CONSTRAINED BENDING PHENOMENON OF A SINGLE LAYERED CABLE ASSEMBLY

International Journal of Advanced Research in Engineering and Technology (IJARET) Volume 9, Issue 2, March April 2018, pp. 58 67, Article ID: IJARET_09_02_008 Available online at http://www.iaeme.com/ijaret/issues.asp?jtype=ijaret&vtype=9&itype=2 ISSN Print: 0976-6480 and ISSN Online: 0976-6499 IAEME Publication CONSTRAINED BENDING PHENOMENON OF A SINGLE LAYERED CABLE ASSEMBLY Rajesh Kumar P Assistant Professor, Mechanical Engineering Department, MVJ College of Engineering, Bangalore-67 Parthasarathy N.S Professor, Mechanical Engineering Department, Christ (Deemed to be University), Bangalore-29 ABSTRACT The stiffness response relations of a helical cable assembly have been formulated, when the cable wraps around a pulley/sheave. The constrained bending phenomenon of such a cable has been investigated, by considering all the forces and moments relations of the helical wire, while the hitherto researchers had partially accounted them or had neglected a few terms. A numerical study has been carried out for a single layered galvanized steel cable, over a steel pulley. A single wire contact mode has been assumed and the stiffness elements of the cable are evaluated and compared with the existing literature. Cite this Article: Rajesh Kumar P and Parthasarathy N.S, Constrained Bending Phenomenon of A Single Layered Cable Assembly. International Journal of Advanced Research in Engineering and Technology, 9(2), 2018, pp 58 67. http://www.iaeme.com/ijaret/issues.asp?jtype=ijaret&vtype=9&itype=2 1. INTRODUCTION Over the years many types of stranded cables have been designed and tested for various applications, where a tensile load has to be transmitted by a flexible member. Apart from the axial response, the bending response becomes vital when the cable passes over a sheave/pulley and is imposed with a transverse loading arising due to its contact with the cable. Theoretical analysis of such a constrained stranded cable is inherently rather complex. The analysis of the cable running over a sheave/pulley has been studied by different authors over a period of time and a few significant works are reviewed as under. Philips & Costello (1985) formulated an analytical model to study the associated forces in the individual wires of the helical strand. The model was constructed for sheave bending case but neglected the shear force in the bi-normal direction. LeClair and Costello (1988) had given an improved treatment to the wire centreline which had taken the shape of the deformed helix. Hobbs and Nabijou (1995) studied the bending strain in the wires of frictionless ropes, by accounting the change in curvature in single and double helices, when the rope is bent into circular arc and derived it from the first principles. The bending strains were observed to be more in the innermost layer in contact with the strand than the outer layer. Ridge M L (2000) investigated the variations in cyclic bending http://www.iaeme.com/ijaret/index.asp 58 editor@iaeme.com

Rajesh Kumar P and Parthasarathy N.S strain, for the stranded cable over the pulley, using electrical resistance strain gauges but with no analytical support. Usabiaga (2006) formulated a new mathematical model considering the classical Coulomb friction law and employed an iterative procedure to solve the analytical model. The theory seemed to be efficient and reliable in most of the cases, but causes errors when the diameter of the sheave becomes very small. Gopinath D et al (2012) have studied the stiffness response of a cable over the flat drum