Are minimum reinforcements additive? 6

[darkjmf1]

Hello,
I am designing a RC beam with a section which, due to architectonical reasons, is much larger than required, resulting in minimum reinforcement both for bending and torsion.
Do I have to calculate both As,min for bending and longitudinal Asl,min for torsion separately and eventually adding them up in the section?
Or could I just calculate both As and Asl required by analysis and just check if the sum of them complies with As,min and Asl,min?
I am using ACI, but I guess this discussion could be applicable for any concrete code.
Thank you all in advance.

 

[KootK]

As I said in the original post, the problem here is that the beam is oversized due to architectural reasons.

Yeah, I get that. And I still don't buy it. How about you just tell us:

1) the size of this beam?

2) the size of the rebar that you've been considering?

3) the applied torsion acting on the beam?

No need to be coy. Let us tinker with some real numbers.

 

[BAretired]

I'm not familiar with the current code, but I do recall that the Canadian code had a provision similar to the one I mentioned above for architecturally oversized members with regard to flexural reinforcement only...i.e. provide the calculated reinforcement As required to resist the moment or minimum As[sub]min[/sub] as stipulated in the code but in no case more than As(1 + 1/3). I do not recall any such statement regarding combined flexural/torsional reinforcement.

BA

 

[Tomfh]

Kootk,

Any particular area of concrete has a given tensile capacity and can only resist that tension once prior to cracking.

The steel is to ensure the concrete in any given area cracks before the steel begins to yield.

If the minimum steel requirements are genuinely additive this implies the concrete has additive tensile cracking capacity when subjected to flexure and torsion.

I agree from a plain reading of the codes it hard to argue against simply adding the steel, given the references to adding. In my opinion the codes are referring to actual steel requirements due to member forces, not minimum requirements.

Interestingly there is a note in the Australian code which says minimum shear steel and minimum torsion steel need not be added, which is what I am saying about the concrete only needing to crack once.

 

[KootK]

provide the calculated reinforcement As required to resist the moment or minimum Asmin as stipulated in the code but in no case more than As(1 + 1/3).

ACI 318-19 has that and OP could certainly use that to take his flexural reinforcing down to 7 bars.

 

[KootK]

Any particular area of concrete has a given tensile capacity and can only resist that tension once prior to cracking.

Hold the phone... are you telling me that air doesn't actually have tensile capacity?!?

If you review the sketches that I posted, you'll see that I've not proposed that a section cracked in torsion can carry moment vial concrete tensile capacity on that same, torsion crack. Rather, I'm saying that the torsional cracks and the flexural cracks are different cracks. Or, at the least, one cannot guarantee that they are the same crack.

If the minimum steel requirements are genuinely additive this implies the concrete has additive tensile cracking capacity when subjected to flexure and torsion.

I don't agree. Firstly, you have the different crack business that I just mentioned. Secondly you've got this which I posted this morning:

This is really something of a moot point in a way anyhow. Try it another way, for the sake of argument:

1) Let's imagine, for sporting fun, that there is only one kind of crack in play: the combined, flexural-torsion crack (the CFTC).

2) When the CFTC let's go, there will be both torsion and flexure acting across that crack. And both actions must be resisted.

3) If the CFTC is only reinforced for the worst of the minimum torsional requirement or the minimum flexural requirement, there's no way to guarantee that will be enough to satisfy the demand where some level of both torsion and flexure is present on the section.

4) So we're back to having to design for additive requirements if we want to be able to guarantee ductile failure in bending.

Interestingly there is a note in the Australian code which says minimum shear steel and minimum torsion steel need not be added, which is what I am saying about the concrete only needing to crack once.

And in that case, it makes perfect sense because torsion cracks are essentially shear cracks. And that means that torsion cracks and shear cracks are the same cracks unlike the case with torsion and flexural cracks which may be different cracks.

 

[darkjmf1]

Yeah, I get that. And I still don't buy it. How about you just tell us:
1) the size of this beam?
2) the size of the rebar that you've been considering?
3) the applied torsion acting on the beam?
No need to be coy. Let us tinker with some real numbers.

