# Stresses due to combined loadings in a bent shaft supported at both ends

I was hoping someone might be able to help clarify where I might be going wrong with this question. I am being asked to essentially determine the maximum normal and shear stresses at the rigid support C due to load $$F$$ at the static weights.

I considered there would be a torsional shear stress induced by the torque, calculated by $$\tau_{xy} = \frac{Tr}{J}$$ where $$T = Fb$$, $$r$$ is the radius of the shaft (the point of maximum torsional shear), and $$J$$ is the polar moment of area. There would also be vertical shear, maximised at the vertical centre of the beam, calculated using: $$\tau = \frac{FQ}{Id}$$ where $$Q$$ is the first moment of area for a semicircle, and $$d$$ is the diameter.

Where I'm having the most trouble is the bending moment at C. I chose to model the beam as a straight beam (which it clearly is not), supported (cantilever) at both ends with a load in the middle, giving a maximum bending moment of $$M_{C} = \frac{FL}{8}$$ from a table in my text, where $$L = 2ac$$. I am almost certain this is incorrect, so I was hoping someone might be able to provide some insight into where I am going wrong. I am assuming that the points that will experience the greatest stresses are most likely the top and bottom edges of the beam, where both bending and torsional stresses are maximised (and vertical shear stresses are zero). I know once I have the correct stress components I can use Mohr's circle to determine the principal stresses and maximum shear stresses that I would use in further analysis.

Full disclosure, this is a homework question, but it is a multiple choice quiz that allows multiple attempts and tells you the number correct, so I already have the "answer" to the question (which relates to material choices after determining the maximum stresses), so this is really only for my own understanding. Appreciate any help :)

The reactions at support "C" are as shown below. From here, you can calculate the normal stresses (due to bending) and shear stress (due to shear force and torsion).

• Thanks for the response, I had assumed the joint at A was rigid, but it does seem to be separate. Aug 29 '21 at 1:14
• As long as the elements are in touch, how they are connected will not affect the result unless joint A is a support joint.
– r13
Aug 29 '21 at 1:22

Why don't you model the beam as a cantilever which it is? Or maybe I am confused and will try to modify my answer if you clarify in a comment.

In that case, the bending moment at C will simply be:

$$M_c = \frac{F}{2}C$$

• Cheers, the formula I found was for a beam cantilevered at both ends, but I didn't make that clear :) Aug 28 '21 at 14:53
• @scrams, but in this case your beam is not cantilevered at both ends. it is 2 separate beams each cantilevered at one end. and the L for the sake of bending moment at C is just C. Aug 28 '21 at 15:15