composite beam moment capacity 1

[Lion06]

I have a question about the calculation of the nominal moment capacities for steel beams with composite concrete filled deck.
The tables in the 13th edition manual are based on the idea that the entire steel section will yield and reach Fy regardless of where the PNA exists. The question is this:
Doesn't the deformation required for the entire steel beam to reach Fy almost guarantee that the slab will crush before the entire steel section yields?
The failure strain of concrete is 0.003, right?

 

[NS4U]

When you calc the nominal moment capacity, you assume that the concrete is at the specified compressive strength (say 4000 psi). Which corresponds to an E of about 3605 ksi

Using Hooke's law the strain (sigma/E)=0.001

 

[Lion06]

Well, if the concrete strain is only 0.001 how in the world is the steel developing Fy throughout its entire section?
The strain of concrete at failure is accepted to be 0.003, or am I missing something?

 

[ash060]

The concrete crushes which is fine since you are doing a plastic analysis. The maximum strain in infinte. But it does not matter how much you strain the concrete after you pass the crush point the most stress you can get in the concrete is like 0.85f'c. The steel yields and you get a constant 50 (or whatever) ksi throughout the whole cross-section. If the PNA fails within the steel section than some of the section will yield in compression some will yield in tension. Neglect the concrete that is below the PNA if that is where it happens to be located. Your idealizing the concrete as providing it max strength regardless of strain.

 

[Lion06]

ash060-
I appreciate the response. The specific question I am asking, though, is won't the concrete crush BEFORE the steel section is able to fully yield?
Once the concrete crushes, teh section is no longer a composite section, right?
I can see some (probably most) of the beam yielding toward the bottom, but I am having a hard time seeing the steel near the ENA (well, between the ENA and PNA) reaching yield before the concrete crushes.

 

[271828]

My Salmon & Johnson talks about this. They show a stress diagram with the concrete max strain set at 0.003, the concrete at 0.85f'c and the steel at Fy.

Without really digging into the subject, I guess it always works out that way, or perhaps close enough, since 50/29000=0.00172<<0.003.

 

[NS4U]

StructuralEIT,

Perhaps I am missing you're question. Why do you say that a strain of 0.001 in the concrete means the steel won't yield? Depending on where the PNA falls the steel can most certainly yield.

Judging from you're last post you are referring to yielding at the PNA. In which case for the steel near the PNA to yield, you are correct the strain in the concrete would need to be greater than 0.001.

With that said, you raise an interesting point, I've never really thought about this in terms of strains. This is definitely something you could could run the numbers on and see what the strain in the concrete is when the steel at the PNA yields.

 

[NS4U]

Damn you really got me thinking on this... check this out

http://www.sussex.ac.uk/engineering/documents/sm_lecture_23.pdf

I think page 3 helps explains it. While we assume that a hair above the PNA is compression, and a hair below is tension... that is never the case... but is just simplified for design...

So you're right you will not yield at the PNA before concrete crushes... you just assume you do...

I guess that's a good reason for phi=0.9

 

[PMR06]

You can set up a strain diagram with a height of your slab, and with compression strain 0.003, and tensile 50/29000 = 0.00172 (as e pointed out). "Connect the dots" and there's your PNA with depth "c". Then calculate the maximum compressive force the slab could develop 0.85 * f'c * a * b (a=0.85c). If that force is not greater than AsFy, then there is no way the slab can fully develop the steel beam in tension, and the neutral axis is lower. If the neutral axis is lower, then part of the top flange of the beam will not have strained enough to yield in tension.

 

[Lion06]

star for mikehughes - nice link.

 

[miecz]

StructuralEIT

I think you are correct except for the following:

Once the concrete crushes, the section is no longer a composite section, right?

While the concrete may strain beyond it's compressive strength, I think it can still act as a composite section, at least up to the "assumed strain limit" of 0.003. Beyond that, the concrete loses strength. PCA Notes on ACI 318 has stress strain curves for various strength of conretes. If the concrete strains beyond .003, I think you are right, you lose composite section. That may be why AISC has a recommended limit for the location of the PNA.

 

[271828]

For what it's worth: One thing to note is that the concrete stress block is very rarely maxed out from a stress standpoint. It seems like almost every time I've calced these manually, and I've done a lot of them, the stress block depth, a, is tiny. It's possible to come up with a counterexample, but I think it would be pretty weird--could be wrong, though.