Direct and shear stresses acting on the outer surface of a hollow shaft

[janine]

Homework Statement

High strength steel hollow shaft which can withstand complex stresses generated by a combo of loading conditions. Tresca yield criterion is to be used as the basis of design with a factor of safety of 1.5. Shaft is of 50mm and 40mm outer and inner diameters and is subjected to the following loading condition:
Torque: 2.5kNm about axis
An axial tensile load of 50kN offset vertically by e (15mm) from axis.
A bending moment of 3.6kNm acting on vertical plane containing cylinder axis and eccentric load line of action.
1) Determine the sizes of direct and shear stresses acting at point A on the outer surface of the shaft (right through the middle)and mark them on the element.

Homework Equations

for a hollow shaft I found these equations: J = pi (D^4 - d^4) / 32
tau max = Torque*radius / J, I = pi(D^4-d^4)/64

The Attempt at a Solution

I couldn't even get started because I got myself really confused as I was taught and examples I was shown in my lectures only focused on either stress transformation or yield criteria not both at the same time and especially not for hollow objects! I would really appreciate some help here as I've spent days on this and got no where.

 

[Mapes]

Hi janine, welcome to PF. The best approach is to decouple the various parts of this problem. First off, you've got several loads that will superpose (i.e., you can add their contributions). For each, you should find the resulting stress state (normal and shear stresses) at point A. Do this one at a time, with a mechanics of materials book handy. Once you know the stress state at A, you can analyze it via the Tresca criterion.

It looks like you've got the torque load figured out. It will be helpful to decouple the eccentric axial load into an on-axis load and a bending moment. This should be described in any mechanics of materials book, along with ways of calculating the resulting stresses at point A. Does this make sense?

 

[janine]

hey mapes, thanks for writing back, and sorry I've took so long to reply. I understand what you're saying and I bought a mechanics of materials book and I've had a look through and I've found somethings that you've pointed out. So I've got to separate the the eccentric axial loading into a bending moment and an on-axis load, then calculate the stresses on these separately and then add them together to get the resultant stresses?

 

[Mapes]

Sounds good!