1 Fatigue under Bimodal Loads Zhen Gao Torgeir Moan Wenbo Huang March 23, 2006.
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Transcript of 1 Fatigue under Bimodal Loads Zhen Gao Torgeir Moan Wenbo Huang March 23, 2006.
1
Fatigue under Bimodal Loads
Zhen Gao
Torgeir Moan
Wenbo Huang
March 23, 2006
2
Contents
• Bimodal random process
• Methods for bimodal fatigue damage assessment
• Case study of mooring system of a semi-submersible
3
Bimodal random process
• A wide-band process with a bimodal spectral density.
• Examples:– Torque of propellers
(or thrusters) in waves
– Mooring line tension
4
Fatigue based on S-N curve and Miner rule• Gaussian narrow-band fatigue damage
• Fatigue damage of a bimodal process
0
0
0( ) ( 2 ) (1 )2
m mS
T T mD s f s ds
K K
where
is the mean zero up-crossing rate.0 is the standard deviation of the process.
( ) ( ) ( )LF HFX t X t X t
LF HF NBD D D D
5
Methods for bimodal fatigue (1)• Time domain methods
– Peak counting– Range counting
• Spectral methods for a general wide-band Gaussian process
– Wirsching & Light (1980)– Zhao & Baker (1992)
• Spectral methods for a bimodal Gaussian process– Single moment method– Sakai & Okamura (1995)– DNV formula (2005)
• Non-Gaussian process– Transformation (Winterstein,1988; Sarkani et al.,1994)
– Level crossing counting– Rainflow counting
– Dirlik (1985)– Benasciutti & Tovo (2003)
– Jiao & Moan (1990)– Fu & Cebon (2000)– Huang & Moan (2006)
6
Methods for bimodal fatigue (2)• Jiao & Moan (1990)
• DNV (2005)
P HFD D D where is the envelope process of
( ) ( ) ( )LF HFP t X t R t Assume
Then
( )HFR t
( ) ( ) ( )LF HFX t X t X t
( )HFX t
( )mLF HF LFLF HF HF
LF HF
D
nD D D
n
For Gaussian processes, analytical formula can be obtained.
7
Methods for bimodal fatigue (3)• Fatigue damage estimation of a Gaussian
bimodal process
Jiao & Moan (1990) DNV (2005)
DNV
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
0
2
4
6
8
10
12
14
16
18
200.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6
v0HF / v0LF
LFLF
WF2
2 2LF
LF WF
0
0
HF
LF
Jian and Moan with Bandwidth 0.1
LFLF
WF
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
v0HF / v0LF
0
2
4
6
8
10
12
14
16
18
200.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6
2
2 2LF
LF WF
0
0
HF
LF
NB
D
D
8
Case study of mooring system
• A semi-submersibleMain particulars of the semi-submersible
Displacement (ton) 52500
Length O.A. (m) 124
Breadth (m) 95.3
Draught (m) 21
Operational water depth (m) 340
• Mooring system– Line No.10– Pre-tension of 1320 kN– Studless chain link with a dia
meter of 125 mm
Horizontal projection of the mooring system
9
Mooring line tension components
• Pre-tension and mean tension due to steady wind, wave and current forces (time-invariant)
• LF line tension (quasi-static, long period (e.g. 1 min))
• WF line tension (dynamic, short period (e.g. 15 sec))
• Both LF and WF tension are narrow-band.
• Bimodal with well-separated low and wave frequencies
• Independent assumption between LF and WF tension
( ) ( ) ( )P M LF WFT t T T T t T t
10
Low frequency (LF) line tension
• Distribution of slowly-varying wave force and motion can be expressed by a sum of exponential distributions given by an eigenvalue problem (Næss, 1986)
• The LF line tension can be quasi-statically determined the line characteristic (cubic polynomial, even linear)
• Distribution of the amplitude of LF tension depends on the fundamental tension process and its time-derivative.
