Probability Of Brittle Fracture For A Cracked Ship
JS Kent and J DG Sumpter
QinetiQ, Rosyth Business Park, Dunfermline KY11 2XR
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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 MPam) 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 counter-intuitive 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.