composite steel design question 1

[Lion06]

I have a question that I've been unable to find an answer for on my own.
When you do a check to see how much of the slab is in compression, you just AsFy/(0.85F'c*beff). That depth is generally considered to be where the PNA lies.
Can someone tell me why this is different from reinforced concrete, where that would just be the compression block depth, a, and not the neutral axis depth, c?
It seems to me like there shouldn't be any difference, but there obviously is.

 

[Taro]

There is no difference from the usual concrete design method if the NA is within the depth of the slab.

 

[Lion06]

I always learned to check if the PNA is in the slab by seeing if the entire slab in compression is less than the entire beam in tension. If they happen to be equal, then we would say the PNA is at the slab/beam interface.
But we can never count on the entire slab in compression if the PNA is truly at that location, correct?
I guess what I am saying is that we could never have a situation where the entire slab is considered in compression (with the rectangular stress block) AND the entire beam be in tension, right?
If the entire slab could be considered to be in compression (using the rectangular stress block), then the PNA would have to be in the steel, which means the entire beam can not be in tension, right?

 

[WillisV]

If it works out that that a=AsFy/(0.85F'c*beff) is equal to the full thickness of the slab (a=ts) then yes the depth of the compressive stress block is equal to the full thickness of the slab. In terms of traditional concrete design for this case then yes the location of the neutral axis (c) would be within the steel (c = a/Beta) - but that does not affect the results of this problem because it is the PLASTIC distribution at ultimate that we are worried about, not the elastic distribution where c would matter. The neutral axis location is irrelevant here.

 

[Lion06]

I don't think the neutral axis is irrelevant. If the PNA is in the steel (not at the slab/beam interface) then that is less steel that is in tension.
I contend that the only way the entire slab can be in compression (using the rectangular stress block) is if the PNA is in the steel, not at the interface.

 

[WillisV]

PNA is different from the (elastic) neutral axis. I did not say the PNA was in the steel - in this case it is at the slab beam interface (depth of PNA from top = a = ts).

The (elastic) neutral axis for calculating strength would be irrelevant, only the transformed (elastic) neutral axis would matter for deflection calcs.

 

[Lion06]

I am not talking about the elastic neutral axis. You wouldn't use 0.85F'c for an elastic check.
What I am saying is that a=ts that the PNA can not be at the slab/beam interface.
We can only use the 0.85f'c for the rectangular stress block, not the entire portion of the slab in compression. I agree that the entire slab would be in compression, but if you are using the rectangular stress block, you can't use the full neutral axis depth, you have to use a.
If the PNA is at the slab/beam interface, then you can't count on the full slab (a=ts) in compression. If the full slab is in compression (a=ts) then the PNA can't be at teh slab/beam interface. It has to be one way or the other, it can't be the best of both worlds.

 

[Lion06]

The neutral axis depth, c, in RC design is a PNA, not an elastic neutral axis.

 

[WillisV]

Take a look at an example - see Table 3-19 of 13th edition manual, W10x12 for instance (p3-188)

Use a PNA of TFL, which means the PNA is at the interface, and Y1 is 0. So the sumQn for full composite should equal AsFy = 3.54x50=177k - Yup.

Now Y2 is equal to the distance from the top of the steel beam to concrete flange force. And Y2 = tconc - (a/2). So lets say I have a 6" slab, and my beff is such that a is equal to the full slab depth 6" as well (in this odd case, with f'c=3ksi, b would be equal to 11.57") - so I'm using the full slab for compression. So Y2 = 6 - 6/2 = 3.

So my LRFD strength should equal 0.9*AsFy[(d/2)+(a/2)] = 0.9*177*[(9.87/2)+(6/2)] / 12 = 105'-k - which is what is in the Table for the Y2=3 case.

So for this example the PNA is at the interface, and the entire slab depth was used in compression.

 

[frv]

StructuralEIT-

How do you figure? c is the depth used to determine the strain in the extreme tension steel through similar triangles. If this were a plastic neutral axis, you would essentially have the entire tension zone at a strain of .005 (yielding of the steel).

