Monte Carlo prediction o f extreme values of the combined load effects and simplified probabilistic design of ocean going ships

Transcription

1 Harbin Engineering University Monte Carlo prediction o f extreme values of the combined load effects and simplified probabilistic design of ocean going ships Wenbo Huang College of Space and Civil Engineering Harbin Engineering University

2 Illustration of linear combination of Constants, deterministic functions of time, random variables and stochastic processes

3 Combination of stochastic processes

4 The purpose of the paper M = W σ M margin min s ct,max M = max{ M () t + M ()(0 t < t < T)} ct,max stillwater wave 1. load combination M ct,max (1) Numerical solution based on stochastic processes (2) Monte Carlo simulation (3) the theoretical probabilistic mod el for M 2. The strength of a ship beam W = min{ Wt ( ) (0 < t< T)} min based on allowable damage 3 Reliability analysis ct, max an effective method is established for reliability-based analysis and design of a ship hull beam

5 List of contents 1. Prediction of extreme values of the combined global longitudinal vertical still water and wave bending moments admiship Theoretical model Monte Carlo simulation The proposed theoretical probabilistic model suitable for the reliability analysis of a ship hull beam. 2 Strength model based on the allowable fatigue damage 3. Numerical Analysis: Prediction of extremes and Simplified reliability analysis of a ship hull girder 4. Conclusions

6 1.Combination of still water and wave loads Still water and wave loads :global vertical longitudinal still water and wave induced bending moments amidships Why: As for loads acting on a ship hull girder, the most important one is the combined global longitudinal vertical still-water and wave induced bending moments amidships because it is directly related to the reliability-based analysis and design of a ship hull beam. Hence, it firstly is necessary to develop an effective probabilistic model for the combined load.

7 Review of relevant works Global vertical longitudinal still water and wave induced bending moments Moan & Jiao [1988], Guedes Saores [1992], Wang & Moan [1996], Wenbo Huang & Moan [2005,2008,2009], M = max{ M () t + M ()(0 t < t < T)} ct,max stillwater wave load combination ψ = ( M M )/ M M M + ψ M ψ = ( M M )/ M M M + ψ M M = W σ M sw ct, wt, swt, ct, max swt, max w wt, max w c, T sw, T w, T c, T max w, T max sw sw, T max margin min s ct,max M Mmargin = Wminσ s M M = W σ M margin min s c, T max + ψ M swt, max w wt, max + ψ M w, T max sw sw, T max

8 1.1 Still water load models Distribution & Characteristic value M sw (t) t v t F ( m) exp{ ν Tλ[1 F ( m)]} M sw M sw,max F m N µ σ ( ): (, ) M sw sw sw v Tλ[1 F ( M )] = ln(1 p) sw sw sw, T sw

9 1.2 Wave load models Distribution & Characteristic value M sw (t) t v t F ( m) exp{ ν Tλ[1 F ( m)]} M w M F w,max M w m ( m) = 1 exp g v Tp [1 F ( M )] = ln(1 p) w e M wt, w q w

10 1.3 Combination model To combine SW & WL M w (t) M w,tv t w t v t f Mc,tv (m) M c,tv t v t

11 1.3 Combination model M w (t) M w,tv t v t w t Numerical solution F M ct, v ( m) = F ( m x) f ( x) dx M w 1 + N [1 F ( m x)] v M w sw t v M c,tv t { } F ( m) exp v Tλ[1 F ( m)] M sw M c ct, v v Tλ[1 F ( M )] = ln(1 p) sw M c T ct, v,

12 1.4 Monte Carlo simulation M w (t) M w,tv f Mc,tv (m) t v t w t M c,tv t v t

13 1.5 A theoretical probabilistic model Generalized extreme value (GEV) distribution { M m} = { M m M m} Pr Pr,, c,max c1 cn { F ( )} M m F mkbc = km b c 1/ k c,max (,,, ) exp{ [1 ( ) / ] } Gumbel k = 0 c F ( mbc,, ) = exp{ exp[ ( m b) / c]} c,max To fit extremes based on Numerical solution and Monte Carlo results to a theoretical probabilistic model n

14 2. Strength model based on the allowable fatigue damage η = N A T ES m [ ] FS ( s) = 1 exp (ln N0) s s NT m ( ) mq / m η = sc ln N0 Γ 1+ A q c q N T the number of stress cycles, A & m material parameters related to SN curve, s c the characteristic value of bending stress range in T 0 (N 0 ), q the shape parameter.

