extreme loads and load combinations

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


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.


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.”


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.


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


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.


Short-term Methods

An exact expression for the short-term extreme load combination (Mansour 1995):

LOAD COMBINATION FACTOR

where:

CORRELATION COEFFICIENT

Short-Term Methods


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


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


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


Ship Routes and Marsden Zones

NA

NP

EA

English Channel


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/trim (ballast) [m] 8.16/2.21



Deadweight (ballast) [t] 89044

BULK CARRIER

Length B. P. [m] 283.00

Breadth [m] 45.00


Draught/trim (ballast) [m] 7.81/2.91



Deadweight (ballast) [t] 77991

CATAMARAN FERRY

Length B. P. [m] 126.60

Breadth [m] 40.00

Draught 4.80


Deadweight [t] 11588

Hull separation [m] 29.00

speed [kn] 35.00


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


Statistical Description of the Ocean


Rejection Sampling


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


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:


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


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);


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:


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.


Results (High Speed Catamaran Ferry)

Catamaran hull form used in this work.

Prototype vessel – HSS 1500 Stena Voyager.

Source: Wikipedia



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


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.



(a) (b)

(a) VBM time series; (b) Scatter diagram of VBM vs. HBM


Results (Containership - NA)



PS SB Comparison of the longitudinal stress from the VBM and the

HBM in the shear strake on PS and SB


Results (Containership - NA):

Comparison of the long-term probability of exceedance of individual VBM

peaks from Jensen et. al. and from simulation


Results and comparisons:

Comparison of the long-term correlation coefficients between

the VBM and the HBM


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.


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.


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.


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.


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.


Thank you for your attention!


extreme loads and load combinations

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