21 and 22 that the predicted values andexperimental results are in good agreement. The average wear coefficients of upper and lower specimens in tests with

Fig. 17. Fretting contact scar of steel wire after the test with the crossing angle of 18� [32].

ical properties and experimental parameters of steel wire.

hanical properties Wirediameter(mm)

Experimental parameters

ile strengthm2)

Ultimate strength(N/mm2)

Young’s modulus(N/mm2)

Hardness(HV)

Relativedisplacement(lm)

Contactload (N)

Frequency(Hz)

Frettingtime (h)

9281 2.08 � 105 365 1 108–140 9–30 3.6 0–20

Displacement

Wire radius

Cycles

Contact load

R

Wire radius

w0

Wear depth

Wear coefficient k

Subtract1

In1 Out1

Subsystem

Product1

Product

uv

Math Func.5

uv

Math Func.4

u2

Math Func.2

uv

Math Func

pi

2

Gain

Divide2

4

Constant9

N

detx

R

F

1/2

Cons.3

4

Cons.2

4

Add

Fig. 18. Simulink simulation model for determination of k.

0

50

100

150

200

Wea

r dep

th (μ

m)

Upper specimen Lower specimen

F=14N F=19N

Fretting time (h)


10 15 20 8 12 16 20 4 8 12 16 20

F=24N

Upper specimenLower specimen

Fig. 19. Evolutions of fretting wear depth with fretting time with the relative displacement of 140 lm under different contact loads.

14 19 240.00E+000

1.00E-014

2.00E-014

3.00E-014

k (m

3 /N⋅m

)

Contact load (N)


Fig. 20. Average wear coefficients of upper and lower specimens during fretting tests under different contact loads.


different relative displacements are presented in Fig. 23 according to fretting wear tests [40]. It’s found that the average wearcoefficients of upper and lower specimens both decrease with increasing relative displacement.

3.2.2. u = 18�The crossing angle between neighboring wires in the rope ranges from 0� to 90� and thereby the evolution of fretting wear

depth of steel wires crossed with different angles during fretting wear should be discussed to understand the actual wear

5 10 15 20 25 300

50

100

150

Lower specimen

Experimental Prediction

Wea

r dep

th (μ

m)

Contact load (N)

0

50

100

150 Experimental Prediction

Wea

r dep

th (μ

m)

Upper specimen

Fig. 21. Predicted and experimental fretting wear depths of steel wires during fretting tests after 15 h with the relative displacement of 140 lm underdifferent contact loads.

10 15 20 25 300

60

120


Contact load (N)

0

60

120


Wea

r dep

th (μ

m)

Upper specimen

Lower specimen

Fig. 22. Predicted and experimental fretting wear depths of steel wires during fretting tests after 20 h with the relative displacement of 140 lm underdifferent contact loads.

108 122 1400.00E+000

1.00E-014

2.00E-014

3.00E-014

4.00E-014

k (m

3 /N⋅m

)

Relative displacement (μm)


Fig. 23. Variation of average wear coefficients of the specimens with the relative displacement.


severity of steel wires in the rope. Fretting wear tests of contacting wires with the crossing angle of 18� were conducted at1.2 Hz with the relative displacement of 180 lm under the normal contact load of 60 N under ambient lab conditions [32].The calculated average wear coefficients of upper and lower specimens are 3.30 � 10�15 m3/(N m) and 3.52 � 10�15 m3/(N m), respectively. The predicted values of wear depth after 1.3 � 104 cycles under various contact loads according to Eq.(14) are shown in Fig. 24, which are in good agreement with the experimental results.

8 16 24 320

2

4

6

8

10

Wea

r dep

th (μ

m)

Experimental Prediction

Lower specimen

Contact load (N)

0

2

4

6

8


Upper specimen

Fig. 24. Experimental and predicted fretting wear depths of upper and lower specimens under different contact loads.


4. Fatigue life prediction of steel wires with pre-cracks

4.1. Fatigue life prediction based on the linear theory of elastic fracture mechanics

To investigate the effect of fretting wear on fatigue fracture properties of the steel wire, the theory for prediction of thefatigue life of fretted wire is discussed as follows. The fatigue life under cyclic loading consists of two phases, the crack ini-tiation stage followed by a crack growth period until failure [21]. However, the fatigue life of wires is primarily governed bythe rate of subcritical crack propagation assuming that the fretted scar on the wire surface is the initial pre-crack. It’s sug-gested that some crack increment occurs in every load cycle and the relation between dh/dN and DK can be described by apower function according to Paris equation as follows.

dhdNf¼ CðDKÞm ð15Þ

where C and the exponent m are material constants, Nf is the fatigue cycles and DK the stress intensity factor range.It is observed in Fig. 25 that the function da/dN = fR(DK) covers three different parts, i.e. (i) region I, the threshold DK-re-

gion, (ii) region II, the Paris-DK-region, (iii) region III, the stable tearing crack growth region. It is shown that two verticalasymptotes occur in the da/dN graph. The left asymptote at DK = DKth indicates that DK values below this threshold levelare too low to cause crack growth. The other asymptote at the right-hand side with Kmax = Kc means that Kmax reaches a crit-ical value which leads to complete failure of the specimen [41]. Therefore, Eq. (15) does not account for the effect of stressratio on crack growth, neither for the asymptotic behavior in regions I and III. Forman [21] proposed the following equationto overcome this problem.

dhdNf¼ CðDKÞm

ð1� RsÞðKc � KmaxÞð16Þ

where Rs is the stress ratio, Kc is the critical stress intensity factor and Kmax is the maximum stress intensity factor in onefatigue cycle.

