Chapter 5.5, Problem 71E

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 $w{\displaystyle {\int }_0^{12}xdx}.$]

 b. If all three variables (X₁, X₂, and W) are normally distributed, what is the probability that the bending moment will be at most 200 kip-ft?