MT, MEK, DTU Peter 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 workshop March 23, 24, Trondheim
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MT, MEK, DTU Peter Friis-Hansen : Design waves 1 Model correction factor & Design waves Peter Friis-Hansen, Luca Garré, Jesper D. Dietz, Anders V. Søborg,
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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
MT, MEK, DTUPeter Friis-Hansen: Design waves 2
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
MT, MEK, DTUPeter Friis-Hansen: Design waves 3
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
MT, MEK, DTUPeter Friis-Hansen: Design waves 4
ExampleExample
T-stiffened plate panel
Subjected to axial andlateral loads
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Limit state and uncertainty Limit state and uncertainty modellingmodelling
Failure is defined when the axial load exceeds the axial capacity )(XR
MT, MEK, DTUPeter Friis-Hansen: Design waves 6
MT, MEK, DTUPeter Friis-Hansen: Design waves 7
Simplified modelsSimplified models
DNV Classification Notes from 1992
Simple plastic hinge model
is axial stress
is bending stressx
b
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Using the DNV class rulesUsing the DNV class rules
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The simple plastic hinge modelThe simple plastic hinge model
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Comparing resultsComparing results
Obtained design points are within 1%
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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
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Design waves for ultimate Design waves for ultimate failure of marine structuresfailure of marine structures
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] ” ?
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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
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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
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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
MT, MEK, DTUPeter Friis-Hansen: Design waves 17
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
MT, MEK, DTUPeter Friis-Hansen: Design waves 18
Wave induced response for shipsWave induced response for ships
Extreme ship responses not driven by large amplitudes
Suitable combination of wave length and amplitude
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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]
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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
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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
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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:
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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
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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
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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
Rigid hull girder Zero speed and head sea The entire scatter
diagram (Hs, Tz) is applied
Bias factors in combination with the MLRW model
MT, MEK, DTUPeter Friis-Hansen: Design waves 32
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
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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