How Do You Calculate Deflection and Shear Stress in a Tapered Shaft?

[optzero]

TL;DR

Tapered shaft equations for max shear stress and deflection

So i need a little bit of help with a problem.

I have a tapered shaft connected to a gearbox and i need to caluclate the deflection of the shaft and the max shear stress of the shaft.

the total length of the shaft is 500mm the diameter is 35mm and tapers to 28mm with the taper length of 90mm (so a standard 35mm taper). the material I am using is 304 stainless steel. as for the torque the torque on the shaft is 248 Nm and its spinning at 200rpm. for further context its a propeller shaft.
as for the deflection equation I am thinking of using the
d(theta) =(T/(J(X)*G)) dx
J is the polar moment of inertia and G shear modulus T is torque.
the above equation will give me the deflection in radians; is there an equation that will give me this deflection in mm?
what equation will i need to calculate the max shear stress the shaft can handle before failure?

any help would be great, and thanks in advance

 

[jrmichler]

Is this homework? If so, we can move it to the homework forum.

Is the shaft straight, with a taper at the very end for mounting a propeller? If so, analyze it as a straight shaft because the taper has minimum effect on the deflection.

Instead of looking for an equation with the right units, you need to:
1) Better define what you are trying to find out,
then
2) Find the appropriate equation.

There is a big difference between torsional deflection and bending deflection. You need to be clear on that in order to solve this problem.

 

[FEAnalyst]

Right, you will have to clarify whether you want to solve a bending or torsion problem (or maybe both). It all sounds like torsion but then you say that you are looking for deflection in mm which doesn't make sense in the case of torsion (there you only calculate the angle of twist). Torsion of a tapered shaft is a quite common topic in mechanics of materials literature so it won't be a problem. Bending of variable section beams is also discussed in various books so you should be able to calculate whatever you need rather easily.