Extreme Loads and Load Combinations
Alaa Mansour
Martin Petricic
Mechanical Engineering Department
University of California
Berkeley, California, 94720
Email: [email protected]
In the presentation, the first author will discuss the importance of determining extreme load
combinations for design purposes, and how to estimate them for the design of high speed catamarans
as well as ships. One of the most important factors for estimating the extreme load combinations is the
correlation coefficient which can be indirectly used for determining the magnitude of one load when the
other is at its maximum (extreme) value. The presentation will include a newly developed procedure for
determining the life time correlation coefficients of wave loads with applications to a high speed
catamaran and other ships. Beside their importance for determining the combined design loads, these
correlation coefficients also represent an important input for Classification Rules when determining load
combination factors upon which a new design can be based. The presentation will also address issues
related to combining the effects of high frequency loads such as slamming and springing with low
frequency wave induced loads. The general procedure is based on rejection sampling of sea states and
direct load simulation from the corresponding output response spectra.
The usually very long computer time required for the life time simulation of the wave loads has been
drastically reduced to a single voyage simulation requiring only a short run-time on an ordinary PC. This
has been achieved by uniformly spreading the random effects of different routes and different seasons
throughout the entire life time of the marine vehicle. It can be shown that this procedure does not
affect the estimate of the correlation coefficient which converges in probability to the true (population)
correlation coefficient even for a one voyage simulation. Since the actual time records of the loads are
simulated, nonlinear combinations of loads can also be investigated by simply combining the point-in-
time values of different loads.
As a background information, the first author will also present a number of short- and long-term load
combination methods currently in use. Short-term methods are usually used to calculate the extreme
load combinations in a particular design sea state and are limited to linear systems with stationary input
IX HSMV Naples 25 - 27 May 2011: Keynote 1
process suitable for a sea state of duration of up to three hours. Long-term methods are used in the
fatigue assessment and reliability calculations. They can only handle linear load combinations and
usually rely on the assumption that the response spectra are narrow-banded. Additionally, the long-
term correlation coefficients are usually determined either from the very long load time records
measured on board a vessel (very expensive), or they are approximated combining several short-term
correlation coefficients. No effects of the ship type, size, route or ship loading condition on the
correlation coefficients are taken into account in these currently used methods.
The simulation method used in this paper provides a fast and more accurate way of obtaining the long-
term correlation coefficients in comparison to the few existing methods. This is because it is very flexible
in terms of the randomness that can be simulated. During the simulation, non-stationary wave
elevations are treated as a sequence of stationary Gaussian stochastic processes. Sea states in different
ocean areas along the ship’s route are statistically represented by the joint probability density functions
(JPDF) of significant wave height (HS), zero crossing period (T0) and the prevailing wind/wave direction
(a). JPDF(HS, T0, a) for each area and season are obtained by fitting the JPDF(HS, T0) to scatter diagrams
from the Global Wave Statistics. Database is developed for every wind/wave direction using the
maximum likelihood estimation. This enables the calculation of explicit dependence of each parameter
in the JPDF(HS, T0) on the wave direction “a” and, thus, the calculation of JPDF(HS, T0, a). Various sea
states represented by HS, T0 and “a” are sampled from the JPDF(HS, T0, a) using rejection sampling.
Depending on HS and the relative heading between the ship and the waves, ship’s speed is determined
from the pre-specified speed/heading profile that takes into account both the involuntary and voluntary
speed reduction in high sea states. Consequently, the ship’s progress along the route is determined
based on its current speed and the duration (two hours) of the stationary sea conditions.
For each simulated sea state, the wave elevations are represented by the ISSC two parameter spectrum
and the response spectra for each load is determined using the linear filter analysis and the cosine
squared spreading function. The effects of forward speed, relative heading and the loading conditions
are taken into account by the pre-calculated linear transfer functions. The actual time series of various
loads are simulated from their respective response spectra making sure that the correct input-output
phase relations are preserved. The estimates of the long-term correlation coefficients between any two
loads are calculated directly from their respective time series.
