Random Heading Angle in Reliability Analyses

1. Random Heading Angle in Reliability Analyses <Jan Mathisen> <March 23 2006>

2. Motivation • Typical goal of a reliability analysis is to calculate an annual probability of failure • Wind, waves and current are randomly distributed over direction • Offshore structures have directional properties • wrt. load susceptibility, stiffness, capacity • fixed, weathervaning or directionally controlled structures • (Ship heading directions are controlled) • Usual practical approaches • Consider most unfavourable direction, or • Sum probabilities over a set of discrete headings • Approach to treating directions as continuous random variables • emphasis on ULS Version

3. Typical probabilistic model for ULS • Piecewise stationary model of stochastic processes • Short term stationary conditions, extreme response (or LS) distribution, conditional on time independent random variables and: • directional wave spectrum (main wave direction, Hs, Tp, ...) • wind spectrum (wind direction, V10, ...) • current speed and profile (current direction, surface speed, ...) • computed mean heading for weathervaning structure • (ship heading and speed) • Long term reponse (or LS) by probability integral over joint distribution of environmental variables, still conditional on time independent random variables • Allowance for number of short term states in a year • Probability of failure by probability integral over time independent random variables Version

4. Linear, short term response in waves • Usual practice • Long-crested(unidirectional) or short-crested (directional) wave spectrum • Linear transfer function for set of discrete wave directions • Short term response computed for same set of discrete directions • Short-crested – simple extension to arbitrary wave directions • Adjust weighting factors on contributions of discrete directions to response variance • Long-crested – extension to arbitrary wave directions • Calculate short term response for available discrete directions • Fit interpolation function – Fourier series or taut splines • Interpolate for short term response in required direction • Seems that discrete directions need to be fairly closely spaced for acceptable accuracy • Ref. Mathisen, Birknes, “Statistics of Short Term Response to Waves, First and Second Order Modules for Use with PROBAN,” DNV report 2003-0051, rev.02. Version

5. Computationally expensive short term response • Response surface approach to allow long term probability integral • Heading angles as interpolation variables on response surface • With non-periodic interpolation model • Vary limits on heading angle such that they are distant from each interpolation point • Ref. Mathisen, "A Polynomial Response Surface Module for Use in Structural Reliability Computations", DNV, report no.93-2030. • Or use periodic interpolation function for angle variable • Fourier series Version

6. Periodic problem • Heading angles are periodic variables 0°  360 °  720 °  1080 ° ... • Difficulty with probability density & distribution • Resolve by limiting valid headings to one period • Fine for probability density • Cumulative probability tends to bemisleading, especially near limits • Unfortunate choice of rangecan cause multiple design points Version

7. Simple example Version

8. g=0 g=0 safe unsafe safe unsafe Version

9. Version

10. Jacket example – still simplified • Approach to including environmental heading  as a random variable in reliability analysis of a jacket • Ref. OMAE2004-51227 • Highly simplified load L and resistance r model • main characteristics typical of an 8-legged jacket in about 80 m water depth, in South China Sea • with one or two planes of symmetry • not a detailed analysis of an actual platform • Basic directional limit state function Version

11. 90 0 Resistance Version

12. 90 0 Load coefficient Version

13. 90 0 Cumulative prob. & density func. for environmental dir. Version

14. 90 0 Environmental intensity Version

15. Short term extreme load • Have mean and std.dev. • Assume narrow-banded Gaussian dstn. of load • Rayleigh dstn. of load maxima follows • Transform load maxima to an auxillary exponential dstn. • Short term extreme maximum of auxillary variable obtained as a Gumbel dstn. • 3hours duration with 8s mean period • assuming independent maxima • extreme auxillary variable transformed back to extreme load Version

16. Short term probability of failure Version

17. Annual probability of failure Version

18. Omitted features of complete problem • Inherent uncertainties in resistance • e.g. soil properties • Model uncertainties on load & resistance • These uncertainties are usually time-independent • do not vary between short term states • Simplified formulation needs to condition on time-independent variables • Outer probability integral needed to handle time-independent variables Version

19. Annual probability of failure Version

20. Design points for environmental direction Version

21. Conclusion • Detailed treatment of heading as a random variable looks interesting/worthwhile in some cases • non-axisymmetric environment • non-axisymmetric load susceptibility or resistance • Care needed with distribution function of heading (periodic variables) • Not much extra work in load and capacity distribution • may need response surface suitable for periodic variables • Some work needed to develop joint distribution of usual metocean variables together with headings • usually conditional on discrete headings • extend to continuous headings • Inaccuracy of FORM demonstrated for problems with heading • SORM seems adequate • Median direction should be close to design point • maybe some difficulty with SORM for inner layer of nested probability integrals Version

22. Version