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International Journal of Mechanical Engineering and Technology (IJMET)

Volume 9, Issue 4, April 2018, pp. 156–168, Article ID: IJMET_09_04_018

Available online at http://www.iaeme.com/ijmet/issues.asp?JType=IJMET&VType=9&IType=4

ISSN Print: 0976-6340 and ISSN Online: 0976-6359

© IAEME Publication Scopus Indexed

STIFFNESS OF A SINGLE LAYERED CABLE

ASSEMBLY OVER A SHEAVE WITH INTERNAL

FRICTION

Rajesh Kumar P

Assistant Professor, Mechanical Engineering Department,

MVJ College of Engineering, Bangalore, India

Parthasarathy N.S

Professor, Mechanical Engineering Department,

Christ (Deemed to be University), Bangalore, India

ABSTRACT

The stiffness response of a single layered helical strand with a straight core

surrounded by a layer of six helical wires has been made with improved relations of

wire curvatures & twist and with internal friction considerations. The stranded cable

undergoes a constant curvature bending over a sheave/pulley under static loading

conditions and experiences the combinations of tension, torsion and bending loadings.

A new analytical model has been developed for the cable in contact with the

pulley/sheave using thin rod theory under linear elastic conditions. The stiffness

coefficients of the cable are evaluated in free bending and constrained bending modes.

The resulting wire strains are evaluated and compared with the experimental results.

Keywords: Free bending, stiffness analysis, interfacial force, frictional force.

Cite this Article: Rajesh Kumar P and Parthasarathy N.S, Stiffness of a Single

Layered Cable Assembly over a Sheave with Internal Friction, International Journal of

Mechanical Engineering and Technology, 9(4), 2018, pp. 156–168.

http://www.iaeme.com/IJMET/issues.asp?JType=IJMET&VType=9&IType=4

1. INTRODUCTION

Several analytical models are available for predicting the mechanical behaviour of a helical

stranded cable under general loaded conditions. The stiffness of the strand depends on the

parameters like i) the constructional geometry ii) the loading type: axial or bending, iii) the

nature of contact among the internal wires, iv) the nature of contact of the cable with an

external surface like pulley or sheave etc. Majority of cable research has been carried out

under the axial loads of tension and torsion, which are the primary loads in any cable. When

the cables are in dynamic oscillation, as in overhead power transmission lines or when the

cable is bent and applied over the pulley/sheave, as in material handing applications they are

subjected to bending, in addition to the axial loads of tension and torsion. The former bending

Stiffness of a Single Layered Cable Assembly over a Sheave with Internal Friction


of the cable is referred as free bending, while the later bending mode is referred as constrained

bending. The salient research works of a strand subjected to the combinations of axial tension,

torsion and bending loads in the cases of free bending are presented below: Costello and

Buston [1982] proposed a free bending model which also included the twisting of the wires

based on Love’s equation for a curved rod [1944]. Lanteigne [1985] proposed a free bending

model in which the bending moment contribution was mostly from the fibre force and the

contribution of bi-normal wire force had been included without wire twist and wire bending

effect. Vinogradov and Atatekin [1986] used a cantilever approach where the study revealed

that the partial twist slip of wire phenomenon was due to strand bending alone and this

twisting moment was considered as the sole factor for energy dissipation, whereas the

dominant axial slip was neglected. LeClair and Costello [1988] had given an improved

treatment to the wire centreline which had taken the shape of the deformed helix. Papailiou

[1995] have developed a model where the slip starts at a point on the strand neutral axis and

that the stress is arising from axial load only. The variations of the wire curvature after

bending of the strand and the wire stiffness in twist have been neglected. Parthasarathy

[1998] formulated the model to include the axial and twist slip of the wire during the axial

loading of the strand and extended them to predict the effective stiffness of the cable during a

free bending phenomenon. Sathikh et al [2000] formulated a free bending model having

wire to core contacts under constant curvature bending and obtained the pre-slip bending

response of a strand of helical wires, having unlimited Coulomb friction. Hardy and

Leblond [2003] imposed a uniform curvature on the strand and predicted the wire movement

only on the outer layer of the strand considering the micro-slip into account at the contact

region, and predicted the bending stiffness. Hong etal [2005] slightly modified the

Papailiou’s [1995] model and observed an interesting prediction that the full slip would never

occur due to increased curvature. Further it was observed that the slip stopped at certain

bending angle which depended on the axial load and frictional coefficient and the minimum

tangent bending stiffness was found to be much larger than the lower bound value. Inagaki

et al [2007] predicted the bending response of the cable by including the frictional forces

resulting from the jacket and insulating material and this frictional model was extended from

Papailiou’s work but neglected the contact between the adjacent wires in the layer. Boston et

al [2011] developed a cable with no bending stiffness and used a superposition theory of

amplitude dependent friction to predict the damping ratio of the cable. The significant

research works that analysed the constrained bending of a cable are summarised as under.