In the paper, the emphasis is placed on the parametric study of the effects of the marine vehicle type,
size, route and loading condition on the values of the correlation coefficients between six different
sectional loads; vertical, horizontal and twisting moments as well as vertical, horizontal and axial forces.
Several marine vehicle types are considered including a large high speed catamaran, a containership and
a tanker, navigating on busy routes: the English Channel, North America / Europe, Asia / North America
and Asia/Europe.
The results of the parametric study have been summarized in tabular and graphical manner.
They show that the effect of marine vehicle type on the value of the long-term correlation
coefficient is dominant, followed by the effect of the longitudinal position of the load and the
IX HSMV Naples 25 - 27 May 2011: Keynote 2
ship route. However, all these three parameters significantly affect the long-term correlation
coefficient between vertical and horizontal bending moments.
In conclusion, the presenter will discuss several issues important for future research in
connection with extreme load estimation for design. He will also point out that without
accurate estimation of extreme loads and load combinations, the full benefits of finite element
analysis and other sophisticated tools such as risk and reliability analyses will not be achieved.
IX HSMV Naples 25 - 27 May 2011: Keynote 3
HSMV Symposium, May, 2011
University of California at Berkeley
HSMV 2011
Naples, Italy
Extreme Loads and Load Combinations
Alaa Mansour
Martin Petricic
4
Introduction
Loads acting on the marine vehicles are random in nature.
Two distinct groups: - Low-frequency wave-induced loads
- High-frequency loads
Both are caused by the same stochastic process (ocean waves) which is
a big source of their correlation.
In order to properly combine these stochastic loads, method of classical
statistics and time series analysis have to be used.
According to Stewart and Melchers, “structural design activities with the
highest error rates are the load combination and reduction factor
assessments.”
IX HSMV Naples 25 - 27 May 2011: Keynote 5
Introduction (Cont.)
Short-term (~3 hours) and long-term (~25 years) methods require the knowledge of the relationship between the loads which is represented by their correlation coefficients.
Exact mathematical solutions exist for the short-term load combinations.
Currently, long-term load combination methods involve many assumptions, biggest of which pertain to the calculations of the long-term correlation coefficient.
IX HSMV Naples 25 - 27 May 2011: Keynote 6
Load Classification
1. QUASI-STATIONARY LOADS
• Stillwater loads
• Thermal loads
2. LOW-FREQUENCY NON-STATIONARY LOADS
• Wave induced loads
3. HIGH-FREQUENCY NON-STATIONARY LOADS
• Springing loads
• Slamming/whipping loads
• Low speed machinery-induced vibrational loads
IX HSMV Naples 25 - 27 May 2011: Keynote 7
Slamming
Based on the slamming observations and records, the slams can be temporally represented as a train of Poisson impulses of random intensity occurring at random time intervals.
Accumulation of slamming responses depends on speed, pitch and heave motions of the vessel.
IX HSMV Naples 25 - 27 May 2011: Keynote 8
Short-term Methods
An exact expression for the short-term extreme load combination (Mansour 1995):
LOAD COMBINATION FACTOR
where:
CORRELATION COEFFICIENT
Short-Term Methods
IX HSMV Naples 25 - 27 May 2011: Keynote 9
Short-term Methods
If:
Square Root of the Sum of Squares method
(SRSS) assumes ρ= 0:
Peak Coincidence method (PC) assumes
K =ρ= 1:
Turkstra’s Rule (TR) assumes K = ρ:
Short-Term Methods
IX HSMV Naples 25 - 27 May 2011: Keynote 10
Long-term Methods
Long-term CDF of the combined response peaks:
Drawbacks:
Difficulties in expressing analytically the JPDF of the HS, T0, V and α for all seasons and all areas of navigation.
