Amy, an entry-level geotechnical engineer, is facing a dilemma. She is in charge of estimating the probability of failure of a foundation for a building due to the actual shear stress being higher than the peak shear capacity of the soil at the site. Based on initial testing and some results from her old geotech.book, Amy believes that the peak shear strength of the soil has a Normal distribution with a mean of114kPa and a standard deviation of 10kPa. However, she decided to run a test to get more accurate information.

The test that Amy has run only indicates whether or not the peak shear strength of the soil is above or below 110kPa. If the peak shear strength of the soil is actually below 108kPa, there is a 95% chance that the test will indicate that the peak shear strength is below 110kPa. If the actual peak shear strength is between 108kPa and 121kPa, there is an 80% chance that the test will indicate that the peak shear strength is below 110kPa. If the actual peak shear strength is above 121kPa there is only a 20% chance that the test will indicate that the peak shear strength is below 110kPa.

As a first approximation, Amy thinks that the actual shear stress once the foundation has been placed will be 121kPa.

a) Find the initial probability of the shear strength being in each of the three categories below108kPa, between 108kPa and 121kPa, and above 121kPa based on the Normal distribution(think back to your introductory statistics course).

b) Find the probability of each of the three shear strength categories if the test returns a result saying that the peak shear strength is below 110kPa.

c) Find the probability of each of the three shear strength categories if the test returns a result saying that the peak shear strength is above 110kPa.

d) If the test returns a result saying that the peak shear strength is below 110kPa, what is the probability of failure of the foundation? Assume that foundation failure occurs (deterministically)if stress exceeds capacity.

e) If the actual needed shear stress due to the loading on the foundation is normally distributed with a mean of 110kPa and a standard deviation of 5kPa and Amy had not done the test (i.e., she would still think the peak shear strength was ~N(114,10)) what is the probability of failure of thefoundation if the test returns a result saying that the peak shear strength is below 110kPa?