Stepped Cantilever Deflection

paddy-boy66

Can anybody tell me how to work out the deflection of a cantilever beam with a stepped diameter? I have looked through Roark's Formulas but can't seem to be able to do it

The diameter of the beam would vary from 5mm to 3mm and back up to 5mm.
I have attached a pic to explain.
Thanks for your help
Paddy

Attached Thumbnails

 

Q_Goest

Hi Paddy. If you're just trying to determine deflection, you can simply divide the sections of the beam up into pieces that are identical or similar in cross section. Then do a separate analysis on each piece. Where one piece ends, you will use that deflection and angle on the adjoining section.

For example, the 3 mm diam shown has some shear and moment on the left end. You can draw a freebody diagram for just that part of the beam and use the equations for deflection on just that part. On the right end of this 3 mm diam beam, you can consider it fixed and find the resultant force and moment needed for equilibrium. Note that the forces and moments on the right end should match the forces and moments calculated for the entire beam. With the FBD in hand, you can then apply equations for deflection and angle.

Do this for all three sections. Then use the deflections and angles to geometrically determine the total amount of deflection.

Note that there will be a stress concentration at each change in diameter which will be ignored.

mwwoodm

I agree w/ Q_Goest, however I have a follow-up question:

Do the stress concentrations at the steps affect the overall deflection of the shaft? For instance, if we step up the diameter of a cantilevered shaft (such as that shown by paddy-boy66), does the shaft deflection reflect the full stiffness of the thickened section or do we lose some effective stiffness due to the stress concentations at the fillets? I suspect the latter is true, but stress concentration theory tends to relate to fatigue (maximum stress) and not to defomation (stress profile) so it's hard to infer the effect on curvature.

Q_Goest

Hi mwwoodm, I don't know of any equations that correlate deflection and stress concentration factor. Generally, the affect is small and neglected.

Right at the point where the shaft gets larger, that larger section has little stress on the outside edge at the corner. The nominal stress rises quickly as you move away from the step, but exactly how quickly is not something I've seen any analytical treatment of. Nevertheless, this points out that the deflection, which is a function of the stress along the entire beam, is going to vary some very small amount from the case I've indicated above, because of the steps in the shaft.

DePurpereWolf

The complex yet physical correct way to do it is by setting up a system of differential equations in which the boundaray conditions line up.
The cantilever can be divided in three sections.
from x0 to x1 to x3 to x4.
The first sections is an ordinary differential equation for a cantilever in which x0 is set to zero, the second is one as well, but the boundary condition for x1 is set to x1 of the first differential equation. In this way a system of differential equations can be derived.

You probably can find more help on the physics section of this forum.
About diff eq: http://eqworld.ipmnet.ru/ this is a good reference.

rocksy

hey do u kno about moment area method. try that one