Ok, KootK, I didn't want to assault you all with the details, but as you insist, there you are:

Beam dimensions: 2010 x 1550 mm (width x height)
Concrete: f'c = 25 MPa
Steel: Fy = 400 MPa

Ultimate forces acting on the beam (the complete set, not only torsion), at midspan:

M vertical = +3200 KNm​

M horizontal = +-1630 KNm​

Axial (tension) = 2500 KN​

Torsion = 1100 KNm​

V vertical = 1400 KN​

V horizontal = 400 KN​

Reinforcement schedules (these were my preliminary estimations, previous to all this discussion):
Longitudinal:

Lower face: As,req = 81 cm2 (M+N) + 15 cm2 (T) = 96 cm2 ; As,min = 96 cm2 (M+N) + 27 cm2 (T) = 123 cm2 (-->16#10)​

Upper face: As,req = 17 cm2 (N) + 15 cm2 (T) = 30 cm2 ; As,min = 17 cm2 (N) + 27 cm2 (T) = 44 cm2 (-->6#10)​

Side faces: As,req = 41 cm2 (M+N) + 15 cm2 (T) = 56 cm2 ; As,min = 55 cm2 (N) + 27 cm2 (T) = 82 cm2 (-->11#10)​

(As,min for M+N try to take advantage of clause 9.6.1.3, which basically limits As,min for M to 1.33 x As,req)​

Transverse:

Inner stirrups for vertical shear: As,req = 2 cm2/m; As,min = 16 cm2/m​

External stirrup for torsion: As,req = 10 cm2/m; (Av+2At),min = 17 cm2/m​

Let's play with real numbers!

 

[KootK]

Fun....

Simple span?

Loads mostly uniform or concentrated?

What are the architect's reasons for dictating the size?.

What supports this beam and would absorb it's torsional end reactions? And what are that thing's proportions.

 

[Tomfh]

Rather, I'm saying that the torsional cracks and the flexural cracks are different cracks

They're not. The premise of the original question is torsion and flexure together, so we are talking about flexure and torsion in the same location. It's the same concrete which carries the torsion tension and the flexure tension. Once that concrete is cracked in torsion, it's cracked, and can no longer carry either torsion or flexure. If you crack it in torsion you can no longer keep loading it in flexure (your hypothetical "step B"), because that concrete is now cracked and cannot keep contributing to flexural resistance.

 

[darkjmf1]

Simple span?
Loads mostly uniform or concentrated?
What are the architect's reasons for dictating the size?.
What supports this beam and would absorb it's torsional end reactions? And what are that thing's proportions.

You are so inquisitive!

I have been talking all the time about architectural reasons to make it simple, but there is no meddling architect.
There are two of these beams which support a huge generator, and our client do not want to modify any dimension of the design provided by the supplier of the equipment.
I guess the supplier is overly conservative with the sizing of the structure, probably because he is concerned for the dynamic behaviour (although we have checked that we are really far from critical frequencies as well), but let's not make the story too long...

They are single span (8.5 m) beams supported at both ends in two concrete columns (2010x2500 mm) 4 meters high. The columns start from a foundation slab 2 m thick.
Such big columns provide fixed supports and therefore negative moments.
Loads are mostly uniform and certainly eccentrical as the generator is supported at the inner edges of the beams, hence the torsion.

They're not. The premise of the original question in torsion and flexure together, so we are talking about flexure and torsion in the same location. It's the same concrete which carries the torsion tension and the flexure tension. Once that concrete is cracked in torsion, it's cracked, and can no longer carry either torsion or flexure. If you break it in torsion you can no longer keep loading it in flexure (your hypothetical "step B"), because that concrete is now cracked and cannot keep contributing to flexural resistance.

Than sounds logical to me...

 

[KootK]

Rather, I'm saying that the torsional cracks and the flexural cracks are different cracks

They're not. The premise of the original question is torsion and flexure together, so we are talking about flexure and torsion in the same location.

I contend that they are. See below for an example of a flexural crack and a torsional crack at the same location (location [A])

It's the same concrete which carries the torsion tension and the flexure tension. Once that concrete is cracked in torsion, it's cracked, and can no longer carry either torsion or flexure. If you crack it in torsion you can no longer keep loading it in flexure (your hypothetical "step B"), because that concrete is now cracked and cannot keep contributing to flexural resistance.

What desperately needs to be acknowledged here, I think, is:

1) That a torsional crack needs to make it around the entire section, 360 degrees, before it becomes a flexural crack and;

2) Just how damn long of a crack #1 is as measured along the axis of the member.

Below, I've shown OP's beam, drawn to scale, with a plausible torsional crack. That crack doesn't even make it once around the beam over the span of the beam. Were I to have drawn the crack at 45 degrees, it still would have taken twenty feet for the crack to run full cycle and open up a full section, flexural crack. At location [A], with the torsion crack stitched up with reinforcing, are you really telling us that you're confident that a flexure crack is not possible?

