# Convert solid shaft to hollow shaft

I have a "bending stress" question, which is causing me "actual stress"...
I have a solid 30m Aluminium shaft, which I need to replace with an equivalent hollow shaft, but I am having difficulty in calculating the internal and external diameters.

The solid shaft diameter is 0.118m, and all I know about the hollow shaft is that the external diameter should be 2.5 times greater than the internal diameter.

I worked out the second moment of area for the solid shaft (9.5*10^-6), but can't seem to find the right way of using this to find the hollow shaft dimensions.

Any help with this would be much appreciated.

CWatters
Homework Helper
Gold Member
Been a long time since I did anything like this so I might be wrong but..

Have you got equations for the bending stress (F) developed in terms of applied bending moment(M) for both hollow and solid?
Presumably the ratio of F/M for both needs to be the same so equate.
See what cancels.
Substitute ID = OD/2.5

Something like that anyway.

rock.freak667
Homework Helper
Sounds like you'd just be inevitably conserving the second moment of area for the shaft such that Isolid = Ihollow and for a shaft I = πD4/4

CWatters
Homework Helper
Gold Member
I found..

For solid
FS = 32M/(pi * Ds3)
and for hollow
FH = 32MOD/(Pi(OD4 - ID4)

where
F = bending stress
M = bending moment
Ds = diameter of solid
OD = Outside Diamater of hollow shaft
ID = Inside Diamater of hollow shaft

If you rearrange these to give the ratio F/M for both and equate it simplifies to :

OD = (OD4 - ID4) / Ds3

Can then substitute for ID = OD / 2.5 to give an equation for the OD of the hollow shaft in terms of the diameter for the solid.

but as I said it's been a long time since I did this. Does it even make sense to preserve the ratio of F/M ?

Edit: Oops I'd better check my working as it looks like the OD of the hollow shaft gets smaller as the solid one gets bigger.

What was the final relationship between solid shaft dia.and hollow shaft outside dia