Calculating Stresses and Principal Stresses on a Shaft: Homework Solution

[mikex24]

Homework Statement

Shaft of diameter 0.3m, has to withstand an estimated axial thrust (compressive load) of 500KN and a torque of 250KNm.
i) Find the magnitudes of the direct and shear stresses at a point on surface.
ii) Determine the principal stresses and the maximum shear stress at that point, and the angle that maximum principal stress makes with the shaft axis(using standard formula for stress transformation).

Homework Equations

The Attempt at a Solution

I don't know if the way i try to solve it was the right, so i am waiting to tell me if i am wrong to something.

For the i)
Shear stress:
I use the tosrion theory: τ=(TR)/J= (250000*0.3)/((π*0,15^4)/2)= 94MN/m^2

Direct stress:

σ (direct)= LOAD/AREA= -500000/(π*0,1^2)= -15,91MN/m^2

For the ii)

I use the σ (1,2) equation and i found the principal stresses.

Is that the right way or i am doing something wrong. There are also another 3 questions for this including tresca yield critirion and von mises tield critirion, and i will try to solve them. If anyone can help me i will appreciate it. Thank you

 

[Mapes]

I haven't checked your numbers, but your approach looks pretty good. (Check your torsion radius, though.)

 

[mikex24]

I haven't checked your numbers, but your approach looks pretty good. (Check your torsion radius, though.)

Thank you mapes for the reply. I am trying to find the angle of maximum principal stress with the shaft axis but i can't. Can you help me on this?

 

[Mapes]

What have you tried so far, using either stress transformation equations or Mohr's circle?

 

[mikex24]

What have you tried so far, using either stress transformation equations or Mohr's circle?

I have to find those information to construct the Mohr's circle repressenting the stresses acting at the point and also to verify my results obtained above. I think that i have to use other way than the mohrs cicle to find them. So i try τ= 1/2 * (σ1-σ2) to find the max shear stress. I also find an equation to find the angle which is tanθ= (σp-σx)/ τxy but i am not sure.

 

[Mapes]

Check these equations. The first is for principal stresses, but you're not in a principal stress state. The second looks off.

 

[mikex24]

Check these equations. The first is for principal stresses, but you're not in a principal stress state. The second looks off.

Is there other equation for the angle without using the mohr's circle to find the angle? Thanks

 

[Mapes]

I don't have a mechanics of materials textbook handy, but they generally cover all the stress transformation equations.