Sung-Yun, Kim 2017. 5. 10 - KAISTcmss.kaist.ac.kr/cmss/PhD_Defense/Ph.D_defence_SY.Kim.pdf ·...

Sung-Yun, Kim

2017. 5. 10

-1-

Fixed platform

Spar FPSO

• URF system :

Umbilical, Riser & Flowline

Land↔Island, Country↔Country

Ocean Platform

Offshore Wind Farm

• Power Transmission system :

Submarine Power Cable System

High system cost, High maintenance fee, High risk when breakdown

Analysis of mechanical behavior & Robust design for cable are very important !!

-2-

Cu

XLPE

Lead alloy sheath

HDPE

Bedding(PP yarn)

Steel wire armor(PE Coated)

Outer serving(PP yarn)

Optical fiber cable (SM 24fibers)

• Cables are Complex Structures

A variety of geometries & materials

• Complex Behavior of Cable

Geometrical nonlinearities

• Large deformations

Material nonlinearities

• Nonlinear stress-strain curves

Contact

• Wire slip (effect of internal friction)

• Main sources for Cable design

Experiments

• Expensive & difficult to conduct

Analytical or Numerical model

• If sufficiently accurate,

it can be a good solution.

-3-

Model Features List

AS

I S

Analytical model (Hruska, Costello, etc.)

- Wire theory

- Considering stretch, twist & bending

Only macro behavior is possible.

- Simple & easy to use

- Validity is limited

Ring Element (Knapp)

- Cable CAD

Concise FEM (Jiang, Yao)

Only 2D or Plain strain is possible.

- Not sufficient to solve 3D behavior

Simple beam model with contact

TO

BE

Cable FE model

- Multiple beam FE model

Efficient & Easy to use

- Less computational time

- Reasonable accuracy

3D Solid Element (Judge, Yujie, Stanova)

- Full 3D model

Great amount of time required.

- Accurate solution

- Long solving time

- Not easy to carry out parametric study

- Element error occurs frequently

-4-

Suggestion of multiple beam FE models to predict efficiently the

mechanical behavior of the cable

Evaluation of the validity & limitation of analytical models using

stiffness comparison method

Proposition of the procedure of torque balance design & Verification

of the design result

Suggestion of the generalized torque balance curves

-6-

hK

K

K

K

M

F

T

T

/

A A

TM

TF

cr

wr

rr

h

hh ll

)( r

(a) (b) (c)

l

(RHL)

s

xfyf

zf

xm ym

zm

1x

2x

3x

(d)

xF

yFzF

xM

yM

zM

x

y

z

• Kinematic relations

A A

TM

TF

cr

wr

rr

h

hh ll

)( r

(a) (b) (c)

l

(RHL)

s

xfyf

zf

xm ym

zm

1x

2x

3x

(d)

xF

yFzF

xM

yM

zM

x

y

z

h

h

tan)(

hr

22 cossin l

lw

)(cossin2sincos2

0

r

)(cossin2cos)cos(sin

0

r

- Axial & shear strain of cable

- Axial strain of wire

- Final curvature of wire

- Final twist of wire

-7-

A A

TM

TF

cr

wr

rr

h

hh ll

)( r

(a) (b) (c)

l

(RHL)

s

xfyf

zf

xm ym

zm

1x

2x

3x

(d)

xF

yFzF

xM

yM

zM

x

y

z

0 xyzyx fFF

ds

dF

0 yxxz

yfFF

ds

dF

• Equilibrium eq. of wire

0 zxyyxz fFF

ds

dF

0 xyzyx mMM

ds

dM

0 yxxz

ymMM

ds

dM

0 zxyyxz mMM

ds

dM

EIM y GJM zwz EAF

yyzy MMF

- Tension, twisting & bending can be simply expressed :

(only external tension applied)

-8-

ccyzT AEFFnF )cossin(

ccyzyzT JGrFrFMMnM )sincoscossin(

• Equilibrium eq. of stranded cable

• Stiffness components of various analytical models

- Hruska’s model

cc AEEAnK )sin( 3

)cossin( 2 EArnKK

cc JGEArnK )cossin( 22

- Machida’s model

cc AEEAnK )sin( 3

)cossin( 2 EArnK

r

EI

r

GJEAr

nK

2

22

cossin2sin

cossin2coscossin

cc JGEI

GJEArnK

cossin2sin

sin2coscossin

2

322

-9-

• Analytical models with their principal features

- Costello’s model

cc AEr

EI

r

GJEAnK

2

23

2

43 sincos2sinsincos2cos

sin

r

EI

r

GJEArnK


cossin

r

EI

r

GJEArnK

)sin1(sincos2sincossin2coscossin

2242

ccJGEIGJEArnK )sin1(sincos2sinsin2coscossin 22522

Model Behavior of wires

Poisson’s effect Tension Torsion Bending

Hruska O × × ×

Machida O O O ×

Costello O O O O

-10-

• Evaluation & Comparison of stiffness coefficients

Layer

No.

