A REFINED SLIP MODEL FOR PREDICTING THE VARIABLE BENDING STIFFNESS IN A SINGLE LAYERED CABLE

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1 International Journal of Mechanical Engineering and Technology (IJMET) Volume 9, Issue 3, Decemer 8, pp. 3, Article ID: IJMET_9_3_ Availale online at ISSN Print: and ISSN Online: IAEME Pulication Scopus Indexed A REFINED SLIP MODEL FOR PREDICTING THE VARIABLE BENDING STIFFNESS IN A SINGLE LAYERED CABLE Shankar. G Associate Professor, Dept of Mechanical of Engineering, MVJ College of Engg, Bangalore, India Dr.N. S Parthasarathy Retired Professor, Dept of Mechanical of Engineering, MVJ College of Engg, Bangalore, India ABSTRACT A refined slip hypothesis that considers all the contriuting factors for the strand ending moment is proposed in the present work to predict the ending stiffness in a pretensioned cale. The model adopts the more familiar Coulom friction hypothesis and Hertzian contact hypothesis to study the slip ehaviour in the helically wound cale under ending load. The ending stiffness variation is studied for an ACSR single layer Racoon Conductor, as a function of curvature. Keywords: Bending stiffness, Slip, Interwire friction, Cale. Cite this Article: Shankar. G and Dr.N. S Parthasarathy, a Refined Slip Model for Predicting the Variale Bending Stiffness in a Single Layered Cale., International Journal of Mechanical Engineering and Technology, 9(3), 8, pp. 3 INTRODUCTION Helically wound cales find major application as conductors in the overhead electrical power transmission lines. The ehavior of power transmission lines during wind induced viratory status, impose ending loads, apart from the usual axial loads of tension and torsion, which are used to keep them taut. Many analytical models ased on the thin rod theory are availale to predict the response of cales under axial tension and torsional loads ut when cales are sujected to ending loads, the analysis ecomes much more complex particularly due to the role of friction effects etween the wires. The analytical models availale currently to predict the ending response focus on the extreme conditions like no-slip where the friction is unlimited and gross slip, where no friction exists. The analytical models which consider the complete transition of wires in a layer of a strand from stick state to slip state under ending load are very limited. Even the few existing ending models relate to the slippage of wires ased on either Coulom friction stick gross slip hypothesis or Hertzian contact Coulom friction microslip hypothesis. As per the former hypothesis, all the wires are initially assumed to e intact and ehaving in a monolithic way with a higher ending stiffness. The Hertzian contact microslip condition 3 editor@iaeme.com