Some assumptions are almost always necessary;
We need to know the short-term conditional CDF of the combined response peaks which limits the analysis to linear
load combinations only;
If the response is not narrow bended then the calculations become much more extensive;
This methods cannot be used to find the correlation coefficients between individual loads.
Long-Term Methods
IX HSMV Naples 25 - 27 May 2011: Keynote 11
Time and Frequency Domains
INPUT
LINEAR
SYSTEM
OUTPUTS
2( , , , ) HBMH v LC
0( , , , , , ) VBM SS H T v LC
0( , , )X SS H T
Time domain Frequency domain
2( , , , ) VBMH v LC
0( , , , , , ) HBM SS H T v LC
IX HSMV Naples 25 - 27 May 2011: Keynote 12
Ship Routes and Marsden Zones
NA
NP
EA
English Channel
IX HSMV Naples 25 - 27 May 2011: Keynote 13
Ship Types Ship Types
CONTAINERSHIP
Length B. P. [m] 283.30
Breadth [m] 32.20
Draught (full load) [m] 11.26
Block Coefficient 0.70
Deadweight (full load) [t] 68240
Deck section modulus [m3] 50.00
Side section modulus [m3] 85.00
TANKER
Length B. P. [m] 282.89
Breadth [m] 49.00
Draught (full load) [m] 15.00
Draught/trim (ballast) [m] 8.16/2.21
Block Coefficient 0.84
Deadweight (full load) [t] 172007
Deadweight (ballast) [t] 89044
BULK CARRIER
Length B. P. [m] 283.00
Breadth [m] 45.00
Draught (full load) [m] 16.00
Draught/trim (ballast) [m] 7.81/2.91
Block Coefficient 0.81
Deadweight (full load) [t] 168743
Deadweight (ballast) [t] 77991
CATAMARAN FERRY
Length B. P. [m] 126.60
Breadth [m] 40.00
Draught 4.80
Block Coefficient 0.45
Deadweight [t] 11588
Hull separation [m] 29.00
speed [kn] 35.00
IX HSMV Naples 25 - 27 May 2011: Keynote 14
Statistical Description of the Ocean
For each Marsden zone, we have 32 scatter tables (4 seasons and 8 directions)
Conditional JPDF of HS and T0 given wave direction for each area of the ocean and each
season is given by (Ochi):
0 0, , , , , , ,S Sf H T Dir A S f H T Dir A S f Dir A S
IX HSMV Naples 25 - 27 May 2011: Keynote 15
Statistical Description of the Ocean
IX HSMV Naples 25 - 27 May 2011: Keynote 16
Rejection Sampling
IX HSMV Naples 25 - 27 May 2011: Keynote 17
Load Spectra
OCEN WAVE DATA (GLOBAL WAVE
STATISTICS)
CALCULATE THE INPUT SPECTRUM
FIND THE RAOs FOR EACH LOAD
CALCULATE THE OUTPUT
SPECTRUM FOR EACH SEA STATE
2( , , , )w aH v LC
Ship position, velocity profile, loading condition
(LC), speed (v), relative heading (α)
FIND HS, T0, β USING THE
REJECTION SAMPLING
FIT THE STATISTICAL
MODEL USING MLE
IX HSMV Naples 25 - 27 May 2011: Keynote 18
Load Spectra
For each sea state (HS, T0), speed, v, loading condition and
relative heading, α, the load spectrum is given as:
The output (load) 2D spectra has been calculated using the cosine squared spreading function.
Where the encounter frequency is given as:
IX HSMV Naples 25 - 27 May 2011: Keynote 19
Load Simulation
For each sea state, a simulated time series of each load is given by the approximation to the
Fourier integral:
The correct phase relation between all the
loads has been established by superposing the
phase difference between waves (input) and
the load (output), obtained from the load
transfer function, to the uniformly distributed
random phase, φ, that characterizes the
randomness of the wave components. For
example:
Load Simulation
IX HSMV Naples 25 - 27 May 2011: Keynote 20
Simulation
Flowchart
The MLE calculations have to be performed only once for each Marsden zone. The resulting fitted JPDF is stored for each zone (each prevailing wave direction and each season);
IX HSMV Naples 25 - 27 May 2011: Keynote 21
Simulation Length
1
2 2
1 1
( )( )
( ) ( )
n
i i
i
n n
j k
j k
x x y y
R
x x y y
Where the unbiased estimate of
the correlation coefficient is:
It was shown in this work that if the seasonal variations are artificially
simulated within a single voyage, then the correlation coefficient can be
estimated based on a single voyage simulation.