considering the effects due to tension, torsion and bending. While establishing the stiffness for the helical wire in contact with the drum it was noticed that the author probably omitted certain parameters, and hence a refined treatment is required. Ya fei Lu etal (2015) developed an analytical method to predict the precise cable drive as a function of bending rigidity. Though it was discussed in detail as an alternative to various existing transmission capability, the author concluded that further research is needed on the evaluation of cable stiffness parameters. Yunapei C etal (2016) established a mathematical model to study the inter-wire phenomenon of the strand subjected to cyclic bending load and concluded that the wear indent takes place in the normal direction of the contacting wires having zero slippage. But when the cables are subjected to cyclic bending, the contribution in the bi-normal direction had been neglected. The above literature survey in the constrained bending phenomenon indicates that the estimation of stiffness of the cable assembly has not been done with all the wire forces and moments in the normal, bi-normal and axial directions. Though omission of one or two parameters have not significantly affected the overall results of the cable, consideration of all the parameters become vital to predict the local wire behavior well. Hence this paper addresses to estimate the stiffness of the cable on the sheave with all the parameters of wire forces and moments mentioned above. Further it considers the wire stretch on the changes in the curvature and twist relations. Though the works of Gopinath (2012) have studied these considerations, it is observed that some terms got omitted in the derivation by the author, probably due to oversight. Hence this paper predicts the stiffness of a cable assembly over a sheave, in a holistic way, setting right the terms omitted by Gopinath (2012). The difference in results are analysed for a single layered cable assembly, with a central core. 2. BASIC MEHCANICS RELATIONS OF A BENT WIRE The deformed conditions of a strand with six helical wires and a central straight core can be obtained from the developed geometry of a single deformed helical wire as shown in the Fig 1 below. Figure 1 Wire geometry Fig 1(a) shows the axial loading of a strand and Fig 1(b) shows the developed geometry of the helical wire in a layer as, The change in helix angle and change in length of the wire can be obtained from Fig 1(b) δα=[δhcosα/(r) sinαδχ]/(l/r) (01) δl=[lcosαδχ+(l/r)sinαδh](l/r) (02) http://www.iaeme.com/ijaret/index.asp 59 editor@iaeme.com

Constrained Bending Phenomenon of A Single Layered Cable Assembly where h is the strand length, and l & r, are the wire length and helix radius, and α &, are the helix angle and the swept angle of the helical wire. During the axial loading, the strand undergoes an axial strain (ϵ) and rotational strain (γ), as a result of which, the axial strain in the wire can be expressed as ϵ () =ϵsin α+γcos α (03) where γ= = rtanα (04)! When the strand is additionally bent, the axial strain in the wire can be computed by superimposing the axial and bending results. Fig 2 shows the geometry of the strand under free bending, with a constant radius ρ as Figure 2 Geometry of bent strand The wire axial strain $ % per unit length of a wire at a position angle ϕ can then be expressed ε (()*+*,) = -.*/ 012.3 (05) 4 Under the combination of axial and pure bending, the net axial strain of the wire (ε ) can be expressed by adding equation (03) and (05) ε =εsin α+rsinαcosα + -.*/ 012.3 (06)! 