Composite Floor Beam Behaviour - Consequences of Uniformly Distributed Studs 1

I've seen many projects where there were more shear studs on the outer ends than in the middle.

AISC addresses shear stud spacing in Section 16.1 I3.2(6) if I've read their designation correctly. But it seems to be a bit ambiguously worded in any case.

I have gone through your attachment yet but I'll try to soon.

Interesting question. I don't know the answer. But, I wanted to point out 2 problems with the idealized resisting moment diagram in the attached sketch (assume plastic/strength moment capacity).
1. At the ends of the beams, the resisting moment is not zero. It is the moment capacity of the non-composite steel beam.
2. I don't think the resisting moment capacity curve is not linear. As the number of shear studs decreases from being fully composite, the plastic neutral axis moves down. Some of the steel beam is in compression, the centroid of the steel in tension moves down, and the centroid of the compression force in the concrete moves up.

Thanks for the input guys.

@ Archie: I see it in section 16.1 I3.2d, both in the body and commentary.

@ Wannabe: In response to your comments:

1. Assuming that we're talking about the diagram at the top of the page, my intent was to graph applied moment (parabola) versus available, composite action, moment capacity (straight-ish line). I agree that the floor value for moment capacity anywhere along the beam is Mp for the bare steel section.

2. I agree -- excellent point. The line/curve representing composite action moment capacity would be curved and would more closely match the applied moment diagram. That being said, I've assumed that composite action moment capacity will grow more slowly than applied moment as one moves from the ends of the beam inwards. The accuracy of this assumption may depend on the relative proportions of the steel & concrete sections I suppose. This brings up another issue for me: is it even appropriate to discuss plastic capacity at any location other than midspan? Multiple locations of plasticity would obviously create a mechanism.

@ Everyone: a colleague pointed out that in the diagram at the bottom right, the non-composite moment would technically be shared between the steel section and the concrete slab, non-compositely, and based on relative stiffness. I agree with that of course.




The greatest trick that bond stress ever pulled was convincing the world it didn't exist.

Per my Steel textbooks it appears that early on AISC developed composite design based on ultimate strength concepts rather than elastic (ASD) concepts.

As a result, since they weren't working in the elastic range of the studs, the resulting slippage due to concrete and stud deformation results in a redistribution of the shears along the length from max. moment point to the end of the beam.

From a steel book: "observations of experiments simulating both static and dynamic loadings indicate that the shear connectors will not fail if the average load per connector is kepth below a "limiting load" defined as that load which causes 0.003 in. slip."

and:

:When ultimate flexural capacity of the composite section is the basis for design, the connectors must be adequate to satisfy equilibrium of the concrete slab between the points of maximum and zero moment.....the magnitude of slip will not reduce the ultimate moment provided that (1) the equilibrium condition is satisfied, and (2) the magnitude of slip is no greater than the lowest value of slip at which an individual connector might fail."

(From Salmon & Johnson - Steel Structures, Design and Behavior)

KootK,
It ends up that I have no idea what you are trying to show in the attachment. What am I missing?

If elastic design is used to analyze the beam, the shear flow demand is VQ/I. Since Q and I are constant, the shape of the diagram would be the same as the normal shear diagram. The capacity for uniformly spaced connectors is constant and equal to the connector capacity/spacing. The units for both are force/length (e.g. kips/in).

Yeah, it's a tricky thing to try to explain to someone who's not in the room with me. Heck, it's been tricky to explain to the guy on the other side of my cube wall. I'll try to clarify.

I find it odd that, in a composite beam, the distribution of horizontal shear resistance (studs) does not need to reflect the distribution of elastic horizontal shear demand (VQ/It business). I get that we design for the plastic state and I realize that this stuff has been tested and works. However, I've never been able to construct a rational mechanical model for how it works. And I'd like to satisfy my own intellectual curiosity with this if nothing else. Besides, almost every junior engineer that I've ever worked with has eventually asked me about this. I'd like to have an answer.

