1. 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, J.J. Jensen Technical University of Denmark CeSOS workshop March 23, 24, Trondheim

2. MT, MEK, DTU Peter Friis-Hansen : Design waves 2 The model correction factor method u State of the art realistic response models are often time consuming and rarely feasible in a reliability analysis u 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 u The simplified model is not realistic with respect to the physical conditions, or with respect to capturing all second-order bending effects. u The probabilistic calibration procedure assures that the simplified model is made “realistic” – at least around the design point

3. MT, MEK, DTU Peter Friis-Hansen : Design waves 3 The model correction factor u Ditlevsen & Arnbjerg-Nielsen (1991, 1994) u The simplified model is everywhere corrected by a random model correction factor such that u Establish a Taylor expansion of around the design point

4. MT, MEK, DTU Peter Friis-Hansen : Design waves 4 Example u T-stiffened plate panel u Subjected to axial and lateral loads

5. MT, MEK, DTU Peter Friis-Hansen : Design waves 5 Limit state and uncertainty modelling u Failure is defined when the axial load exceeds the axial capacity

6. MT, MEK, DTU Peter Friis-Hansen : Design waves 6

7. MT, MEK, DTU Peter Friis-Hansen : Design waves 7 Simplified models u DNV Classification Notes from 1992 u Simple plastic hinge model is axial stress is bending stress

8. MT, MEK, DTU Peter Friis-Hansen : Design waves 8 Using the DNV class rules

9. MT, MEK, DTU Peter Friis-Hansen : Design waves 9 The simple plastic hinge model

10. MT, MEK, DTU Peter Friis-Hansen : Design waves 10 Comparing results Obtained design points are within  1%

11. MT, MEK, DTU Peter Friis-Hansen : Design waves 11 Summary u Compared to the FEM model the DNV model has a higher degree of model realism than the plastic hinge model u This implies fast convergence of the series of design points u Using the DNV model as idealised model requires 2-3 FEM analyses u Using the plastic hinge model requires 3 x 2 FEM analyses u Resulting design points are almost identical u Plastic hinge model does not contain the information about Young's modulus it requires two FEM

12. MT, MEK, DTU Peter Friis-Hansen : Design waves 12 Design waves for ultimate failure of marine structures

13. MT, MEK, DTU Peter Friis-Hansen : Design waves 13 Why design waves … or critical wave episodes ? u Critical wave episodes: a wave pattern that will result in an unwanted event u The physical wave pattern that causes the problem drives the design u Allows the designer to evaluate better the problem u Can lead to new and innovative solution alternatives u Can lead to safer and more competitive structures u How may we identify critical wave episodes? u How may we calculate: “ P[Wave patterns > critical wave episodes] ” ?

14. MT, MEK, DTU Peter Friis-Hansen : Design waves 14 G.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

15. MT, MEK, DTU Peter Friis-Hansen : Design waves 15 The stochastic modelling u 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 u 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? Numerical code Wave model White noise Wave elevation Response signal Considered point in time

16. MT, MEK, DTU Peter Friis-Hansen : Design waves 16 How to solve ? u Task: Find up-crossing rate,, of a given critical level,, of the considered response. The underlying stochastic variable is the wave process, u The critical wave episode is defined as the most likely wave pattern,, that results in the up-crossing u Mathematical formulation of the up-crossing problem u Rewritten using Madsen’s formula and effectively solved using FORM- SORM. can be extracted as a bi-product of this analysis output from stability code

17. MT, MEK, DTU Peter Friis-Hansen : Design waves 17 Outcome of analysis u Up-crossing rate of selected levels u Short and long term distributions may be calculated u Probability of unwanted event (capsize, moment, slamming, …) is obtained u To obtain long term distribution we need to perform the analysis over multiple sea states u Can we speed-up the calculation of the long term distribution by reusing results from other sea states ? (I think so) u Can we identify a ”design wave pattern” for stability calculations and other highly non-linear problems ? (I hope so) u But, how may we decide on what magnitude of the event is critical ? Calls for risk analysis – calculating the expected loss: R=p·C

18. MT, MEK, DTU Peter Friis-Hansen : Design waves 18 Wave induced response for ships u Extreme ship responses not driven by large amplitudes u Suitable combination of wave length and amplitude

19. MT, MEK, DTU Peter Friis-Hansen : Design waves 19 Identifying 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]

20. MT, MEK, DTU Peter Friis-Hansen : Design waves 20 The model correction factor u Identify an idealised model that captures part of the real model u Model correct the idealised model such that it is made equivalent to the real model u is only established as a zero order expansion at carefully selected points

21. MT, MEK, DTU Peter Friis-Hansen : Design waves 21 The MLRW Model Z(t) is the unconditional wave profile: V n and W n 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

22. MT, MEK, DTU Peter Friis-Hansen : Design waves 22 The CRRW Model CRRW Model: Derived from a Slepian model process Linear regression of V = (V n, W n ) on Y = (Y 1, Y 2, Y 3, Y 4 ) The conditional vector:

23. MT, MEK, DTU Peter Friis-Hansen : Design waves 23 The 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, H s The zero-upcrossing wave period, T z

24. MT, MEK, DTU Peter Friis-Hansen : Design waves 24 Application of CRRW Application of the CRRW given a conditional linear response: Select a stationary sea state and operational profile Derive the constrained coefficients V c,n and W c,n Use the CRRW in a non-linear time- domain code

25. MT, MEK, DTU Peter Friis-Hansen : Design waves 25 The Vessel u PanMax Container Ship: Length, L pp = 276.38 m Breadth, B mld = 32.2 m Draught, T= 11.2 m Displacement= 63350 t Service Speed = 24.8 kn u ShipStar non-linear (2D) strip theory code

26. MT, MEK, DTU Peter Friis-Hansen : Design waves 26 Short-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

27. MT, MEK, DTU Peter Friis-Hansen : Design waves 27 Effect of an Elastic hull girder Wave- and whipping-induced response Low frequency part  Wave-induced High frequency part  Whipping-induced Filtering

28. MT, MEK, DTU Peter Friis-Hansen : Design waves 28 The Slamming Problem, MLRW u Sagging

29. MT, MEK, DTU Peter Friis-Hansen : Design waves 29 Short-term Response Statistics MLRW  Good first approach but less accurate than CRRW CRRW  Accurate prediction of the wave- and whipping- induced response

30. MT, MEK, DTU Peter Friis-Hansen : Design waves 30 Summary – Flexible Hull Girder u MLRW: Results are biased as compared to the CRRW or brute force model, up to 1.25 Hogging is not as well predicted u Recommended: Applied the CRRW model for short-term statistics Captures non-linear effects well for both hogging and sagging

31. MT, MEK, DTU Peter Friis-Hansen : Design waves 31 Long-Term Response Statistics Long-term Response statistics (Wave-induced response) Rigid hull girder Zero speed and head sea The entire scatter diagram (H s, T z ) is applied Bias factors in combination with the MLRW model

32. MT, MEK, DTU Peter Friis-Hansen : Design waves 32 Areas of Contribution Two areas observed: One that contributes significantly One that hardly influences the results Concentration of energy Hull length ~ wave length

33. MT, MEK, DTU Peter Friis-Hansen : Design waves 33 Conclusions u 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) u 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