4 The change in curvatures and twist expressions of the helical wire subjected to combination of axial deflection and bending of the strand can be obtained as k=.*0(7812./ 0).* 4 k : = ;.*/ 012. / 0 ϵ+sinαcosα(1+sin α) +.*= 012. -! 4 (07) (08) τ=.*012.? 0 ϵ+sin @ α.*? 012.012. (09) -! 4 It can be noted that these relations are due to the refinements that took into account the wire stretch effect, as an additional contributory parameter. This feature though introduced in axial case, is accounted fully in bending now. The resulting forces and couples in the helical wire for the bent strand can be obtained from the following mechanics relations. T=EAD E (10) G=EI k (11) http://www.iaeme.com/ijaret/index.asp 60 editor@iaeme.com

Rajesh Kumar P and Parthasarathy N.S G =EI k (12) H=CJ τ (13) where T is the axial force & G, G are the bending moment in the normal and bi-normal directions & H is the twisting moment in the axial direction of the wire respectively. The axial, bending (flexural) & torsional stiffness of the wire are EA, EI & CJ respectively. WIRE EQUILIBRIUM EQUATIONS The equilibrium equations for the helical wire in the normal, bi-normal and axial directions are given by +L +. +L: +. +Q +. +S +. US: UV UX UV N τ+tk +X=0 (14) Tk+Nτ+Y =0 (15) Nk +N k+z =0 (16) G τ+hk N +K =0 (17) HK+GΤ+N+K =0 (18) GK +G K+Θ=0 (19) where N, N are the shear forces in the normal and bi-normal directions of the wire and k, k & τ are the curvatures and twist of the helical wire at any instant. 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 the normal, bi-normal and axial directions respectively. Using the equilibrium equations from (14) to (19) the shear forces in the normal and binormal directions are given by N=-Gτ (20) N =Hk G τ (21) The above relations express the state of the wire in a free bending mode 3. CABLE BENDING OVER A SHEAVE/PULLEY When the cable is bent over a sheave/pulley, that portion of the wire in contact with the pulley receives additional forces p & q in the normal and bi-normal directions of the wire, which can be duly accounted with the fictional effects at the contacting wire and pulley interface. Figure 3 Wire in contact with the sheave http://www.iaeme.com/ijaret/index.asp 61 editor@iaeme.com

Constrained Bending Phenomenon of A Single Layered Cable Assembly Fig 3 shows a contacting wire (bottom wire) with the pulley surface and the corresponding forces p & q introduced by the pulley on it. The position angle of the contact wire at this instant is ϕ=180 as per the angular notations shown in Fig 3. While the wire that makes contact with the pulley will be treated with constrained bending relations, the remaining wires in the outer layer will be treated with the relations shown for free bending. As a result of the constrained bending of the contacting wire (at position angle ϕ =180 ) with the sheave, the axial and bi-normal forces in the wire are modified as under T 7-) =ε EAsin α+earsinαcosα + \]-.*/ 012.! 4 +μpρ (22) N 7-) =Hk₀ G τ₀+q (23) 4. CABLE EQUILIBRIUM EQUATIONS Resolving the wire forces and moments along the cable axis, The axial force, twisting moment and bending moment of the cable are given as under F =E 1 A 1 +(m 1)(Tsinα+N cosα)+t 7-) sinα+n 7-) cosα (24) M f =G 1 J 1 +(m 1)(Hsinα+G : cosα+trcosα N rsinα)+(hsinα+g cosα+ T 7-) rcosα N 7-) rsinα) (25) M ( =E 1 I 1 +(m 1)[(Tsinα+N cosα)rcos +Gsin +G sinαcos Hcosαcos Ncotαrsin ]+[(T 7-) sinα+n 7-) cosα)rcos +Gsin +G sinαcos Hcosαcos Ncotαrsin ] (26) where m is number of wires in the outer layer The above expressions are accounted with the contribution of the core parameters, having