Question about max stress on circular cross section with two moments

[grotiare]

Homework Statement

This is more of a conceptual understanding. HW problems given do not deal with this specific scenario

Relevant Equations

σmax = (M*c)/I

I couldn't fit in the title, but this is with a hollow circular cross section

So currently I am trying to figure what occurs when two, perpendicular bending moments are applied to a hollow circular cross section (one about the z axis, and the other about y). I know that if I was dealing with a square cross-section, the stress caused by bending moment will be greatest at the corners and you simply add the max bending stress caused by each moment. But, for a circular cross section, I am unsure of the location of when the stresses are greatest. I attached a photo of what I am visualizing, thanks.

 

[onatirec]

For a linear-elastic, prismatic body with symmetric cross-section, the normal stress varies linearly with the distance from the neutral axis and the flexure formula is simply sigma=M*y/I... Just as in the case of a rectangular cross-section, the two resulting sets of normal stresses (from the applied moments) can simply be added together - or subtracted if, for instance, one produces compression and the other tension...

Assuming Mz=My and Iz=Iy, the highest stress would occur where y+z is a maximum... Without thinking too much about it, I would expect this to occur at 45 degrees.

 

[Lnewqban]

Certain amount of those two moments will cancel each other and you will have one single equivalent moment acting on your circular section.

 

[PhanthomJay]

It is best to find the resultant moment and use the radius of the circle as the c value. The angle of the resultant moment can also be calculated using vector analysis.

 

[grotiare]

Thanks guys for all the tips! I figured it out shortly after, but yes like what everyone above said I had to find the resultant moment of both, which can simply be found be treating the two moments as vectors and utilizing magnitude formula sqrt(x^2 +y^2) to get the magnitude, and then plugging that resultant magnitude into the stress from bending moment formula.