2 Shankar. G and Dr.N. S Parthasarathy considers the complete transition of the wires in a cale from an initial full stick condition to the final full slip condition. The gradual transition of the wires in the cale can e monitored y using this theory. The prediction of the exact ending stiffness at every stage of slipping of wires in a cale is very essential to know apriori the actual failure of cales in reality. McConnell and Zemke (98), conducted experimental tests on different conductor samples and had explained the ranges of their stiffness values. Costello and Butson (98) developed a cale ending model which incorporated wire twist ased on the Love s equation (944) of thin rod ut the model neglected the effect of friction and slipping in the wires. Considering the predominant wire axial force, Lanteigne (985) otained the ending moment of the cale. A stair stepping effect was oserved during the ending stiffness prediction in the stick -slip regime. According to this study, the slip process starts in the wires lying farthest from the neutral axis. Using the kinematic models for the deformed and undeformed wire, Knapp (988) proposed the ending and twisting strains, which accounted only the extreme conditions of no slip and no friction states. Sathikh and Parthasarathy (988) studied the ending response of a strand under sinusoidal transverse displacement. The model proposed accounted for the predominant axial force along with the three wire moments in a resting lay condition. Papailiou (995,997) proposed a model ased on thin rod approximation, in which the individual wire ehaviour under tensile and ending loads were studied. The model considered the presence of interlayer friction and slip in the conductor during ending. A smooth transition of the ending stiffness from stick to slip state was oserved in this work.cardou and Jolicoeur(997) summarised the existing cale models. The author emphasized for the development of new cale model for predicting the local ehaviour and gloal ehaviour of wires more efficiently, particularly under ending loads, applicale for more complex situations like multi-layered cales etc. Sathikh () proposed a preslip model for predicting the ending response of helically wound cales under core-wire contact. The study in the preslip stage was ased on the Unlimited Coulom friction hypothesis. The model accounted for the major wire axial force and the moments in the wires. The model considered the effect of wire stretch on the wire curvature and twist expressions, taking clue from Ramsey (988). Hong et al (5) extended the model proposed y Papailiou for a multi-layered cale under free ending.the slip phenomenon in the wires were studied y considering the non linear response of the cales under frictional forces. Inagaki K et al (7) developed a new model to evaluate the response of cale under ending, taking frictional effects into account. Spak (3) reviewed the recent models availale to predict the cale damping. The need for further study in the direction of interwire friction mechanism, variale ending stiffness was emphasized y the author. Foti (6) predicted the mechanical ehaviour of metallic strands using Euler Bernouli eam theory. The model neglected the deformation of contact surfaces, ut accounted for the friction using Couloms law for studying the stick slip condition. Though many models are availale to predict the ending stiffness and stick slip ehaviour, a complete model considering all the aspects together is still wanting and hence the present work attempts in predicting of the cale ending stiffness y duly accounting the slip ehaviour at each and every stages of the ending phenomenon. A refined slip hypothesis, which comines the Coulom friction law and the Hertzian contact phenomenon is adopted to explain the slip at the contact interfaces. The resulting loss of stiffness along the cale is studied as a function of axial load and the ending curvature at any location. Inclusion of all the wire forces and wire couples with refined expressions for the wire curvatures and twist are some of the specific features introduced in this work, proaly for the first time in ending studies.. REFINED SLIP HYPOTHESIS The present work evaluates the displacement of the helical wire δ as per Couloms friction stick gross slip hypothesis and that from Hertzian contact microslip hypothesis δ and tries to comine 4 editor@iaeme.com

3 A Refined Slip Model for Predicting the Variale Bending Stiffness in a Single Layered Cale. them as a two series spring assemly,since the displacements are caused y the same wire axial force. The total displacement in a series spring comination is the sum of the individual displacements, it can e given as δ = δ + δ () Since the displacements that occur in ending can e related proportionally to the respective curvatures κ and κ,the total curvature of the eam, when oth the hypotheses are comined can e given as κ = κ + κ,which can e rewritten as κ δ κ = κ + = κ + κ δ Where κ is the curvature of the eam undergoing a fire effect. 3. MATHEMATICAL RELATIONS FOR A BENT STRAND UNDER REFINED SLIP HYPOTHESIS. The configuration of a ent strand along with the position of the helical wires is shown in Figure. () Figure.: Geometry of ent strand From Figure, the strain of a helical wire, placed at a helical radius r and position angle φ,can e otained in the axial direction of the strand as, ε = κ r cos φ (3) Where κ is the ending curvature. The resulting strain in the axial direction of the wire using the lay angle of the wire β, can e otained as, ξ = κ r cos β cos φ (4) w 3.. Displacement of the helical wire as per Coulom friction hypothesis The displacement δ of the helical wire in its axial direction, can e otained from the strain relation as 5 editor@iaeme.com

4 Shankar. G and Dr.N. S Parthasarathy δ = ds ξ w (5) And can e related with the equilirium equation of the helical wire, which relates the wire force as T = EAξ To get, (6) w δ = BZκ sin φ (7) Where B (r / R) Eπsin β = (8) And Z = EA cos β sin β (9) 3.. Displacement of the helical wire as per Hertzian contact hypothesis The displacement of the wire in the axial direction ( δ ) due to the contact of the core and the helical wire can e otained from the wire compliance equations in the normal direction and in the tangential direction and the friction force arising at the contact location. Applying the contact stress theories for the curved helical wire and its underneath core, δ can e otained as /3 3 µ XS Z δ = ν µ X () Since there are m wires located at different φ values in the cross section of a strand, the discrete slip/displacement phenomenon can e studied aove for one single wire along its length and can e averaged over a continuous range π,and multiplied y the numer of wires as under. δ = BZ κ π () Similarly, the averaged value of δ can e otained as π/ 3 Zκ sin φ δ = D dφ π µ X () 3.3. Evaluation of ending curvature at various slip stages The ending curvature κ for a helical wire at a position angle φis related with the fire axial force Z in Equation (9), which can e rearranged as Z = Zκ sin φ (3) Equating this wire axial force Z,to the axial friction force caused y the radial clenching force X,the curvature at which the slip will e initiated at the helical wire at the outermost layer can e given as, 6 editor@iaeme.com

