Question 1 (40 Marks)
Sixty items were put on a reliability test and the time to failure, in hours, was recorded for each item. You will be supplied with a data set that will list, for each item, their time to failure or the time they were removed from the test. Each item will be identified as:
− F: failed. Time denotes recorded failure time.
− I: removed for inspection. Time denotes the time that item was removed from test for inspection. The items had not failed at this time.
− R: still running at end of test. These items were still working at the end of the reliability test and the time denotes the end of the test period.
a) Estimate the reliability of the items using the Median Rank method and show the results in the form of a reliability time graph. [30%]
b) Select an appropriate failure probability distribution model (such as the normal, lognormal or Weibull distributions) that gives the best fit to the estimated reliability data using the straight line plotting method and then determine the values of the model parameters. [70%]
Where a spreadsheet or other computer tool is used, a copy of the results must be included in the report along with a clear explanation of the formulae used to generate the results.
Question 2 (60 Marks)
a) Long, thin-walled, circular tubes are subject to a uniform compressive longitudinal stress, σ. The maximum value of σ in any 10-year period may be represented by a log-normal distribution with a mean 130 MPa µσ = and a coefficient of variation
V =0.20 σ .
The critical local buckling stress, σ cr , of a long thin-walled circular tube under compressive longitudinal stress is:
where t is the wall thickness, r is the tube radius, C is the model uncertainty, E and ν are the material elastic modulus and Poisson’s ratio respectively.
There is no uncertainty in the Poisson’s ratio for the material, which is 0.33. The values of r, C and E are uncertain and are log-normally distributed with the means and coefficients of variation as given below:
Assuming that there is no uncertainty associated with the wall thickness, determine the required value of t such that the tubes have a reliability of 0.995 for a 10-year period. [25%]
b) For the tubes described in part (a), perform a Monte Carlo simulation to estimate the failure probability of the tubes for the thickness value determined in part (a). In your answer you must include:
• A graph to show the convergence of the estimated failure probability with increasing sample size;
• A graph to show the convergence of the estimated standard deviation with the increasing sample size.
• Justification for the final sample size used. [35%]
c) Further investigation of the maximum value of σ determines that it is better represented by a three parameter Weibull distribution with the following parameters:
σ = 80 MPa; η = 55 MPa; β = 2.2.
In practice there is also variation in the tube thicknesses from the manufacturing process.
Assuming that the tube thickness is normally distributed with a mean equal to the thickness calculated in part (a) and a standard deviation of 0.02 mm; perform a Monte Carlo simulation to estimate the updated failure probability of the tube. [40%]
A complete listing of the MATLAB scripts used to answer parts (b) and (c) of this question must be included in an appendix of the assignment report.