Shear Flow 15

[Stillerz]

I am sure most here are familiar with the concept of shear flow as it related to horizontal shear stresses in beams.
When designing a I-Shaped plate girder, most references, if not all, will design the weld between the flange and web using the shear formula VQ/I to determine the force on the welds.
My question is, isn't there bending stress on the weld as well in the form of MC/I?
If one had a simply support girder with a uniformly distributed load, the shear at the center of the beam would be = zero and the moment at a maximum. This would imply that no weld would be needed at the center, yet this is the section where having the entire cross-section engaged in bending is most critical.
What am I missing here?

 

[cntw1953]

I think many have getting tired on this, but I couldn't hold on to my DUMB question:

A assembly of two same size rectangular beams subjected to pure bending, one simply stacks on top the other, zig-zag would result at the beams' ends. However, if we glue/weld/nail these two beams together, the zig-zags at the ends would disappear, now the ends would be two slanted straight lines.

My question is: what is the stress in the bonding/binding agent (glue/weld/nails), if the joint is right on the N.A.?
(Ignore friction between the beams)

 

[KootK]

Yes, the idealized model would have to preclude buckling failures as well.

cntw1953: that should just be straight up VQ/It

 

[cntw1953]

Under pure bending (end moments), ignore selfweight, does VQ/It still hold?

 

[KootK]

Sure does. It'll just yield a value of zero for your case is all. You have to remember: if your member is loaded the way that you say it is, you shouldn't have ANY "zig-zag" at the ends. Every differential element of every cross section will wind up exactly where it ought to be via axial strain alone. There will be no need for shearing stresses to keep plane sections plane.

The application of the end moments in manner consitent with these assumptions is a big part of what makes that scenario so hypothetical.

 

[cntw1953]

The explanation on beam ends makes sense. Thanks.

Now, still under pure bending, we all know the stress distribution differs for the 2 beams assembly and the single (smae size) beam, because "I" differs for the two system.

My question now is what makes the latter stronger than the former if there is no stress at the N.A. ?

Am I missing something again?

 

[KootK]

Now THAT is an interesting question. Again, I think that the crux of the matter is in how those end moments are applied. Try this:

To get the most out of your combined beam cross section with respect to flexural capacity, you want a stress distribution where the fibers furthest from the neutral axis are the most heavily stressed. Of course, strain compatibility must still be maintained.

In the case of general loading, this can only be accomplished by providing a horizontal shear connection between the constituent parts of the cross section. It is precisely that shear connection that allows the axial stresses and strains to be transferred to the outskirts of the cross section where they will be most effective.

In your end moment scenario, there is only ONE way to apply those moments without inducing localized horizontal shears at the ends and thus violating the assumptions inherent in the problem. The end moments have to be applied as axial stresses distributions consistent with Mc/I of the composite section. This will yield linearly varying stresses and strains across the depth of the section at the ends.

With the end moments applied in this way, the stresses don't need to be transferred to the outskirts of the cross section because that's where they have originated from in the first place. Consequently, no horizontal shear stresses are required to facilitate the transfer and it does not matter if the constituent parts of cross section are connected.

In summary, under the specific case of equal end moment loading, you get the composite stress distribution for "free". You get it without having to do do the work of providing a mechanism (horizontal shear) for transferring axial stresses to the outskirts of the cross section.

Having the stress distribution of a composite section also gives you the bending strength of a composite section.

 

[cntw1953]

Would there be a possibility that the two systems have identical "I" when subjected to pure bending? The reason is when bending is introduced at the ends, there must be a clamping force couple that introduce strains on the extreme fibers. Due to linear elasic behavior of the beams, that would resulting in identical linear strain and stress distribution for both systems.

Makes sense?

 

[KootK]

Yes, given the moment application stipulations described above, the composite and non-composite systems would have identical "I" values under pure bending. For a system to be composite with respect to moment of inertia, all that is required is that plane sections remain plain across the combined section. The system doesn't care whether this is enforced through horizontal shear or serendipitous loading.

I'm afraid that I don't follow with the clamping force. Can you elaborate?

 

[cntw1953]

In my mind, the bending can be introduced by several different devices, all will result in linear strain and stress in the systems. One is the simplest - hold the beam ends by two fingers and apply bending force, now the top most fiber would be in axial compression, and the lower extreme fiber would be in axial tension. If we can control the forces precisely, through inear elastic behavior, both beam systems would deform identically under the same amount of force (bending moment) applied.

I was trying to depict the action of "hold the beam ends and bend", obviously "clamping force" was a false expression for that. Sorry.

Thank you very much for walk with me on this seemly non-sense excercise. Your comments/insights on this is sincerely appreciated.

 

[miecz]

cntw1953-

The beams that are welded at the neutral axis will have 4x the moment of intertia of the stacked beams, so for the same end moment the welded system will see only 1/4 of the deflection of the stacked system.

 

[miecz]

cntw1953.-

There is a shear across the "weld" at the neutral axis of the composite beam. But the shear is not a result of VQ/It. The shear is a result of the applied moment, which must be applied as a couple.

 

[cntw1953]

Miecz:

I agree it is the case for general loading cases. But for the rare event of pure bending, there maybe more things need to be sorted out before we can confidently claim it follows the pattern of the general case.

Please review the 9 posts above yours, pay close attention to KootenayKids' responses on the question I posted on 22 DEC 09, 12:08.

Have a joyful holiday season.

 

[Stillerz]

This post has taken over the entire forum!!!!!

 

[cntw1953]

Stillerz:

Sorry to have your post hijacked, I think the phenonmenon of the current discussion and the understanding on your case are somehow interrelated. Hope you don't mind. Thanks.

 

[Stillerz]

No need to apologize as i am only kidding.
I think the post is great.

 

[miecz]

cntw1953-

I have read the previous posts. My posts apply to the special case. That's why I mentioned an "end moment." I was so intrigued by your question that I went ahead and built a finite element model of stacked beams, with and without a weld at the interface, with end moments. The shear pattern in the composite beam indicated a shear across the weld, and the non-composite section deflected 4x that of the composite section.

 

[cntw1953]

As I have suggested before, you may want to compile it into a book dedicated to dhengr, so he would never run out supply again

Have fun in the holidays.

 

[BAretired]

miecz,

This has been a long drawn out discussion and maybe I'm missing your intent. Suppose we have a beam of span L with a cantilever of length A at each end. The beams are stacked but the cantilever sections are welded together so they act as a composite unit for a distance A at each end. The load consists of a point load, P at the tip of each cantilever.

Are you saying this beam would behave differently than the same stacked beam in which welding is continued across the entire length, L + 2A? If you are, I disagree.

BA

 

[cntw1953]

miecz:

Isn't that (shear) intriguing? Pure bendind - shear at N.A. ?

Thanks for the update. I would need time to get my head kick-start

 

[miecz]

BA-

That's not what I was saying. The model I had in mind came from cntw's 22 Dec post that had the zigzag pattern at the beam end, and that's not the case for a cantilevered beam. I don't honestly know how a cantilevered beam would behave. I'll see if I can model it up.