MODEL FOR THE STRUCTURE OF ROUND-STRAND WIRE ROPES

2INTRODUCTIONWire ropes are used for transmitting tensile forces. Themain characteristics that make them so well suited to this function are flexibility and strength. Wire ropes are used in manyapplications involving the safety of people, such as elevators, skitows, and cranes. In mine hoisting systems, wire rope is used totransport personnel, product, and supplies between surface andunderground. The condition of the rope deteriorates during usedue to fatigue, wear, and corrosion. Because the failure of a wirerope can be catastrophic, periodic inspection is needed so that thedecision can be made as to whether the rope should be retired.Mine safety research has long been concerned withimprovingthe understanding of rope behavior to forestall thehazardous use of a degraded rope, yet prevent theuneconomical, premature retirement of still-useful rope.Becausethere is widespread disagreement among specialistswithregard to the indicators and methods now used to assesshoistrope condition, the National Institute for OccupationalSafetyand Health (NIOSH), Pittsburgh Research Laboratory,studiedthe factors responsible for the degradation of hoistropesso that a better understanding of rope performance canbe developed.The current Federal retirement criteria for wire ropesused in mine hoisting specify the allowable reductions ofropediameter and outside wire diameter and the location andnumber of broken wires [30 CFR4 56.19024, 57.19024,75.1434,77.1434 (1997)]. However, their effects on strengthloss for ropes of different constructions have not beenproperly considered.This will lead to removing wire ropesfrom service at different stages where the actual loss ofstrengthis either less or more than what is anticipated. Toremedythis deficiency, the knowledge of how the total loadis distributed among the wires in different rope constructionsneeds to be acquired. In general, the load distribution isdependent not only on thecross-sectional area of wires, butalso on the specific arrangement of wires in a rope.The wire stresses in an independent wire rope core(IWRC) were compared by Costello [1990]. It was foundthat,for 17,379 N (3,907 lb) of load applied to the IWRC, thenormal stress was 310,264 kPa (45,000 psi) in the centralwireof the center strand and 279,196 kPa (40,494 psi) in thecentralwire of the outside strand. They were not only significantly different, but also considerably higher than 247,591kPa (35,910 psi), the nominal stresscomputed by taking thetotal load and dividing it by the total metallic area. It isthereforebelieved that the load distribution must be considered for different rope constructions to preventcatastrophic failure of ropes in service. To do this, anunderstandingof the wire geometry that affects the loaddistribution mustfirst be acquired. Although mathematicalmodels have been used to study wire geometry by manyresearchersin the past, these models can be used only forsingle helical wires. Lee [1991] presented two sets ofCartesiancoordinate equations in matrix form for doublehelical wires, but did not give detailed derivation of theequations.One set of the equations was for regular lay ropes;the other was for lang lay ropes.In this study, the rope structure was analyzed, and a generalizedmathematical model describing the wire geometry inanyrope construction with round strands was developed. Themodelcontains a rotation ratio, termed "relative rotation" inthis report, which characterizes the relationship between thewireand the strand helices. In the use of rope with the endsrestrained from rotating, this relative rotation remainsconstant,thus reducing the parameters in the models to theangle of wire rotation only. The model is general enoughthat any combination of wire and strand lay directions can behandledif the stated sign conventions for the angles of strandand wire rotation and the relative rotation are followed.DESCRIPTION OF ROPE STRUCTURESTRUCTURAL ELEMENTSA wire rope is a structure composed of many individualwires.A typical wire rope is composed of two major structuralelements,as shown in figure 1. One is the strand that isformed byhelically winding or laying wires around a centralwire or a strand core. Different shapes of strand may beformeddepending on the shape of the core. For example, intriangular strand constructions, triangular cores may becomposed of triangular wires or of round wires laid intriangles. Only a wire rope made of round strands wasanalyzed inthis report. The other major structural element isthe core around which the strands are4helicallywound.The .coreis madeof natural fibers, polySee CFRin references.Code of FederalRegulationspropylene,or steel that provide proper support for the strandsunder bending and loading in normal use. The mostcommonlyused cores are fiber core, independent wire ropecore (IWRC), and wire strand core (WSC).Although the strand can be laid in any one of manyspecificgeometric arrangements and composed of any numberof wires, the rope also can have any number of strands.Therefore,wire rope generally is identified by the number ofstrands, the nominal or exact numberof wires in each strand,

3and its specific geometric arrangement. When wires are laidin the direction opposite to that of the strands, the rope iscalled a regular lay rope. When wires are laid in the samedirection as