Nabijou S and Hobbs R.E [1995] studied the bending strain in the wires of frictionless ropes

and accounted for the change in curvature in single and double helices, when the rope is bent

into circular arc and applied over a sheave and derived the relations from the first principles.

However the helical wire stiffness variation over the curvature has not been accounted.

Marco Giglio, Andrea Manes [2005] estimated of the state of stress in the inner and external

wires of rope, by the bending moment models that accounted for the radius of curvature of a

rope bent over a small sheave. The model contributed to calculate the local stresses in all the

wires, along the rope by knowing the curvature difference between the unreformed and

deformed rope, and found to be useful in successive fatigue analysis of the wires. Usabiaga

[2008] formulated a new mathematical model for the mechanical interaction of the wire rope

on a sheave, by considering the Coulomb friction at the sheave interface and employed an

iterative procedure to solve the analytical equations. The disagreements between the

analytical and experimental results have thrown the study to investigate local wire effects.

Gopinath [2012] considered the constant curvature bending of a strand in a constrained

bending mode and formulated mathematical relations to include the wire effects due to axial,

torsional and bending loads on the cable. The frictional forces arising at the strand-sheave

interface and its effects at the contacting wire are formulated as a function of bending radius.

Rajesh Kumar P and Parthasarathy N.S


The stiffness coefficients of the strand have been estimated. It has been observed that the

stiffness coefficient expressions of the contacting wire with the sheave are inadequately

represented, probably with omission of some relevant terms. Hence this paper attempts to

adopt refined relations for the stiffness coefficients of the contacting wire and predict the

resulting stiffness of the cable, on the sheave. In addition, this paper considers the contact

forces among the wires in the cable and the associated internal frictional effects at all the wire

interfaces, probably for the first time in this nature of research on the constrained bending

behaviour. The analytical results that are predicted with these refinements are compared with

experimental work, which is performed exclusively for this study.

2. EXTRACTION OF THE MODEL

It has been assumed that the strand is radially rigid and free from Poisson’s effect of the

wires. In the following analysis, each wire is considered in the strand as a long slender curved

rod. Figs 1(a) and 1(b) show the cross section of a seven-wire strand and the developed

geometry of a single helical wire. The strand initially consists of a straight centre core of

radius of Rc surrounded by ‘m’ helical wires of wire radius Rw and the helix angle α

Figure 1 Wire geometry

From Fig. 1b the geometric relations can be obtained as

h=l sinα (01)

χ=l cosα (02)

⁄ (03)

where ‘h’ is the strand height, and l & r, are the wire length and helix radius, and α & ,

are the helix angle and the swept angle of the helical wire.

For a radially rigid strand the following relations are valid

δ δ δ δ (04)

The change in helix angle and change in length of the wire are given by

(05)

(06)

During the axial loading, the strand undergoes an axial strain ( ) and rotational strain ( as a result of which, the axial strain in the wire can be expressed as

(07)

where

(08)



When the strand is additionally bent, the axial strain can be computed as under. Fig 2

shows the geometry of the strand under free bending, with a constant radius of ρ

Any elemental length of the strand about neutral axis can be shown as

(09)

Figure 2 Geometry of bent strand

The elemental length at a point located by the angle at a distance from the

neutral axis would then be

(10)

The strand axial strain can be obtained from (9) and (10) for any position angle

(11)

The wire axial strain per unit length of wire inclined at will then be,

(12)

Under the combination of axial and pure bending, the net axial strain of the wire is

expressed as

(13)

Combining Eqn 7,8 & 12, Eqn 13 can be written as

(14)

3. CURVATURE AND TWIST OF THE HELICAL WIRE

For an initially straight strand configuration, the components of wire curvatures and wire twist

(without slip) in the normal, bi-normal and wire axial directions are given as

(15)

The final curvature and twist of helical wire under combined effect of axial load and

bending are given by

(16)

The change in curvatures and twist of helical wire subjected to combination of axial

deflection and bending of strand are expressed as under

(17)

(18)



(19)

The refinement in the curvature twist expressions above are as a result of wire stretch, a

special feature considered in recent researches and it is accounted in bending now.