5 A Refined Slip Model for Predicting the Variale Bending Stiffness in a Single Layered Cale. µ X κ = Z (4) When the curvature of the ent cale exceeds that shown in Equation (4), the wire undergoes slipping and reaches a final state of full gross slip. The helical wire egins to slip at its outer most position φ = 9 and reaches a full slip state when φ = and in any intermittent position the wire undergoes the partial state of slip. Sustituting curvature at full slip is given y ( / ) φ = when it crosses the neutral axis, the κ f = π κ (5) The curvature at any intermittent partial slip state can e otained y sustituting that position angle of the wire φ = to 9 in Equation (4) and otained as κ = κ π φ cos φ 4. STRAND BENDING MOMENT The strand ending moment is otained y considering all the three wire forces and three wire couples in this paper. Since the shear force N lies in the same plane of ending, its effect is null. The shear force in the inormal direction of the helical wire N is negligily small compared to the wire axial force, T and the effect of N is also neglected in the calculation of strand ending moment. Hence the strand ending moment is evaluated y accounting the effect of the wire axial force Tand the wire moments G, G and H,as follows. It can e noted that, since this work addresses the slip phenomenon along the axial direction of the wire, the different slip stages caused y the wire axial force Tis addressed, while the wire moments G, G and H are not affected y the slip phenomenon. The components of the wire moments G and H along the transverse direction of the cale are accounted for the strand ending moment. 4.. Layer ending moment due to wire axial force Stage : No slip stage: In the no slip stage, the ending curvature is less than the initial curvature and hence, the moment in the layer due to axial force T is given y M. π/ MT = m Tr cosβ cos φdφ π Stage : Intermittent stage/mixed stage: In the intermittent stage, the slip happens in two phases as partial slip and full slip. In the partial slip region, the wire axial force is given y TP,where the value of φ ranges from to φ i.e. ( φ φ ) axial force is given y T s where the value of φ ranges from φ to The wire axial force in the partially slipped region is given y T (6) (7). In the fully slipped region, the wire π i.e. ( ) π φ φ. Tp EAr cos cos = κ β φ π r While that in the fully slipped region is given y T s = µ X φ sin β (8) (9) 7 editor@iaeme.com

6 Shankar. G and Dr.N. S Parthasarathy Accordingly, the layer ending moment in the partial and fully slipped regions are given y φ MT = m T p pr cosβ cos φdφ π () π MT = m T S sr cosβcos φdφ π φ The total ending moment in the intermittent stage is given y M = MT + M Ti p TS Stage 3: Fully slipped stage: In the fully slipped state, the oundary φ = and hence the final curvature is given y κ = κ = ( π ) stage is given y M Tf f () () κ and the layer ending moment in the fully slipped µ Xr cosβ = m π sin β (3) The layer ending moment M T due to wire axial force T is given y the per the existing slip status of the helical wire, in that position angle. 4.. Layer ending moment due to the rotation of the wires The layer ending moment due to the contriution of the wire couples G, respectively as π/ MG = m EI κ sin φdφ π π/ M m EI cos cos d G π = κ β φ φ π/ MH = m GJ τsin βcos φdφ π MT or M or M as T i T f G and H are given Where κ, κ are the curvatures of the helical wire along normal, inormal direction and τ is the twist of the wire and are given y, cos β ( + sin β) sin φ κ = ρ, 4 cos βcos κ = 4.. Strand ending moment φ ρ and (4) (5) (6) 3 sin βcos βcos φ τ = ρ (7) The strand ending moment M B can e otained y adding the ending moment arising from the layer of helical wires and that of the central core. The ending moment from the layer is the sum of the moments caused y wire axial tension and wire couples and is given as M = M + M + M + M L T G G H ( 8) 8 editor@iaeme.com