4that of the strands, the rope is called a lang lay rope. If thestrandsare laid into the rope to the right in a fashion similar tothe threading in a right-hand bolt, they are right lay ropestrands. Conversely, strands laid into the rope to the left areleft lay rope strands. Different combinations of these wireropelays are shown in figure 2. The WireRope Users Manual[Wire Rope Technical Board 1993] contains more detailedinformationon wire rope identification and construction. Mostof the rope produced today is preformed; this means that thewiresare permanently shaped into the helical form they willassumein the rope. This manufacturing process eliminates thetendencyof the wires to unlay, usually hazardously, when theyare unrestrained or when the rope is cut.CLASSIFICATION OF WIRESIt is assumed in this study that all wires have a circularcrosssection and remain circular when they are stretched orbent. The centroidal axis, which lies along the center of awire,is selected to represent the path of the wire and used tostudy its geometric characteristics that are related to wirestress. The centroidalaxis of the central wire of a strand alsorepresents the path of the strand.Based on the structural elements in a wire rope asdescribedabove, there is at most one straight wire in a straightrope. It is located in the center of a WSC or IWRC rope. Theremaining wires can be classified geometrically into twogroups: single helices or double helices. The outer wires in astraightstrand used as the WSC have a single helical formbecause they are helically wound only once around a straightaxis. When a strand is helically wound into a rope, the centralwirealso has a single helical form. All of the other wires havea double helical form because they are wound twice, oncearound the strand axis and another around the rope axis.However,they remain single helices relative to the central wireof the strand in which they are wound. This relationship isimportant in the modeling of a double helical wire.Figure 1.C Structural elements in a typical wire rope.STRUCTURAL PARAMETERSThe following basic parameters specifying the helicalstructure are defined; the symbols representing them in themathematical model are shown in parentheses.1.Strandhelix axis (Z): The axis of the rope aroundwhichthe strands are helically wound to form a rope or aroundwhichthe wires are helically wound to form the center strandof a rope. The positive direction of the axis is defined to bethe direction that the helix advances.2.Wirehelix axis (W): The centroidal axis of the helical wire around which other wires are helically wound to forma strand. It is also the centroidal axis of a helical strand. Thepositivedirection of the axis is defined to be the direction thatthe helix advances.Figure 2.C Comparison of typical wire rope lays.

53.is formed and negative if a left lay rope is formed. The angleRadiusof strand helix (rs): The perpendicular distance between the centroidal axis of the strand and the strandis expressed in radians, unless specified otherwise.helix axis.9.Angleof wire rotation (2w): The angle at which the4.centroidalaxis of a helical wire sweeps out in a plane perRadius of wire helix (rw): The perpendiculardistancebetween the centroidal axis of the wire and the wirependicular to the wire helix axis. The angle is defined to behelix axis.positivein a right-handed coordinate system if a right-hand5.: The strand helix having a constantstrand isformed and negative if a left-hand strand is formed.Circular helixhelical radius is a circular helix. Similarly, the wire helixThe angle is expressedin radians, unless specified otherwise.having a constant helical radius is also a circular helix.10. Lay length of strand (Ls): The distance measured6.parallelto the axis of the rope around which the centroidal axisAngle of strand helix ("s): The angle of a strandhelix at any point along the centroidal axis of the strand is theof a strand or wire makes one complete helical convolution.angle between the tangent vector at that point, headingthein11. Laylength (pitch) of wire (Lw): The distance measdirectionthat the strand helix advances, and the plane that isuredparallel to the wire helix axis around which the centroidalperpendicular to thestrand helix axis and passes through thataxis of a wire makes one complete helical convolution.point.12. Lengthof rope (Sr or z): The length measured along7.the strand helix axis. It represents the distance that a strandAngle ofwire helix ("w): The angle of a wire helixat any point along the centroidal axis of the wire is the anglehelix has advanced on the axis of the rope.between the tangent vector at that point, heading in the13. Lengthof strand (Ss or w): The length measureddirectionthat the wire helix advances, and the plane that isalongthe wire helix axis. It represents the distance that a wireperpendicular to the wire helix axis and passes through thathelix has advanced on the centroidal axis of the strand.point.14. Length of wire (Sw): The path length measured8.along the centroidal axis of the wire.Angleof strand rotation (2s): The angle at which thecentroidalaxis of a helical strand sweeps out in a plane perpendicularto the strand helix axis. The angle is defined to bepositivein a right-handed coordinate system if a right lay ropeMATHEMATICAL MODELINGobtainedby using the developed view of the wire helix and areexpressed below.BASIC RELATIONSHIPSIn circular helices, the centroidal axes of both the wireand the strand maybe considered to be lying on right circularcylinders.Because the surface of a cylinder can be developedinto a plane, some basic relationships between each of thecentroidalaxes and the other parameters can be established byusing the developed views shown in figure 3.Therelationships between the length of rope and the angleof strand rotation and between the length of strand and theangle of strand rotation can be obtained by using thepreviouslydefined parameters and from the developed view ofthe strand helix and are expressed below.Sr ' rs2s tan("s)Ss 'r s2scos("s )(1)(2)The length of rope, Sr, in equation 1 becomes the lay length ofstrand,Ls, when 2s ' 2B. Similarly, the relationships betweenthe length of strand and the angle of wire rotation and betweenthe length of wire and the angle of wire rotation also can beSs ' rw2w tan("w)(3)rw2wcos("w )(4)Sw 'The lengthof strand, Ss, in equation 3 becomes the lay lengthof wire, Lw, when2w ' 2B.Becausethe length of strand obtained from the wire helixmust equal that obtained from the strand helix for a givenlengthof rope, a new term "n" is defined to be the ratio of theangleof wire rotation to the angle of strand rotation, which canbe obtained from equations 2 and 3.n '2w '2srsrw tan("w ) cos((5)Thisratio is dependent on the angles of both helices when bothhelicalradii are fixed. It is considered to be important in characterizing the rope structure, specifically the relationship