No. of

Wires

Helical

direction

& angle

Wire

diameter

[mm]

Pitch

length

[mm]

Young’s

Modulus

[GPa]

Plastic

Modulus

[GPa]

Yield

Strength

[GPa]

Poisson’s

ratio

Core 1 - 3.94 - 188 24.60 1.54 0.3

1 6 RHL, 72.97º 3.72 78.67 188 24.60 1.54 0.3

A A

TM

TF

cr

wr

rr

h

hh ll

)( r

(a) (b) (c)

l

(RHL)

s

xfyf

zf

xm ym

zm

1x

2x

3x

(d)

xF

yFzF

xM

yM

zM

x

y

z

- 7-wire helically stranded cable

-11-


Model Helix angle [deg] [MN] [MN-mm] [MN-mm] [MN-mm2]

Hruska

50 7.80 17.71 17.71 58.63

60 10.26 17.61 17.61 40.65

70 12.46 14.18 14.18 21.48

80 14.00 7.91 7.91 7.05

Machida

50 7.80 17.71 16.71 61.57

60 10.26 17.61 16.69 45.10

70 12.46 14.18 13.49 27.08

80 14.00 7.91 7.54 13.34

Costello

50 7.90 17.28 16.26 60.36

60 10.30 17.33 16.40 42.74

70 12.48 14.07 13.38 23.69

80 14.00 7.89 7.53 9.22

K K K K

The difference in each stiffness coefficient is relatively small.

-12-

- Relative significance of individual contributions (for Costello’s model)

40 50 60 70 80 90

Helix angle [deg]

0

4

8

12

16

K [M

N]

Stretch

Twist

Bending

40 50 60 70 80 90

Helix angle [deg]

0

4

8

12

16

20

K

[M

N-m

m]

Stretch

Twist

Bending

2

23

2


sinr

EI

r

GJEAnK

r

EI

r

GJEArnK


cossin

Stretch Twist Bending



-13-

Stretch terms are dominant in every stiffness coefficient !!

40 50 60 70 80 90

Helix angle [deg]

0

5

10

15

20

K

[M

N-m

m]

Stretch

Twist

Bending

40 50 60 70 80 90

Helix angle [deg]

0

20

40

60

K [M

N-m

m2]

Stretch

Twist

Bending

r

EI

r

GJEArnK

)sin1(sincos2sincossin2coscossin

2242

)sin1(sincos2sinsin2coscossin 22522 EIGJEArnK




-15-

• Finite element models for 7-wire helically stranded cable

(b)

(a)

(c)

8-node solid element

solid node

2-node beam element

beam node

solid element beam element

- Three different model : 8-node solid FE model, 2-node beam FE model and

mixed model (by commercial software CATIA & Marc)

Solid FE model

Beam FE model

Mixed FE model

-16-

• Boundary & Loading condition

BF0 zyx uuu

0 zyx

0 ,0 ,0 zyx uuu

0 zyx

tension)-Pre(TF

0 zyx uuu

0 zyx 0 ,0 zyx uuu

0 zyx

Fixed end Loading end

x

z

y

TF

(a)

(b)

BF0 zyx uuu

0 zyx

0 ,0 ,0 zyx uuu

0 zyx

tension)-Pre(TF

0 zyx uuu

0 zyx 0 ,0 zyx uuu

0 zyx


x

z

y

TF

(a)

(b)

BF0 zyx uuu

0 zyx

0 ,0 ,0 zyx uuu

0 zyx

tension)-Pre(TF

0 zyx uuu

0 zyx 0 ,0 zyx uuu

0 zyx


x

z

y

TF

(a)

(b)

- Axial load case

- Transverse load case

-17-

• Full solid finite element model

Center Point

1 Pitch =78.67mm

Max. Point

(Contact Point)

(a)

- Extrusion of the wire cross-section along the helix corresponding to the centroid line

- Construction of solid elements of each wire positioned around the core

- To accurately capture the radial contact, a relatively large number (28 elements) of solid

elements were used on the circumference direction.