Fig. 25. Three regions of the crack growth rate as a function of DK [41].


The crack growth threshold is included by the following so-called Priddle equation [41]:

dhdNf¼ C

DK � DKth

Kc � Kmax

� �m

ð17Þ

where DKth is the crack growth threshold, DKth = A(1 � R)b. A and the exponent b are both material constants.It is suggested in the literature that such equations are useful in view of prediction algorithms but this argument is not

really appropriate [41]. However, a useful indication can be drawn from Eq. (15) if the exponent m is known from experi-ments and K substituted by Eqs. (19)–(23) as discussed below. Integrating Eq. (15), we will obtain the relationship betweenthe crack propagation life of steel wires, N1 and the fretting wear depth, h1, as given by

Nf ¼Z hf

h1

dh=CðDKÞm ð18Þ

where D is the wire diameter, h1 is the fretting wear depth and hf is the crack depth at fracture failure. As introduced in Sec-tion 4.2, hf is equal to D/2.

The stress intensity factor K is a parameter indicating the severity of the stress distribution around the tip of a crack. Thefatigue load on a cracked specimen introduces a cyclic stress intensity at the crack tip varying between Kmax and Kmin.According to the linear elastic fracture mechanics (LEFM), the stress intensity factor at the crack tip can be expressed as fol-lows [42]:

K ¼ YðcÞrffiffiffiffiffiffiphp

ð19Þ

where Y(c) is the associated normalized stress intensity factor, c is the ratio of crack depth to wire diameter, i.e. h/D and r isthe tensile stress.

The stress intensity factor range is

DK ¼ KmaxKmin ¼ YðcÞrmax

ffiffiffiffiffiffiphp

� YðcÞrmin

ffiffiffiffiffiffiphp

¼ YðcÞDrffiffiffiffiffiffiphp

ð20Þ

where Kmin is the minimum stress intensity factor in one fatigue cycle and rmax and rmin are maximum and minimum axialstresses in one fatigue cycle, respectively.

4.2. The evolution of normalized stress intensity factor for crack growth of steel wire in pure tension

The pre-cracks of wire surfaces induced by fretting damage propagate under cyclic loading and the front of surface cracksundergoes a shape change. The crack tends to assume a semicircular front very early in the crack growth process. As thecrack extends, its front tends to flatten and assumes a straight shape with c approaching 0.5 [30,43]. Then the steel wire frac-tures because the remaining cross-sectional area cannot withstand the cyclic load any longer. The normalized stress inten-sity factor varies at different stages of the crack propagation as discussed below.

The evolution equations of the normalized stress intensity factor with c for the semicircular surface crack front under ten-sion and the straight crack front under tension are given by [43]

YðcÞ ¼ 0:7037� 1:0589cþ 5:8771c2 ð21Þ

YðcÞ ¼ 0:8149þ 2:8225c� 28:885c2 þ 159:75c3 � 334:05c4 þ 263:28c5 ð22Þ

0.0 0.2 0.4 0.6 0.80

2

4

6

Nor

mal

ized

str

ess

inte

nsity

fact

or (

)

Normalized crack length (h/D)

Semi-circular crack Straight crack Transitional stage

h

h

Fig. 26. One-regime model for the normalized stress intensity factor for crack growth in pure tension.


Through the polynomial fit for the transition between the semicircular crack front and the straight crack front, the one-regime model for the normalized stress intensity factor during different stages of crack growth under tension is given by(Fig. 26) [43]

YðcÞ ¼ 0:7282� 2:1425cþ 18:082c2 � 49:385c3 þ 66:114c4 ð23Þ

5. Conclusions

6 � 19 + IWS wire rope and its three-layered straight strand are mainly focused on in the present study to investigate thestress states of wires and fretting fatigue parameters between contacting wires. Abaqus/CAE and Pro/Engineer Wildfire 5.0are employed for the stranded rope construction applying the equations of locations along the centerlines of different wires.The axially loaded rope is analyzed employing the elastic finite element model, whereas the finite element analysis of thestraight strand under axial extension takes into account friction, all possible interwire contacts, plastic deformation and con-cise boundary conditions. Global responses of the rope and three-layered straight strand employing finite element analysesboth predict very closely responses in the linear region as given by Costello’s theory. The finite element analysis of the rope(straight strand) indicates different stress distributions at distinct locations of the rope (straight strand) cross-section anddifferences in fretting fatigue parameters between different contacting wires. The effect of material model and axial exten-sion strain on the stress distributions of the strand cross-section and fretting fatigue parameters is also discussed.

In order to study the wear severity of steel wires during fretting wear, Archard’s wear coefficient is introduced. The rela-tions of fretting wear depths of contacting wires crossed at different angles and fretting parameters are established accordingto the wear coefficient and theory of contact mechanics, which is validated by the experimental results. To investigate theeffect of fretting wear on fatigue fracture properties of the steel wire, the relationship between the crack propagation life ofsteel wire and the initial fretting wear depth is established based on the linear elastic fracture mechanics, Paris equation andthe evolution of dimensionless stress intensity factor for crack growth in pure tension.

Acknowledgments

The research reported here was supported by National Natural Science Foundation of China (Grant No. 50875252), Jiang-su College Postgraduate Research Innovation Plan Project of 2011 (CXLX11-0315), Fundamental Research Funds for the Cen-tral Universities (China University of Mining and Technology, 2011JQP03).

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Finite element analysis of hoisting rope and fretting wear ...

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