15 Derive the allowable characteristic stress range based on the allowable damage NT m ( ) mq / m η = sc ln N0 Γ 1+ = ηl A q s s cl NT F 1/ m 1/ m 1/ q m ( A/ NT ) snt 0 ξ 1/ m q γf = ( ηl) γ F = (ln N ) Γ 1+ = ( A/ N ) ( η ) L T 1/ m 1/ m 1/ q ξ = (ln N0) Γ 1+ γ = = m q 1/ m the fatigue allowable stress range in T years, a random load factor, a fatigue safety factor,

16 s s c c = M s cl Minimum of section modulus + M wc, hog wc, sag W M + M s ξ W γ wc, hog wc, sag NT γ W W0 = ( Mwc, hog + Mwc, sag ) ξ s F F NT γ Wmin = W0 = ( Mwc, hog + Mwc, sag ) ξ s F NT M c,hog M c,sag characteristic bending moments related to hogging & sagging; The minumum of section modulus of a ship beam is the function of M c,hog M c,sag and the fatigue allowable stress range in T years, a random load factor,a fatigue safety factor,

17 Safe Margin Equation M F mkbc km b c c,max 1/ k c,max (,,, ) = exp{ [1 ( ) / ] } γ Wmin = ( Mwc, hog + Mwc, sag ) ξ s M = W σ M margin min s c,max F NT p = km b c fm beam p f = p f ( ) fmbeam M m beam beam dm beam m = W σ beam 1/ k 1 exp{ [1 ( beam ) / ] } min s

18 3. NUMERICAL ANALYSES a) Analysis of the Monte Carlo simulation results A lot of simulations are carried out for the different kinds of ships. In each case, 200 samples are produced for extreme values of individual and combined loads in the design time of 1 and 20 years. The simulation time is different for each case, which can vary from about 5 to 40 minutes because the mean voyage times are different for different ships.

19 Parameters of loads Ship type Load conditions Bending moments Voyage durations (days) Time in port (days) sw v p Container hog Suezmax oil tanker hog sag µ sw Large tanker hog Small tanker hog Bulk carrier hog σ µ σv µ σ p The mean wave period is 8s. The shape parameter is 1, for a Suezmax oil tanker The mean wave period is 10s. The shape parameter of the Weibull distribution of the wave load is 0.89 for other ships

20 The statistics of extreme values of still water loads for 20 year Table 3. The statistics of extreme values of SWL for 20 year Statistics 20 years Container Suezmax Tanker hogging Large tanker Small tanker Bulk carrier Min Max Mean Std. Dev Skew Kurtosis

21 The statistics of extreme values of Statistics 20 years wave loads for 20 year Table 5 The statistics of extreme values of WL of 20 years Container Suezmax Tanker hogging Large tanker Small tanker Bulk carrier Min Max Mean Std. Dev Skew Kurtosis

22 The statistics of extreme values of Statistics 20 years combined loads for 20 year Table 7 The statistics of extreme values of CL 20 years Contain-er Suezmax Tanker hogging Large tanker Small tanker Bulk carrier Min Max Mean Std. Dev Skew Kurtosis

23 Fig. 3 CDF of the combined extreme value of container ship based on different models L Num. 1year R Num. 20year L Ana.G 1year R Ana.G 20year L Ana.Ev2 1yearl R Ana.Ev2 20year L Monte. 1year R Monte. 20year

24 Fig. 4 CDF of the combined extreme value of bulk carrier based on different models (upper talils) L Num. 1year R Num. 20year L Ana.G 1year R Ana.G 20year L Ana.Ev2 1year R Ana.Ev2 20year L Monte. 1year R Monte. 20year

25 Fig. 5 CDF of the combined extreme value of large tanker based on different models (upper talils) L Num. 1year R Num. 20year L Ana.G 1year R Ana.G 20year L Ana.Ev2 1year R Ana.Ev2 20year R Monte. 1year L Monte. 20year

26 L Num. 1year R Num. 20year L Ana.G 1year R Ana.G 20year L Ana.Ev2 1year R Ana.Ev2 20year L Monte. 1year R Monte. 20year Fig. 6 CDF of combined hogging extreme values of Suezmax oil tanker based on different models

27 Fig. 7 CDF of the combined sagging extreme value of Suezmax oil tanker based on different models L Num. 1year R Num. 20year L Ana.G 1year R Ana.G 20year L Ana.Ev2 1year R Ana.Ev2 20year L Monte. 1year R Monte. 20year

28 Fig. 8 CDF of the combined extreme value of small tanker based on different models L Num. 1year R Num. 20year L Ana.G 1year R Ana.G 20year L Ana.Ev2 1year R Ana.Ev2 20year L Monte. 1year R Monte. 20year

29 TypeⅡ:Recommended theoretical probabilistic model for the combined extreme loads: It is very hard to give a good reason why TypeⅡextreme value distribution is much better than TypeⅠmodel for the example cases. The possible reasons are as follows; a) the domain of definition of TypeⅡmodel is much suitable for modelling the distribution of the combined extreme value than TypeⅠmodel; b) the parent distribution of the combined extreme value may have a polynomial tail.

