Find the principal stresses in a shaft with torque applied

[Kyle Grayston]

Homework Statement

Homework Equations

M = Fd
σ/y = M/I
T/J = τ/r
σalternating (σa) = σmax - σmin / 2
σmean (σm) = σmax + σmin / 2
soderberg: σa/σ'e + σm/σy = 1/FoS

The Attempt at a Solution

I am still unsure whether my progress so far is correct but..
I have calculated J to be 38.35e-9 m⁴ and I to be 19.17e-9 m⁴. I then use these values in the shear stress and normal stress equations to find that:

σ = 82.16 MPa
τ = 41.07 MPa

I apply these values to a loading element diagram (am I right in thinking there is no force in the y direction??) and then used Mohrs circle to find that the maximum and minimum principal stresses are:

σmax = 99.17 MPa
σmin = -17.01 MPa

When applying these values to the σalternating and σmean equations, I am not sure whether it is expressed as:

σa = σmax - σmin / 2
σa = 99.17 - (-17.01) / 2 OR
σa = 99.17 - 17.01 / 2

After I get the correct σa and σm values and I need to find a suitable factor of safety, I'll have σa, σm, σy but I am not sure how I am to find the σ'e, or is there a different method to finding a suitable factor of safety?

 

[Chestermiller]

I'm not sure whether they really intended for you to be considering the bending of the shaft in addition to the torque. What do you think?

 

[Kyle Grayston]

I'm not sure whether they really intended for you to be considering the bending of the shaft in addition to the torque. What do you think?

I see what you're saying, I guess when I saw 'principal stresses' I thought about what we had learned in class and to draw mohrs circle. I am not sure how I would go about finding the maximum and minimum principal stresses without applying a bending stress? I have drawn the following loading element:

Im just unsure as to whether I am on the right track with this..