This significantly reduces the simulation time to a single voyage simulation.
Using Slutsky’s lemma one can prove that for n large:
IX HSMV Naples 25 - 27 May 2011: Keynote 22
Simulation Resolution
According to the Shannon’s Theorem, a continuous time series is completely described if the values
are generated with the frequency that is at least twice as large as the maximum frequency,
ωe,MAX, of a periodic component that is present in the series. 2ωe,MAX is called the Nyquist
frequency. It has been found in this work that for all load encounter spectra, ωe,MAX < π rad/s.
Therefore, generating load values at the frequency of 2π rad/s or 1 Hz, has been found
sufficient.
Another consideration is the number of encounter frequency intervals N. This number must be
sufficiently large to avoid any periodicities in the simulated time series and to ensure its
approximate normality according to the central limit theorem. This can be checked by means of a
normal q-q plot. N=100 has been found to satisfy both criteria.
IX HSMV Naples 25 - 27 May 2011: Keynote 23
Results (High Speed Catamaran Ferry)
Catamaran hull form used in this work.
Prototype vessel – HSS 1500 Stena Voyager.
Source: Wikipedia
IX HSMV Naples 25 - 27 May 2011: Keynote 24
Results (High Speed Catamaran Ferry)
Correlation matrix – catamaran on the
English Channel route
LF HF SF T VBM HBM
LF 1.00 -0.04 -0.43 0.01 0.80 0.07
HF -0.04 1.00 -0.21 -0.02 0.00 0.00
SF -0.43 -0.21 1.00 0.00 -0.45 0.00
T 0.01 -0.02 0.00 1.00 0.01 0.02
VBM 0.80 0.00 -0.45 0.01 1.00 0.17
HBM 0.07 0.00 0.00 0.02 0.17 1.00
CATAMARAN FERRY
Length B. P. [m] 126.60
Breadth [m] 40.00
Draught 4.80
Block Coefficient 0.45
Deadweight [t] 11588
Hull separation [m] 29.00
speed [kn] 35.00
Note: The correlation matrix is based on
the average of 50 time series, each of which is 200 voyages long.
IX HSMV Naples 25 - 27 May 2011: Keynote 25
Results (High Speed Catamaran Ferry)
(a) (b)
(a) VBM time series; (b) Scatter diagram of VBM vs. HBM
IX HSMV Naples 25 - 27 May 2011: Keynote 26
Results (Containership - NA)
IX HSMV Naples 25 - 27 May 2011: Keynote 27
Results (Containership - NA)
IX HSMV Naples 25 - 27 May 2011: Keynote 28
Results (Containership - NA)
PS SB Comparison of the longitudinal stress from the VBM and the
HBM in the shear strake on PS and SB
IX HSMV Naples 25 - 27 May 2011: Keynote 29
Results (Containership - NA):
Comparison of the long-term probability of exceedance of individual VBM
peaks from Jensen et. al. and from simulation
IX HSMV Naples 25 - 27 May 2011: Keynote 30
Results and comparisons:
Comparison of the long-term correlation coefficients between
the VBM and the HBM
IX HSMV Naples 25 - 27 May 2011: Keynote 31
Results (Ship Route Effect)
Correlation matrix – containership NA
Correlation matrix – containership NP
Correlation matrix – containership EA
Selected correlation coefficient estimates for different
ship routes. All three cases are for the containership.