7 A Refined Slip Model for Predicting the Variale Bending Stiffness in a Single Layered Cale. The ending moment from the core is given y M C = E I κ C C Where E C & IC are the Young s modulus and moment of inertia of the core. Therefore, cale or strand ending moment is given y MS = ML + MC (9) (3) 4.3. Strand ending stiffness The cale or strand stiffness can e evaluated from the ending moment ( M S ) and ending curvature ( κ ) relations discussed in the previous sections, y plotting a graph and evaluating its slope at each curvature. The effective stiffness of the cale is given y (EI) eff = M κ S 5. NUMERICAL RESULTS The revised slip hypothesis discussed in the earlier section gives a scheme for the prediction of helical wire displacements ased on the Coulom friction and the Hertzian contact theories. The resulting stiffness of the cale/strand is otained as a function of strand ending curvature ( κ ). The aove theoretical formulations have een worked out numerically, for a single layer composite ACSR Racoon conductor, the specifications are shown elow in Tale. Tale : Geometric and material properties of Racoon conductor Parameter Value Core diameter, D, mm 4.9 c Wire diameter, D, mm 4.9 w Helix angle of the wire, α, Degree Elasticity modulus of the core, E, GPa 7 Elasticity modulus of the helical wire, E, GPa 79 Poisson s ratio of the core, γ c.3 Poisson s ratio of the helical wire, γ w.33 Numer of wires, m 6 Coefficient of friction, µ.3 Cale length, L,mm 3 For an applied axial load ( P ) on a straight conductor strand, the cale axial strain ( ε ) is calculated from the axial stiffness of the strand assemly, reported in the literature. The wire axial strain ξw and the wire axial force T,inormal force N and the radial clenching force X are evaluated as cited in the equilirium equations of Love s (944). The friction force caused y this radial clenching force X is evaluated and the fire axial force Z present in a helical wire at the core-wire interface is evaluated as per its position angle φ and compared with the friction force for different ending curvatures. It was oserved that up to 5 a ending curvature κ =.54E,the frictional force was greater than fire axial force and hence the cale exhiited no slip status. The ending moment due to fire effect MT from Equation (7) and the layer ending moment of the strand is evaluated from Equation (8) after accounting 9 editor@iaeme.com c w (3)

8 Shankar. G and Dr.N. S Parthasarathy the moments from the three wire moments. Considering the effects of the central core wire, the total ending moment of the strand is evaluated from Equation (3) and the stiffness of the strand in no slip stage is evaluated from Equation (3), for different curvatures. When the curvatures are further increased, slip starts at different locations in a cross section, as per the position angle of the wires and progresses. At this stage, some wires undergo partial slip, while some other tend to attain full slip. These are identified wire y wire with respect to their position angles and the respective curvatures are identified as per the two slip hypotheses. The resulting ending moments in partial slip and full slip stages of the wires are evaluated as in Equations (8) and (9).The fully slipped state of all the wires, in the cale, can e otained as per the curvature relations outlined in Equations() and the corresponding moment relations in Equation (),along with the layer,core and strand ending moment equation in (8),(9)and(3).The strand ending stiffness at this stage is evaluated from Equation(3),as in other cases. In the fully slipped stage, neither the fire effect nor the contact theory holds good and the ending moment that a helical wire can sustain is purely from its wire axil force T as in Equation (3).The moments due to three wire couples G,G and H will also prevail on each wire as in Equations (4) to (6), apart from that contriuted y the core as in Equation (9). The plot in Figure indicates the reduction of strand stiffness due to the various slip stages that occur as the curvatures are increased, for conductor axial loads of 3,4,6 and 7kg.The limiting values of cale ending stiffness in the monolithic and loose wire assemly are evaluated for the ACSR Racoon conductor using Mc Connell (98) approach and are plotted in Figure, as two straight lines. From the Figure, it can e oserved that the cale maintains monolithic ehaviour up to certain curvature loads, then turns to slip states as explained in the revised slip theory presented in the paper and then attains the loose wire ehaviour and reaches the lower limit of stiffness. Figure Bending Stiffness-Curvature plot The ending moment and curvature values otained as per the revised slip model are also plotted in Figure 3, for the four axial loads adopted in the numerical computations. editor@iaeme.com