30 Comparison of extremes by numerical and Monte Carlo simulation 1yrs. Cer. hog SuT hog SuT sag LT hog ST hog BC hog MC M ct Num M ct

31 Comparison of extremes by numerical and Monte Carlo simulation 20 year Cer. hog SuT hog SuT sag LT hog ST hog BC hog MC M ct Num M ct

32 Reliability based analysis and design of a ship hull beam Table 4 Probabilities of failure vs σ s η L η L σ s 250Mpa 300Mpa 390Mpa e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e-007

33 4. CONCLUSIONS Based on Poisson models, a systematic Monte Carlo simulation is developed to estimate extreme values of the combined load effects of ocean-going ships. The extreme values simulated are compared with those based on the theoretical methods (Huang & Moan 2008). The numerical analyses show that the results based on the two methods agree with very well but the numerical solutions are on the conservative side. Moreover, the empirical distribution of the combined extreme values based on the Monte Carlo simulation can be well fitted to the GEV distribution (the theoretical distribution model of TypeⅡ), which will be very convenient for the reliability analyses of the ship hull beam.

34 4. CONCLUSIONS By considering the effect of fatigue damage on the bending capacity of a ship hull beam, the strength model of a ship hull beam is established, which is very convenient to use in the reliability-based analysis and design of a ship hull beam because it is simply based on the main dimension of a ship under design. Finally, with the developed load and strength modes, the reliability analysis and design of a ship hull beam are carried out. Hence, an effective method is established for reliabilitybased analysis and design of a ship hull beam

35 REFERENCES [1] Mansour, A.E. & Thayamballi, A. 1993, Probability based ship design loads and load combinations. Ship Structure Committee, SSC-373, Washington, DC, USA [2] Moan, T., Jiao, G., 1988, Characteristic still water load effects of production ships, Report MK/R 104/88, Norwegian institute of technology, Trondheim Norway. [3] Guedes Soares, C., 1992, Combination of primary load effects in ship structures, Probabilistic Engineering Mechanics 7: [4] Wang, X. & Moan, T., 1996, Stochastic and deterministic of still water and wave bending moments in ships, Marine Structures 9: [5] Huang, W. & Moan, T., 2005, Combination of global still-water and wave load effects for reliability-based design of floating production, storage and offloading (FPSO) vessels, Applied Ocean Research 27: [6] Huang, W. & Moan, T., 2008, Analytical method of combining global longitudinal loads for ocean-going ships, Probabilistic Engineering Mechanics 23: [7] Huang W., Moan T., 2009, Load combination factors suitable for probabilistic and semiprobabilistic design of ocean-going ships, Proceedings of the tenth international conference on structural safety and reliability (ICOSSAR2009), OSAKA, JAPAN, SEPTEMBER. [8] Huang, Wenbo, 2010, Simplified Combination of still water and wave loads with the emphasis in considering a general Weibull Characteristic of wave loads, Proceedings Of OMAE2010, June 6-11, Shanghai, China.

36 REFERENCES [9] O.,F., Hughes, 1988, Ship structure design-a rationally-based, computer-aided, optimization approach, Wiley, USA. [10] Guedes Soares, C. & Moan, T., 1988, Statistical analysis of still water load effects in ship structures, SNAME 96: [11] Wen, Y.K., 1990, Structural load modelling and combination for performance and safety evaluation, Amsterdam: Elsevier. [12] Guedes Soares, C. & Moan, T., 1991, Model uncertainty in the long-term distribution of waveinduced bending moments for fatigue design of ship structures, Marine Structures 4: [13] Naess, A., 1999, Extreme response of nonlinear structures with low damping subjected to stochastic loadings, Journal of offshore mechanics and arctic engineering 121: [14] Turkstra, C.J. & Madsen, H.O., 1980, Load combination in codified structural design, J. Struct. Div. Proc. ASCE 106(12): [15] Larrabee, R.D. & Cornell, C.A., 1981, Combination of various load processes, J Struct Div. ASCE 107: [16] Rackwitz, R., Fiessler, B., 1976, Note on discrete safety checking when using non-normal stochastic models for basic variables, Load Project Working Session, MIT, Cambridge, MA, June. [17] Coles, S., 2001, An introduction to statistical modelling of extreme value, London:Springer- Verlag. [18] Gaspar B. Guedes Soares G. 2012, Hull girder reliability using a Monte Carlo based simulation method, Probabilistic Engineering Mechanics,

37 Harbin Engineering University Thank you very much! Wenbo Huang College of Space and Civil Engineering Harbin Engineering University