RI 9644 REPORT OF INVESTIGATIONS/1998

Model for the Structure of Round-Strand

Wire Ropes

U.S. DEPARTMENT OF HEALTH AND HUMAN SERVICES

Public Health Service

Centers for Disease Control and Prevention

National Institute for Occupational Safety and Health
Cover: Paths of three types of double helical wires in

a 6×19 Seale, IWRC, right regular lay wire rope

looking down the rope axis.

Report of Investigations 9644

Model for the Structure of Round-Strand Wire Ropes

Richard C. Wang, Anthony J. Miscoe, and William M. McKewan

U.S. DEPARTMENT OF HEALTH AND HUMAN SERVICES

Public Health Service

Centers for Disease Control and Prevention

National Institute for Occupational Safety and Health

Pittsburgh Research Laboratory

Pittsburgh, PA

September 1998
International Standard Serial Number

ISSN 1066-5552
CONTENTS

Page

Abstract................................................................................... 1

Introduction................................................................................ 2

Description of rope structure................................................................... 2

Structural elements........................................................................ 2

Classification of wires..................................................................... 3

Structural parameters...................................................................... 3

Mathematical modeling....................................................................... 4

Basic relationships........................................................................ 4

Coordinate systems....................................................................... 5

Vector equations for single and double helices ................................................... 5

Single helix model...................................................................... 5

Double helix model..................................................................... 6

Example for a specific wire rope............................................................. 7

Model applications.......................................................................... 10

Generation of wire paths................................................................... 10

Evaluation of geometric properties ............................................................ 12

Path length per lay of strand............................................................... 12

Curvature and torsion.................................................................... 12

Analysis of deformations................................................................... 15

Axial elongation and lateral contraction...................................................... 16

Wire strain.......................................................................... 16

Reduced wire radius.................................................................. 16

Radius of deformed helix............................................................... 17

Angle of deformed helix.................................................................. 18

Bending and twisting.................................................................... 19

Conclusions................................................................................ 19

References ................................................................................ 19

ILLUSTRATIONS

1. Structural elements in a typical wire rope................................................... 3

2. Comparison of typical wire rope lays...................................................... 3

3. Developed views of strand and wire helices ................................................. 5

4. Coordinate systems for single helix........................................................ 6

5. Coordinate system for double helix........................................................ 7

6. Strand arrangement of 33-mm 6×19 Seale, IWRC, right regular lay wire rope ....................... 8

7. Cross section of IWRC strand S1 .......................................................... 8

8. Cross section of IWRC strand S2 .......................................................... 9

9. Cross section of Seale strand S3.......................................................... 9

10. Paths of single helical wires W11, W20, W30 ................................................ 11

11. Path of double helical wire W21.......................................................... 11

12. Path of double helical wire W31.......................................................... 11

13. Path of double helical wire W32.......................................................... 11

14. The moving trihedral................................................................... 12

15. Curvature of IWRC and S3 wires......................................................... 14

16. Torsion of IWRC and S3 wires........................................................... 15
UNIT OF MEASURE ABBREVIATIONS USED IN THIS REPORT

kPa kilopascal psi pound (force) per square inch

lb pound rad radian

mm millimeter E or deg degree

N newton

SYMBOLS USED IN THIS REPORT

Vectors

b unit binormal vector Q vector that traces single helix in strand 

 coordinate system

f, g, h unit vectors in strand coordinate

 system q vector that traces wire rotation and lies on

 U-V plane

i, j, k unit vectors in rope coordinate

 system R vector that traces single helix in rope

 coordinate system

n unit principal normal vector

t unit tangent vector

P vector that traces double helix

 in rope coordinate system

ii

TABLES

Page

1. Structural parameters of strands in 33-mm 6×19 Seale wire rope .................................. 8

2. Structural parameters of wires in 33-mm 6×19 Seale wire rope ................................... 8

3. Path length of wires per lay of strand ....................................................... 12

4. Curvature and torsion of single helical wires ................................................. 13

5. Curvature and torsion of double helical wires ................................................ 14
Other Symbols

c1, c2, ..., c6coefficients in equations 44 and 45 xu, yu, zuprojections of u on X, Y, and Z

 on U-V plane

xv, yv, zvprojections of v on X, Y, and Z

c7common term in coefficients c 1

 through c6 "sangle of strand helix

Lslay length of strand "wangle of wire hexlix

Lwlay length of wire ,rrope strain

n relative rotation ,ssingle helical wire strain

Rsradius of single helical wire ,wdouble helical wire strain

Rw

radius of double helical wire 2sangle of strand rotation

r

s

radius of strand helix 2wangle of wire rotation

rwradius of wire helix 6scurvature of single helical wire

Sr or z length of rope 6wcurvature of double helical wire

Ss or w length of strand F Poisson’s ratio

Swlength of wire Jstorsion of single helical wire

U, V, W axes of strand coordinate system Jwtorsion of double helical wire

u, v, w coordinates along U, V, and W N difference in angle of wire rotation

 between wires W31 and W32

uc, vccontact point coordinates between

 cross sections on U-V plane 10 subscript denoting straight wire W10

X, Y, Z axes of rope coordinate system 11, 20, 30 subscripts denoting single helical wires

 W11, W20, and W30

xs, ys, zscomponent functions in single helix

 model along X, Y, and Z 21, 31, 32 subscripts denoting double helical wires 

 W21, W31, and W32

xw, yw, zw ,component functions in double helix

 model along X, Y, and Z dsubscript denoting deformed rope structure

iii

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To receive other information about occupational safety and health problems, call 1-800-35-NIOSH (1-800-356-4674), or visit

the NIOSH Home Page on the World Wide Web at http://www.cdc.gov/niosh
MODEL FOR THE STRUCTURE OF ROUND-STRAND WIRE ROPES

By Richard C. Wang, 1 Anthony J. Miscoe, 2 and William M. McKewan 3

ABSTRACT

The behavior of wire ropes used in hoisting is not well understood. In an effort to improve this

understanding, the structure of round-strand wire ropes was analyzed. This report provides a generalized

mathematical model that completely describes the geometry of the wires. It consists of two sets of vector

equations and is valid for any round-strand wire rope. One set of equations is used to trace the paths of wires

that have the form of a single helix; the other is used for the paths of double helical wires. The specific

model for a 33-mm 6×19 Seale, independent wire rope core (IWRC), right regular lay wire rope was

presented as an example. The paths and the geometric properties of the wires, which include the path length

per lay of strand, the curvature, and the torsion, were determined from this model. The effects of strain

deformation were analyzed, resulting in a system of equations for determining the structural parameters of

the deformed rope. In future work, the model could be used to analyze wire ropes of different constructions

so that a more scientifically based understanding of rope behavior can be established.

1Mechanical engineer.

2Mechanical engineer (retired).

3

Physical scientist.

Pittsburgh Research Laboratory, National Institute for Occupational Safety and Health, Pittsburgh, PA.
2

INTRODUCTION

Wire ropes are used for transmitting tensile forces. The

main characteristics that make them so well suited to this function are flexibility and strength. Wire ropes are used in many

applications involving the safety of people, such as elevators, ski

tows, and cranes. In mine hoisting systems, wire rope is used to

transport personnel, product, and supplies between surface and

underground. The condition of the rope deteriorates during use

due to fatigue, wear, and corrosion. Because the failure of a wire

rope can be catastrophic, periodic inspection is needed so that the

decision can be made as to whether the rope should be retired.