 

[Lion06]

WillisV-
I understand what you are saying and I am not necessarily disagreeing. What I am asking is why. How can you have "a" and "c" be at the same location.
When you use an actual concrete stress-strain curve (Hognestad model for example) the portion of concrete near the neutral axis is strained little and, consequently, stressed little.
The reason we can use a rectangular stress block is because we don't use the entire portion of the concrete that is in compression, but a portion of it (Beta1).
What I am asking is why, if the neutral axis is at the interface, you can use the entire slab for the compression block a. It seems to me you should use Beta1*ts.

frv-
Even for a steel only beam, a true plastic stress distribution is almost impossible to achieve. It would have to rotate almost uncontrollably.
I am not sure PNA means that the entire tension zone needs to get to a strain of 0.005 (this is well above yielding - this is what ACI defines for underreinforce. I am not sure this is relevant to this discussion) as long as all the steel has yielded. Similar to a case of metal deck on a steel beam... does it matter if the "air" gap between the slab beam reaches a strain of 0.005 (or 0.00207 - actual yielding)? Of course not, as long as the steel yields who cares about the rest.

 

[shin25]

StructuralEIT,

You are right, to have the full depth of the concrete to be at .85f'c stress, the NA has to be in the steel beam.

Structural steel (36 ksi) beams usually yields at relatively lower strain than rebars. At about .0012. This means that the steel will start to yield at a very short distance from the NA.

For our design purpose, we consider that the stress strain curve to be flat after yielding. But this will not be the case due to strain hardening and etc. The stress will continue to increase with the increased strain in the atructural steel.

so at the extreme most tension fiber of the steel, the stress is going to be well above the yield stress of the steel. If some one to average out all these stresses in the tension side of the NA, then I think considering the entire steel section yielding will be more conservative.

 

[frv]

Structural EIT-

You are correct. I only mentioned the .005 strain because that is what ACI requires for beams. BTW, for a composite deck on a wide flange, yielding would occur at a strain of .00172. Wide Flanges are 50ksi, not 60.

As far as a "true" plastic stress distribution, according to Salmon & Johnson, "Extensive testing has adequately verified that plastification of the entire cross-section does occur" (4th Ed., p.373).

In any case, I still fail to see why you say that the c value is a PNA. The distribution of stress is assumed to vary linearly with distance from the neutral axis- this is by definition elastic. (now, this is not what actually happens, as the true stress distribution is parabolic in the compression block, but for purposes of analysis it is well within reason). I don't know if you're mixing stress distribution with the Whitney Stress Block.

 

[Lion06]

frv-
The stress distribution in the concrete at failure does not vary linearly with distance from the NA. It does at service loads, not at failure. At failure, it is parabolic, as you note. If it varied linearly, Beta1 would be different and we might not even need whitney's stress block.
The NA depth is also different at service loads than at failure.
That is the difference between PNA and ENA.

 

[frv]

StructuralEIT-

I think we are disagreeng about separate issues. I agree- at service loads, stress is (mostly) linear above the neutral axis. At failure it is not.

What I am disagreeing with is why you say that c is a PNA. In reality, at failure (the point at which c is calculated for purposes of determining strain) it is not plastic any more than it is elastic- it is simply the point of zero strain.

 

[Lion06]

frv-
Agreed that it may not truly be a PNA (it is also not an elastic neutral axis) like in a steel only beam, but I think it is just as much of a PNA as referenced in composite steel design. I am not seeing the difference between the As of the steel beam and the As of reinforcing bars.

 

[Lion06]

shin-
Your explanation makes sense. Is this true even though part of the top flange would actually be in compression?

 

[shin25]

In a well designed structure, the effective compression area of the concrete will be more than enough to equal the yield force in the steel section. In other words, regularly the C will lie in the concrete deck.

 

[Lion06]

I am uploading a sketch to try to get my point across a little better.
My question is why would we treat this differently than a RC beam (for analysis of behavior purposes, not for design purposes)? Couldn't we just consider this a T-beam with a very large As with the centroid of the steel located at the centroid of the WF? If you look at the two stress diagrams I drew, you can see in the one on the right that you would clearly not use the entire 5" depth when trading out the Hognestad model for the Whitney Stress Block. Why can we do this for composite steel beams?
I've shown in red (on each of the stress diagrams) the opposing stress diagram. For the one on the left, if we use Whitney's stress block, why wouldn't the actual depth of compressive stress reach further down in the section?
For the one on the left, the depth of Whitney's block is clearly smaller than the depth using the Hognestad model.
I am just trying to understand.
I can see for the one on the left, that the compressive stress would actually be a uniform Fy below what I've shown as the PNA (what I have shown in red).
I also understand that most composite beams wil have the PNA in the slab, but AISC gives values with the PNA in the beam for a reason.... it happens ocasionally.