Random Heading Angle in Reliability Analyses

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Random Heading Angle in Reliability Analyses
ltJan Mathisengt ltMarch 23 2006gt
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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

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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

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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.

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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

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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

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Simple example
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safe
unsafe
safe
unsafe
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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

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Resistance
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Load coefficient
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Cumulative prob. density func. for
environmental dir.
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Environmental intensity
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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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Short term probability of failure
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Annual probability of failure
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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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Annual probability of failure
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Design points for environmental direction
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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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