Mine safety research has long been concerned with

improving the understanding of rope behavior to forestall the

hazardous use of a degraded rope, yet prevent the

uneconomical, premature retirement of still-useful rope.

Because there is widespread disagreement among specialists

with regard to the indicators and methods now used to assess

hoist rope condition, the National Institute for Occupational

Safety and Health (NIOSH), Pittsburgh Research Laboratory,

studied the factors responsible for the degradation of hoist

ropes so that a better understanding of rope performance can

be developed.

The current Federal retirement criteria for wire ropes

used in mine hoisting specify the allowable reductions of

rope diameter and outside wire diameter and the location and

number of broken wires [30 CFR4

 56.19024, 57.19024,

75.1434, 77.1434 (1997)]. However, their effects on strength

loss for ropes of different constructions have not been

properly considered. This will lead to removing wire ropes

from service at different stages where the actual loss of

strength is either less or more than what is anticipated. To

remedy this deficiency, the knowledge of how the total load

is distributed among the wires in different rope constructions

needs to be acquired. In general, the load distribution is

dependent not only on the cross-sectional area of wires, but

also 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 found

that, for 17,379 N (3,907 lb) of load applied to the IWRC, the

normal stress was 310,264 kPa (45,000 psi) in the central

wire of the center strand and 279,196 kPa (40,494 psi) in the

central wire of the outside strand. They were not only significantly different, but also considerably higher than 247,591

kPa (35,910 psi), the nominal stress computed by taking the

total load and dividing it by the total metallic area. It is

therefore believed that the load distribution must be considered for different rope constructions to prevent

catastrophic failure of ropes in service. To do this, an

understanding of the wire geometry that affects the load

distribution must first be acquired. Although mathematical

models have been used to study wire geometry by many

researchers in the past, these models can be used only for

single helical wires. Lee [1991] presented two sets of

Cartesian coordinate equations in matrix form for double

helical wires, but did not give detailed derivation of the

equations. 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 generalized mathematical model describing the wire geometry in

any rope construction with round strands was developed. The

model contains a rotation ratio, termed "relative rotation" in

this report, which characterizes the relationship between the

wire and the strand helices. In the use of rope with the ends

restrained from rotating, this relative rotation remains

constant, thus reducing the parameters in the models to the

angle of wire rotation only. The model is general enough

that any combination of wire and strand lay directions can be

handled if the stated sign conventions for the angles of strand

and wire rotation and the relative rotation are followed.

DESCRIPTION OF ROPE STRUCTURE

STRUCTURAL ELEMENTS

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 be

formed depending on the shape of the core. For example, in

triangular strand constructions, triangular cores may be

composed of triangular wires or of round wires laid in

triangles. Only a wire rope made of round strands was

analyzed in this report. The other major structural element is

the core around which the strands are

4

helically wound. The core is made of natural fibers, poly- Code of Federal Regulations . See CFR in references.

propylene, or steel that provide proper support for the strands

under bending and loading in normal use. The most

commonly used cores are fiber core, independent wire rope

core (IWRC), and wire strand core (WSC).

Although the strand can be laid in any one of many

specific geometric arrangements and composed of any number

of wires, the rope also can have any number of strands.

Therefore, wire rope generally is identified by the number of

strands, the nominal or exact number of wires in each strand,
3

and its specific geometric arrangement. When wires are laid

in the direction opposite to that of the strands, the rope is

called a regular lay rope. When wires are laid in the same

direction as
4

Figure 1.CStructural elements in a typical wire rope.

Figure 2.CComparison of typical wire rope lays.

that of the strands, the rope is called a lang lay rope. If the

strands are laid into the rope to the right in a fashion similar to

the threading in a right-hand bolt, they are right lay rope

strands. Conversely, strands laid into the rope to the left are

left lay rope strands. Different combinations of these wire

rope lays are shown in figure 2. The Wire Rope Users Manual

[Wire Rope Technical Board 1993] contains more detailed

information on wire rope identification and construction. Most

of the rope produced today is preformed; this means that the

wires are permanently shaped into the helical form they will

assume in the rope. This manufacturing process eliminates the

tendency of the wires to unlay, usually hazardously, when they

are unrestrained or when the rope is cut.

CLASSIFICATION OF WIRES

It is assumed in this study that all wires have a circular

cross section and remain circular when they are stretched or

bent. The centroidal axis, which lies along the center of a

wire, is selected to represent the path of the wire and used to

study its geometric characteristics that are related to wire

stress. The centroidal axis of the central wire of a strand also

represents the path of the strand.

Based on the structural elements in a wire rope as

described above, there is at most one straight wire in a straight

rope. It is located in the center of a WSC or IWRC rope. The

remaining wires can be classified geometrically into two

groups: single helices or double helices. The outer wires in a

straight strand used as the WSC have a single helical form

because they are helically wound only once around a straight

axis. When a strand is helically wound into a rope, the central

wire also has a single helical form. All of the other wires have

a double helical form because they are wound twice, once

around the strand axis and another around the rope axis.

However, they remain single helices relative to the central wire

of the strand in which they are wound. This relationship is

important in the modeling of a double helical wire.

STRUCTURAL PARAMETERS

The following basic parameters specifying the helical

structure are defined; the symbols representing them in the

mathematical model are shown in parentheses.

1. Strand helix axis (Z): The axis of the rope around

which the strands are helically wound to form a rope or around

which the wires are helically wound to form the center strand

of a rope. The positive direction of the axis is defined to be

the direction that the helix advances.

2. Wire helix axis (W): The centroidal axis of the helical wire around which other wires are helically wound to form

a strand. It is also the centroidal axis of a helical strand. The

positive direction of the axis is defined to be the direction that

the helix advances.
5

Ss'

r

s

2s

cos("s)

(2)

Sw '

rw2w

cos("w)

(4)

n '

2w

2s

'

rs

rw tan("w)cos(

(5)

3. Radius of strand helix (rs): The perpendicular distance between the centroidal axis of the strand and the strand

helix axis.

4. Radius of wire helix (rw): The perpendicular

distance between the centroidal axis of the wire and the wire

helix axis.

5. Circular helix: The strand helix having a constant

helical radius is a circular helix. Similarly, the wire helix

having a constant helical radius is also a circular helix.

6. Angle of strand helix ("s): The angle of a strand

helix at any point along the centroidal axis of the strand is the

angle between the tangent vector at that point, heading in the

direction that the strand helix advances, and the plane that is

perpendicular to the strand helix axis and passes through that

point.

7. Angle of wire helix ("w): The angle of a wire helix

at any point along the centroidal axis of the wire is the angle

between the tangent vector at that point, heading in the

direction that the wire helix advances, and the plane that is

perpendicular to the wire helix axis and passes through that

point.

