Question

The maximum wind speed in a year in a region, V, is lognormally distributed with a mean of 50 km/h and a standard deviation of 10 km/h. The FOS of a building under the wind can be calculated as follows

Fs= M/V
where M is a parameter measuring the resistance of building to winds.
(1) Determine the wind speed with a return period of 100 years, V100.
(2) Suppose the Fs of the building is 1.2 when V = V100. What is the value of M ?
(3) What is the failure probability of the above building in one year?

Answer 1

To determine the wind speed with a return period of 100 years (V100), use the formula: V100 = mean + (standard deviation * z). The value of M can be calculated using the formula Fs = M / V. The failure probability can be determined by finding the z-score and calculating the area under the normal distribution curve.

Step-by-step explanation:

(1) To determine the wind speed with a return period of 100 years, V100, we can use the formula for the return period:

V100 = mean + (standard deviation * z)

Where z is the z-score corresponding to the return period. For a return period of 100 years, the z-score is approximately 2.326. Plugging in the values, we get:
V100 = 50 + (10 * 2.326) = 72.326 km/h

(2) Given Fs = M / V and Fs = 1.2 when V = V100, we can rearrange the formula to solve for M:
M = Fs * V = 1.2 * 72.326 = 86.79 km/h

(3) The failure probability can be calculated using the standard normal distribution table by finding the z-score corresponding to a wind speed of V100, which is 72.326. From the table, we can find the area under the normal distribution curve to the left of this z-score, which represents the failure probability. The failure probability can be calculated as 1 - area under the curve, so P(failure) = 1 - 0.6665 = 0.3335.