3. WIRE FORCES AND COUPLES

The forces and couples in the helical wire for the bent strand can be obtained from the Eqns

(14), (17), (18) & (19)

The forces in the axial direction of the wire can be given by

(20)

The bending moment in the normal direction of the wire can be related as

(21)

The bending moment in the bi-normal direction of the wire can be obtained as

(22)

The twisting moment of the helical wire can be related as

(23)

Figure 3 Forces and moments on helical wires

The above forces and couples acting on a helical wire are shown in Fig 3. The wire

normal and bi-normal forces N, N´ are also indicated in Fig 3.The components of the

distributed forces per unit length of the wire are X, Y, Z and the components of distributed

moments acting per length of the wire are K, K', Θ in normal bi-normal and axial directions

respectively are as shown in Fig 3

4. WIRE EQUILIBRIUM EQUATIONS

The equilibrium equations for the curved rod model in the normal, bi-normal and axial

directions are as follows

(24)

(25)

(26)

(27)

(28)

(29)

Using the equilibrium Eqns from (24) to (29) the shear forces in the normal and bi-normal

directions can be obtained as



N= -G (30)

(31)

5. CONTACT MODES

In the assembled state, the behavior of the cable is governed by the nature of the contact

between the core and the helical wires. The core and the wires initially retain themselves in a

combined contact mode, where all the wires make contact among themselves and the core.

During the axial pull, the unequal radial deformation of the core and the helical wires, turn the

wires to come in contact among themselves alone and loose contact with the core, especially

when the core is soft. When all the wires of the cable are of same material and are of same

size, all the helical wires maintain contact with the core wire only and this is often referred as

the core-wire contact. During successive loading the cable geometry maintains this contact

mode only generally and hence this paper considers the core- wire contact mode. In this

contact mode, all the helical wires exert a radial clenching contact force on the core, as

denoted by S in Fig 4

Figure 4 Core-Wire contact mode

6. FRICTIONAL FORCES

When the friction at the core-wire interphase is considered, the limiting force of friction can

be obtained by the Coulomb’s law of dry friction. When this limiting friction is overcome by

the axial force in the helical wire, the wire will slip or move in the axial direction. When the

helical wire rolls on the core, this limiting force causes a twisting moment about the axis of

the core, which is responsible for the twist slip of the wire. For clarity, these two limiting

forces are indicated in Fig 4 as P and Q, though they are the same, magnitude wise. The

limiting friction forces can be expressed by the following relations.

(32)

Where is the frictional coefficient between core and helical wire

Accordingly the following equations for the normal, bi-normal shear forces N, N' and the

contact force S are established as under

((

)

) ((

)

)

(

) (33)

(34)

7. CONSTRAINED BENDING

A constrained bending case of the cable and the pulley has been considered in this study. It is

assumed that at any instant only one wire from outer layer makes contact with the sheave. Fig

5 shows the contact of the bottom wire with the constrained surface, with line loads p & q



acting in the normal & bi-normal directions of the contacting wire, at its position angle

. The five wires that do not make a contact with the pulley will be treated as in free

bending, whereas the one which establishes contact with the pulley at a position angle 180º

will be treated with constrained bending, in the formulation of global stiffness of the

stand.The radial force is the distributed load per unit length which the pulley exerts on the

contacting wires, and in conjunction with the friction coefficient ‘ ’ between the stranded

cable and the pulley, gives rise to a friction force ‘ ’ which modifies the wire tension about

the axial direction as:

Figure 5 Wire in contact with the sheave

(35)

The bi-normal direction force q exerted by the pulley on the wire which is in contact, modifies

the wire shear force about bi-normal direction as:

(33)

8. EQUILIBRIUM EQUATIONS OF THE STRANDED CABLE

The resultant strand axial force twisting moment and bending moment of the

cable can be obtained from the following equations, when the core deformations are also

added.

(36)

(37)

(38)

where m is number of wires in the helical layers

The above equations cab be represented in a matrix form as under,

{

} [

] {

⁄

⁄} (39)

where , and are the effective strand axial stiffness, strand torsional rigidities

and flexural rigidities. and are the tension-torsion, and are the tension-

bending. and are the torsion bending coupling parameters, respectively and , ⁄

and ⁄ are the strand axial strain, rotation per unit length and curvature respectively. The

Eqn (39) gives the stiffness coefficients of the cable over a sheave. It can be noted that the

above stiffness coefficients represent the sum of the stiffness of the non-contacting wires and

that of the contacting wire with the pulley/sheave. These expressions are shown in Appendix

Part A & B respectively. It can be noted that the stiffness coefficients of the contacting wire

shown in Part B are as per the refined formulation. The respective stiffness coefficients of

Gopinath model can be deduced from Part B, as follows, that is ; ;



; ; ; ; when , in A, B, C, D, E, F & D, E, F are multiplied

by . In the case of free bending, pulley contact does not exist, and hence Appendix Part A

alone is applicable to all the wires.