IX HSMV Naples 25 - 27 May 2011: Keynote 32
Results (Ship Route Effect)
Comparison of the sagging VBM peaks for containership navigating on three different
routes. Green line represents the EA route, blue line NP route and the red line
represents the NA route. IX HSMV Naples 25 - 27 May 2011: Keynote 33
Results (Ship Type Effect)
Correlation matrix – containership EA
Correlation matrix – tanker EA
Correlation matrix – bulk carrier EA
Selected correlation coefficient estimates for different
ship types. All three cases are for the EA route.
IX HSMV Naples 25 - 27 May 2011: Keynote 34
Results (Longitudinal Position Effect)
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0 50 100
Percent of the ship's length
ρ
VBM - HBM
VBM - VF
VBM - Torsion
NOTE: The accuracy of these results is still being verified.
IX HSMV Naples 25 - 27 May 2011: Keynote 35
Springing Effect on the VBM
0 0.5 1 1.50
0.5
1
1.5
2
2.5x 10
6 180
RA
O
omega
0 0.5 1 1.50
0.5
1
1.5
2
2.5x 10
6 170
RA
O
omega
0 0.5 1 1.50
0.5
1
1.5
2
2.5x 10
6 160
RA
O
omega
0 0.5 1 1.50
0.5
1
1.5
2x 10
6 150
RA
O
omega
0 1 20
0.5
1
1.5
2
2.5x 10
6 140
RA
O
omega
0 1 20
1
2
3
4x 10
6 130
RA
O
omega
0 1 20
1
2
3
4x 10
6 120
RA
O
omega
0 1 2 30
1
2
3
4
5x 10
6 110
RA
O
omega
0 1 2 30
2
4
6
8x 10
6 100
RA
O
omega
0 2 40
1
2
3
4
5
6x 10
5 90
RA
O
omega
0 5 100
0.5
1
1.5
2
2.5x 10
5 80
RA
O
omega
0 2 4 60
1
2
3
4x 10
5 70
RA
O
omega
0 2 40
1
2
3
4x 10
5 60R
AO
omega
0 1 2 30
1
2
3
4x 10
5 50
RA
O
omega
0 1 2 30
2
4
6
8x 10
5 40
RA
O
omega
0 1 2 30
2
4
6
8x 10
5 30
RA
O
omega
0 1 2 30
2
4
6
8x 10
5 20
RA
O
omega
0 1 2 30
2
4
6
8x 10
5 10
RA
O
omega
0 1 2 30
2
4
6
8x 10
5 0
RA
O
omega
VBM RAO for containership at v=25.6 kn for various headings (obtained by program SOST) IX HSMV Naples 25 - 27 May 2011: Keynote 36
Springing Effect on the VBM
VBM [kNm]
Springing included
Springing not
included
Long
-Term
Pro
bability o
f Ex
ceedanc
e
Note: These are preliminary results obtained by Martin Petricic and Jelena Vidic-Perunovic for a containership on the EA route. IX HSMV Naples 25 - 27 May 2011: Keynote 37
Conclusion
Advantages of time domain simulations:
One advantage is that nonlinear combinations of extreme loads can be determined by directly combining their point-in-time values;
The procedure is flexible in terms of the randomness that can be modeled.
No assumptions are needed regarding the bandwidth of the input or the output spectra.
Different routes and loading conditions can easily be included in the simulation.
The developed procedure is efficient and fast needing only a few seconds to generate the entire voyage time series for all loads.
IX HSMV Naples 25 - 27 May 2011: Keynote 38
Research Issues for Future Work
Non-linear loads and their combinations.
For high speed marine vehicles:
1. Slamming combination with L.F. wave induced loads.
2. Springing importance as speed increases.
The correlation structure between the sectional loads and
wave pressure (primary and secondary or tertiary loads).
The effects of weather routing.
Importance of extreme loads and load combinations.
IX HSMV Naples 25 - 27 May 2011: Keynote 39
Thank you for your attention!
IX HSMV Naples 25 - 27 May 2011: Keynote 40