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Model correction factor & Design waves

CeSOS workshop March 23, 24, Trondheim. Model correction factor & Design waves . Peter Friis-Hansen, Luca Garré, Jesper D. Dietz, Anders V. Søborg, J.J. Jensen Technical University of Denmark. The model correction factor method.

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Model correction factor & Design waves

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  1. CeSOS workshopMarch 23, 24, Trondheim Model correction factor & Design waves Peter Friis-Hansen, Luca Garré, Jesper D. Dietz, Anders V. Søborg, J.J. Jensen Technical University of Denmark

  2. The model correction factor method • 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 • Formulate a simplified structural model • 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

  3. The 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

  4. Example • T-stiffened plate panel • Subjected to axial andlateral loads

  5. Limit state and uncertainty modelling • Failure is defined when the axial load exceeds the axial capacity

  6. Simplified models • DNV Classification Notes from 1992 • Simple plastic hinge model is axial stress is bending stress

  7. Using the DNV class rules

  8. The simple plastic hinge model

  9. Comparing results Obtained design points are within 1%

  10. Summary • 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

  11. Design waves for ultimate failure of marine structures

  12. Why design waves …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] ” ?

  13. 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

  14. Wave elevation Response signal Wavemodel Numericalcode White noise Considered point in time The 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?

  15. 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 output from stability code

  16. Outcome 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

  17. Wave induced response for ships • Extreme ship responses not driven by large amplitudes • Suitable combination of wave length and amplitude

  18. 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]

  19. The 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

  20. The MLRW Model The MLRW profile, c(t) conditioning on a given linear response amplitude 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: a is obtained from the response spectrum  is the corresponding phase

  21. The 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:

  22. 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, Hs • The zero-upcrossing wave period, Tz

  23. Application 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

  24. The 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

  25. Short-Term Response Statistics Linear and non-linear results: • Head sea and v = 10 m/s 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 Simul Linear MLRW CRRW

  26. Effect of an Elastic hull girder Wave- and whipping-induced response • Low frequency part  Wave-induced • High frequency part  Whipping-induced Filtering

  27. The Slamming Problem, MLRW • Sagging

  28. Short-term Response Statistics MLRW  Good first approach but less accurate than CRRW CRRW  Accurate prediction of the wave- and whipping-induced response

  29. Summary – 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

  30. Long-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

  31. Areas of Contribution Two areas observed: • One that contributes significantly • One that hardly influences the results Concentration of energy • Hull length ~ wave length

  32. Conclusions • 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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