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

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

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

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

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

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

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

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

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

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

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

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

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] ” ?

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) 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

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

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

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

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

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]

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

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

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:

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

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

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

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

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

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

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

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

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

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

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