the suffix c. 5. CABLE STIFFNESS RELATIONS The equilibrium Eqns (24) to (26) mentioned above can be expressed in matrix form as under h А o pp o pq o pr $ gj k m=no qp o qq o rq sgt hm (27) j l o rp o rq o rr 1 w where ε, δχ h and 1 ρ are the strand axial strain, rotation per unit length and curvature respectively. K, K ff and K (( are effective strand axial stiffness, strand torsional rigidity and flexural rigidity. K f and K f are the tension-torsion, K ( and K ( are the tension-bending. K (f and K f( are the torsion bending coupling parameters, respectively. The stiffness parameters in the above equation are shown in the Appendix in Part A for free bending and in Part B for constrained bending. 6. RESULTS AND DISCUSSIONS: The cable stiffness parameters mentioned in the previous section and in the Appendix are numerically evaluated for a single layered galvanized strand assembly with a straight central core whose specifications are shown in Table 1 http://www.iaeme.com/ijaret/index.asp 62 editor@iaeme.com

Rajesh Kumar P and Parthasarathy N.S Table 1 Specifications of galvanized strand Parameters Symbols Values Number of helical wires m 6 Radius of core R c 3.2mm Radius of helical wire R w 3.15mm Helix angle α 83º Young s modulus for core and wire E c & E w 210000N/mm 2 Poisson s ratio x 0.5 Coefficient of friction y between sheave-wire z;% 0.5 Radius of curvature of the strand w 120mm Position angle of helical wires in the strand ɸ 0º - 360º The numerical values of the stiffness parameters are computed as per the equations of LeClair & Costello (1988), who have adopted the basic mechanics relations and whose model has been used as a reference in many research works. The resulting values are tabulated in the 2 nd column of the Table 2. It can be noted that LeClair & Costello model has defined the phenomenon of free bending of the cable without contact with the pulley. Since the present paper has refined the normally used wire curvature and twist relations with inclusion of wire stretch, the corresponding stiffness parameters are evaluated and tabulated in column 3 of Table 2, for the free bending mode. Though the overall individual stiffness values between LeClair & Costello and the present model, in free bending mode, do not significantly change, consideration of the refinement with the wire stretch effect, has yielded significant change in the wire twist and bending components that made up the total stiffness value. As a sample explanation, the breakup of the bending stiffness coefficient (K (, K (f & K (( ) for the bottom wire (wire no 4 at position angle ϕ=180 ) is shown in Table 3, for the LeClair & Costello and present model with the wire stretch effect. It can be observed that consideration of wire stretch effect, has significantly changed the twist and bending component of the bending stiffness coefficient for the wire no 4, while axial component remains unaltered. Though this may not affect the overall wire axial elongation due to the predominant effect of axial component, its local twisting moment and bending moment are altered, thereby contributing for exact evaluation of local effects, like slip initiation, growth etc. Computations are also made for evaluating the stiffness coefficients of the galvanized cable ( shown in Table 1), when it is passed over pulley, thereby making contact with the wire, and contributing to the constrained bending phenomenon, as per the formulations made in this paper in Section 3. The stiffness coefficients are evaluated using the theoretical relations shown in Appendix. The relations shown in Part A