- Convergence study was performed for model length and mesh refinements.

-18-

• Convergence study of solid FE model

- Model length

0 1 2 3 4

Model Length [Number of Pitches]

1430

1432

1434

1436

1438

1440

Eq

uiv

ale

nt

Str

ess

[M

Pa]

Center Point

Max. Point

0 40 80 120

Number of Elements per Pitch Length

1424

1428

1432

1436

Eq

uiv

ale

nt

Str

ess

[M

Pa]

Center Point

Max. Point

Min. 2 pitches Min. 60 elements per pitch

- Mesh refinements

-19-

(a)

Beam

90º d

Beam b

a

e

ar

br

(b)

contact area

• Beam finite element model

(a)

Beam

90º d

Beam b

a

e

ar

br

(b)

contact area

- Beam to beam contact

. Not only node to node, but node to element contact

(algorithm creates extra node at contact points)

. Coulomb friction model with a friction coefficient of 0.115

. The friction contact conditions between the wires and between the wires and the core

0 errdg ba

- Contact condition of circular beam

Contact penetration function :

-20-

• Convergence study of beam FE model

1 Pitch =78.67mm Center Point

(a)- Model length

0 1 2 3 4

Model Length [Number of Pitches]

1428

1432

1436

1440

Eq

uiv

ale

nt

Str

ess

[M

Pa]

0 40 80 120

Number of Elements per Pitch Length

1320

1360

1400

1440

Eq

uiv

ale

nt

Str

ess

[M

Pa]

Min. 2 pitches Min. 40 elements per pitch

- Mesh refinements

-21-

• Mixed finite element model

(b)

(a)

(c)

8-node solid element

solid node

2-node beam element

beam node

solid element beam element

- Obtaining of the advantages of both the solid FE and beam FE models

- Consideration of the contact between the solid elements and the beam elements

- Solid elements : 11,932, Beam elements : 942 , DOFs : 48,825

Master node

Rigid link

• Special B.C : Master node & Rigid link

- Loads are applied at the master node and then conveyed to the whole section.

- The end effects can be greatly reduced.

-23-

• Axial-loading analysis

0 0.004 0.008 0.012 0.016 0.02

Axial Strain

0

50

100

150

200

250

Axia

l L

oad

[k

N]

Experiment

Analytical model

Solid model

Beam model

Mixed model

0 20 40 60 80 100

Torque [N-m]

0

20

40

60

80

100

Axia

l L

oad

[k

N]

Experiment

Analytical model

Beam model

- Axial load/strain curves - Axial load/torque curves

- All of the FE models are in sound agreement with the experimental results.

(Analytical results agreed with experimental results only in the linear-elastic range)

-24-

• Von-Mises stress contours under axial loading

- The maximum stress occurs in the core (first at yield & the ultimate stress).

: Core is subjected to both axial and transverse contact stress.

: Wires bear less stress because of unwinding.

- Beam model and mixed model have a good agreement with 3D solid model

Von-Mises stress @ ε=0.019, all of wires already yielded (Sy=1540MPa)

(b)

(a)

(c)

Solid FE model

Beam FE model

Mixed FE model

-25-

• Transverse-loading

0 20 40 60 80

Transverse Displacement [mm]

-8

-6

-4

-2

0

Tran

sverse

Load

[k

N]

Solid model

Beam model

Mixed model

10kN20kN30kNPretension

(b)

(c)

(a)

- Transverse load/displacement curves - Von-Mises stress (pretension 20kN)

- Three different pre-tensions : 10kN, 20kN, and 30kN. The transverse loading : up to 7kN

- Almost same results are obtained with all FE models from load/displacement curves.

- The global stress level of beam model is similar with the other FE models (except boundary end).

Solid FE model

Beam FE model

Mixed FE model

-26-

• Influence of friction

- Influence of friction model (Coulomb friction)

nt ff tnt ff : stick : slip

tf

rv

Stick

Slip

tuor

nf

nftf : tangential force nf : normal force

t

: friction coefficient

: tangential direction vector

Arctangent model Stick-slip model Bilinear model

tf

rv

rv vR 01.0

rv vR

rv vR 1.0

nf

nf

tu

nf

tf tf

tu

nf

-27-

• Influence of friction

0 0.004 0.008 0.012 0.016 0.02

Axial Strain

0

40

80

120

160

Axia

l L

oad

[k

N]

Arctangent model

Stick-slip model

Bilinear model

- Axial load/strain

- Computation time with friction model

Friction model Arctangent model Stick-slip model Bilinear model

Computation time [sec] 26,308 [123%] 23,916 [112%] 21,450 [100%]

- The global response is almost the same.