8. Angle of strand rotation (2s): The angle at which the

centroidal axis of a helical strand sweeps out in a plane perpendicular to the strand helix axis. The angle is defined to be

positive in a right-handed coordinate system if a right lay rope

is formed and negative if a left lay rope is formed. The angle

is expressed in radians, unless specified otherwise.

9. Angle of wire rotation (2w): The angle at which the

centroidal axis of a helical wire sweeps out in a plane perpendicular to the wire helix axis. The angle is defined to be

positive in a right-handed coordinate system if a right-hand

strand is formed and negative if a left-hand strand is formed.

The angle is expressed in radians, unless specified otherwise.

10. Lay length of strand (Ls): The distance measured

parallel to the axis of the rope around which the centroidal axis

of a strand or wire makes one complete helical convolution.

11. Lay length (pitch) of wire (Lw): The distance measured parallel to the wire helix axis around which the centroidal

axis of a wire makes one complete helical convolution.

12. Length of rope (Sr or z): The length measured along

the strand helix axis. It represents the distance that a strand

helix has advanced on the axis of the rope.

13. Length of strand (Ss or w): The length measured

along the wire helix axis. It represents the distance that a wire

helix has advanced on the centroidal axis of the strand.

14. Length of wire (Sw): The path length measured

along the centroidal axis of the wire.

MATHEMATICAL MODELING

BASIC RELATIONSHIPS

In circular helices, the centroidal axes of both the wire

and the strand may be considered to be lying on right circular

cylinders. Because the surface of a cylinder can be developed

into a plane, some basic relationships between each of the

centroidal axes and the other parameters can be established by

using the developed views shown in figure 3.

The relationships between the length of rope and the angle

of strand rotation and between the length of strand and the

angle of strand rotation can be obtained by using the

previously defined parameters and from the developed view of

the strand helix and are expressed below.

Sr ' rs2s

 tan("s) (1)

The length of rope, Sr, in equation 1 becomes the lay length of

strand, Ls, when 2s ' 2B. Similarly, the relationships between

the length of strand and the angle of wire rotation and between

the length of wire and the angle of wire rotation also can be

obtained by using the developed view of the wire helix and are

expressed below.

Ss ' rw2w tan("w) (3)

The length of strand, Ss, in equation 3 becomes the lay length

of wire, Lw

, when 2w ' 2B.

Because the length of strand obtained from the wire helix

must equal that obtained from the strand helix for a given

length of rope, a new term "n" is defined to be the ratio of the

angle of wire rotation to the angle of strand rotation, which can

be obtained from equations 2 and 3.

This ratio is dependent on the angles of both helices when both

helical radii are fixed. It is considered to be important in characterizing the rope structure, specifically the relationship
6

Figure 3.CDeveloped views of strand and wire helices.

between the wire and strand helices, and is termed the "relative

rotation" in this report. The relative rotation will be positive

for lang lay ropes and negative for regular lay ropes.

COORDINATE SYSTEMS

Because several geometric characteristics of helices that

are related to load distribution and wire stresses can be easily

evaluated through vector analysis, vector equations describing

these helices are used to model the different wires in a rope.

To distinguish vectors from scalars, boldface type is used for

vectors in the equations. Two three-dimensional, right-handed,

rectangular Cartesian coordinate systems are selected to

analyze the strand and wire helices.

One is a global fixed system called, for convenience, the

rope coordinate system (figure 4A). The coordinates are X, Y,

and Z with the origin at the center of the rope and the Z-axis

coinciding with the rope axis. The X-Y plane is perpendicular

to the rope axis and is the plane where the angle of strand

rotation is measured. The X-axis is arbitrarily selected so that

it intersects, in its positive direction, with the centroidal axis

of a strand. The X-axis is also used as the reference line from

which the angle of strand rotation, 2s

, is measured. The unit

vectors directed along the positive directions of X, Y, and Z

are i, j, and k, respectively.

The other, a local coordinate system, is the strand coordinate

system (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 is

in the direction of the tangent vector to the centroidal axis of the

strand. The U-V plane is perpendicular to the centroidal axis of

the strand and is the plane where the angle of wire rotation, 2w, is

measured. The U-axis is parallel to the X-Y plane and is also

parallel to the line on the X-Y plane that specifies the angle of

strand 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 DOUBLE

HELICES

The model describing the centroidal axis of the central

wire of a strand in the rope using the rope coordinate system

is a single helix model. The model describing the centroidal

axis of a wire in a strand using the strand coordinate system is

also a single helix model. Once these single helix models are

formed, they will be used to develop a double helix model

describing the centroidal axis of a double helical wire in either

a regular lay rope or a lang lay rope in the rope coordinate

system.

Single Helix Model

When the rope coordinate system is placed at the center of

the wire rope and a certain strand is specified to have an initial

strand rotation angle of 0 at z ' 0, as shown in figure 4A, the

vector equation of the helix for the centroidal axis of this strand is

R ' xsi % ysj % zsk. (6)

The subscript "s" indicates variables that are associated with a

single helix. The parametric equations of R for a circular

strand helix are

xs ' rs

 cos(2s), (7)

ys ' rs sin(2s), (8)

and zs ' rs2s tan("s). (9)

The strand rotation angle 2s in equations 7, 8, and 9 is positive

for a right lay rope and negative for a left lay rope.

Similarly, when the strand coordinate system is initially

placed on the centroidal axis of a certain strand at 2s ' 0, a

certain wire is specified to have an initial angle of wire

rotation of 0 at w ' 0, as shown in figure 4B. The vector

equation of the circular helix for the centroidal axis of this

wire is similar to equation 6 in the rope coordinate system and

can be written as
7

Q' u f% v g% w h. (10)
8

Figure 4.CCoordinate systems for single helix. A, rope coordinate system; B, strand coordinate system.

The parametric equations of Q for a circular wire helix in a

strand are

u ' rw cos(2w), (11)

v ' rw sin(2w), (12)

and w ' rw2w tan("w). (13)

The wire rotation angle 2w in equations 11, 12, and 13 is

positive if it forms a right-hand strand and negative if it forms

a left-hand strand. Because the coordinate system is moving

along the centroidal axis of the strand, w simply represents the

path length along the centroidal axis that the system has

traveled for a wire rotation angle of 2w

.

Double Helix Model

The double helix model can be developed by properly

combining the vector R in the rope coordinate system and a

vector q on the U-V plane of the strand coordinate system, as

shown in figure 5. It is assumed that, in the rope coordinate

system, a position vector P with the head of the vector located

at (u,v,w) of the strand coordinate system traces the centroidal

axis of a double helical wire and has a general form

P ' xwi % ywj % zwk, (14)

where xw, yw, and z w are the component functions. The subscript "w" indicates variables that are associated a double

helix.

The vector q in the strand coordinate system is a position

vector that traces the centroidal axis of a double helical wire

on the U-V plane at a certain value of w in the strand

coordinate system. The vector equation for q may be

written as

q ' u f % v g. (15)

The w component is not needed in specifying the location of

the centroidal axis of a double helical wire because q is always

on the U-V plane. The parametric equations for u and v are

identical to equations 11 and 12.