are used for the wires that are not in contact with the pulley and that shown in Part B are used for the wires that make contact with the pulley. The stiffness coefficients are evaluated for the cable at an instant as shown in Fig 3, where the bottom wire no 4 at position angle ϕ =180, makes contact with the pulley thereby undergoing constrained bending and the other wires are free from contact with the pulley, undergoing free bending phenomenon and are tabulated in the 5 th column of the Table 2. It can be noted that formulations of the constrained bending case mentioned in this paper have accounted all the terms for the contacting wire, as presented in Appendix Part B, while the formulations used by Gopinath (2012), have considered them partially. The respective stiffness coefficient of the contacting wire (wire no 4) for Gopinath model can be obtained from Appendix Part B, as K @ =A ; K f@ =B ; K (@ =Ċ ; K f@ = D ; K ff@ =E ; K (f@ =F ; when R = 1, in A, B, C, D, E, F & D, E, F are multiplied by ρ. http://www.iaeme.com/ijaret/index.asp 63 editor@iaeme.com

Constrained Bending Phenomenon of A Single Layered Cable Assembly Model Table 2 Comparisons of cable stiffness coefficient for free bending and constrained bending Stiffness coefficient Free bending model LeClair and Costello (1988) Present model Constrained bending model Gopinath model (2012) Present model k (N) 112,90,432 112,90,462 128,57,944 128,68,563 k f (N mm) 37,38,826 37,38,814 43.48,904 43,54,344 k ( (N mm) 0 0-49,70,977-50,06,199 k f (N mm) 35,60,117 37,38,814 43,48,752 43,33,424 k ff (N mm ) 66,66,235 70,89,637 73,29,824 73,08,243 k f( (N mm ) 0 0-19,26,609-18,83,064 k ( (N mm) 0 0-49,72,345-50,10,567 k (f (N mm ) 0 0-19,35,518-19,54,307 k (( (N mm ) 608,98,541 609,00,246 648,10,948 757,14,018 The stiffness coefficients for the cable in Table 1 are evaluated as per Gopinath formulations and are presented in column 4 of Table 2, to compare the results in column 5 with that of the present model, where the formulations are complete with all the terms. 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 increase in the bending stiffness coefficient of the present constrained bending model is justified with the breakup of individual components of axial, twisting & bending of the contacting wire as shown in Table 3. It can be observed that the axial component of the bending stiffness of the contacting wire has a significant increase, due to its contact with the pulley. The variation of the bending stiffness of the contacting wire has been studied as a function of the spatial locations of the wire on the pulley (pitch length) for the present model and Gopinath model and have been plotted in Fig 4. It can be observed that the present model yields a higher bending stiffness than that of Gopinath model, as it has accounted all the terms during the formulation process, justifying the refinement made in this study. Table 3 Break up of stiffness coefficient for wire 4(at position angle = ) for free bending and constrained bending models Bending-Tension (N-mm), component Bending-Torsion (N-mm 2 ), component Bending-Bending (N-mm 2 ), component Axial Twist Bending Axial Twist Bending Axial Twist Bending LCM -5080730 3464 9146-1980676 89596-241643 16131318 11337 977653 PFBM -5080730 53 4573-1980676 90967-238298` 16131318 11169 963133 PCBM -101,56,156 49 4,505-39,33,986-94,696-2,34,947 320,36,537 10,711 9,49,586 LCM: LeClair & Costello Model, PFBM: Present Free Bending Model, PCBM: Present Constrained Bending Model. http://www.iaeme.com/ijaret/index.asp 64 editor@iaeme.com