9. EXPERIMENTAL SETUP

Figure 6a Schematic diagram of the Experimental set up

Figure 6b Actual experimental set up

The schematic diagram of the experimental setup of strain measurement of a cable over a

pulley is shown in Fig 6a. The cable pulley assembly is fixed to a loading frame, which

consists of a screw and nut hand-wheel assembly, which imparts axial load on the cable. The

axial load can be measured by a load cell and associated load indicator. The cable consists of

a galvanized strand of seven wires with a central core whose specifications are shown in

Table 1. The axial strains in the contact wire, over the pulley are recorded by affixing strain

gauges on each wire. The surface of the wires was thoroughly cleaned at the location where

the gauges had to be mounted, to remove any grease or dirt present. The cable is allowed to

pass through a U-grooved pulley of 120mm radius. A graduated scale affixed on the

circumference of the pulley rim records, the angular locations of the contacting wires. The

complete experimental setup is shown in Fig 6b. The maximum tensile load applied on the

cable is restricted to 25% of its breaking strength, as adopted in the practical work of such

strands. The measured strain values are tabulated in the Table 3 for various axial loads.

10. RESULTS AND DISCUSSIONS:

The analysis of the cable is carried out using the specifications shown in the Table 1

Table 1 Specifications of galvanized strand

Parameters Symbols Values

Number of helical wires m 6

Radius of core Rc 3.2mm

Radius of helical wire Rw 3.15mm

Helix angle α 83º

Young’s modulus for core and wire Ec & Ew 210000N/mm2

Poisson’s ratio 0.3

Coefficient of friction between core-wire 0.5

Coefficient of friction between sheave-wire 0.5

Radius of curvature 120mm

Position angle of helical wires strand ɸ 0º - 360º



The stiffness coe`fficients (Kaa…Kbb) of the cable are computed for two different cases

namely a) the free bending & b) the constrained bending over a sheave and are tabulated in

the Table 2. In the free bending case, the stiffness coefficients are computed from Part A of

Appendix extending the summation for all the six wires in the outer layer with core. The

results are compared with LeClair and Costello [5] (fundamental research work quoted by all

researchers) who have not considered the wire stretch effect and internal friction among the

wires. Computations made in the free bending case, with and without internal friction among

the wires are tabulated in columns 3 & 4 of Table 2. Since these results include the wire

stretch effect, tabulations become a ready reference with that of LeClair and Costello, who

have not handled these parameters. The second part of the Table 2 presents the results of the

constrained case ie the bottom wire4 in contact with the sheave. The stiffness coefficients of

the non-contact wires are calculated from Part A of the Appendix and that of the contact wire

from Part B of the Appendix and the combined values are tabulated in the last column of

Table 2. The stiffness coefficients as derived by Gopinath [2] are calculated and presented in

column 5 while the results as per the present refined formulations are made in column 6. The

difference in the values of column 5 and 6 indicate the refined expressions, which probably

got omitted in Gopinath’s work but considered in this paper. The last column of Table 2

indicates the results of the present model, with the refined expressions for the constrained

bending and inclusion of internal friction among the wires, which is a new contribution in

this paper.

10.1. Stiffness response

It can be observed from the results in free bending that the consideration of wire stretch effect

has refined all the stiffness coefficients and has shown an increase of 6% for the torsional

stiffness from that of LeClair and Costello. When frictional effect at the wire interfaces are

considered, the stiffness values register a significant reduction, due to the slip of the wires at

the contact interfaces, as shown in the column 4 of Table 2. In the constrained bending case,

with the present model accounting all the terms, the stiffness coefficients are believed to be

exact and complete. Though the overall results of the present model and that of Gopinath

model seem to be closer as in column 5 and 6, a significant increase of 14% in bending

stiffness is noticed in the present model, justifying the exact derivations made in this paper.

The axial force and the bending moment in the cable can be calculated using the stiffness

values in Eqn (39) for various cable axial strains. Figs 7a & 7b show the results of the axial

stiffness and bending stiffness of the cable in free bending and constrained bending modes, as

per the present refined models. The axial stiffness of the cable is found to increase as a

function of axial strain and the maximum increase is registered as 10% between constrained

and unconstrained (free) modes. The results of the bending stiffness indicate a reduction in

their values as a function of axial strain and the maximum reduction is registered as 14%

between constrained and unconstrained modes. The results of the present constrained model,

with consideration of internal friction among the wires (column 7 of Table 2) show a

significant reduction in all stiffness elements, justifying the inclusion of internal friction

among the wires as a necessary refinement in the analysis.

