A pile is designed to have a mean capacity of 20 tons howeve

A pile is designed to have a mean capacity of 20 tons; however, because of uncertainties, the pile capacity is lognormally distributed with a coefficient of variation (cov) of 20%. Suppose the pile is subject to a maximum lifetime load that is also lognormally distributed with a mean of 10 tons and a cov of 30%. Assume that pile load and capacity are statistically independent.

(a) Determine the probability of failure of the pile.

(b) A number of piles may be tied together to form a pile group to resist an external load. Suppose the capacity of the pile group is the sum of the capacities of the individual piles. Consider a pile group that consists of two single piles as described above. Because of proximity, the capacities between the two piles are correlated with a correlation coefficient of 0.8. Let T denote the capacity of this pile group.

Determine the mean value and cov of T.

Solution

a) we know coefficient of variation for capacity, from there we can find sigma. and we have mean. so we can find any probaabilty by standardizing. similar for pile load. Then use multiplicative rule for independent random variable as P(A and B)= P(A) *P(B).