- The influence of friction models on the global

behavior of the cable is small.

- The bilinear friction model is an effective

in terms of computational efficiency.

-28-

• Influence of friction coefficients (under axial load)

- Axial load/strain curves

0 0.004 0.008 0.012 0.016 0.02

Axial Strain

0

40

80

120

160

Axia

l L

oad

[k

N]

Friction 0.0

Friction 0.05

Friction 0.115

Friction 0.2

Friction 0.4

0 0.1 0.2 0.3 0.4

Friction Coefficient

1410

1420

1430

1440

1450

Eq

uiv

ale

nt

Str

ess

[M

Pa]

Center Point

Max. Point

- The influence of friction coefficient on the global behavior of the cable is small.

- The friction coefficient increases, the equivalent stresses decrease.

- Von-Mises stress

-29-

0 20 40 60 80

Transverse Displacement [mm]

-8

-6

-4

-2

0

Tran

sverse

Load

[k

N]

Friction 0.0

Friction 0.05

Friction 0.115

Friction 0.2

Friction 0.4

10kNPretension 30kN 20kN

(b)

0 0.1 0.2 0.3 0.4


150

160

170

180

190

155

165

175

185

195

Center Point

0 0.1 0.2 0.3 0.4


800

840

880

780

820

860

Max. Point

Eq

uiv

ale

nt

stre

ss [

MP

a]

- Axial load/strain curves - Von-Mises stress

• Influence of friction coefficients (under transverse load)

- The friction effect plays a small role in global behavior.

-30-

• Remarks on computational cost and accuracy

- Computational time

FE model Number of

Elements

Number of

DOFs

Computation time [sec]

Axial load Transverse load

Solid model 96,556 [100%] 343,104 [100%] 21,450 [100%] 13,445 [100%]

Beam model 1,099 [1.1%] 6,636 [1.9%] 999 [4.7%] 1,121 [8.3%]

Mixed model 12,874 [13.3 %] 48,825 [14.2 %] 3,193 [14.9 %] 1,745 [13.0 %]

- Equivalent stresses (Von-Mises stress)

FE model

Center point [MPa] Maximum point [MPa]

Axial load

[100 kN]

Transverse load

[7 kN]

Axial load

[100 kN]

Transverse load

[7 kN]

Solid model 1431 162 1434 860

Beam model 1428 159 1428 851

Mixed model 1431 161 1433 858

-31-

• Validity & Limitation of analytical solutions

50 60 70 80 90

Helix angle [deg]

6

8

10

12

14

16

K

[M

N]

Beam model

Hruska

Machida

Costello

50 60 70 80 90

Helix angle [deg]

0

10

20

30

40

50

60

K

[M

N-m

m2]

Beam model

Hruska

Machida

Costello

(within 2.2%) 75

hK

K

K

K

M

F

T

T

/

0TF 0 0TM 0

(except Hruska’s model) 5.62

-32-

• Validity & Limitation of analytical solutions

0TF

within 5% 75

0 0TM 0

50 60 70 80 90

Helix angle [deg]

0

4

8

12

16

20

K

[M

N-m

m]

Beam model

Hruska

Machida

Costello

50 60 70 80 90

Helix angle [deg]

0

4

8

12

16

20

K

[MN

-mm

]

Beam model

Hruska

Machida

Costello

70 within 5%

-34-

• Torque balance design

- During axial loading, the helical wires induce a twisting of the stranded cable.

- Cable twisting may loosen some wires and tighten others depending on the helix directions.

(Some layers are stressed at higher levels and the breaking strength is considerably reduced.)

– Slight relaxations of the cable tension can result in hockling (looping) due to the instability.

Method to prevent : external torque, minimization of twisting in design stage

Design method

Finding the proper helical

angles to make “0” twisting

α2 (LHL)

α3 (RHL)

A A

α1 (RHL)

r1

r2

r3

TF

(a) (b)

0 zyx uuu

0 zyx 0 ,0 zyx uuu

0 zyx

x

z

y

TF

(c)

TM

-35-

Liu F.C. R.P. Christian R.H. Knapp & Shibu G.