10.2. Strain response

The axial stain in the contacting wire around the pulley can be estimated using Eqn (14) and

Eqn (39) for different applied axial loads, without and with internal friction in the wires. The

respective results are presented in Table 3 in column 2 and column 3, for different axial

loads. The measured values of strain as per the detailed experimental procedure on this

contacting wire, are tabulated in column 4 for these applied axial loads. The computed results

with consideration of internal friction in the wire are relatively closer to the experimental



observations, justifying the inclusion of internal friction as a desirable parameter in the

constrained bending research.

Table 2 Comparisons of cable stiffness coefficient for free bending and constrained bending

Stiffness

coefficient

Free bending Constrained bending

LeClair

and

Costello

(1988)

Present model

with no

internal

friction among

the wires

Present model

with internal

friction among

the wires

Gopinath

model

(2012)

Present model

with no

internal

friction among

the wires

Present model

with internal

friction among

the wires

112,90,432 112,90,462 104,54,567 128,57,944 128,68,563 120,33,791

37,38,826 37,38,814 37,71,116 43.48,904 43,54,344 43,91,809

0 0 0 -49,70,977 -50,06,199 -24,69,282

35,60,117 37,38,814 36,71,019 43,48,752 43,33,424 42,70,267

66,66,235 70,89,637 64,84,968 73,29,824 73,08,243 67,10,194

0 0 0 -19,26,609 -18,83,064 -19,01,141

0 0 0 -49,72,345 -50,10,567 -69,69,041

0 0 0 -19,35,518 -19,54,307 -9,85,351

608,98,541 609,00,246 555,20,775 669,29,416 757,14,018 708,86,518

Figure 7a Cable axial force Vs cable axial strain

Figure 7b Cable bending moment Vs cable axial strain

Table 3 Contacting wire strain values

Force N

Axial strain in wire number 4

Computed values as per theory Measured values

from the experiment Present model with no internal

friction among the wires

Present model with internal

friction among the wires

0 0 0 0

4000 0.0254 0.0241 0.0203

8000 0.0251 0.0238 0.0200

12000 0.0248 0.0235 0.0198

16000 0.0245 0.0232 0.0169

20000 0.0242 0.0229 0.0168



11. CONCLUSIONS

A single layered helical strand with a straight central core has been investigated for its

performance around a pulley /sheave. Analytical expressions are derived for the stiffness

coefficients of the cable models represented without and with internal friction among the

wires. The stiffness responses and strain responses are predicted and the results are verified

with strain measurements on the wire in contact with the pulley/sheave.

REFERENCES

[1] Boston, C., Weber, F., and Guzzella, L., Optimal Semiactive Damping of Cables With

Bending Stiffness, Smart Material Structure, 2011, 20, p. 055005.

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Engineering Mechanics, 1982, Vol 108, No 219-227.

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torsion and bending, Journal of Applied Mechanics, 1985 Vol 52, No 423-432.

[8] LeClair R.A and Costello G.A., Axial bending and torsional loading of a strand with

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[9] Love, A.E.H. A treatise on mathematical theory of elasticity, Dover Publications, New

York 1944.

[10] Papailiou, K. O., 1995, Bending of Helically Twisted Cables Under Variable Bending

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Eidgenossische Technische Hochschule Zurich, Zurich, Switzerland.

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1998

[12] Sathikh, S., Rajasekaran, S., Jayakumar, C.V. and Jebaraj, C. General thin rod model for

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[13] Vinogradov, O.G. and Atatekin, I.S., Internal friction due to wire twist in bent cable,

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[14] S. Karthik Ramnarayan, S. Ramasundaram and Aishwarya Anil, A Comparative Study on

Lateral Stiffness of Plating and Plan Braces in Offshore Decks. International Journal of

Civil Engineering and Technology, 8(8), 2017, pp. 1213–1217

[15] Sheethal Mary Jose, Asha U Rao, Dr.Abubaker KA Comparitive Study on The Effect of

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http://dx.doi.org/10.1088/0964-1726/20/5/055005

http://dx.doi.org/10.1016/j.ijsolstr.2006.06.045



APPENDIX

Part A

Stiffness coefficients for the non-contacting case

(

)

)

Part B

Stiffness Coefficients of the bottom contact wire at position angle



(

)

(

)

(

)

(

)

(

)

(

)

(

)

(

)

(

)

stiffness of a single layered cable …...and accounted for the change in curvature in single and...

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