Because the head of R is located exactly at the tail of q,

the vector P can be readily obtained through vector addition

once the vector q in the strand coordinate system is projected

to the rope coordinate system. Using the fact that the U-axis

is parallel to the X-Y plane and the line that specifies the angle

of strand rotation (2s

) and that the U-V plane is perpendicular

to the W-axis, which has an angle of strand helix ("s

), the

individual projections of u and v on the X-, Y-, and Z-axes are
9

Figure 5.CCoordinate system for double helix.

zw'r

s

tan("s)

2w

n

&rwcos("s)sin(2w) (26)

yw'r

s

sin

2w

n

% rwcos(2w)sin

2w

n

%rwsin("s)sin(2w)cos

2w

n (25)

' rs cos 2w

n

% rw cos(2w)cos

& rw sin("s)sin(2w)sin 2w

n

(24)

xu ' 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)

and zv ' & v cos("s). (21)

The vector q now can be expressed in the rope coordinate

system as

q ' (xu % xv) i % (yu % yv) j % (zu % zv) k. (22)

Because the vector Pis the sum of R and q, the general form

of the vector equation for P now can be written as

P ' (xs % xu % xv) i % (ys % yu % yv) j % (zs % zu % zv) k.(23)

By introducing the relative rotation n (defined by equation 5)

into equations 7, 8, 9, 16, 17, 19, and 20, replacing u and v

with equations 11 and 12, and substituting them into equation

23, the following component functions for the double helix

model in terms of only wire rotation angle are obtained.

The sign for 2w is positive when it rotates counterclockwise

and negative when it rotates clockwise. The lay type determines the sign for n as defined by equation 5. The component

zw is always positive and increases in the direction that the

helix advances.

EXAMPLE FOR A SPECIFIC WIRE ROPE

The circular helix models presented above are applicable

to any type of rope construction as long as its strands are

round, i.e., the wires are laid in circular pattern. As an

example, the structural parameters of a 33-mm 6×19 Seale,

IWRC, right regular lay wire rope are used to show how the

model for a specific rope is obtained.

The basic strand arrangement of a 6×19 Seale wire rope is

shown in figure 6. The structural parameters of different

strands are presented in table 1. The strand cross sections perpendicular to the strand or wire helix axis are shown in

figures 7 through 9. The structural parameters of the single

and double helical wires in each strand are presented in table 2.

Some of the parameters, such as the lay length of strand, the

lay length of wire, and the relative rotation, were calculated

based on the basic relationships given by equations 1, 3, and 5.

The model for single helical wires can be obtained by

simply substituting the values of the parameters rs

 and " s, as

indicated in table 2, into equations 7, 8, and 9. The model for

double helical wires can be obtained by substituting the values

of the parameters rs , rw , "s, and n as shown in table 2 into equations 24, 25, and 26. As examples, the component functions of

the model for each type of wire in the rope are presented

below.
10

Figure 6.CStrand arrangement of 33-mm 6×19 Seale, IWRC,

right regular lay wire rope.

Figure 7.CCross section of IWRC strand S1.

Table 1.CStructural parameters of strands in 33-mm 6×19 Seale wire rope

Strand Form No. of

strands

Strand

radius,

 mm

Helix parameters

 rs

, 

mm 

"s, 

rad 

Ls, 

mm 

IWRC:

 S1.......... Straight..... 1 2.271 NAp NAp NAp

 S2.......... Single helix. . 6 2.016 4.287 1.2362 77.48

Seale:

 S3.......... Single helix. . 6 5.110

11.41

3 1.2259 199.61

NAp Not applicable.

Table 2.CStructural parameters of wires in 33-mm 6×19 Seale wire rope

Wire

No. of 

wires per

strand

Wire

radius,

mm

Helix parameters

 rs, 

 mm

 "s,

 rad

Ls,

 mm

rw,

 mm

"w,

 rad

Lw,

mm

n 

Straight:

 W10........... 1 0.801 NAp NAp NAp NAp NAp NAp NAp

Single helical:

 W11........... 6 0.735 1.536 1.2864 33.02 NAp NAp NAp NAp

 W20........... 1 0.704 4.287 1.2362 77.48 NAp NAp NAp NAp

 W30........... 1 1.456 11.413 1.2259 199.61 NAp NAp NAp NAp

Double helical:

 W21........... 6 0.656 4.287 1.2362 77.48 1.360 1.4149 54.37 1.5087

 W31.......... 9 0.712 11.413 1.2259 199.61 2.168 1.7849 62.65 &3.3855

 W32........... 9 1.243 11.413 1.2259 199.61 3.867 1.9408 62.65 &3.3855

NAp Not applicable.
11

Figure 8.CCross section of IWRC strand S2.

Figure 9.CCross section of Seale strand S3.

xw ' 4.287 cos 2

1.5087

% 1.360 cos(2w)

& 1.285 sin(2w)

(24a)

yw ' 4.287 sin 2

1.5087

% 1.360 cos(2w)

% 1.285 sin(2w)

(25a)

1. Wire W10

Because this is the center wire of the rope, its path is always

on the Z-axis. The vector equation has only one component in

the direction of k, and the magnitude is simply the rope length.

2. Wire W11

xs ' 1.536 cos(2s) (7a)

ys ' 1.536 sin(2s

) (8a)

zs ' 5.255 2s(9a)

3. Wire 20

xs ' 4.287 cos(2s) (7b)

ys ' 4.287 sin(2s) (8b)

zs ' 12.331 2s

(9b)

4. Wire W21

zw ' 8.173 2w & 0.447 sin(2w

) (26a)

5. Wire W30

xs ' 11.413 cos(2s

) (7c)

ys ' 11.413 sin(2s) (8c)

zs ' 31.768 2s

(9c)
12

xw ' 11.413 cos 2w

&3.3855

% 2.168 cos(2w) cos

& 2.040 sin(2w) sin

(24b)

yw ' 11.413 sin

&

% 2.168 cos(2

% 2.040 sin(2

(25b)

xw ' 11.413 cos

&3.

% 3.867 cos(2w)

& 3.639 sin(2w)

(24c) 

yw ' 11.413 sin 2w

&3.3855

% 3.867 cos(2w) sin

% 3.639 sin(2w) cos

(25c) 

6. Wire W31

z

w ' &9.384 2w & 0.733 sin(2w

) (26b)

7. Wire W32

 z

w ' &9.384 2w & 1.307 sin(2w

) (26c)

MODEL APPLICATIONS

The model for a specific wire rope can be easily obtained,

as shown in the example above, and has many practical applications. It can be used to generate the wire paths and to evaluate the geometric properties of the wires. The effect of deformation can be determined by substituting the structural

parameters of the deformed rope into the original model. The

model also has other applications, such as predicting damage

patterns through external and internal wear, examining and

improving the design of a rope construction prior to manufacturing, and producing three-dimensional pictures of the

wires for computer analysis.

GENERATION OF WIRE PATHS

As described earlier, the wires in wire ropes have three

forms: straight, single helix, or double helix. The only

straight wire in a rope is the center wire in an IWRC. The

wires around the center wire forming the center strand and the

center wires in the outer strands of the core and in the surface

strands have the shape of a single helix. The wires forming the

outer strand s

of the core and the surface strands except their center wires

have double helical paths that are very complex in their

configurations. Using the model developed in this report, the

wire paths can be easily generated by a computer. They not

only reveal the shapes of the wire paths, but also are useful in

locating the places where a wire will be rubbed by the other

wires and in determining the interval at which an outer wire

will be exposed on the rope surface.