• Concept of torque balance

Concept of Torque ratio

0cossin1

2

n

i

ii

c

iii

L

rEAK

0/ h 0 K0TM

hK

K

K

K

M

F

T

T

/

• Previous study for torque balance

11111 sin1

rEAT w

iiii

oooot

rdn

rdnR

sin

sin2

2

22222 sin

2 rEAT w

Torque balance curve

Torque balance design can be used in the same manner regardless of the static or

dynamic condition

-36-

• Two layer cable

- Geometric and material properties

Layer

No.

No. of

wires

Helical

angle [α]

Wire

Diameter [mm]

Pitch

length [mm]

Model

Length [mm]

Young’s

Modulus [GPa]

Poisson’s

ratio

Core 1 - 3.66 - 334.56 198 0.3

1 6 RHL, 75.37º 3.33 84.11 334.56 198 0.3

2 12 RHL, 75.9º 3.33 167.28 334.56 198 0.3

- Geometry

-37-

• Two layer cable

0 0.002 0.004 0.006

Axial Strain

0

40

80

120

Axia

l L

oad

[k

N]

Experiment

Analytical model

Beam model

0 0.002 0.004 0.006

Axial Strain

0

40

80

120

Axia

l L

oad

[k

N]

Experiment

Analytical model

Beam model

Strain

0.003

86.3

76.2

72.1

- Axial load/axial strain curves - Axial load/torque curve

0 40 80 120 160

Torque [N-m]

0

40

80

120

Axia

l L

oad

[k

N]

Experiment

Analytical model

Beam model

- The differences from the experimental results

are 20 % and 5.7 % for the analytical model

and the beam FE model, respectively.

- Cable produces a net torque of 140 N-m at a

working load of 120 kN ; that is, the torque in

this cable is not balanced.

-38-

• Two layer cable

72 76 80 84 88

Layer 2 [, LHL]

-160

-120

-80

-40

0

40

Torq

ue

[N-m

]

=85

=82.5

=80

=77.5

=75

=72.5

Torque balance line &

points (Torque=0) from

beam model analysis

- Torque balance points

Perform torque analyses, find zero torque

points and plot the torque balance curve

Compute the flexural rigidity and plot the

flexural rigidity curve

Determine the torque balance point that

satisfies the design stress limit and

minimizes the flexural rigidity

Select helix angles of each layer

(1 and 2 )

Compute the equivalent stress and plot

the equivalent stress curve

[Step 1]

[Step 2]

[Step 3]

[Step 4]

[Step 5]

- Torque balance design procedure

- Six values (72.5°, 75°, 77.5°, 80°, 82.5°, and 85° RHL)

and five values (72.5°, 75°, 80°, 85°, and 90° LHL) are selected.

1

2

- 30 cases (six × five )of torque analyses were performed using beam FE model 1 2

-39-

• Two layer cable

- Bending B.C and deformation - Flexural rigidity curve

- A lower F.R. More effective cable handling capability

- Nonlinear & friction dependent behavior

- A smaller helix angle A smaller flexural rigidity

M

F.R. (Flexural rigidity) before cable installation is very important factor to handle the cable.

0 1 2 3 4 5

Curvature [m-1]

20

20.5

21

21.5

22

22.5

23

23.5

Fle

xu

ral

rig

idit

y [

N-m

2]

72.5-86.3

75-86.7

77.5-87.1

80-87.6

82.5-88.2

85-88.8

Bend radius : 0.6 [m]

-40-

• Two layer cable

①

②

③

72 76 80 84

Layer 1 [, RHL]

86

86.5

87

87.5

88

88.5

89

Layer 2

[

2, L

HL

]


Equivalent stress curve

Flexural rigidity curve

700

800

900

1000

1100

1200

1300

Eq

uiv

ale

nt

Str

ess

[M

Pa]

21.5

21.6

21.7

21.8

21.9

22

22.1

Fle

xu

ral

Rig

idit

y [

N-m

2]

Design stress limit [924 MPa]

- Torque balance

2. Beyond Point ①

satisfies design stress limit

85.781

3. Point ② has minimum flexural

rigidity

4. Point ③

finally determined

85.781 41.872

1. Torque balance curves are plotted

(the relations between and ) 1 2

-41-

• Three layer cable

α2 (LHL)

α3 (RHL)

A A

α1 (RHL)

r1

r2

r3

TF

(a) (b)

0 zyx uuu

0 zyx 0 ,0 zyx uuu

0 zyx

x

z

y

TF

(c)

Layer

No.

