# Engineering science- Replacing hollow shaft with solid shaft

## Homework Statement

Hi, thanks for looking! Appreciate it!

Right, the question is

For a HOLLOW shaft determine, max torque induced, pilot second moment of area, max shear stress at outer edge, and max angle of twist induced in the shaft, with a material modulus of rigidity of 110 GN/m2. Given that-
Shaft diameters, 160mm and 140mm, length 650mm, power 350kw at 3500rpm.

I have calculated all of these and got what look to be reasonable answers, but the second part of the question is, if you were to replace the hollow shaft will a solid shaft, what would the diameter of the new shaft be? If it were to perform the same duty of max torque and max shear stress.

I have no formula for working this out??

The only info I have is-
T\J = ζ/r

And

∏Ds4
(over)
32 Ds/2

Hope someone can help, thank you!

## The Attempt at a Solution

For torque problems with hollow circular shafts you are working with the polar moment of inertia given by:

J = (pi/32)(Do^4 - Di^4)

If the shaft is solid, set Di = 0 and proceed.

can i use this to calculate the required diameter of the solid shaft?

Size it by making the shearing stress the same as it was for hollow shafts.

sorry I still don't understand???

Shearing stress in a shaft under torsional at radius r load is given by

stress = Tr/J

J is the polar moment of inertia is given by

J = pi(Do^4 - Di^4)/32

where Do and Di are outer and inner diameters, respectively and T is torque.

For a solid shaft, set Di to zero.

The shear modulus G

G = S/y
where S is stress and y is the shearing strain which is r*theta/L
where r is radius of beam
theta is angle of twist
L is length of beam

Theta, the angle of twist is
theta = T*L/(G*J)

Put all this together so that the solid shaft has the same shearing stress in its outer fibers as the hollow shafts.

Thanks again for replying, so I work out the value for J by setting Di to zero, then I use this in the calculation for stress using a value of radius that makes the stress the same as it was for the hollow shaft, and that value is my solid shaft radius? Just to confirm? Or, can the equation for stress be transposed to get radius. Thanks agen!

Thanks again for replying, so I work out the value for J by setting Di to zero, then I use this in the calculation for stress using a value of radius that makes the stress the same as it was for the hollow shaft, and that value is my solid shaft radius? Just to confirm? Or, can the equation for stress be transposed to get radius. Thanks agen!

You can transpose the equation to solve for radius. Aside from getting students familiar with computing stress in shafts due to torque, this problem will demonstrate that hollow shafts are nearly as strong as solid shafts due to the 4th power in the calculation for J. Hollow shafts have considerably higher natural frequencies than solid ones and this is quite important in the design of spinning shafts.

So I'm thinking it would transpose to r x T/j ?

So I'm thinking it would transpose to r x T/j ?

The maximum stress occurs at the outermost surface where r = Do/2. Use the values for stress for the hollow shafts and determine a new Do from above equation. Do not forget that J is a function of the diameter.