Jtam-A4.dvi


JOURNAL OF THEORETICAL

AND APPLIED MECHANICS

56, 4, pp. 951-960, Warsaw 2018
DOI: 10.15632/jtam-pl.56.4.951

MODEL CONSTRUCTION AND EXPERIMENTAL VERIFICATION OF THE

EQUIVALENT ELASTIC MODULUS OF A DOUBLE-HELIX WIRE ROPE

Hong Yue Chen, Kun Zhang, Yang Xi Bai

School of Mechanical Engineering, Liaoning Technical University, Fuxin, China; e-mail: zhangkunliaoning@163.com

Ying Ma

Coal Mining and Designing Department, Tiandi Science and Technology Co., Ltd., Beijing, China

Han Zhong Deng

College of Materials Science and Engineering, Liaoning Technical University, Fuxin, China

To accurately describe mechanical properties of a complex wire rope, a double-helix wire
rope is used as an example in this study. According to the spatial structure characteristics
of the central helical line of each wire rope, the spatial configuration curve for the double-
-helix wire rope is obtained by using differential geometry theory.On the basis of this curve,
the mathematical model of the equivalent elastic modulus of the wire rope is developed,
and the elastic modulus of a 6×7+IWS wire rope is measured using a universal tensile
testingmachine. The experimental results are comparedwith the predicted results to verify
correctness of the elastic modulus prediction of the double-helix wire rope.

Keywords: mechanical properties, wire rope, double-helix wire rope, spatial configuration
curve, equivalent elastic modulus

1. Introduction

Wire ropes are widely used inmaterial-handlingmachinery because they possess high strength,
are light inweight and provide stable and reliable operation. Their safety and reliability directly
affect production efficiency andpersonnel safety. Since the 1950s, numerousdomestic and foreign
scholars have conducted research onwire ropes.However, because of the complex spiral structure
of awire rope, the theoretical basis required for accuratepredictionof itsmechanical performance
is not yet completely established. Some existing research studies related to this topic are as
follows. Stanova et al. (2011) fully considered the spatial spiral structure of a single wire and
strandedropesanddevelopedaparametricmathematicalmodel of thewire rope.Ma et al. (2015)
deduced a function expression for the central line of awire ropebased on the Serret-Frenet frame
theory.Using thedifferential geometry theory as the theoretical basis,Hobbs andNabijou (1995)
andNabijou andHobs (1995) providedapath expression for awire on a rope sheave basedon the
mathematical model of a vertical wire rope.Wu and Cao (2016) deduced the equivalent elastic
modulus of a wire rope based on the theory of slender elastic rods. Prawoto andMazlan (2012)
studiedmechanical properties of a steel wire ropeunder tensile load bynumerical simulation and
experimentalmethodsandobtained themicrostructure of fracture in the steelwire rope.Stanová
(2013) derived mathematical models of oval strand ropes and used Pro/E software to build
geometric models of the ropes. Sathikh et al. (1996) used theWempner-Ramsey theory to study
the asymmetryof the stiffnessmatrix in awire-rope elasticitymodel underbendingand torsional
loads and validated the model experimentally. Erdönmez and Erdem Imrak (2009) developed a
three-dimensional structural model of a double-helix wire rope that took into account friction
and slip between the rope cores. Machida andDurelli (1973) defined an expression for the axial
force, bendingmomentand torsionalmoment inahelix.Elata et al. (2004) proposedanewmodel