No. of

wires

Helical

direction

and angle [α]

Wire

diameter

[mm]

Pitch

length

[mm]

Young’s

Modulus

[GPa]

Poisson’s

ratio

Core 1 - 1.09 - 190 0.3

1 6 RHL, 79.23º 1.00 34.52 190 0.3

2 12 LHL, 79.23º 1.00 67.55 190 0.3

3 18 RHL, 79.23º 1.00 100.58 190 0.3

(a)

(b)

(c)

- Beam FE model - Geometry

Core & layer 1

Layer 2

Layer 3

-42-


0 0.002 0.004 0.006 0.008

Axial Strain

0

10

20

30

40

Axia

l L

oad

[k

N]

Experiment

Analytical model

Beam model

Torque balance line &

points (Torque=0) from

beam model analysis

72 76 80 84 88

Layer 3 [, RHL]

-4

0

4

8

12

Torq

ue

[N-m

]

=85

=82.5

=80

=77.5

=75

=72.5

- Axial load/strain curves - Torque balance point (@ ) 851

- Six values (72.5°, 75°, 77.5°, 80°, 82.5°, and 85° RHL), Six values (72.5°, 75°, 77.5°, 80°, 82.5°, and 85° LHL),

and five values (72.5°, 75°, 80°, 85°, and 90° RHL) are selected.

1

3

2

- 180 cases (six × six × five ) of torque analyses were performed using beam FE model. 1 2 3

-43-


72 76 80 84 88

Layer 2 [, LHL]

83

84

85

86

87

88

89

90

Layer 3

[

3, R

HL

]

600

800

1000

1200

1400

1600

1800

Eq

uiv

ale

nt

Str

ess

[M

Pa]

Design stress limit

①

②

③

α1=72.5°α1=75°α1=77.5°α1=80°α1=82.5°α1=85°

α1=72.5°

α1=75°

α1=77.5°

α1=80°

α1=82.5°

α1=85°

A

B

2. Stress limit can be reached three

(80º, 82.5º and 85º) 1

4. Point ① has minimum flexural

rigidity (0.342 N-m2)

5.

finally determined

851 1.802

1. Torque balance curves are plotted

(the relations among , and ) 1 2 3

3. Corresponding points in the torque

balance curves point ① ② ③

3.863

-44-

- Nicolas K. suggests analytical model to minimize the torque up to five layers’ cable (‘15)

. Difference from previous analytical model : the radial deformation is considered

- Torque balance can be achieved by designing the total torque before the last layer is

same as that of last layer with opposite helical direction regardless of the numbers of layers.

- When using beam FE model, for practical and simple method, it could be a good method to

determine only two helical angles, and , for multilayer torque balance.

hK

K

K

K

M

F

T

T

/

0)(......)()( _21 nlayerlayerlayer KKK

n

n

i

i TT

1

1

0........ _321 nlayerlayerlayerlayer TTTT

ni

n

i

KK )()( 1

1

1

• Multilayered helically-stranded cables (four or more layers)

1n n

-46-

• ACSR(Aluminum conductor steel reinforced) Cable

r1

r2

r3

(b)

α2 (LHL)

α3 (RHL)

A A

α1 (RHL)

TF

(a)

Layer

No.

No. of

wires

Helical

direction

and angle [α]

Wire

diameter

[mm]

Pitch

length

[mm]

Young’s

Modulus

[GPa]

Poisson’s

ratio

Core 1 - 3.25 - 162 0.3

1 6 RHL, 85º 3.25 233.4 162 0.3

2 12 LHL, 75.5º 3.25 157.9 69 0.33

3 18 RHL, 85.4º 3.25 761.4 69 0.33

1. Core & Layer 1 : Main structural part

(aluminum-clad steel )

2. Layer 2 & 3 : Conductor part

(aluminum alloy)

-47-

• Design of ACSR Cable

72 76 80 84 88

Layer 3 [, RHL]

-40

0

40

80

120

Torq

ue [

N-m

]

=85

=82.5

=80

=77.5

=75

=72.5

72 76 80 84

Layer 2 [, LHL]

84

85

86

87

88

89

90

Layer

3 [

3, R

HL

]




