Computing stress in hollow shaft

[togo]

Homework Statement

An alloy steel shaft has an outside diameter of 100 mm. A central hole of 60 mm diameter is bored in part of its length as shown. Compute the shearing stress in the hollow section if the stress in the solid section is 200 MPa.

Homework Equations

angle = TL/GJ

t = torque
L = length of shaft
G = Shearing strain, megapascals
J = [piD^4/32] = 987477 - 1272345 = -284868 ? (this doesn't seem right)

The Attempt at a Solution

Above, and looking for some general direction on that attempt. My guess is to generate two formulas for the shaft out of the above formula, with one side of the formula including the unknown "G" shearing strain which is what we're looking for?

Or should the formula Tmax = Tc/J be used? (T= torque, c = radius, J = piD^4/32)

thanks

 

[PhanthomJay]

You are looking for shear stress, not shear strain, so it is not necessary to know G. You can use Tc/J to calculate shear stress, but you first need to calculate T for the solid part using the J of a solid shaft you must calculate, then calculate the shear stress for the hollow part using the appropriate calculated value of J for a hollow cross section.

 

[togo]

thats strange because my instructor used the above formula to solve the problem.

He used

T1L1/G1J1 = T2L2/G2J2

T1J2/J1 - T2 = 0

 

[PhanthomJay]

thats strange because my instructor used the above formula to solve the problem.

He used

T1L1/G1J1 = T2L2/G2J2

T1J2/J1 - T2 = 0

Very strange. This equation states that T1 and T2 are not equal. But a free body diagram cut through any section will show that T1 = T2 = T

 

[togo]

Ok. I just don't know where to go with it. This is what I have so far:

Two J values:
J solid = 9817477 mm^4
J hollow = 8545132 mm^4

200 MPa = T (50 mm) / 9817477 = T (20 mm) / 8545132

are those numbers right? Something tells me this should be algebraically solved and then have numbers pumped into it. Thanks

(T should cancel out right?)

 

[PhanthomJay]

Ok. I just don't know where to go with it. This is what I have so far:

Two J values:
J solid = 9817477 mm^4
J hollow = 8545132 mm^4

200 MPa = T (50 mm) / 9817477 = T (20 mm) / 8545132

are those numbers right? Something tells me this should be algebraically solved and then have numbers pumped into it. Thanks

(T should cancel out right?)

I haven't checked your math for the J values, but beyond that, you have a couple of errors. The 'c' value for the hollow shaft is not 20 mm. The value of 'c' is the distance from the centroid of the section to the outermost fibers. And the max stresses in each section are not equal. You should solve for T in the solid shaft, then solve for max stress in the hollow shaft using that value of T.