What I've tried to show in my sketch is the best rational mechanical model that I can think of. The important features of which are:

1) Once you specify the distribution of horizontal shear transfer, you lock yourself into a maximum rate of moment capacity growth. This, at least, is true for the portion of the moment resistance attributable to the section acting compositely. This is the diagram at the top of the page.

2) According to my analysis there will be a "remainder" portion of the applied moment which cannot be resisted by composite flexure and needs to be resisted by some other mechanism. This is represented by the area labelled as "B" at the bottom of the page. I postulate that the "B" portion of the moment diagram must be resisted by non-composite flexure in the steel beam.

Know that I'm not presenting anything as being "correct" here. I'm not at all confident in what I've proposed which is why I've tabled my ideas here for discussion. I'd be happy to crumple my sketch up and eat it if someone can replace it with another, better one. The diagram at the bottom right of the page seems to imply some kind of phantom interior support. It's messed up. Yet, analytically, that's what I come up with.



The greatest trick that bond stress ever pulled was convincing the world it didn't exist.

Once the slippage happens (and it is not large in terms of distance) all your diagrams go out the window and the shear force is balanced across the half-span.

Think of the concrete/studs as a bunch of springs. The "mushiness" (a technical term) spreads the shear load out more or less evenly.

I hesitantly disagree JAE. Plane sections remaining plane goes out the window. Elastic stress distributions and VQ/IT go out the window. However, all of my sketches are based purely on equilibrium which never goes out of style.

I've attached an amended version of my sketch incorporating wannabe's comments.

There is a bit more to this story that I'll share now. Originally, I thought that it would just muddy the waters needlessly. However, the waters are clearly pretty opaque as it is so what the hell. Attempt this mental experiment:

1) Isolate half of the steel beam, sans slab, as a free body diagram.

2) along with the other load effects, apply the horizontal shear at the interface between beam and slab as a uniformly applied shear load.

3) Pick a few sections along the beam segment and try to draw vertical shear stress diagrams that are consistent with the fact that tau_x = tau_y and tau_x is a constant, non-zero value at the top of the beam representing the uniform action of the shear studs.

I'd post a sketch of said FBD myself if I were able to produce one. I can't figure it out. Among other things, at midspan, I get a situation where vertical applied shear is zero but horizontal shear stress at the interface is not, implying that horizontal shear stress must change sign somewhere on the steel section. Physically, I don't even know that means. This, fundamentally, is where things go off the rails for me.

The greatest trick that bond stress ever pulled was convincing the world it didn't exist.

I agree with JAE, and the philosophy is explained in the AISC Specification Commentary, at least it was in the editions I used in the 1970's. There are exceptions where closer spacings are required due to different loading, e.g. concentrated loads.

I'm home for the day so I'm not able to look in my steel manual, but I agree with JAE. The statics of the problem still works out, but you have some slippage. In my mind, the only real consequence of this is a reduce Ieff. I believe the commentary provides for a reduced I of like 0.85Ieff. The moment capacity is unchanged.

I analogize this to a bolted moment connection with an idealized setup of slightly oversized holes (top and bottom) that allow some initial slippage of the plates relative to the beam flanges and magically all go into bearing at precisely the same moment. The connection still has all the capacity it was designed for, but has some additional rotation.

I need to do a better job of asking this question. I think that the code clauses and complex mechanics surrounding stud behaviour is obfuscating the point that I'm trying to focus on. Please stick with me on this. I think that there is still a very interesting nugget of engi-truth to be gleaned here.

I'd like to temporarily shift the focus of the discussion to the flat truss shown in the attached sketch. It's a special truss that I consider to be almost completely analogous to the composite beams that we've been discussing, at least with respect to the point that I'm trying hone in on.

THINGS TO KNOW ABOUT THE SPECIAL TRUSS:

1) The top chord, bottom chord, and webs are intended to be analogous to the concrete slab, steel beam, and shear studs respectively.