952 H.Y. Chen et al.

for simulating themechanical response of wire ropeswith individual steel cord cores. Themodel
fullly considered the double-helix structure of a single wire in the wound strand and provided
the stress in the wire layer to estimate global characteristics of the wire rope. Hu et al. (2016)
derived initial parameters of different general wire-rope models and developed IWRC6/36WS
wire-ropemodel usingMATLAB andPro/E. The elastic properties of the wire rope under axial
tension were analyzed by Abaqus/Explicit. Wang et al. (2015) developed a parametric model
for arbitrary centerline wire-rope structures and derived a series of recursive formulas for the
spatial enwinding equations of wires and strands. Liang et al. (2011) calculated the equivalent
elastic modulus of a wire rope in different positions based on a linear strengthening model.
Bai (2011) derived a mathematical expression for the equivalent elastic modulus of a wire-rope
conveyor belt based on pendency.Ma (2014) analyzed the equivalent elastic modulus of a wire-
-rope using static theory and validated the correctness of the analysis process by using ANSYS
andby experimentalmeasurements.Xu et al. (2012, 2015) proposedamethod for calculating the
equivalent elasticmodulusof a single-fibermultidirectional winding tubebased on the laminated
plate theory and a theoretical estimation method for calculating the three-dimensional elastic
modulus of a multifiber hybrid multidirectional winding tube considering the mixed effect.

In this study, based on the micromechanical wire-rope model, a theoretical formula for the
equivalent elastic modulus of a wire rope is deduced using the differential geometry theory
according to spatial characteristics of the central helical lines in the double-helix wire rope. The
formula is expected to serve as a theoretical basis for dynamic analysis and optimization of
mechanical systems containing wire ropes.

2. Three-dimensional geometrical model of the wire rope

The object of the study is a double-helix wire rope which is shown in Fig. 1. It comprises
stranded ropeswound around the central strand helix according to certain rules. Eachwire rope
is a helical wire bundlewith a high load-carrying capacity and consists ofmultiple wires twisted
around the corewire according to a spatial spiral relationship. In this study, the spatial geometry
of thewire, whose basic units are strands and ropes, is divided into the following four types: the
central strand corewire, central strand sidewire, lateral strand corewire, and lateral strand side
wire. The central strand core wire is mostly straight or has a simple curve. The central strand
side wire and the lateral strand side wire are first-degree spatial helical lines wound around the
central strand core wire. The lateral strand side wire is a second-degree spatial helical line with
respect to the central strand core wire. The cross section of the double-helix wire rope is shown
in Fig. 1b.

To facilitate the analysis, a unit length of the twisted wire rope is intercepted. The central
helical lines of the central strand core wire, central strand side wire, lateral strand core wire
and the lateral strand side wire are considered separately to establish the Cartesian coordinate
system, as shown in Fig. 2.

Consider a random point on the central strand side wire P1(x,y,z). The projection of this
point on theXOY plane is P ′1, whose coordinates are

x=R1cosθ1 y=R1 sinθ1 z=
θ1
2π
L1 (2.1)

whereR1 is the twisted circle radius of the central strand side wire (R1 =(d0+d1)/2, where d0
and d1 are the radii of the central strand corewire and the central strand sidewire, respectively,
in millimeters), θ1 is the polar angle corresponding to P1 in radians, and L1 is the twist pitch
of the central strand side wire in millimeters.



Model construction and experimental verification of the equivalent elastic modulus... 953

Fig. 1. Schematic diagram of the wire rope; (a) structure diagram of the wire rope,
(b) cross section of the wire rope

Fig. 2. Spatial coordinate system of the wire rope

Consider a small arc P1Q1 with P1 as the starting point. Then, the coordinates of Q1 are
(x+dx,y+dy,z+dz), and thus

dx=−R1 sinθ1dθ1 dy=R1cosθ1dθ1 dz=
L1
2π
dθ1 (2.2)

The length of the small arc P1Q1 can be obtained from equations (2.1) and (2.2)

dS1 =

√

R21+
(L1
2π

)2
dθ1 (2.3)

Thus, the length of the central strand side wire is

S1 =

2π
∫

0

√

R21+
(L1
2π

)2
dθ1 =

√

4π2R21+L
2
1 (2.4)

The tangent inclination cosine of the small arc P1Q1 at P1 on the central strand side wire is

cosα1 =
L1

√

4π2R21+L
2
1

=
L1
S1

(2.5)



