Probability Of Brittle Fracture For A Cracked Ship
JS Kent and J DG Sumpter
QinetiQ, Rosyth Business Park, Dunfermline KY11 2XR
Email: jdsumpter@qinetiq.com
Fatigue cracking is a common problem in ships. Ship Classification Society rules
state that all fatigue cracks should be repaired as soon as they are discovered; but,
in practice, it may be necessary to operate a cracked ship for an extended period
before a repair can be made. Brittle fracture can potentially cause sudden and
catastrophic failure of the ship, so it is very important to have a clear understanding
of the risk that is being run by leaving the crack un-repaired. The factors affecting
brittle fracture risk are the length of the crack, steel grade, sea state, and
temperature. If the crack is not repaired it will continue to increase in length by
fatigue, so that risk increases with time.
The most important variable affecting brittle fracture risk is scatter in toughness.
QinetiQ have a large data bank of steel toughness measurements on ship steel. The
data have been fitted to a modified ‘Master Curve’ approach linking fracture
mechanics toughness (characterised by KJc in MPam) to Charpy 27 Joule
temperature. Particular issues discussed are the selection of T0, the reference
temperature in the Master Curve approach, and the inclusion of a lower bound cut-off
to the toughness distribution. It is found that leaving the lower bound to be
determined by a Weibull distribution gives an unrealistically high frequency of very
low fracture toughness values.
Probability of ship fracture as a function of time at sea is predicted by a Monte Carlo
simulation combining the toughness distribution with a distribution representing the
peak applied load in a given time period. An estimate of the long term fluctuating load
is used to estimate the fatigue crack growth rate. The calculation is repeated for
successive time steps to determine the increase in probability of fracture with
elapsed time at sea.
Previous work by the authors used Bayesian updating to discard failed samples from
subsequent time steps. The logic of this feature was that the program was
addressing the future survival of a ship which had been discovered to have a crack
sometime into its life. It was reasoned that if the crack had been in brittle steel, it
would never have been able to reach that length without failing, and that some credit
should be claimed for this in future survival rate. Unfortunately, this led to counterintuitive trends in the subsequent probability of failure predictions. Reasons for this
are explained.
The current work uses a toughness distribution which is invariant with time step. This
leads to well behaved trends of probability of fracture with temperature and steel
grade. Some possible target probabilities of fracture are discussed. Using a figure of
5 x 10-4 events per ship year as broadly typical of the observed frequency of fracture,
the simulation suggests that this is equivalent to a crack length of 250 mm in grade A
steel at 0oC.
One major uncertainty remaining with the current model is that the probability of
fracture increases as the sampling frequency is increased (e.g. from once a month to
once a day). This is a consequence of the assumption that the toughness is assumed
to be constant for the duration of the sampling period. The more often the toughness
distribution is sampled the greater the probability of encountering a low result. It is felt
that this problem cannot be fully resolved without a better understanding of the length
scale over which fracture toughness fluctuates in a given steel plate.