7. A stress element is cut by an oblique plane with a normal  at an angle  counterclockwise from the axis. If , , , and ,   and    are independent, what is the distribution of stresses  and  ?

Solution

Considering the force equilibrium with respect to the  axis, we have

                                                (1)

Considering the force equilibrium with respect to the  axis, we have

                                                (2)

The cross shears are equal, and hence

                                                                     (3)

Combining Eqs. (1) - (3), we find the stresses  and  given below.

Because ,  and  are independently and normally distributed, their linear combinations,  and , are also normally distributed. The mean and standard deviation of   and  are given

So the distributions of the stresses are  and .                                        Ans.