1
1
exp( )2 2
( )
exp( )2 2
Mj
j j j
Z Nj
j M j j
l z
f zl z
0
0,
z
z
1
1
(1 )N
kj
k jk j
l
11
Wave frequency (WF) line tension (1)
• Simplified dynamic model (Larsen & Sandvik, 1990)
• Distribution of the amplitude of WF line tension (Combined Rayleigh and exponential distribution)(Borgman, 1965)
* 2 *( ) ( ) ( ) ( ) ( )TUEWF G WFT t c u t u t k u t m x t
Basically, it is a Morison formula with a drag term and an equivalent inertia term.
max
22 2
2 20
(3 1) exp( (3 1) )2
( )3 1 3 1
exp( ( ))2 2 2
WFT
yk y k
f yyk k
yk k
0
0
0
,
y y
y y
is a measure of the relative importance of the drag term and the equivalent inertia term.
k
12
Wave frequency (WF) line tension (2)
• Morison force ( ) ( ) ( ) ( )d mf t k u t u t k u t 2
d u
m u
kk
k
Fatigue damage due to normalized Morison force (Madsen,1986)
10-2
10-1
100
101
102
0
0.5
1
1.5
2
2.5
3
k
0kGaussian
D
D
2
( ) ( ) ( )( )N
u u
u t u t u tf t k
Normalized:
13
Amplitude distribution of LF and WF line tension
• The amplitude distribution of LF line tension shows a higher upper tail, which indicates a larger extreme value.
• While that of WF line tension is quite close to a Rayleigh distribution. Because in this case, the equivalent inertia term is dominating.
Scaled by the standard deviation of the fundamental process
( )max
X
X
pdf of the maximum of the variables
x0 1 2 3 4 5 6
f X(x
)
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7Standard Rayleigh variableScaled maximum of LF mooring line tensionScaled maximum of WF mooring line tension
14
Fatigue damage due to combined LF and WF line tension• The amplitude distribution of the process
with Gaussian and non-Gaussian cases
• The mean zero up-crossing rate can be obtained by the Rice formula.
( )P t
,0
(0) (0, )P P Ppf p dp
Scaled by the standard deviation of the fundamental process
( )max
X
X
( ) ( ) ( )LF WFP t X t R t
pdf of the maximum of the combined variables
x0 1 2 3 4 5 6 7 8
f X(x
)
0.0
0.1
0.2
0.3
0.4
0.5
0.6Standard Rayleigh+Rayleigh variableScaled maximum of LF+WF mooring line tension
15
Short-term and long-term fatigue
LF WF Combined
RFC 0.040 0.855 1Non-Gaussian 0.059 0.854 1.030
Gaussian 0.046 0.921 1.080
• Short-term fatigue damage– A 3-hour sea state with
Hs=6.25m, Tp=12.5s, Uwind=7.5m/s, Ucurrent=0.5m/sConditional short-term fatigue damages
• Long-term fatigue damage– A smoothed northern North Sea
scatter diagram
29-year Joint PDF of Hs and Tp
Tp (sec)
0 5 10 15 20H
s (m
)0
2
4
6
8
100.000 0.005 0.010 0.015 0.020 0.025 0.030 0.035 0.040 0.045 0.050 0.055 0.060 0.065
LF WF Combined
Non-Gaussian 0.065 0.812 1Gaussian 0.053 0.851 1.017
Total long-term fatigue damages
Smoothed scatter diagram(Joint density function)
16
Long-term fatigue contribution
Non-Gaussian
Contour Graph 1
TP (s)0 5 10 15 20 25 30
HS (
m)
0
2
4
6
8
10
12
14
16
180.0000 0.0001 0.0002 0.0003 0.0004 0.0010 0.0012 0.0014 0.0016 0.0018
Contour Graph 1
TP (s)0 5 10 15 20 25 30
HS (
m)
0
2
4
6
8
10
12
14
16
180.0000 0.0002 0.0004 0.0006 0.0008 0.0010 0.0012 0.0014 0.0016 0.0018
Contour Graph 1
TP (s)0 5 10 15 20 25 30
HS (
m)
0
2
4
6
8
10
12
14
16
180.000 0.002 0.004 0.006 0.008 0.010 0.012 0.014 0.016 0.018
Contour Graph 1
TP (s)0 5 10 15 20 25 30
HS (
m)
0
2
4
6
8
10
12
14
16
180.000 0.002 0.004 0.006 0.008 0.010 0.012 0.014 0.016 0.018
Contour Graph 1
TP (s)0 5 10 15 20 25 30
HS (
m)
0
2
4
6
8
10
12
14
16
180.000 0.002 0.004 0.006 0.008 0.010 0.012 0.014 0.016 0.018
Contour Graph 1
TP (s)0 5 10 15 20 25 30
HS (
m)
0
2
4
6
8
10
12
14
16
180.000 0.002 0.004 0.006 0.008 0.010 0.012 0.014 0.016 0.018
Gaussian
LF WF Combined
D=0.065 D=0.812 D=1
D=0.053 D=0.851 D=1.017
(sec)PT
( )SH m
(sec)PT
( )SH m
17
Thank you !