Typical paths of single helical wires generated by equations 7a-8a-9a, 7b-8b-9b, and 7c-8c-9c for wires W11, W20

and W30, respectively, are shown in figure 10. The paths

shown are in about one lay of strand S3. Typical paths of

double helical wires generated by equations 24a-25a-26a, 24b25b-26b, and 24c-25c-26c for wires W21, W31, and W32, respectively, are shown in figures 11 through 13. The paths

shown are in about two lays of the strand formed by each wire.

The reason for the major difference between the shape of the

W21 path and that of the W31 and W32 paths is that strand S2,

which contains wire W21, is a lang lay strand, whereas strand

S3, which contains W31 and W32, is a regular lay strand. The

side views of all wire paths show much sharper turns than the

actual wire paths. This is because much larger scales have

been selected on the Y coordinate than those on the

Z coordinate of these figures so that more of their paths can be

viewed. 
13

Figure 10.CPaths of single helical wires W11, W20, W30.

Figure 11.CPath of double helical wire W21.

Figure 12.CPath of double helical wire W31.

Figure 13.CPath of double helical wire W32.
14

Ss ' 2Brs

cos("s)

(27)

Sw ' 2Brs

cos("s) sin("w)

(28) 

Figure 14.CThe moving trihedral.

EVALUATION OF GEOMETRIC PROPERTIES

To learn how the tensile load is distributed among the wires

in a rope and to calculate wire stresses, several geometric properties of each wire must be evaluated before and after the application

of load. These properties are the path length, curvature, and

torsion.

Path Length Per Lay of Strand

The equations for evaluation of the path length of single and

double helical wires in each lay of the strand can be derived from

equations 2, 3, and 4 and are shown in equations 27 and 28.

Path length of single helical wires in each lay of strand:

Path length of double helical wires in each lay of strand:

Table 3 shows the path lengths per lay of strand for a 6×19

Seale rope as calculated by equations 27 and 28. The table also

shows the wire-to-rope length ratios by comparing the wire paths

to the strand lay lengths.

Table 3.CPath length of wires per lay of strand

 Wire

Lay length

of strand 

referred to

Lay length 

of strand Ls,

 mm 

Path length

per lay 

of strand,

mm 

Wire-to-rope

length 

 ratio 

Straight:

 W10 ...... NAp NAp NAp 1.000

Single helical:

 W11 ...... L11 33.02 34.40 1.042

 W20 ...... L20 77.48 82.03 1.059

 W30 ...... L30 199.61 212.10 1.063

Double helical:

 W21 ...... L20 77.48 83.04 1.072

 W31 ...... L30 199.61 217.06 1.087

 W32 ...... L30 199.61 227.50 1.140

NAp Not applicable.

Curvature and Torsion

A special moving frame of reference, similar to the strand

coordinate system adapted in the mathematical modeling, has been

used in the generation of the basic equations for evaluation of the

curvature and torsion of a curve in three-dimensional space

[Sokolnikoff and Redheffer 1958; Leithold 1986]. This frame is

formed by a three-dimensional, right-handed set of orthogonal unit

vectors, as shown in figure 14. The origin of the frame is located

at the head of any position vector that may be R or P of the models

just developed. The three unit vectors t, n, and b are called the

unit tangent, the unit principal normal, and the unit binormal

vectors, respectively. The frame is sometimes referred to as the

"moving trihedral" associated with the curve.

The curvature at a certain point of a curve is a measure of how

quickly the curve changes direction at that point. It is the

reciprocal of the radius of the curve at that point and expressed in

the unit of 1/mm in this report. In a wire, a change of the

curvature is produced by bending moments that act on the wire

cross section. Curvature not only is related to the shearing stress,

but also affects the distribution of the tensile stress on the cross

section. Therefore, the deformation of the rope structure in terms

of the curvature change needs to be specified to determine its

effect on shearing and normal stresses.
15

6s'

*R! × R"*

*R!*

3

(29)

6w'

*P! × P"*

*P!*

3

(30)

6s'

(ys!zs"& zs!ys")

2 % (zs

!xs"& xs!zs")

2 % (xs

!ys"& ys!xs")

2

(xs!)

2 % (ys

!)

2 % (zs

!)

2

3

(29a)

6s'

cos2("s)

r

s

(29b)

6w'

(yw! zw"& zw!yw")

2 % (zw

!xw"& xw!zw")

2 % (xw

!yw"& yw!xw")

2

(xw!)

2 % (yw

!)

2 % (zw

!)

2

3

(30a)

The curvature vector is defined to be the first derivative of t

in the moving trihedral, as shown in figure 14, with respect to arc

length of a curve. It can be expressed as 6n, where 6 is a scalar

multiplier. The curvature vector is in the same direction of the

principal normal vector. The magnitude of this vector is called the

curvature of the curve and is simply equal to 6 because the

magnitude of n is unity. The curvature of a straight line is always

zero because the tangent vector is constant. The curvature of the

centroidal axis of either a single or a double helical wire, therefore,

may be specifically defined to be the magnitude of the rate of

change of the unit tangent vector with respect to arc length of

the wire.

Basic formulas derived for evaluation of curvature are

expressed in terms of the position vector with arc length as the

parameter, because arc length arises naturally from the shape of

the curve [Sokolnikoff and Redheffer 1958]. To directly apply the

model developed in this report, it is more convenient to use

equations 29 and 30 for computation of the curvature [Stewart

1991]. The vectors R and P denoted with single and double

primes in the equations represent the first and second derivatives,

respectively, with regard to 2s

 or 2w, just as for real-valued functions. Similarly, the vectors denoted with triple primes, to be used

later in this report, represent the third derivatives with respect to 2s

or 2w. The symbol "| |", by which the vector is bounded,

represents the magnitude of the vector, which is generally used in

vector analysis. R was given by equation 6, and its component

functions were given in equations 7, 8, and 9. P was given by

equation 14, and its component functions were given in

equations 24, 25, and 26.

Curvature of single helical wires:

Curvature of double helical wires:

Substituting the first and second derivatives of each vector

into equations 29 and 30 and performing the cross and dot products

and other operations, the expanded forms of these equations

expressed in terms of the component functions are shown below.

Expanded curvature equation for single helical wires:

After substituting the functions into equation 29a, it can be reduced

to its simplest form, as shown in equation 29b. This expression

indicates that the curvature of a single helical wire is independent

of the angle of strand rotation and is constant for a given helix

angle.

Expanded curvature equation for double helical wires:

The curvatures were computed for both single and double

helical wires in a 6×19 Seale wire rope. The results for single

helical wires are shown in table 4. The results for double helical

wires are shown in table 5, with the absolute value of 2w increasing

from 0° to 360° at increments of 15°.

The curvatures of the IWRC and S3 helical wires were plotted

against the absolute value of the angle of strand or wire rotation.

