In Exercise 66, the weight of the beam itself contributes to the bending moment. Assume...

In Exercise 66, the weight of the beam itself contributes to the bending moment. Assume that the beam is of uniform thickness and density so that the resulting load is uniformly distributed on the beam. If the weight of the beam is random, the resulting load from the weight is also random; denote this load by W (kip-ft).

a. If the beam is 12 ft long, W has mean 1.5 and standard deviation .25, and the fixed loads are as described in part (a) of Exercise 66, what are the expected value and variance of the bending moment? [Hint: If the load due to the beam were w kip-ft, the contribution to the bending moment would be .]

b. If all three variables (X₁, X₂, and W) are normally distributed, what is the probability that the bending moment will be at most200 kip-ft?

Reference Exercise 66

If two loads are applied to a cantilever beam as shown in the accompanying drawing, the bending moment at 0 due to the loads is

a. Suppose that X1 and X2 are independent rv’s with means 2 and 4 kips, respectively, and standard deviations .5 and 1.0 kip, respectively. If a₁ = 5 ft and a₂ _ 10 ft, what is the expected bending moment and what is the standard deviation of the bending moment?

b. If X1 and X₂ are normally distributed, what is the probability that the bending moment will exceed 75 kip-ft?

c. Suppose the positions of the two loads are random variables. Denoting them by A₁ and A₂, assume that these variables have means of 5 and 10 ft, respectively, that each has a standard deviation of .5, and that all A_i’s and X_i’s are independent of one another. What is the expected moment now?

d. For the situation of part (c), what is the variance of the bending moment?

e. If the situation is as described in part (a) except that Corr(X₁, X₂) = .5 (so that the two loads are not independent), what is the variance of the bending moment?