954 H.Y. Chen et al.

Similarly, consider a randompoint on the lateral strand core wireP2(x,y,z). The projection
of this point on theXOY plane is P ′2, whose coordinates are

x=R2cosθ2 y=R2 sinθ2 z=
θ2
2π
L2 (2.6)

whereR2 is the twisted circle radius of the lateral strand core wire (R2 =(d0+d2)/2+d1+d3,
where d2 and d3 are the radii of the lateral strand core wire and the lateral strand side wire,
respectively, in millimeters), θ2 is the polar angle corresponding to P2 in radians, andL2 is the
twist pitch of the lateral strand core wire in millimeters.
Consider a small arc P2Q2 with P2 as the starting point. Then, the coordinates of Q2 are

(x+dx,y+dy,z+dz), thus

dx=−R2 sinθ2dθ2 dy=R2cosθ2dθ2 dz=
L2
2π
dθ2 (2.7)

The length of the small arc P2Q2 can be obtained from equations (2.6) and (2.7)

dS2 =

√

R22+
(L2
2π

)2
dθ2 (2.8)

The length of the lateral strand core wire is

S2 =

2π
∫

0

√

R22+
(L2
2π

)2
dθ2 =

√

4π2R22+L
2
2 (2.9)

The tangent inclination cosine of the small arc P2Q2 at P2 on the central strand side wire is

cosα2 =
L2

√

4π2R22+L
2
2

=
L2
S2

(2.10)

Consider a random point on the lateral strand side wire P3(x,y,z). The projection of this
point on theXOY plane is P ′3, whose coordinates are

x=R2cosθ2−R3cosθ2cosθ3+R3 sinβ2 sinθ2 sinθ3

y=R2 sinθ2−R3 sinθ2cosθ3−R3 sinβ2cosθ2 sinθ3

z=
θ2
2π
L2+R3cosβ2 sinθ3

(2.11)

Consider a small arc P3Q3 with P3 as the starting point. Then, the coordinates of Q3 are
(x+dx,y+dy,z+dz), and thus

dx=
[

−

1

n
R2 sin

θ3
n
+
(1

n
+sinβ2

)

R3 sin
θ3
n
cosθ3+

(

1+
1

n
sinβ2

)

R3cos
θ3
n
sinθ3

]

dθ3

dy=
[1

n
R2cos

θ3
n
−

(1

n
+sinβ2

)

R3cos
θ3
n
cosθ3+

(

1+
1

n
sinβ2

)

R3 sin
θ3
n
sinθ3

]

dθ3

dz=
( L2
2πn
+R3cosβ2cosθ3

)

dθ3

(2.12)

whereR3 is the twisted circle radius of the lateral strand side wire, in millimeters (R3 =(d0+
d1+d3)/2), θ3 is the polar angle corresponding to P3 in radians, β2 is the helical ascent angle
of the lateral strand side wire in radians, and n is the number of twisted rounds of the lateral
strand side wires about the lateral strand core wire.



Model construction and experimental verification of the equivalent elastic modulus... 955

Next, set

C1 =
1

n
R2 C2 =

(1

n
+sinβ2

)

R3 C3 =
(

1+
1

n
sinβ2

)

R3

C4 =
L2
2πn

C5 =R3cosβ2

(2.13)

The length of the small arc P3Q3 can be obtained from equations (2.11)-(2.13)

dS3 =
√

(C1−C2cosθ3)2+(C3 sinθ3)2+(C4+C5cosθ3)2 dθ3 (2.14)

The length of the lateral strand side wire is

S3 =

2π
∫

0

√

(C1−C2cosθ3)2+(C3 sinθ3)2+(C4+C5cosθ3)2 dθ3 (2.15)

The tangent inclination cosine of the small arc P3Q3 at P3 on the central strand side wire is

cosα3 =
C4+C5cosθ3

√

(C1−C2cosθ3)2+(C3 sinθ3)2+(C4+C5cosθ3)2
(2.16)

Combining equations (2.5), (2.10) and (2.16), the nominal area of the wire rope can be
obtained

A∗ =A0+6
A1
cosα1

+6
A2
cosα2

+36
A3
cosα3

(2.17)

where A0, A1, A2 and A3 are the cross-sectional areas of the central strand core wire, central
strand side wire, lateral strand core wire and lateral strand side wire, respectively.