6The other, a local coordinate system, is the strand coordinatesystem (figure 4B). Its coordinates are U, V, and W with the origin on the centroidal axis of a strand. This local coordinate system moves along the centroidal axis of the strand. The W-axis isin the direction of the tangent vector to the centroidal axis of thestrand. The U-V plane is perpendicular to the centroidal axis ofthe strand and is the plane where the angle of wire rotation, 2w, ismeasured. The U-axis is parallel to the X-Y plane and is alsoparallel to the line on the X-Y plane that specifies the angle ofstrand rotation. The unit vectors directed along the positive directions of U, V, and W are f, g, and h, respectively.VECTOR EQUATIONS FOR SINGLE AND DOUBLEHELICESThe model describing the centroidal axis of the centralwire of a strand in the rope using the rope coordinate systemis a single helix model. The model describing the centroidalaxisof a wire in a strand using the strand coordinate system isalso a single helix model. Once these single helix models areformed,they will be used to develop a double helix modeldescribingthe centroidal axis of a double helical wire in eithera regular lay rope or a lang lay rope in the rope coordinatesystem.Single Helix ModelFigure 3.C Developed views of strand and wire helices.betweenthe wire and strand helices, and is termed the "relativerotation"in this report. The relative rotation will be positivefor lang lay ropes and negative for regular lay ropes.When the rope coordinate system is placed at the center ofthe wire rope and a certain strand is specified to have an initialstrand rotation angle of 0 at z ' 0, as shown in figure 4A, thevector equation of the helix for the centroidal axis of this strand isR ' xs i % ys j % zs k.(6)COORDINATE SYSTEMSBecause severalgeometric characteristics of helices thatare related to load distribution and wire stresses can be easilyevaluatedthrough vector analysis, vector equations describingthese helices are used to model the different wires in a rope.boldface typeis used forTo distinguish vectors from scalars,vectorsin the equations. Two three-dimensional, right-handed,rectangular Cartesian coordinate systems are selected toanalyze the strand and wire helices.Oneis a global fixed system called, for convenience, theropecoordinate system (figure 4A). The coordinates are X, Y,and Z with the originat the center of the rope and the Z-axiscoincidingwith the rope axis. The X-Y plane is perpendicularto the rope axis and is the plane where the angle of strandrotationis measured. The X-axis is arbitrarily selected so thatit intersects, in its positive direction, with the centroidal axisof a strand. The X-axis is also used as the reference line fromwhichthe angle of strand rotation, 2s, is measured. The unitvectorsdirected along the positive directions of X, Y, and Zare i, j, and k, respectively.The subscript "s" indicates variables that are associated with asingle helix. The parametric equations of R for a circularstrand helix areandxs ' rs cos(2s),(7)ys ' rs sin(2s),(8)zs ' rs2s tan("s).(9)The strand rotation angle 2s in equations 7, 8, and 9 is positivefor a right lay rope and negative for a left lay rope.Similarly,when the strand coordinate system is initiallyplacedon the centroidal axis of a certain strand at 2s ' 0, acertain wire is specified to have an initial angle of wirerotation of 0 at w ' 0, as shown in figure 4B. The vectorequation of the circular helix for the centroidal axis of thiswireis similar to equation 6 in the rope coordinate system andcan be written as