 

[KootK]

@Tomfh: I've been hoping that you'd speak to the critique below which I tabled twice above (perhaps it wasn't clear that I was seeking a response). Can you speak to that now? Basically, if an arbitrary combination of flexure and torsion will be present at first crack (flexral, torsional, or flexural-torsional), then I don't see how one could guarantee that both minimums would be satisfied if one has only designed for one minimum or the other. And this aspect is entirely independent of the whole multiple kinds of cracks business. For example, say that you had the following force set just prior to first cracking:

- 80% of the flexural cracking moment present.

- 80% of the torsional cracking moment present.

When that first crack forms, is it still reasonable to assume that satisfying only one of the minimums would provide sufficient ductility to both actions when this much torsion and flexure are present simultaneously? I don't know how to prove that rigorously. Perhaps you do.

This is really something of a moot point in a way anyhow. Try it another way, for the sake of argument:

1) Let's imagine, for sporting fun, that there is only one kind of crack in play: the combined, flexural-torsion crack (the CFTC).

2) When the CFTC lets go, there will be both torsion and flexure acting across that crack. And both actions must be resisted.

3) If the CFTC is only reinforced for the worst of the minimum torsional requirement or the minimum flexural requirement, there's no way to guarantee that will be enough to satisfy the demand where some level of both torsion and flexure is present on the section.

4) So we're back to having to design for additive requirements if we want to be able to guarantee ductile failure in bending.

 

[KootK]

You are so inquisitive!

Indeed I am. That's not really what I'm doing here however. Questions asked on this forum often proceed like this::

1) OP tells us the bare minimum of what he or she thinks we need to know in order to help.

2) We flail around trying to help with only the information initially provided by the OP.

3) We start to wonder if we might be able to do a better job of helping if we had more context for the problem than was originally provided.

So that was me at step #3. Firstly, I've checked that your torsion makes sense for your loading and that you are indeed above 0.25 Tcr. So we can set that aside now and focus on that which truly matters. The next step will be for us to use what we now know of your situation to see if we can devise an appropriate "cheat" for you that reduces the number of longitudinal bars. Working on it...

No offence but it's not as though I was just so profoundly interested in your beam that I just had to know more. Rather, asking for more was part of a strategic effort to help you more effectively.

 

[Tomfh]

For example, say that you had the following force set just prior to first cracking:

- 80% of the flexural cracking moment present.

- 80% of the torsional cracking moment present.

When that first crack forms, is it still reasonable to assume that satisfying only one of the minimums would provide sufficient ductility to both actions when this much torsion and flexure are present simultaneously?

If you had those forces the tensile face would have already cracked.

You're double counting concrete's tensile resistance. You can't load it to 80%, and then load it another 80%.

 

[JoshPlumSE]

Gosh, I didn't expect to see so much discussion on this topic. I have a couple of things to add that I don't think anyone has explicitly said:
1) The OP later asked why columns don't have a minimum steel for torsion or something to that effect. Not sure that's entirely true (though I haven't specifically looked it up). But, I wanted to point out that column have requirements for closed shear ties. Beams often have open stirrups that don't really help with torsion at all. But, the closed ties definitely help out your columns.... especially when you've got seismic tie spacing where the hoops are there not so much for shear, but for confinement.

2) There are two types of torsion in a beam. a) Torsion that can be relieve due to force redistribution(i.e. a beam that is supported on both ends can impart some torque into it's supports due to the fixed end condition). b) Torsion that cannot be relieved due to moment redistribution (i.e. a cantilever beam imparting a torsion onto a girder that supports it).

Therefore, for case a) once the the beam cracks for torsion (and presumably re-distributes it's torque towards the center of the beam), the beam should be okay for the bending demand. Not so much for case b).... Or, even for a variation of a) where the beam center span reinforcing isn't sufficient to handle the re-distribution.

My point with this 2nd item is that if you've designed your beam to be capable of handling re-distribution, then I don't believe you need these minimums to be additive. The structure is still stable under full bending and full torque. It's just that the torque gets re-distributed.

 

[KootK]

If you had those forces the tensile face would have already cracked. You're double counting concrete's tensile resistance. You can't load it to 80%, and then load it another 80%.

I don't agree. The sketch below shows how I believe that you could arrive at a [T + M] combination that would produce a minimum reinforcing demand in excess of either the flexural min or the torsional min alone. I don't want to get hung up on this, however, given that I consider it a moot point because I'm confident that the torsion and flexural cracks are actually different cracks. My previous post that included the second sketch below contained the bulk of what I consider to be my salient arguments for that theory. Do you not find any of that convincing? The fact that a full section torsional crack runs the full length of the beam here?