3. Calculation of the elastic modulus of the wire rope

Because only the equivalent elastic modulus of the double-helix wire rope is calculated in this
study, the following assumptions aremade: (1) The cross section of the wire is perpendicular to
the tangent line corresponding to the central helix. (2) Friction between the wires is negligible.
(3) Thewires elongate without twisting, and the elongation amount is recorded as∆L. Tension
and elongation of the central strand core wire, central strand side wire, lateral strand core wire
and the lateral strand side wire are deduced separately.

Ad. (1) For the central strand core wire, after an elongation of ∆L in theZ-axis direction, the
corresponding tensile stress ε and tension T0 are respectively given as

ε=E
∆L

L1
T0 = ε0A0 =

πEd20
4L1
∆L (3.1)

Because materials of the wire rope are mostly nonalloy carbon steels, the elastic modulus is
assumed to beE =183.9GPa (Liu, 2014).

Ad. (2) According to the helical spatial structure of the central strand side wire, the elongation
can be deduced by the whole differential as follows

∆S1 =
4π2R1
S1
∆R1+

L1
S1
∆L=

L1
S1
∆L
(

1+
4π2R1
S1

∆R1
∆L

)

(3.2)



956 H.Y. Chen et al.

According toWang et al. (2004)

1+
4π2R1
S1

∆R1
∆L
≈ 1 (3.3)

Thus

∆S1 ≈
L1
S1
∆L (3.4)

Based on material mechanics, the tension of a single central strand side wire in the Z-axis
direction can be expressed as

T1 =E
∆S1
S1
A1cosα1 =

πEd21L
2
1

4S31
∆L (3.5)

Ad. (3) Because the lateral strand core wire has the same spatial structure as the central strand
side wire, the samemethod can be used to obtain expressions for the elongation and tension of
the lateral strand core wire. The elongation ∆S2 can be expressed as

∆S2 ≈
L2
S2
∆L (3.6)

The tension T2 in theZ-axis direction is

T2 =E
∆S2
S2
A2cosα2 =

πEd22L
2
2

4S32
∆L (3.7)

Ad. (4) The lateral strand side wire is the second-degree spatial helical line with respect to the
central strand core wire; thus

∆S3 =
∂S3
∂R3
+
∂S3
∂L3

T3 =E
∆S3
S3
A3cosα3 =

EA3cosα3
S3

(∂S3
∂R3
+
∂S3
∂L3

)

(3.8)

According to the above equations, the tension of the wire rope T∗ can be expressed as

T∗ =T0+6T1+6T2+36T3 (3.9)

The elastic modulus of the wire rope can be obtained according to equations (2.17) and (3.9)

E∗ =
T∗

A∗X
(3.10)

Thus

E∗ =LE

d
2

0

L1
+
6d2
1
L1 cosα1
S
2

1

+
6d2
2
L2cosα2
S
2

2

+
36

(

∂S3

∂R3
+
∂S3

∂L3

)

d
2

3
cosα3

S3

d20+
6d2
1

cosα1
+
6d2
2

cosα2
+
36d2
3

cosα3

(3.11)

whereX =∆L/L (L is length of the wire rope in millimeters), S1 =
√

4π2R21+L
2
1 is length of

the central strand sidewire inmillimeters,S2 =
√

4π2R22+L
2
2 is length of the lateral strand side

wire in millimeters, S3 (see Eq. (2.15)) is length of the lateral strand side wire in millimeters,
cosα1 = L1/S1 is tangent inclination cosine of the small arc on the central strand core wire,
cosα2 =L2/S2 is tangent inclination cosine of the small arc on the lateral strand core wire, and
cosα3 (see Eq. (2.16)) is tangent inclination cosine of the small arc on the lateral strand side
wire. In addition,Ci (i=1, . . . ,5) see Eqs. (2.13).



