Convert solid shaft to hollow shaft

[spikeybrummy]

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]

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]

Sounds like you'd just be inevitably conserving the second moment of area for the shaft such that I_solid = I_hollow and for a shaft I = πD⁴/4

 

[CWatters]

I found..

For solid
F_S = 32M/(pi * D_s³)
and for hollow
F_H = 32MO_D/(Pi(O_D⁴ - I_D⁴)

where
F = bending stress
M = bending moment
D_s = diameter of solid
O_D = Outside Diamater of hollow shaft
I_D = Inside Diamater of hollow shaft

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

O_D = (O_D⁴ - I_D⁴) / D_s³

Can then substitute for I_D = O_D / 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.

 

[mutuku]

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