Their relationships are shown in figure 15 for two periods. They

indicate that (1) the curvature of single helical wires is independent

of the angle of strand rotation, as expressed by equation 29b,

(2) the curvature of double helical wires is a periodical function of

the angle of wire rotation with a period of 360°, and (3) the

curvature of double helical wires is at maximum when the wires

are farthest from the rope center and at minimum when the wires

are nearest to the rope center because the angle of wire rotation is

measured from the positive U-axis, which points away from the

rope center, as shown in figure 5.

Table 4.CCurvature and torsion

of single helical wires

Wire Curvature,

1/mm

Torsion,

1/mm

W11 ...... 0.0513 0.1753

W20 ...... 0.0252 0.0724

W30 ...... 0.0100 0.0179
16

Js '

R! × R" R

*R! × R"*

2

(31)

Figure 15.CCurvature of IWRC and S3 wires.

Table 5.CCurvature and torsion of double helical wires

Angle of

wire

rotation, o

Curvature, 1/mm Torsion, 1/mm

W21 W31 W32 W21 W31 W32 

 0 ...... 0.0652 0.0202 0.0265 0.1249 &0.0198 &0.0268

 15 ...... 0.0649 0.0201 0.0264 0.1256 &0.0200 &0.0273

 30 ...... 0.0638 0.0197 0.0260 0.1277 &0.0207 &0.0288

 45 ...... 0.0620 0.0190 0.0254 0.1313 &0.0218 &0.0314

 60 ...... 0.0593 0.0180 0.0244 0.1365 &0.0233 &0.0353

 75 ...... 0.0559 0.0167 0.0232 0.1435 &0.0254 &0.0407

 90 ...... 0.0517 0.0151 0.0216 0.1528 &0.0281 &0.0479

105 ...... 0.0467 0.0132 0.0197 0.1653 &0.0319 &0.0580

120 ...... 0.0411 0.0110 0.0175 0.1829 &0.0377 &0.0723

135 ...... 0.0353 0.0086 0.0152 0.0286 &0.0485 &0.0933

150 ...... 0.0297 0.0060 0.0130 0.2460 &0.0757 &0.1234

165 ...... 0.0254 0.0033 0.0113 0.2914 &0.1897 &0.1586

180 ...... 0.0238 0.0016 0.0106 0.3157 &0.7359 &0.1768

195 ...... 0.0254 0.0033 0.0113 0.2914 &0.1897 &0.1586

210 ...... 0.0297 0.0060 0.0130 0.2460 &0.0757 &0.1234

225 ...... 0.0353 0.0086 0.0152 0.2086 &0.0485 &0.0933

240 ...... 0.0411 0.0110 0.0175 0.1829 &0.0377 &0.0723

255 ...... 0.0467 0.0132 0.0197 0.1653 &0.0319 &0.0580

270 ...... 0.0517 0.0151 0.0216 0.1528 &0.0281 &0.0479

285 ...... 0.0559 0.0167 0.0232 0.1435 &0.0254 &0.0407

300 ...... 0.0593 0.0180 0.0244 0.1365 &0.0233 &0.0353

315 ...... 0.0620 0.0190 0.0254 0.1313 &0.0218 &0.0314

330 ...... 0.0638 0.0197 0.0260 0.1277 &0.0207 &0.0288

345 ...... 0.0649 0.0201 0.0264 0.1256 &0.0200 &0.0273

360 ...... 0.0652 0.0202 0.0265 0.1249 &0.0198 &0.0268

The torsion at a certain point of a curve measures the degree

of twisting of a curve at that point. The change of torsion in a wire

is produced by twisting moments that act on the wire cross section.

Angular deformation between cross sections results in shearing

stresses.

The first derivative of b in the moving trihedral, as shown in

figure 14, with respect to arc length of a curve will yield a vector

that is parallel to n and can be expressed as &Jn. The scalar multiplier J is called the torsion of the curve. It measures the rate at

which the centroidal axis of either a single or a double helical wire

twists out of its osculating plane, which is the plane containing the

unit tangent and the unit principal normal vectors. The torsion of

a straight line is defined to be zero. If the curve is a plane curve,

the torsion is always zero, because the osculating plane is the plane

of the curve and the unit binormal vector is constant.

Basic formulas derived for computation of torsion are expressed in terms of the position vector with arc length as the

parameter [Sokolnikoff and Redheffer 1958]. To directly

apply the models developed in this report, it is more

convenient to use equations 31 and 32 for computation of the

torsion [Stewart 1991].

Torsion of single helical wires:
17

Jw'

P! × P" P

*P! × P"*

2

(32)

Js'

xs(ys! zs"& zs!ys") % ys(zs!xs"& xs!zs") % zs(xs!ys"& ys!xs")

(ys!zs"& zs!ys")

2 % (zs

!xs"& xs!zs")

2 % (xs

!ys"& ys!xs")

2

(31a)

Js'

sin("s) cos("s)

r

s

(31b)

Jw'

xw(yw!zw"& zw!yw") % yw(zw!xw"& xw!zw") % zw(xw!yw"& yw!xw")

(yw! zw"&zw!yw")

2 % (zw

!xw"& xw!zw")

2 % (xw

!yw"& yw!xw")

2

(32a)

Figure 16.CTorsion of IWRC and S3 wires.

Torsion of double helical wires:

Substituting the derivatives of each vector into the

equations and performing the cross and dot products and other

operations, the expanded forms of these equations expressed in

terms of the component functions are shown below.

Expanded torsion equation for single helical wires:

After substituting the functions into equation 31a, it can

be reduced to its simplest form, as shown in equation 31b. It

indicates that the torsion of a single helical wire is independent

of the angle of strand rotation and is constant for a given helix

angle.

Expanded torsion equation for double helical wires:

The torsions of both single and double helical wires in a

6×19 Seale wire rope were computed. The results for single

helical wires are shown in table 4. The results for double

helical wires are shown in table 5, with the absolute value of

2w

 increasing from 0° to 360° at increments of 15°. The

negative torsions for W31 and W32 represent the twisting of

the centroidal axes of these wires in a left lay strand, where the

twisting is opposite in direction to that in a right lay strand.

The torsions of the IWRC and S3 helical wires were

plotted against the absolute value of the angle of strand or wire

rotation. Their relationships are shown in figure 16 for two

periods. They indicate that (1) the torsion of single helical

wires is independent of the angle of strand rotation, as

expressed by equation 31b, (2) the torsion of double helical

wires is a periodical function of the angle of wire rotation with

a period of 360°, and (3) the torsion of double helical wires is

at minimum when the wires are farthest from the rope center

and at maximum when the wires are nearest to the rope center.

It is also noted that the minimum torsion occurs at the

locations where the curvature is at maximum, and the

maximum torsion occurs at the locations where the curvature

is at minimum.

