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**Assignment 3 **

**Engineering 9411 **

**More Probability and Simulation Problems **

In a study of possible iceberg damage to an offshore structure, the following information has been gathered from past records. The distribution of the number of icebergs passing through the offshore site per year can be modeled as a Poisson process with a mean arrival rate of

1.4/year. The distribution of the force of an iceberg hit is lognormal with a median of 7MN and a coefficient of 15%. From structural analysis of the offshore structure, the structure is adequate for any iceberg hit that is less than 10 MN. Assume that the force of each hit between iceberg arrivals is statistically independent. a) What is the probability of damage to the structure due to an iceberg hit? b) What is the probability of damage in a year? c) Compare your analytical results to that obtained by Monte Carlo simulation.

This problem concerns the design of an offshore drilling platform as seen in Figure a. it is known that the annual extreme water level Z depends on (i) the annual extreme original sea level Y, and (ii) the annual extreme height of the wave, X, above the original sea level. In other words, when a storm occurs, the platform may or may not be overtopped depending on the original sea level Y. Suppose the annual extreme original sea level is represented by the density function shown in Figure b, and the annual extreme wave heights is represented by the probability density given in Figure c, determine: a) The probability of the platform being overtopped during an extreme storm if the platform level is 50 m above the seabed. b) If the platform was overtopped during an extreme storm, what is the probability that the original sea level was greater than or equal to 30 m? c) If the acceptable risk of failure (failure is defined as when the water level overtops the platform) is 10% over the life time of the platform which is 20 years, what should be the minimum level of the platform?

L. M. Lye (2009)

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Solve Problem 2 by using (a) 11 discrete points, and (b) 21 discrete points, to approximate the pdf of Y. Compare your answers with the analytical solution and Monte Carlo solution.

If the wave heights X is presented by an exponential distribution given by: f ( x )

1

exp

* x x*

0

Where

=8m.

And if the annual extreme sea level is represented by a normal distribution with the same mean and standard deviation as that of the uniform distribution shown in Problem 2, what would be the answers to 2a, 2b, and 2c? Solve this problem using Monte Carlo simulation.

Please attach a printout of your macros or programs and results.

L. M. Lye (2009)