Rajesh Kumar P and Parthasarathy N.S Present constrained model Gopinath model Bending Stiffness N-mm 2 2.E+07 2.E+07 2.E+07 1.E+07 1.E+07 1.E+07 8.E+06 6.E+06 4.E+06 2.E+06 0.E+00 0 50 100 150 200 Pitch Lenght mm Figure 4 Bending stiffness of the contacting wire vs pitch length 7. CONCLUSIONS The stiffness response of a single layered cable with six helical wires around a central straight core has been studied over a sheave. A refined mathematical model has been developed to analyse the contact effects of the sheave over the helical wire, and the analytical expressions are established assuming a single wire contact with the sheave. The present model has been analysed under two different cases of free and constrained bending, and the results are compared with that of LeClair & Costello (for free bending) and with that of Gopinath model (for constrained bending) justifying the improvised formulations. REFERENCE (1) Gopinath D et al Theoretical estimation of stiffness of stranded cable subjected to constrained bending International Journal of Offshore and Polar Engineering,2012, Vol 22 No 4,330-336. (2) Hobbs.R.E and Nabijou.S Changes in wire curvature as a wire rope is bent over a sheave Journal of Strain Analysis, 1995, vol 30, No 271-281. (3) LeClair R.A and Costello G.A., Axial bending and torsional loading of a strand with friction Journal of offshore mechanics and Artic engg, 1988 Vol. 110 No 38-42. (4) Philips J.W & Costello G.A., Analysis of wire ropes with internal-wire rope cores Journal of Applied Mechanics, 1985, Vol 52, No 510-516. (5) Ridge I M L, Zheng J and Chaplin C R, Measurement of cyclic bending strain in steel wire rope Journal of Strain Analysis, 2000, Vol 35(6), No 545-558. (6) Usabiaga H, Ezkurra M, Madoz M and Pagalday J., Mechanical interaction between wire rope and sheavesoipeec Conference 2006, Trends for rope, Athens Greece(2006), No 157-166. (7) Ya fei Lu, Da-peng Fan, Hua Liu, Mo Hei Transmission capability of precise cable drive including bending rigidity Journal of Mechanism and Machine Theory, 2015, Vol 94, No 132-140. http://www.iaeme.com/ijaret/index.asp 65 editor@iaeme.com

Constrained Bending Phenomenon of A Single Layered Cable Assembly (8) Yuanpei C, Fanming M and Xiansheng G Interwire wear and its influence on contact behaviour of wire rope strand subjected to cyclic bending load Wear, 2016, 368-369, No 470-484. APPENDIX Part A Stiffness coefficients for the non-contacting case ƒƒ = ˆ Š Œ+ Žˆ ŠŒ ˆ Œ8 ˆ Š Œ ˆ Œ + ƒ = ( ˆ Š Œ ˆŒ+ Žˆ Š Œ ˆ Œ; ˆ Š Œ ˆ Œ(8ˆ Š Œ) ƒ = ˆ Š Œ ˆ Žˆ Š Œ ˆ Œ ˆ ˆ Š Œ ˆ Œ ˆ ƒ = ( Žˆ Š Œ ˆ Œ+ ˆ Š Œ ˆŒ ˆ Š Œ ˆ Œ ) Žˆ Š Œ ˆ Œ ˆ Š Œ ˆ Œ ) = ( Žˆ Š Œ+ ˆ ŠŒ ˆ Œ{+ˆ Š Œ}+ ˆ ŠŒ ˆ Œ Žˆ Š Œ ˆ Œ+ ˆ Š Œ ˆ Œ{+ˆ Š Œ})+œ Ž =( Žˆ Š Œ ˆŒ ˆ + ˆ Š Œ ˆŒ ˆ + ˆ Š Œ ˆŒ ˆ + Žˆ Š Œ ˆ Œ ˆ + ˆ Š Œ ˆŒ ˆ ) ƒ =( ˆ Š Œ ˆ + Žˆ ŠŒ ˆ Œ ˆ Žˆ ŠŒ ˆ Œ ˆ ) + ˆ Š Œ ˆ Œ ˆ ˆ Š Œ ˆ Œ ˆ =( ˆ Š Œ ˆŒ ˆ + Žˆ Š Œ ˆ Œ ˆ ˆ Š Œ ˆ Œ{+ˆ Š Œ} ˆ + ˆ Š Œ ˆŒ{+ˆ Š Œ} ˆ Žˆ Š Œ ˆ ) =( ˆ Š Œ ˆ Žˆ Š Œ ˆ Œ ˆ ˆ Š Œ ˆ Œ ˆ + ˆ ŠŒ{+ ˆ Œ}ˆ Š + ˆ Š Œ ˆ + Žˆ Š Œ ˆ Œ ˆ + ˆ ŠŒ ˆ Œ{+ ˆ Œ}ˆ Š ) + Part B Stiffness Coefficients of the bottom contact wire at position angle =180 ƒƒ = ƒƒ + ˆˆ ŠŒ+ž ˆŒ ƒ = ƒ +Ÿ ˆˆ ŠŒ+ ˆŒ ƒ = ƒ +Ċ ˆˆ ŠŒ+ ˆŒ ƒ = ƒ + ˆ ˆŒ žˆ ŠŒ = +Ÿ ˆ ˆŒ ˆ ŠŒ = +Ċ ˆ ˆŒ ˆ ŠŒ ƒ = ƒ + ˆˆ ŠŒ ˆ +ž ˆŒ ˆ = +Ÿ ˆˆ ŠŒ ˆ + ˆŒ ˆ = +Ċ ˆˆ ŠŒ ˆ + ˆŒ ˆ http://www.iaeme.com/ijaret/index.asp 66 editor@iaeme.com

Rajesh Kumar P and Parthasarathy N.S = Ÿ = Ċ= ƒƒ E 8 ˆ ŠŒ 8 ƒ ˆŒ (; ˆˆ ŠŒ) E 8 ˆ ŠŒ ; ˆ ˆ Œ ƒ E 8 ˆ ŠŒ 8 ˆŒ (; ˆˆ ŠŒ) E 8 ˆ ŠŒ ; ˆ ˆ Œ ƒ E 8 ˆ ŠŒ 8 ˆŒ (; ˆˆ ŠŒ) E 8 ˆ ŠŒ ; ˆ ˆ Œ ž= ƒ8 ˆ ˆŒ E 8 ˆ ŠŒ = 8Ÿ ˆ ˆŒ E 8 ˆ ŠŒ = 8Ċ ˆ ˆŒ E 8 ˆ ŠŒ http://www.iaeme.com/ijaret/index.asp 67 editor@iaeme.com