19. As shown in the figure, a beam cantilevered at  is subjected to an uniform load of  and a concentrated force . If the cross-section has a diamater of , what is the distirbution of maximum bending stress? Note that  and  are independent.

Solution

Based on the moment equilibrium of beam OC, the bending moment acting at O is

Thus the maximum bending stress is given by

Since  and  are independently and normally distributed, their linear sum,  is also normally distributed. And the mean and standard deviation of  are given by

So the distribution of the maximum bending stress is .