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

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 04 December 2018

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 04 December 2018

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 04 December 2018

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 04 December 2018

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 be misleading, especially near limits Unfortunate choice of range can cause multiple design points Version 04 December 2018

Simple example Version 04 December 2018

safe g=0 unsafe g=0 safe unsafe Version 04 December 2018

Version 04 December 2018

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 04 December 2018

Resistance 90 Version 04 December 2018

Load coefficient 90 Version 04 December 2018

Cumulative prob. & density func. for environmental dir. 90 Version 04 December 2018

Environmental intensity 90 Version 04 December 2018

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 04 December 2018

Short term probability of failure Version 04 December 2018

Annual probability of failure Version 04 December 2018

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 04 December 2018

Annual probability of failure Version 04 December 2018

Design points for environmental direction Version 04 December 2018

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 04 December 2018

Version 04 December 2018