ANALYSIS OF DEFORMATIONS

When a tensile load is applied to a wire rope, each

individual wire will deform. Because of the differences in the

wire lengths and the helix angles of single and double helical

wires, the load will not be distributed among the wires simply

based on the cross-sectional areas. The effect of these wire
18

Ssd ' [Ls(1% ,r)]2 % (2Br

(33)

[Ls(1% ,r)]

2 % (2Brsd)2 % (2 (34)

,s '

Ssd & Ss

Ss

(35)

,w '

Swd & Sw

Sw

(36)

deformations on the geometry of the rope structure needs to be

determined. The major change is axial elongation along the

centroidal axis of a wire. Accompanying the axial elongation

is a lateral contraction of the cross section. In addition,

bending and twisting moments are generated in the wire that

cause changes in the curvature and torsion of the wire. The

combination of all of these individual wire deformations

results in the deformation of the rope structure.

There will be a resultant twisting moment, which will

cause the rope to rotate if the ends are not restrained.

Therefore, deformation of the rope structure will depend on

whether the rope is allowed to rotate or not. In mine hoisting,

the cage or skip and rope are prevented from rotating by the

shaft guides. Some rope manufacturers also produce rotationresistant rope [Wire Rope Technical Board 1993], which is

made with layers of strands laid in opposite directions to

produce counteracting torques. In the use of rope with both

ends restrained, the total number of strand lays and the total

number of wire lays in a rope are kept constant. Thus, the

relative rotation, n, included in the models for double helical

wires, remains constant as the rope structure deforms under

load. Because of the objective of evaluating the retirement

criteria for mine hoists, only the restrained-rotation case will

be considered in this report.

Axial Elongation and Lateral Contraction

The axial elongation and lateral contraction of the wires

will occur simultaneously when the rope is under tension.

Neither can be determined independently. However, a sufficient number of equations may be established to obtain the

solutions for a given rope strain. The structural parameters of

the deformed rope used in deriving the required equations are

the path length of the wire in each lay of the deformed strand,

the wire strain, the reduced wire radius, and the radius of the

deformed helix. Each of them is described below. The

symbols used are the same as those defined previously, except

that the subscript "d" has been added to represent their values

in the deformed rope structure. The variable used to specify

the axial elongation of the rope is the rope strain, which is

represented by ,r

 in the equations. It is determined by dividing

the amount of elongation of the rope by the original length

before the application of load.

Wire Strain

Hooke's law states that, within the elastic range, the

deformation produced by external forces that act on nonrigid

bodies is proportional to the stress. The path lengths of the

wire before and after the application of load, therefore, are

needed to specify the axial deformation and determine the

tensile stress.

To establish the relationship between thelengths of wire

and rope in the deformed rope structure, the developed views,

which are the same as those shown in figure 3 for the

undeformed rope, will be used. The only difference is to

replace the original structural parameters with the

deformed ones, i.e., Sr with Srd, Ss with Ssd, r

s

 with rsd, "s

with "sd, Sw with Swd, r

w

 with rwd, and "w with "wd. When 2s

' 2B, Srd becomes the deformed lay length of strand,

which may be expressed as Ls(1 % ,r). On the basis of a

single lay of a strand, equations 33 and 34 are obtained for

single and double helical wires, respectively. In each lay of

the strand, a double helical wire has an angle of wire

rotation of 2Bn, although the strand only has an angle of

strand rotation of 2B.

Path length of the deformed single helical wire in each lay

of strand:

Path length of the deformed double helical wire in each lay

of strand:

The wire strain is the change of the path length of a wire,

as expressed by equations 35 and 36.

Wire strain for single helical wires:

Wire strain for double helical wires:

Reduced Wire Radius

Experiments show that the axial elongation of a wire in

tension is always accompanied by lateral contraction of the wire.

For any material, the ratio of the unit lateral contraction to the unit

axial elongation is known as Poisson's ratio, F, which is constant

within the elastic range. The unit lateral contraction for a round

wire is determined by dividing the reduction of the wire radius in

the deformed rope structure by the original radius. If Rsand Rw

represent the original wire radii of single and double helical wires,

respectively, the reduced wire radii, Rsd and Rwd, can be expressed

in terms of the wire strain ,sand ,w, respectively, in accordance

with the definition of Poisson's ratio.
19
20

r

32d

& R

2

32d &

[ucL32(1%,30)]

2

[L32(1%,30)]

2 % (2Br

32d

)

2

'

&(c2uc% c5) % (c2uc% c5)

2 & 4c3

(c1u

2

c % c4

uc% c6)

2c3

(44)

Reduced radius of single helical wires:

Rsd ' Rs(1 & F,s) (37)

Reduced radius of double helical wires:

Rwd ' Rw(1 & F,w) (38)

A Poisson's ratio of 0.303 for steel [Avallone and Baumeister

1986] may be used for determining the reduced wire radius at

different wire strains.

Radius of Deformed Helix

The radius of the deformed helix for single helical wires

is a function of the various reduced wire radii. For example,

the radii of the deformed helices for wires W11 and W20,

according to figures 6 through 8, are simply the sum of the

various reduced wire radii, as shown below.

Radius of the deformed strand helix for wire W11:

r

11d' R10d% R11d (39)

Radius of the deformed strand helix for wire W20 or strand S2:

r

20d' R10d% 2R11d% 2R21d% R20d (40)

The radius of the deformed strand helix for wire W30 or strand

S3 cannot be simply expressed as in equations 39 and 40 for

wires W11 and W20. It will depend on the radius of the

deformed wire helix for wire W32, r32d, which is not merely

the sum of the various reduced wire radii. However, both can

be

related by the following equation.

Radius of the deformed strand helix for wire W30 or strand S3:

 r30d' R10d% 2R11d% 4R21d% 2R20d% R32d% r32d 

(41)

The radius of the deformed wire helix for double helical

wires also is a function of the various reduced wire radii. For

example, the radii of the deformed wire helix for wires W21

and W31, according to figures 8 and 9, are simply the sum of

the various reduced wire radii, as shown below.

Radius of the deformed wire helix for wire W21:

r

21d' R20d% R21d (42)

Radius of the deformed wire helix for wire W31:

r

31d' R30d% R31d (43)

The radius r32d will be dependent on the location of the contact

point, as shown in figure 9, of the adjacent elliptical cross

sections of wires W31 and W32 in the deformed rope structure.

Assuming that the contact point between the wire W32 at 2w'

&3B/2 rad and the wire W31 at 2w ' &(3B/2 % B/9) rad is at

(uc,vc) on the U-V plane, which is perpendicular to the

centroidal axis of the deformed strand S3, equations 44 and 45

can be derived. Equation 44 is obtained from the equations of

the two ellipses by letting v31 of the upper quadrants be equal

to v32 of the lower quadrants at u ' uc. Equation 45 is obtained

by letting their respective derivatives be equal, i.e., dv31/du '

dv32/du at u ' uc. The equations represent two conditions that

r

32d must satisfy.

