Middle East Technical University Department of Civil Engineering Fall Semester 2014-2015 CE 589 Structural Reliability Date: Oct. 20, 2014 Due: Nov. 3, 2014 HOMEWORK No: 1 1. The random variable, X takes the value of 5 with probability 0.3 and is triangularly distributed in the range [10, 20] as shown in the figure below. a) Compute the appropriate value of k so that X has a proper probability density function. Write down the expression for the probability distribution for random variable X. b) Find the mean, mode and median of X. c) Calculate the variance, standard deviation and coefficient of variation of X. d) What is the probability that X is greater than 12, if it is known that X will not exceed 15? fx(x) 0.3 k 0 5 10 20 x 2. A tower is subject to a horizontal force by high winds. An important factor that should be taken into account when strengthening the tower is the duration of winds. It has been assumed that the duration T of the wind is a random variable having a normal distribution with a mean of 4 hours and standard deviation of 1 hour. Find: a) The probability that the wind will be blowing for more than 6 hours. b) The probability that the wind will last less than 9 hours given that it has already blown for more than 5 hours. c) Find the duration t such that P(T  t) = 2 P(T > t). 3. A rain gage (an instrument to measure intensity of rainfall) is being designed for the Deriner Dam site. The intensity of rainfall in the Çoruh basin is assumed to have a lognormal distribution. The median of the annual (one year period) rainfall intensity is 3510 mm and the coefficient of variation is 0.35. a) Determine the parameters of the lognormal distribution. b) What must be the minimum capacity of the rain gage (in mm) to ensure a 90% probability that it will not be overloaded in one year period? c) According to this annual rainfall intensity distribution, is it possible to observe rainfall intensity exceeding 800 mm in a given year? Explain. 4. On the average, it is observed that 1.5 cracks every 1 km section of interstate highway need repair. Assuming Poisson distribution for the distribution of cracks compute: a) The probability that there will be no cracks that need repair in 2 km of highway. b) The probability that there will be at least 1 crack needing repair in 1.5 km of highway? c) The coefficient of variation of cracks needing repair in 10 km of highway? 5. Consider the following joint probability density function f XY x, y  c y 2 =0 for 0 ≤ x ≤ 2 ; elsewhere. 0≤y≤1 a) Determine the value of the constant c. b) Find the marginal probability density functions of the random variables X and Y. c) Are X and Y statistically independent? Explain. d) Compute the median value for the random variable X. e) Compute E(X+Y). f) Find the probability that X+ Y ≤ 2. 6. A cantilever structure, shown below, is subjected to a random vertical load S at a random distance X from the support. The probability density functions of S and X are as follows: fS (s) = 1/4 0 ≤ s ≤ 4 kN fX (x) = cx 0≤x≤3m a) Find the value of c so that fX (x) is a proper probability density function. b) Compute the expected value and variance of S and X. c) Let M denote the fixed end moment. Find the expected value and coefficient of variation of M. Assume S and X to be statistically independent. Use the exact formulation. d) Find the expected value and coefficient of variation of M by using the first-order second moment (FOSM) approximation. S M X 3m 2