MT, MEK, DTU Peter Friis-Hansen : Design waves 1 Model correction factor & Design waves Peter...

MT, MEK, DTUPeter Friis-Hansen: Design waves 1

Model correction factor Model correction factor & Design waves & Design waves

Peter Friis-Hansen, Luca Garré, Jesper D. Dietz, Anders V. Søborg, J.J.

Jensen

Technical University of Denmark

CeSOS workshopMarch 23, 24, Trondheim


The model correction factor The model correction factor methodmethod

State of the art realistic response models are often time consuming and rarely feasible in a reliability analysis

MCF: an efficient response surface technique

Principle of the model correction factor method

1. Formulate a simplified structural model

2. Perform a calibration – in a probabilistic sense – to the time consuming, but more realistic, model

The simplified model is not realistic with respect to the physical conditions, or with respect to capturing all second-order bending effects.

The probabilistic calibration procedure assures that the simplified model is made “realistic” – at least around the design point


The model correction factorThe model correction factor

Ditlevsen & Arnbjerg-Nielsen (1991, 1994)

The simplified model is everywhere corrected by a random model correction factor such that

Establish a Taylor expansion of around the design point

)(S XR)(X

)()()( SC XXX RR

)(XX

...)()()( *** XXXXX X T


ExampleExample

T-stiffened plate panel

Subjected to axial andlateral loads


Limit state and uncertainty Limit state and uncertainty modellingmodelling

Failure is defined when the axial load exceeds the axial capacity )(XR


Simplified modelsSimplified models

DNV Classification Notes from 1992

Simple plastic hinge model

is axial stress

is bending stressx

b


Using the DNV class rulesUsing the DNV class rules


The simple plastic hinge modelThe simple plastic hinge model


Comparing resultsComparing results

Obtained design points are within 1%


SummarySummary

Compared to the FEM model the DNV model has a higher degree of model realism than the plastic hinge model

This implies fast convergence of the series of design points

Using the DNV model as idealised model requires 2-3 FEM analyses

Using the plastic hinge model requires 3 x 2 FEM analyses

Resulting design points are almost identical

Plastic hinge model does not contain the information about Young's modulus it requires two FEM


Design waves for ultimate Design waves for ultimate failure of marine structuresfailure of marine structures


Why design waves …Why design waves …or critical wave episodes ?or critical wave episodes ?

Critical wave episodes: a wave pattern that will result in an unwanted event

The physical wave pattern that causes the problem drives the design

Allows the designer to evaluate better the problem Can lead to new and innovative solution alternatives Can lead to safer and more competitive structures

How may we identify critical wave episodes? How may we calculate:

“ P[Wave patterns > critical wave episodes] ” ?


G.F. Clauss: Max Wave resultsG.F. Clauss: Max Wave results”Dramas of the sea: episodic waves and their impact on offshore structures”

Applied Ocean Research 24 (2002) 147-161

G. Clauss identified one wave pattern that always results in capsize.

Different risk reducing initiatives may be studied using this wave.

Problem: No probabilistic information about criticality of wave pattern


The stochastic modellingThe stochastic modelling

Traditional approach Brute force Monte Carlo simulation of white noise, thus wave elevation + : Will always work – : Requires very long time series to predict small probabilities

Critical wave episode approach Find the up-crossing rate of a specified level (say: roll > 50 deg or

m > x MNm) Use ”reverse engineering” to find critical wave episodes (by-product of

procedure) + : It will be fast, independent of probability level, give good results – : Limited experience. Test examples are promising, but will it work?

Numericalcode

Wavemodel

White noise

Wave elevation Response signal

Considered point in time


How to solve ?How to solve ?

Task: Find up-crossing rate, , of a given critical level, , of the considered response. The underlying stochastic variable is the wave process,

The critical wave episode is defined as the most likely wave pattern, , that results in the up-crossing

Mathematical formulation of the up-crossing problem

Rewritten using Madsen’s formula and effectively solved using FORM-SORM. can be extracted as a bi-product of this analysis

Ct

)(

C)(t

)(* t

0

)()()(1

lim0

)(tt

CCt

Ct ttttPt

output from stability code

)(* t


Outcome of analysisOutcome of analysis

Up-crossing rate of selected levels Short and long term distributions may be calculated Probability of unwanted event (capsize, moment, slamming,

