Fixed End Condition Inside Cylindrical Shell

[MDDSI]

I am having some debate with coworkers on the requirements of a fixed end condition on a beam. In theory, a fixed end should allow for a "weaker" beam for any given load as opposed to a simply supported beam. In particular, I am concerned with how the moment at the ends of a fixed beam may impact the surface the beam is attached to - in this case, a thin shell. I have attached a rough sketch of the situation in question. We are considering welding a beam across the diameter of a cylindrical shell. The beam would be midway (vertically) on the cylinder, rather than placed on the top or bottom ends. To size out the beam to support the load we are looking for, I figure we can either use a simply supported beam, or a fixed beam. To make the beam fixed, I maintain we need to strengthen up the ends with some rings to resist the bending moments as I feel the shell (7 gauge steel) will be inadequate to resit bending.

I am trying to ultimately figure out how large the rings need to be to treat the beam as fixed. From the simplified formula for a fixed-fixed beam, the maximum moment at the ends and center is equal to P*L/8. To my mind, I need to size the rings to resist this moment. Because the beam has a height, I simplify the beam to an applied moment (P*L/8, Roark's table 8.1 case 1d) and assume this needs to be resisted in either tension or compression by the rings, each at a distance of half of the beam height, h/2, from the center line of the beam. I can solve for the force taken up by the rings by solving for moments equal to zero. This gives me a radial ring stress that I can check for stress with Roark's ring equations. The vertical load should be taken up by the shell, so I have not considered it. This method sort of resembled Bednar's examples for sizing reinforcing rings for lug supports, if you are familiar with it.

However, when I apply this method to actual values, I get incredibly large rings. This is making my head explode, and makes me think that I may have a flaw in my logic.

Please let me know your thoughts on my methods, and if I am mistaken, suggest where I may make adjustments.

 

[SAIL3]

it may be difficult to obtain a truly fixed condition @ the ends of the bm because of the deflections in the rings...the deflection should be checked and a decision made if it is ok to assume a fixed condition...one method to reduce the size of the rings is to spread the rings apart and by running a vertical member on the outside of the cylinder to carry the moment forces to the rings....

 

[WARose]

With that set up.....you are likely never to get a fully "fixed" reaction. It will be somewhere in between pinned and fixed. A FEA analysis will give you the force(s). A pressure vessel handbook might too.

 

[rb1957]

you're completely correct to wrong about the next link in the change ... the stiffness of the foundation is an important feature in determining the fixity of the support.

it looks like your beam is a diameter of the ring/cylinder. how does the ring respond to this fixed end moment ?

you're correct that a double cantilever has a smaller moment in it than a SS beam. Partial fixity also helps. but this means you have to design for negative moments. If you want to "optimise" the beam design, then tailor the section to the local internal loads (BM) and tailor the local sections to maximise their allowables (eg crippling). I suspect there isn't much difference in weight between a nearly optimum SS beam or a nearly optimum double cantilever (or even a lazily designed (or optimal manufacturing), constant section, double cantilever).

another day in paradise, or is paradise one day closer ?