640

660

680

700

720

740

Eq

uiv

ale

nt

Str

ess

[MP

a]

17

18

19

20

21

Fle

xu

ral

Rig

idit

y [

N-m

2]

Design stress limit [720 MPa]①

②

③

- Torque balance points (@ )

1. Beyond Point ①


5.752 2. Point ② has minimum

flexural rigidity

3. Point ③

finally determined

5.752 4.853

851

-48-

- Experimental test

- Both ends socketed and fixed against rotation

- One end was attached to a torque meter

TF

0 0.001 0.002 0.003 0.004

Axial Strain

0

20

40

60

80

100

120

Axia

l L

oad

[k

N]

Experiment

Beam model

• Comparison between test and design of ACSR Cable

- Axial load/strain curves

-49-

• Comparison between test and design of ACSR Cable

1.97N-m - Axial load applied 100kN, torque generated :

. Experiment : 2.53 [N-m]

. Beam model : 0.56 [N-m]

Difference is due to unequal load sharing

among wires (wires have unequal lengths

between sockets).

The torque value from experiment is

relatively small.

TF

0 4 8 12 16 20

Torque [N-m]

0

20

40

60

80

100

Axia

l L

oad

[k

N]

Experiment

Beam model

- Axial load/torque curves

-50-

• Submarine power cable

Description Details

Conductor Keystone conductor

Conductor Screen Extruded PE

Insulation Extruded XLPE

Insulation Screen Extruded PE

Axial water seal Semi-conducting swelling tape(s)

Metallic Screen Lead alloy sheath

Sheath Extruded polyethylene

Bedding Semi-conducting polyester tape

Armor 1 Copper wire

Armor 2 Copper wire

Serving Polypropylene yarn

Layer No. of

wires Helix angle [α]

Outer Diameter

[mm]

Pitch

[mm]

Young’s

modulus [GPa]

Poisson’s

ratio

Conductor - - 58.8 - 117 0.36

Insulation - - 117.2 - 0.30 0.46

Metallic screen - - 130.6 - 9.78 0.45

Sheath - - 137.0 - 0.88 0.46

Armor 1 85 RHL, 78.8º 148.5 (5mm : wire) 2270 117 0.36

Armor 2 78 LHL, 82.4º 162.7 (6mm : wire) 3629 117 0.36

- Modified model

Equivalent core

Armors

(a) (b)

- Equivalent core properties should be defined

-51-


- Experimental test (tensile test) for core

Strain gage

Actuator Cable core Roller (four direction)

TF

0 0.0002 0.0004 0.0006 0.0008

Axial Strain

0

50

100

150

200

Axia

l L

oad

[k

N]

Experiment

TF

-300 -200 -100 0

Torque [N-m]

0

50

100

150

200

Ax

ial

Load

[k

N]

Experiment

- Axial load/strain curve

- Axial load/torque curve

- Axial rigidity can be measured.

- Torque (281 N-m) was generated due to

conductor structure (stranded flat wire).

-52-


- Experimental test (bending test) for core

HDPEsheath

T8

T9

T10

S3

T7T1

48

3WLEI

0 0.0002 0.0004 0.0006 0.0008

Axial Strain

0

50

100

150

200

Axia

l L

oad

[k

N]

Experiment

Beam model =0.1

Beam model =0.2

Beam model =0.4

Beam model =0.45

- Poisson’s ratio

- The effect of Poisson’s ratio is so small - Bending(Flexural) rigidity

-53-


- Properties for equivalent model

Layer No. of

wires

Helix

angle [α]

Wire

Dia.

[mm]

Pitch

length

[mm]

Young’s

Modulus *

[GPa]

Poisson’s

ratio

Eq.

core - - 138.12 -

19.84 (A)

2.36 (B) 0.4

A 1 85 RHL, 78.8º 5 2270 117 0.36

A 2 78 LHL, 82.4º 6 3629 117 0.36

72 76 80 84 88

Layer 2 [, LHL]

-8000

-4000

0

4000

8000

Torq

ue

[N-m

]

=85

=82.5

=80

=77.5

=75

=72.5

- Torque balance points

* Two Young’s modulus, A : for axial rigidity, B : for flexural

rigidity

-54-


72 76 80 84 88

Layer 1 [, RHL]

78

80

82

84

86

88

Layer 2

[

2, L

HL

]




156

157

158

159

160

161

162

Eq

uiv

ale

nt

Str

ess

[M

Pa]