Model construction and experimental verification of the equivalent elastic modulus... 957

4. Case analysis and verification

To verify the correctness of the expression for the equivalent elastic modulus predicted by the
spatial distribution of the central line of the double-helix wire rope, the 6×7+IWS wire rope is
used as an example. Thebasic structural parameters of this rope are listed inTable 1. According
to equation (3.11), the equivalent elasticmodulus of thewire rope can bedetermined.According
to the measurement method of the actual elastic modulus of a wire rope given in the national
standardGB/T24191-2009, the universal tensile testingmachine (Fig. 3) is used tomeasure the
actual equivalent elastic modulus of the wire rope.

Table 1. Structural parameters of 6×7+IWS wire rope

Wire rope
diameter
[mm]

Twist
pitch
[mm]

Number of twisted rounds
Wire
twist
angle [◦]

Wire-rope diameter
of lateral strand side [mm]
wires about lateral central lateral
strand core wire strand strand

4.5 36 3.1416 10.3848 0.6 0.4

Fig. 3. Universal tensile testing machine

In the analysis, 600mm of the wire rope is intercepted from selected wire-rope samples and
placed at a room temperature of 18◦ for 24h.Both ends of the sample aremechanically clamped
in the universal tensile testing machine. Each end has clamping length of 50mm. The uniaxial
tensile test (Wu et al., 2014) is performed by clamping the wire rope through the jaws of the
testing machine.

According to the requirements of the national standard GB/T24191-2009, the elastic mo-
dulus of the wire-rope sample in the fully stable state should be 10%-30% of the minimum
breaking tension (or nominal breaking load). The loads at 10% and 30% are denoted as F10%
andF30%, respectively. Simultaneously, the elongations of the wire rope atF10% andF30% loads
are recorded as x1 and x2, respectively. According to the national standard GB8918-2006, the
minimumbreaking tension of the 6×7+IWS (2006) wire rope is 11.6kN.According to the above
experimental steps, the corresponding displacement-load deformation curve is obtained for the
wire rope throughmeasurement of the elastic modulus, as shown in Fig. 4.

According to the actual elastic modulus of the wire rope given in GB/T24191-2009

E10−30 = l0
F30%−F10%
A(x2−x1)

(4.1)



958 H.Y. Chen et al.

Fig. 4. Displacement-load deformation curve

whereE10−30 is the actual elasticmodulus of thewire rope in gigapascals: l0 is the initial length
of the wire rope in millimeters; F10% and F30% are the loads at 10% and 30% of the minimum
breaking tension (ornominalbreaking load) of thewire-rope sample, respectively, inkilonewtons;
A is the cross-sectional area of the wire rope in millimeters (in this studyA=A∗); and x1 and
x2 are the elongations of the wire rope at F10% and F30% loads, respectively, in millimeters.
According to equation (4.1), the elastic modulus of the 6×7+IWSwire rope is

E10−30 =500
3480−1160

13.52(11.75−2.39)
= 9.17GPa (4.2)

Compared to the theoretical value of the equivalent elastic modulus of the wire rope
E∗ = 8.75GPa, the error of the calculated is value 4.8%, which is relatively small. The re-
sults show that the value calculated using the model can be used to accurately predict the
actual elastic modulus of the wire rope.

5. Conclusion

In this study, according to the spatial distribution characteristics of each central line of a double-
-helix wire rope, the theoretically predicted expression for the equivalent elastic modulus of the
wire rope is obtained using the differential geometry theory. The 6×7+IWS wire rope is used
as the study object, and an elastic modulusmeasurement experiment is performed on this rope.
The correctness of the theoretical deduction of the equivalent elasticmodulus is verified through
the comparison and analysis of the experimental value and the theoretically predicted one.

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Manuscript received August 18, 2017; accepted for print February 6, 2018