Upload
others
View
0
Download
0
Embed Size (px)
Citation preview
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
-5-
-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
32232 sincos2sinsincos2cos
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-
• Evaluation & Comparison of stiffness coefficients
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
43 sincos2sinsincos2cos
sinr
EI
r
GJEAnK
r
EI
r
GJEArnK
32232 sincos2sinsincos2cos
cossin
Stretch Twist Bending
Stretch Twist Bending
• Evaluation & Comparison of stiffness coefficients
-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
Stretch Twist Bending
Stretch Twist Bending
• Evaluation & Comparison of stiffness coefficients
-14-
-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
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
Fixed end Loading end
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.
-22-
-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
Friction Coefficient
150
160
170
180
190
155
165
175
185
195
Center Point
0 0.1 0.2 0.3 0.4
Friction Coefficient
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%
-33-
-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
]
Torque balance curve
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-
• Three layer cable
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-
• Three layer cable
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
-45-
-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
]
Torque balance curve
Equivalent stress curve
Flexural rigidity curve
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
satisfies design stress limit
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-
• Submarine power cable
- 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-
• Submarine power cable
- 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-
• Submarine power cable
- 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-
• Submarine power cable
72 76 80 84 88
Layer 1 [, RHL]
78
80
82
84
86
88
Layer 2
[
2, L
HL
]
Torque balance curve
Equivalent stress curve
Flexural rigidity curve
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
satisfies design stress limit
8.781
2. Point ② has minimum
flexural rigidity
3. Point ③
finally determined
8.781 4.822
-55-
• Submarine power cable
- 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-
• Submarine power cable
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.
-57-
-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
-62-