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

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
04 December 2018

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
04 December 2018

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 , rev.02.
Version
04 December 2018

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
Or use periodic interpolation function for angle variable
Fourier series
Version
04 December 2018

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 be misleading, especially near limits
Unfortunate choice of range can cause multiple design points
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7. Simple example
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8. safe
g=0
unsafe
g=0
safe
unsafe
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9. Version
04 December 2018

10. Jacket example – still simplified
Approach to including environmental heading  as a random variable in reliability analysis of a jacket
Ref. OMAE
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
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11. Resistance 90
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12. Load coefficient 90
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13. Cumulative prob. & density func. for environmental dir.90
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14. Environmental intensity90
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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
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16. Short term probability of failure
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17. Annual probability of failure
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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
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19. Annual probability of failure
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20. Design points for environmental direction
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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
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04 December 2018

22. Version
04 December 2018