9 A Refined Slip Model for Predicting the Variale Bending Stiffness in a Single Layered Cale. Figure 3: Bending Moment-Curvature plot 6. CONCLUSIONS The refined slip hypothesis, which includes all the factors contriuting for the strand ending moment is studied in the present work for predicting the ending stiffness variation as a function of strand curvature. The variation of the ending stiffness as a function of strand curvature confirmed the complete transition of the cale from a monolithic stick state to a loose wire gross slip state explaining the slip hypothesis adopted in this paper. It is hoped that the slip hypothesis suggested in this work, will enale prediction of realistic cale stiffness at various ending curvatures. REFERENCES [] Costello, G.A. and Butson, G.J. Simplified ending theory for wire rope,asce J Engg Mech Div,Vol.8,pp. 9-7,98. [] Cardou, A. and Jolicoeur, C. Mechanical models of helical strands, Applied Mechanics Review, Vol. 5, No., pp.-4, 997. [3] Foti F, Martinelli, L. Mechanical modeling of metallic strands sujected to tension, torsion and ending. Int J Solids Struct, Vol.9, pp.-7, 6. [4] Hong, K. J. and Kiureghian, A.D. Bending ehavior of helically wrapped cales, Journal of Engineering Mechanics, Vol. 7(), pp.5-5, 5. [5] Inagaki K, Ekh J, Zahrai S. Mechanical analysis of second order helical structure in electrical cale. Int J Solids Struct.Vol.44 (5):657-79, 7. [6] Knapp, R.H. Helical wire stresses in ent cales, Journal of Offshore Mechanics and Arctic Engineering, Vol., pp. 55-6, 988. [7] Love, A.E.H., 944, A treatise on the mathematical theory of elasticity, Dover Pulishers, New York, [8] Lanteigne, J. Theoretical estimation of the response of helically armored cales to tension, torsion and ending, Journal of Applied Mechanics, Vol. 5, pp , 985. [9] McConnell, K. G. and Zemke, W. P. Measurement of flexural stiffness of multistranded electrical conductors while under tension, Experimental Mechanics, Vol. (6), pp. 98-4, editor@iaeme.com

10 Shankar. G and Dr.N. S Parthasarathy [] Papailiou, K.O. (995), Wire rope ending, taking into account ending stiffness variation due to internal friction, imposed axial traction and curvature, Ph.D. Thesis, ETH Zürich. [] Papailiou, K.O. On the Bending Stiffness of Transmission Line Conductors, IEEE Transactions on Power Delivery, Vol., pp , 997. [] Ramsey H. A theory of thin rods with application to helical constituent wires in cales. Int J Mech Sci.Vol.3 (8), pp.559-7, 988. [3] Sathikh, S. and Parthasarathy, N.S. Discussion of: Internal friction due to wire twist in ent cale, Journal of Engineering Mechanics, Vol. 4, No. 4, pp , 988. [4] Sathikh, S. Effect of inter-wire friction on transverse viration of helically stranded cale, in Proc. ASME Design engineering Technical Conference, Montreal PQ, Canada, ASME, DE. Vol.8-4, pp.47-53, 989. [5] Sathikh, S., Rajasekaran, S., Jayakumar, C.V. and Jearaj, C. General Thin Rod Model for Preslip Bending Response of Strand, Journal of Engineering Mechanics, Vol. 6, pp. 3 39,. [6] Spak K, Agnes G, Inman D. Cale modeling and internal damping developments. Appl Mech Rev.Vol.65 () 3. editor@iaeme.com