43.2

43.3

43.4

43.5

43.6

43.7

Fle

xu

ral

Rig

idit

y [

10

3, N

-m2]

Design stress limit [158 MPa]①

②

③

- Torque balance point for cable design

1. Beyond Point ①


8.781

2. Point ② has minimum

flexural rigidity

3. Point ③

finally determined

8.781 4.822

-55-


- Experimental test

TF

0 0.0004 0.0008 0.0012 0.0016

Axial Strain

0

200

400

600

800

1000

Axia

l L

oad

[k

N]

Experiment

Beam model

-56-


TFTF

≒

Cable Core

-1600 -1200 -800 -400 0

Torque [N-m]

0

200

400

600

800

1000

Axia

l L

oad

[k

N]

Experiment_Cable

Experiment_Core

Beam model

100

TF

-300 -200 -100 0

Torque [N-m]

0

50

100

150

200

Ax

ial

Load

[k

N]

Experiment

- Torque 1,381 N-m (1,000kN applied)

- Comparison between design & test considering torque from core

- Torque 281 N-m (200kN applied to core)

- Torque 1,405 N-m may be generated if

1,000kN applied.

-58-

• Dimensionless parameter

2222

1111

rAEn

rAEnRt

Model Layer No. of

wires

Wire

diameter

[mm]

Center line

radius

[mm]

Young’s

Modulus

[GPa]

Original

Core - 138.12 69.06 19.84

0.703 Armor 1 85 5 71.56 117

Armor 2 78 6 77.06 117

Type 1

Core - 138.12 69.06 19.84

0.703 Armor 1 85 5 71.56 117

Armor 2 57 7 77.56 117

Type 2

Core - 138.12 69.06 19.84

0.703 Armor 1 59 6.02 72.07 117

Armor 2 78 6 78.08 117

Type 3

Core - 138.12 69.06 19.84

0.703 Armor 1 85 5 71.56 100

Armor 2 66 6.03 77.08 117

Type 4

Core - 69.06 34.53 19.84

0.703 Armor 1 45 5.09 37.08 117

Armor 2 40 6 42.62 117

tR

)cossin( 2 EArnK

r

EI

r

GJEAr

nK

2

22

cossin2sin

cossin2coscossin

r

EI

r

GJEAr

nK)sin1(sincos2sin

cossin2coscossin

22

42

can be introduced from tRK

Hint : Stretch terms are dominant

-59-

• Comparison of torque balance curves with

72 76 80 84 88

Layer 1 [, RHL]

78

80

82

84

86

88

La

yer

2 [

2, L

HL

]

Original

Type 1

Type 2

Type 3

Type 4

tR

703.0tR

- Same gives

same torque balance curves

- Generalized torque balance

curves can be plotted by

tR

- Torque balance analysis performed using beam model

tR

-60-

• Generalized torque balance curves

No.

1.1 70º 65.8º

0.8 75º 78.6º

0.5 80º 85.2º

21

68 72 76 80 84 88 92

Layer 1 [, RHL]

60

64

68

72

76

80

84

88

92

Layer 2

[

2, L

HL

]

Curve 1 : Rt = 0.1

Curve 2 : Rt = 0.2

Curve 3 : Rt = 0.3

Curve 4 : Rt = 0.4

Curve 5 : Rt = 0.5

Curve 6 : Rt = 0.6

Curve 7 : Rt = 0.7

Curve 8 : Rt = 0.8

Curve 9 : Rt = 0.9

Curve 10 : Rt = 1.0

Curve 11 : Rt = 1.1

Curve 12 : Rt = 1.2

1

2

3

5

6

7

8

9

10

11

12

4

①

②

③

This can be design guidance

instead of

experiments or analysis tools

①

②

③

tR

-61-

Multiple beam FE models was proposed to predict the behavior of the cable.

- Beam FE model can be the most effective solution due to accuracy and computation cost.

- Optimal FE model & friction effect were determined by sensitivity analysis.

- The validity and limitation of analytical models were evaluated.

- This model can be used a variety of cable analyses, dynamic, buckling, birdcaging, etc.

A new procedure for the torque balance design was proposed.

- Helix angles of each of the layers satisfy the design stress limit and the minimum

flexural rigidity as well as the satisfaction of torque balance.

- The design procedure was verified by experimental test.

The generalized torque balance curves were proposed, which can be a good design

guide instead of relying on experimental test or analysis tools.

• Conclusion

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