Radius of the deformed wire helix for wire W32 subject to two

conditions:
21

uc[L32 (1%,30 )]

2

[L32 (1%,30 )]

2% (2Br

32d

)

2

[L32 (1%,30 )]

2% (2Br

32d

)

2

(R

2

32d&u

2

c

)[L32 (1%,30 )]

2% (2Br

32dR32d

)

2

'

1

2c3

&c

2

%

c2(c2uc%c5)&2c3(2c1uc% c4)

(c2uc%c5)

2& 4c3

(c1u

2

c % c4

uc% c6)

(45)
22

c1 ' cos2(N) %

sin2(N)

c7

(46)

2 sin(N) cos(N) % 2 sin(N)

c7

(47)

c3 ' sin2(N) %

cos2(N)

c7

(48)

4 '

&2 sin(N)(R30d % R31d)

c7

(49)

5 '

&2 cos(N) (R30d % R31d)

c7

(50)

[L30(1% ,r)]2 % (2Br30

L30(1% ,r)]2 % (2Br30d)2 %

(52)

c6 '

(R30d % R31d)2 & R

2

31d

c7

(51)

"sd ' tan&1 Ls(1% ,r)

2Brsd

(53)

"sd ' B & tan&1 Ls(1% ,r)

2Brsd

(54)

wd' tan&1 [Ls(1% ,r)]2 % (2Brsd)

(2Bnrwd)2

(55)

wd' B& tan&1 [Ls(1% ,r)]2% (2Brsd

(2Bnrwd)2

(56)

The coefficients c1 through c6 in these equations are described

by the following expressions, where N ' B/9. The common

term in all of the coefficients is c 7, as defined in equation 52.

 Because wire W10 is the straight wire of strand S1, the

wire strain is identical to the rope strain. Its reduced wire

radius, R10d, is simply equivalent to R10(1& F,r). The structural parameters in length measurement for the other helical

wires are interrelated, as indicated by the equations above.

They can be determined by solving the required simultaneous

equations for the wires in each strand. For example, four

simultaneous equations, i.e., equations 33, 35, 37, and 39, may

be solved for the structural parameters of wire W11 under a

certain rope strain. Once the structural parameters of the wires

in strand S1 in the deformed rope become known, the

structural parameters of the wires in strand S2 under the same

rope strain can be determined by solving eight simultaneous

equations, i.e., equations 33, 35, and 37 written for wire W20

with equation 40 and equations 34, 36, and 38 written for wire

W21 with equation 42. Using the solutions obtained for the

wires in strands S1 and S2, the structural parameters of the

wires in strand S3 under the same rope strain can be

determined by solving 20 simultaneous equations, i.e.,

equations 33, 35, and 37 written for wire W30 with

equation 41; equations 34, 36, and 38 written for wire W31

with equation 43; and equations 34, 36, and 38 written for wire

W32 with equations 44 through 52.

Angle of Deformed Helix

When the rope elongates under tensile load and is restrained from rotating, the angles of the strand and wire helices

will also change because of the axial elongation and lateral

contraction of the wire. Using figure 3, the relationship between the angle of the deformed strand or wire helix and the

rope strain, ,r, is given in the following equations.

Angle of the deformed strand helix:

If the original "s is less than B/2 rad, then

If the original "s is greater than B/2 rad, then

Angle of the deformed wire helix:

If the original "w is less than B/2 rad, then

If the original "w is greater than B/2 rad, then

Using the radii of the deformed helix determined previously, the angle of the deformed helix for single helical wires, "sd,

can be calculated with either equation 53 or 54, depending on

whether the original angle, "s, is less than or greater than B/2.

Similarly, the angle of the deformed helix for double helical

wires,"wd, can be calculated with either equation 55 or 56, also

depending on whether the original angle, "w

, is less than or

greater than B/2. The lay length of strand, Ls, in these equations is the original lay length.

The model for the deformed wire rope can now be constructed by replacing the structural parameters in the original

model with the deformed ones obtained from the analysis of

deformations shown above.
23

Bending and Twisting

To determine the bending and twisting of the wires, the

curvature and torsion of the deformed wires must be evaluated.

The equations required for evaluation are the same as those

derived earlier, except that the structural parameters of the deformed rope or the component functions of the deformed

model must now be used in equations 29b and 31b for single

helical wires and equations 30a and 32a for double helical

wires.

If 6sd and 6 wd represent the curvatures of the deformed

single and double helical wires, respectively, then bending

may be expressed as (6sd& 6s) or (6wd& 6w). Similarly, if Jsd

and Jwd

represent the torsions of the deformed single and double

helical wires, respectively, then twisting may be expressed as

(Jsd& Js

) or (Jwd& Jw). As indicated by equations 29b and 31b,

both the curvature and torsion of the deformed single helical

wires also will be independent of the angle of strand rotation.

Therefore, both the bending and twisting of single helical wires

would be uniform along their entire paths when a load is

applied. Both the curvature and torsion of the deformed

double helical wires are periodical functions of the angle of

wire rotation. The period is 2B, just as shown in figures 15

and 16 for the undeformed double helical wires.

CONCLUSIONS

The model developed in this report fully describes the geometry of the structure of wire ropes of any round-strand

construction. It is expressed by vector equations in a threedimensional, right-handed, rectangular Cartesian coordinate

system and is general enough that any combination of wire and

strand lay directions can be handled if the stated sign conventions for the angles of strand and wire rotation and the relative

rotation are followed in the component functions.

The wire paths are defined for the first time by using a

developed model, which not only reveal the shapes of the

various wires, but also are useful for predicting damage

patterns through external and internal wear. The geometric

properties of each wire can be easily evaluated by using this

model. A 33-mm 6×19 Seale, IWRC, right regular lay wire

rope was analyzed to illustrate the model's usefulness.

A system of equations was also established for

determining the structural parameters of the deformed rope at

a given rope strain, with restrained ends, thus obtaining the

model for the deformed rope. The geometric properties of

each deformed wire can be evaluated the same way as shown

in this report for the undeformed rope.

It is recommended for future work that stress analysis be

conducted based on the changes of these geometric properties

of the deformed wires to determine how the load is distributed

among these wires. Furthermore, the model could be used to

study the effect of wear and breaking of wires on strength loss

for the various round-strand wire ropes used in mine hoisting

so that more scientifically based retirement criteria can be

established.

REFERENCES

Avallone EA, Baumeister T III, eds. [1986]. Marks' standard handbook

for mechanical engineers. 9th ed. New York, NY: McGraw-Hill Book Co.,

p. 5-16.

CFR. Code of Federal regulations. Washington, DC: U.S. Government

Printing Office, Office of the Federal Register.

Costello GA [1990]. Theory of wire rope. New York, NY: SpringerVerlag, pp. 53-54.

Lee WK [1991]. An insight into wire rope geometry. Int J Solids

Structures 28(4):471-490.

Leithold L [1986]. The calculus with analytic geometry. 5th ed. New

York, NY: Harper & Row, Publishers, Inc., pp. 1072-1077.

Sokolnikoff IS, Redheffer RM [1958]. Mathematics of physics and modern

engineering. New York, NY: McGraw-Hill Book Co., Inc., pp. 311-315.

Stewart J [1991]. Calculus. 2nd ed. Pacific Grove, CA: Brooks/Cole

Publishing Co., pp. 688-691.

Wire Rope Technical Board [1993]. Wire rope users manual. 3rd ed.

Woodstock, MD: Wire Rope Technical Board.
DHHS (NIOSH) Publication No. 98-148

September 1998

To receive other information about occupational safety and health problems, call

1-800-35-NIOSH (1-800-356-4674), or

visit the NIOSH Home Page on the World Wide Web at 

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