 

[KootK]

My point with this 2nd item is that if you've designed your beam to be capable of handling re-distribution, then I don't believe you need these minimums to be additive.

I disagree with that conclusion because, frankly, I'm baffled by this statement:

Therefore, for case a) once the the beam cracks for torsion (and presumably re-distributes it's torque towards the center of the beam), the beam should be okay for the bending demand

Unless I'm grossly misinformed on this topic, the torque does not redistribute along the span of the beam from the ends towards the center. Rather, it redistributes out in to the the adjacent structure (slab, perpendicular beams) as flexure in those elements.

The overwhelming majority of the time, retribution torsion will result in:

1) Short segments at the ends of a beam that yield in torsion and, effectively, become torsionally free ends from that point forward and;

2) Beam segments in the middle of the beam that aren't torsionally cracked at all.

Because of this, the best argument for the minimums not being additive is where you have a truly simple span beam such that your peak moment and peak torsions do not coincide. I don't believe this applies to OP's case, however, owing to one of his earlier statements, repeated below. It's at the supports where OP needs to be worried about the combined minimums.

Such big columns provide fixed supports and therefore negative moments.

 

[Tomfh]

Kootk,

No, I don't find your analysis convincing because it appears to rely on double counting of the tensile capacity of the concrete. You're counting the tensile capacity of the concrete once towards flexural crack resistance, and once again towards torsional crack resistance. You are treating the flexural crack resistance and torsional crack resistance as distinct things which can both be loaded to 100% without influencing the other, when in reality they both depend entirely on the same concrete in tension.

Your diagram for example shows a torsion crack on the flexural compression face, but no torsion crack on the tensile flexural face. I'm not denying a flexural crack can't happen, I'm saying it WOULD HAVE ALREADY HAPPENED given that the flexural tensile face is the critical face in terms of combined flexural and torsional tension.

 

[Settingsun]

80% cracking flexural moment + 80% cracking torsional moment = 129% of concrete tensile strength if I've done Mohr's circle correctly. 62% of each is equal to the cracking stress. From the 4th post, that gives 0.62*10 bars for flexure and 0.62*11 for torsion = 13 bars as an estimate of minimum consistent with the requirements for pure flexure/torsion.

Other matters to bear in mind are:

- The torsion minimum also fulfils a crack control role. For combined flexure and torsion, some extra reinforcement beyond the torsion minimum would be needed to prevent wider cracks from the additional flexural stresses.

- The code writers aren't super-conservative with these minimum reinforcement quantities. The flexural minimum is based on a low estimate of concrete tensile strength*, with a pretty rough assumption that the difference between this and the actual tensile strength is equal to the tensile stresses due to restraint. Similarly, for torsion, a low-ish estimate of the cracking moment is the basis for the minimum reinforcement. It depends on your personality whether you take this to mean you don't need to worry too much about the exact minimum reinforcement quantity, or whether 13 bars is worryingly low.

* - may not be the case for ACI. I've found my older metric version (so can now understand the numbers...) and the minimum looks a bit more than Australia.

 

[JoshPlumSE]

KootK -

Not sure that you understood my argument. I'm not re-distributing torsion. I'm re-distributing beam end moments that were the cause of torsion in their supporting girder.

The end moment of a fixed end beam framing into a girder causes a torque in the girder, right? When the cracking occurs (flex cracking in beam, or torsional cracking in girder) that fixed end moment gets re-distributed towards the center of the beam. Therefore, the girder no longer experiences the torque.... because the beams end moment has been relieved / redistributed away from the girder.

 

[Celt83]

Here is my crude attempt using JoshPlum's beam to girder connection.
The top steel in the beam (green) and the bottom compression block introduce a point torque on the girder which is resisted by a field of compression struts following the pattern in KootK's sketch. Each of these struts must be balanced out by a tie or two orthogonal ties (closed stirrups + longitudinal bar). These compression struts and ties occur on all faces of the beam. The flexural tension created by the vertical reaction of the beam is resisted solely by the girder main bottom longitudinal steel and the As,min associated with that to govern ductility.

Edit:
Another side of this is ACI doesn't put an upper bound on the strain in the steel, so the additive requirement of the minimums may have the hidden effect of keeping the steel strains below the rupture value.

Open Source Structural Applications:

https://github.com/buddyd16/Structural-Engineering