…) is obtained To obtain long term distribution we need to perform the

analysis over multiple sea states Can we speed-up the calculation of the long term distribution

by reusing results from other sea states ? (I think so)

Can we identify a ”design wave pattern” for stability calculations and other highly non-linear problems ? (I hope so)

But, how may we decide on what magnitude of the event is critical ?Calls for risk analysis – calculating the expected loss: R=p·C


Wave induced response for shipsWave induced response for ships

Extreme ship responses not driven by large amplitudes

Suitable combination of wave length and amplitude


Identifying a Response WaveIdentifying a Response Wave

Idea

• Assume the waves that generates an extreme linear response will also generate the non-linear extremes

The principle

• The response wave is found by conditioning on a given linear response

• This wave profile is subsequently used in a non-linear time domain program

Two Models

• Most Likely Response Wave (MLRW)

• Conditional Random Response Wave (CRRW)

[MLRW is similar to MLER (Most Likely Extreme Response) wave]


The model correction factorThe model correction factor

Identify an idealised model that captures part of the real model

Model correct the idealised model such that it is made equivalent to the real model

is only established as a zero order expansion at carefully selected points

)()()( IdealReal XXX RR

)(X


The MLRW ModelThe MLRW Model

Z(t) is the unconditional wave profile:

Vn and Wn are random Gaussian zero mean variables a represents wave amplitudes from the wave spectrum

The linear response is given as:

The MLRW profile, c(t) conditioning on a given linear response amplitude

a is obtained from the response spectrum is the corresponding phase


The CRRW ModelThe CRRW Model

CRRW Model:• Derived from a Slepian model process

• Linear regression of V = (Vn , Wn ) on Y = (Y1 , Y2 , Y3 , Y4 )

The conditional vector:


The Critical MLRWThe Critical MLRW

What is the shape of these waves?

Sagging: Supported by a wave crest near AP and FP

Hogging: Supported by a wave crest near amidships

For a given response level the shape of the MLRW is not affected by:

The significant wave height, Hs

The zero-upcrossing wave period, Tz


Application of CRRWApplication of CRRW

Application of the CRRW given a conditional linear response:

• Select a stationary sea state and operational profile

• Derive the constrained coefficients Vc,n and Wc,n

• Use the CRRW in a non-linear time-domain code


The VesselThe Vessel

PanMax Container Ship:

Length, Lpp = 276.38 m

Breadth, Bmld = 32.2 m

Draught, T = 11.2 m Displacement = 63350 t Service Speed = 24.8 kn

ShipStar non-linear (2D) strip theory code


Short-Term Response StatisticsShort-Term Response Statistics

Simulation time: The MLRW model

20 simulations of 1 min

The CRRW model

20 x (50 to 100) simulations of 1 min

Brute force

3 weeks of simulations

Linear and non-linear results:• Head sea and v = 10 m/s

CRRW

MLRW

Linear

Simul


Effect of an Elastic hull girder Effect of an Elastic hull girder

Wave- and whipping-induced response Low frequency part Wave-induced High frequency part Whipping-induced

Filtering


The Slamming Problem, MLRWThe Slamming Problem, MLRW Sagging


Short-term Response StatisticsShort-term Response Statistics

MLRW Good first approach but less accurate than CRRW

CRRW Accurate prediction of the wave- and whipping-induced response


Summary – Flexible Hull GirderSummary – Flexible Hull Girder

MLRW: Results are biased as compared to the CRRW or brute

force model, up to 1.25 Hogging is not as well predicted

Recommended: Applied the CRRW model for short-term statistics Captures non-linear effects well for both hogging and

sagging


Long-Term Response StatisticsLong-Term Response Statistics

Long-term Response statistics (Wave-induced response)

Rigid hull girder Zero speed and head sea The entire scatter

diagram (Hs, Tz) is applied

Bias factors in combination with the MLRW model


Areas of ContributionAreas of Contribution

Two areas observed: One that contributes

significantly One that hardly

influences the results

Concentration of energy

Hull length ~ wave length


ConclusionsConclusions

MLRW – Most Likely Response Wave: Independent of the sea state considered Slightly biased compared to results of brute force

simulations, up to 1.15 (present example)

CRRW – Conditional Random Response Wave: Good agreement in comparison to brute force simulation Apply well for both a rigid and flexible hull girder The random background wave is found to be more and

more important as forward speed is introduced

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