Random Heading Angle in Reliability Analyses Jan Mathisen

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

Motivation n Typical goal of a reliability analysis is to calculate an annual probability of failure n Wind, waves and current are randomly distributed over direction n Offshore structures have directional properties - wrt. load susceptibility, stiffness, capacity fixed, weathervaning or directionally controlled structures n (Ship heading directions are controlled) n Usual practical approaches a) Consider most unfavourable direction, or b) Sum probabilities over a set of discrete headings v Approach to treating directions as continuous random variables - Version emphasis on ULS 06/03/2021 2

Typical probabilistic model for ULS n Piecewise stationary model of stochastic processes n 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, V 10, . . . ) current speed and profile (current direction, surface speed, . . . ) computed mean heading for weathervaning structure (ship heading and speed) n Long term reponse (or LS) by probability integral over joint distribution of environmental variables, still conditional on time independent random variables n Allowance for number of short term states in a year n Probability of failure by probability integral over time independent random variables Version 06/03/2021 3

Linear, short term response in waves n 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 n Short-crested – simple extension to arbitrary wave directions - Adjust weighting factors on contributions of discrete directions to response variance n Long-crested – extension to arbitrary wave directions - n Version 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. 06/03/2021 4

Computationally expensive short term response n Response surface approach to allow long term probability integral n Heading angles as interpolation variables on response surface n 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. n Or use periodic interpolation function for angle variable - Fourier series Version 06/03/2021 5

Periodic problem n Heading angles are periodic variables 0° 360 ° 720 ° 1080 °. . . n Difficulty with probability density & distribution n 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 06/03/2021 6

Simple example Version 06/03/2021 7

safe g=0 unsafe Version 06/03/2021 8

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Jacket example – still simplified n Approach to including environmental heading as a random variable in reliability analysis of a jacket - Ref. OMAE 2004 -51227 n 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 n Version Basic directional limit state function 06/03/2021 10

Resistance 90 0 Version 06/03/2021 11

Load coefficient 90 0 Version 06/03/2021 12

Cumulative prob. & density func. for environmental dir. 90 0 Version 06/03/2021 13

Environmental intensity 90 0 Version 06/03/2021 14

Short term extreme load n Have mean and std. dev. n Assume narrow-banded Gaussian dstn. of load n Rayleigh dstn. of load maxima follows n Transform load maxima to an auxillary exponential dstn. n Short term extreme maximum of auxillary variable obtained as a Gumbel dstn. - Version 3 hours duration with 8 s mean period assuming independent maxima extreme auxillary variable transformed back to extreme load 06/03/2021 15

Short term probability of failure Version 06/03/2021 16

Annual probability of failure Version 06/03/2021 17

Omitted features of complete problem n Inherent uncertainties in resistance - e. g. soil properties n Model uncertainties on load & resistance n These uncertainties are usually time-independent - do not vary between short term states n Simplified formulation needs to condition on time-independent variables n Outer probability integral needed to handle time-independent variables Version 06/03/2021 18

Annual probability of failure Version 06/03/2021 19

Design points for environmental direction Version 06/03/2021 20

Conclusion n Detailed treatment of heading as a random variable looks interesting/worthwhile in some cases - non-axisymmetric environment non-axisymmetric load susceptibility or resistance n Care needed with distribution function of heading (periodic variables) n Not much extra work in load and capacity distribution - n Some work needed to develop joint distribution of usual metocean variables together with headings - n usually conditional on discrete headings extend to continuous headings Inaccuracy of FORM demonstrated for problems with heading - Version may need response surface suitable for periodic variables SORM seems adequate Median direction should be close to design point maybe some difficulty with SORM for inner layer of nested probability integrals 06/03/2021 21

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