Stress in a hollow beam bent laterallly

[DMT69]

I would like to calculate the maximum stress in a hollow beam being bent laterally. There has been some discussion in my office about the correct way to view the system in question, and so would like some other opinions on the matter.

I will include pictures below. In simplified terms, the system can be though of as a pipe with small hollow beams welded across the pipe. The system then rests on another solid beam, so that the two hollow beams act as axial stops to prevent axial movement of the pipe.

The problem is that the pipe has a bending moment, which will be transferred in some manner to the axial stop hollow beams. There is concern that they will not be able to handle with load.

Some of the questions that have come up:
1) which moment of inertia to use: should the system be two parallel cantilever beams, add there thickness and then have a longer distance from the center for stress? Or is there some sort of twisting moment of inertia to use instead?

2) how is the pipe's bending moment transferred to the support beams? Does it induce the same moment in the beams? Is it transformed into a point/distributed load on the beams?

3) are both hollow beams affected, or only the one taking the downward load? Does the one on the other side help resist the load at all?

I haven't put in any numbers, since I want formulas. Assume hollow beams have a thickness of t, a square cross section with side length x, and length L. Bending moment from the pipe is M.

Here is the hollow beam and cross section:
https://drive.google.com/file/d/0B7abncbCKejtSmtHVUJoOGE1V1U/view?usp=sharing

Here is a side view. The purple pipe is welded to the blue hollow beams. This also shows how the bending moment is acting:
https://drive.google.com/file/d/0B7abncbCKejtV3ZNSVJybmNGa1E/view?usp=sharing

And here is a top view of the system:
https://drive.google.com/file/d/0B7abncbCKejtV2ItaXpiWDFESDA/view?usp=sharing

 

[SteamKing]

If the pipe is simply resting on the solid support, the only load the support will see is the vertical reaction due to supporting the pipe; i.e., the moment cannot be transferred directly from the pipe to the support.

If the axial stops are not welded to the pipe support, there is no shear connection here, thus no moment can be transferred to the axial stops by applying a bending moment to the pipe, especially if there is a significant gap between the axial stops and the support, to allow for installation, thermal growth, etc. It is possible that a moment might develop as the pipe rotates when a bending moment is applied, and the axial stops rotate with the pipe and come into contact with the supporting beam.

To answer your question about the strength characteristics at the support, I would treat the pipe as a continuous beam resting on however many simple supports as there are pipe supports, using only the area and moment of inertia of the pipe as the structural properties of the beam. Once you have determined the reactions and rotations of the pipe at the supports, then you can analyze the axial stops at each support and see what happens to them. If the axial supports make contact with the pipe support, you may have to apply some additional moments at the supports and repeat your calculations until you have everything in static equilibrium, and all of the internal bending moments in the beam are compatible with the rotations at the supports.

If you are looking for a magic formula which analyzes all of the details of this support, good luck, because I doubt there is one. I think you have to go back to first principles and analyze this construction in a step-by-step manner, as I have described.

 

[DMT69]

Okay thanks.

The axial stops are welded to the pipe. The pipe just rests on the support. The axial stops are flush against the support with no gap.

In this case the pipe can be considered infinitely stiff, and since the axial stops are welded to the pipe and flush against the support, we figure that the axial stop will be pressed into the support as the pipe tries to rotate, creating in effect an bending moment on the axial supports. Exactly how we try to model the stresses caused in the axial stops is where the discussion came up.

 

[SteamKing]

Okay thanks.

The axial stops are welded to the pipe. The pipe just rests on the support. The axial stops are flush against the support with no gap.

In this case the pipe can be considered infinitely stiff, and since the axial stops are welded to the pipe and flush against the support, we figure that the axial stop will be pressed into the support as the pipe tries to rotate, creating in effect an bending moment on the axial supports. Exactly how we try to model the stresses caused in the axial stops is where the discussion came up.

If the pipe is 'infinitely stiff', as you assume, then there can be no rotation of the pipe at the supports. You are making contradictory assumptions about what is happening to the pipe and the axial stops.

 

[DMT69]

What i meant is that we can disregard bending of the pipe itself. Perhaps I phrased it badly. The pipe is trying to rotate, this is prevented mainly by the front axial stop pushing into the support. The relevant question is if the axial stop will collapse or not from stopping the pipes rotation. Stopping the rotation gives rise to bending moment that we can calculate. Exactly how that transfers stress to the axial stop is the question.

 

[pongo38]

I think you might be confusing primary effects (vertical deflections of the blue beams - are they cantilevers?) with secondary effects (lateral bending, if any, of the blue beams). It is usual to ignore secondary effects, especially as construction tolerances can affect your assumptions.I wasn't sure what the grey blob on the drawings was supposed to represent.If it is a vertical support up tight against the purple pipe, then no vertical loads will transfer to the blue beams. If you believe that the blue beams can slide vertically against the grey support, you could use your knowledge of the bending moment diagram in the pipe to calculate the rotation due to M at this support. That rotation can then be applied to the blue beams, one up one down. Maybe that is what you are seeking, or have I completely misunderstood?

 

[DMT69]

The blue beams are welded to the purple pipe. The gray blob in the middle is another beam, much thicker and stronger than the others (so bending in it can be neglected). The purple pipe with the welded blue beams sits (with no attachments or welding) on the gray beam. The spacing is such that the blue beams are flush up against the gray beam. Since the pipe/blue beam construction is just resting on the gray beam, it can move straight up only stopped by gravity. However, rotation should be stopped by the blue beam hitting up against the gray beam.

 

[OldEngr63]

How is the "infinitely stiff" pipe (post 8) going to flex to develop load between the stop and the grey support beam? Infinitely stiff beams (even when they are pipes) do not flex (but, boy are they expensive!).