Bending stress due to 2 bending moments in different planes

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IMO, it's probably a little easier to calculate the bending stresses for each moment and add them together, especially if this is your first problem dealing with moments about different axes. The bending stresses add directly.
 
julian89 said:
Or [do I] take the resultant vector and calculate stresses with the new moment?
julian89: Only because you have a circular cross section, it is much easier (and therefore better) to compute the resultant moment, as you described here, and then compute stress with the resultant moment.