7Q ' u f % v g % w h.(10)

8Figure 4.C Coordinate systems for single helix.A, rope coordinate system;B, strand coordinate system.The parametric equations of Q for a circular wire helix in astrand areandu ' rw cos(2w),(11)v ' rw sin(2w),(12)w ' rw2w tan("w).(13)The wire rotation angle 2w in equations 11, 12, and 13 ispositiveif it forms a right-hand strand and negative if it formsa left-handstrand. Because the coordinate system is movingalongthe centroidal axis of the strand, w simply represents thepath length along the centroidal axis that the system hastraveled for a wire rotation angle 2ofw.Double Helix ModelThe double helix model can be developed by properlycombining the vector R in the rope coordinate system and avectorq on the U-V plane ofthe strand coordinate system, asshownin figure 5. It is assumed that, in the rope coordinatesystem,a position vector P with the head of the vector locatedat (u,v,w) of the strand coordinate system traces the centroidalaxis of a double helical wire and has a general formP ' xw i % yw j % zw k,(14)wherexw, yw, and z w are the component functions. The subscript "w" indicates variables that are associated a doublehelix.The vector q in the strand coordinate system is a positionvectorthat traces the centroidal axis of a double helical wireon the U-V plane at a certain value of w in the strandcoordinate system. The vector equation for q may bewritten asq ' u f % v g.(15)The w component is not needed in specifying the location ofthe centroidal axis of a double helical wire because q is alwayson the U-V plane. The parametric equations for u and v areidentical to equations 11 and 12.Becausethe head of R is located exactly at the tail of q,the vector P can be readily obtained through vector additiononce the vectorq in the strand coordinate system is projectedto the rope coordinate system. Using the fact that the U-axisis parallel to the X-Y plane and the line that specifies the angleof strand rotation (2s) and that the U-V plane is perpendicularto the W-axis, which has an angle of strand helix ( "s), theindividualprojections of u and v on the X-, Y-, and Z-axes are

9By introducing the relative rotation(definednby equation 5)into equations 7, 8, 9, 16, 17, 19, and 20, replacing u and vwithequations 11 and 12, and substituting them into equation23, the following component functions for the double helixmodel in terms of only wire rotation angle are obtained.' rs cos2wn% rw cos(2w) cos& rw sin("s)sin(2w) sin2wn(24)The sign for 2w is positive when it rotates counterclockwiseyw 'r s sin2wn% r w cos(2w) sin% r w sin("s) sin(2w)cosz w ' r s tan("s)2wn2wn2wn& r w cos("s) sin(2w)(25)(26)and negative when it rotates clockwise. The lay type determinesthe sign for n as defined by equation 5. The componentzw is always positive and increases in the direction that thehelix advances.Figure 5.C Coordinate system for double helix.andEXAMPLE FOR A SPECIFIC WIRE ROPExu ' u cos(2s),(16)yu ' u sin(2s),(17)zu ' 0,(18)xv ' & v sin("s) sin(2s),(19)yv ' v sin("s) cos(2s),(20)zv ' & v cos("s).(21)The vector q now can be expressed in the rope coordinatesystem asq ' (xu % xv) i % (yu % yv) j % (zu % zv) k.(22)P is the sum ofR and q, the general formBecause the vectorP now can be written asof the vector equation forP ' (xs % xu % xv) i % (ys % yu % yv) j % (zs % zu % zv) k. (23)The circular helix models presented above are applicableto any type of rope construction as long as its strands areround, i.e., the wires are laid in circular pattern. As anexample,the structural parameters of a 33-mm 6 19 Seale,IWRC,right regular lay wire rope are used to show how themodel for a specific rope is obtained.Thebasic strand arrangement of a 6 19 Seale wire rope isshown in figure 6. The structural parameters of differentstrandsare presented in table 1. The strand cross sections perpendicularto the strand or wire helix axis are shown infigures7 through 9. The structural parameters of the singleanddouble helical wires in each strand are presented in table 2.Some of the parameters, such as the lay length of strand, thelay length of wire, and the relative rotation, were calculatedbasedon th

A wire rope is a structure composed of many individual wires. A typical wire rope is composed of two major structural elements, as shown in figure 1. One is the strand that is formed by helically winding or laying wires around a central wire or a strand core. Different shapes of strand may beFile Size: 324KBPage Count: 32