2) The chords and vertical webs have been designed to preclude failure. The only failure mode available is ductile yielding of the diagonal webs.

3) The diagonal webs have been designed to yield in a ductile fashion at a shear force equal to 50% of the shear demand at the supports. This is what is "special" about this truss. It will produce a horizontal shear capacity that is uniform and capable of redistribution, just like the composite beams that we've been discussing.

4) The truss will possess a plastic moment capacity at midspan that is equal to the maximum applied moment, just like the composite beams.

5) At any section cut along the truss, other than at mid-span, the available truss moment capacity will NOT be sufficient to resist the applied moment at that location. The "remainder" moment needed to enforce equilibrium would have to come from some secondary load path. For the truss, this secondary load path would clearly be non-composite (non-truss action) flexure in the top and bottom chords. For the composite beams, I've proposed that this secondary load path is non-composite flexure in the steel section acting on it's own.

6) If the special truss had truly pinned joints where the chords were not continuous through the panel points, it would fail.

CONCLUSION (OR AVENUE TO SPIRITED DEBATE):

When we supply a horizontal shear capacity in a composite member that is not in synch with the vertical shear demand, equilibrium cannot be satisfied via composite action flexure alone. A secondary flexural load path must exist. For composite beams, this secondary flexural path is the bare steel section acting non-compositely.

The greatest trick that bond stress ever pulled was convincing the world it didn't exist.

I think KootK has a point. Still mulling it over.

One difference between the truss and the composite beam is that the composite beam is capable of carrying the full vertical shear on the web whereas the truss cannot.

BA

Good for brain exercise, but I will rest mine. Design methods usually simplify actual structural behaviour, and I am happy about that.

@ BA: welcome to the thread. I was waiting for you to make an appearance. I agree about the differences in vertical shear capacity.

@ Hokie: Thanks for your contributions. It is very much an extraneous brain teaser.

The greatest trick that bond stress ever pulled was convincing the world it didn't exist.

I would exactly second hokie66's opinion. I know that reality rules and the stresses within these systems do reach equilibrium (i.e. composite beams do work) and that levels of safety are generally ensured by using the analysis/design methods in AISC. I will rest my brain as well. I don't have a problem with your exercise KootK - have fun!

A number of tests carried out at Fritz Engineering Laboratory at Lehigh University in Bethlehem, Pennsylvania seem to confirm that composite beams can attain their ultimate strength using a uniform spacing of shear studs regardless of the shape of the moment curve. For me, that is not an intuitive finding but it appears to be valid based on a few load tests which I found in a Google search. The explanation for this behavior is not clear.



BA

BA's earlier post got me thinking that it would be nice if I had an analogy where the horizontal shear capacity was limited but the vertical shear capacity was not. Also, as Archie mentioned at the top, it would be great if the analogy could accommodate a shifting neutral axis.

I've found such an analogy. See the two pages attached to this post. The analogy is that of a rectangular steel beam with a longitudinal "throat" down the centre that limits the horizontal shear capacity to 6 kip/in. The best thing about this analogy is that the resulting model is simple enough for me to put some numbers to it, which I've done graphically using MathCAD on the second page.

Observations for this -- and only this - example:

1) There is a remainder moment.
2) The absolute magnitude of the remainder moment is very small.
3) At the extreme end, the ratio of composite moment capacity to applied moment is 1.50.

I was desperately hoping to be able to draw some vertical / horizontal shear stress diagrams as well. Unfortunately, it turns out that I don't really know how to draw such diagrams for partially plastified sections. Perhaps that can be a subject of a future post.



The greatest trick that bond stress ever pulled was convincing the world it didn't exist.

Revisions:

-It was WannabeSE who mentioned the shifting NA.
-Applied Moment / composite capacity = 1.50.

The greatest trick that bond stress ever pulled was convincing the world it didn't exist.