Understanding Development Length 1

[ChickenBake]

I'm stuck on a thought experiment regarding development length and I'm wondering if I can get some guidance.

Consider a simple column footing. Assume that the moment at the critical section, Mu, is exactly equal to φMn. Also, assume that the footing section is tension-controlled.

It follows from the assumptions that the strain at the critical section is far beyond the yield strain. Because the moment tapers down to 0 at the tip of the footing, and the yield moment must be some value between Mu and 0, there is some section, between the tip and the critical section, where the steel has yielded.

ACI requires the force in steel reinforcing to be developed on both sides of the plane in all cases. Yet anytime I see calcs for a footing, development is only checked at the critical section. Shouldn't it actually be checked at the point the bar yields? Am I wrong in my assumptions somewhere?

I have attached brief sketch of this below.

 

[phamENG]

Mn is based on yielding of the bars, is it not? So at phi*Mn, the bar hasn't yielded.

 

[ChickenBake]

That’s only true if the section is designed for the balanced condition, or it’s compression controlled. In this example the footing is tension controlled, which means the strain at PhiMn is greater than the yield strain plus 0.002.

So no, the steel will absolutely yield prior to the factored strength being reached.

 

[Greenalleycat]

In our nomenclature, Mn is the point at which Ecu = 0.003 so as per OP the steel is way past yield
Maybe that varies around the world though

OP - in principle, yes you're right
It is possible to have a critical 'not critical' section that relates to underdeveloped bars
There was a famous concrete professor here who used to pose questions in an engineering journal - some related to that type of micro analysis

In reality, and in a situation such as yours, I think the natural robustness of codes comes into play
I think the development lengths are based around being able to fracture the bar before failing the concrete (correct me if I'm way off here)
This is a much larger force than yield so already a factor of safety is introduced

Then you look at the moment profile vs development profile
My memory is hazy on bar development but I think it would either be a relatively linear stress development or clustered at one end
In the case of a hook, it's pretty obvious that most of the development is clustered at one end
How does that help us? Well, if we need to develop 100% FRACTURE capacity at a distance of say 1m (to keep it simple) and the moment profile is linear, parabolic, or higher order, then a development length that is linear or clustered at the low-moment end will outpace the force demands of the moment
So by the point of bar yield you will have more than enough capacity

As I typed that I realised a picture would help... here's what I'm picturing in very general terms

 

[ChickenBake]

Your premise agrees largely with what I’ve read on this issue.

My understanding is that bar development is required to preclude splitting or bond failure in the concrete which is a brittle mode of failure, in lieu of yield which is ductile. So this has nothing to do with actual rupture of the bar, which is a different discussion entirely. Correct me if am wrong however, since I’m not fully aware of the derivation of the development length equation.

And yes, your graph is matching my reading which indicates that flexural strength is linearly proportional to development length. In other words, Mn goes from 0 to Mn linearly over a length of ld.

My problem is I’m questioning how you square this with a tension controlled section. The relationship would indicate the bars can only yield once they’re fully developed, and yet we see in my example that the bars have to yield in a spot where they aren’t fully developed in order for the tension-controlled assumption to hold true.

 

[Greenalleycat]

I've just dug through the commentary for our concrete code
In short, it makes the same key points you have

1) Development is to yield of bar (not fracture as I thought, I think that came from a discussion about couplers/mechanical anchorage that are required to develop fracture)

2) You need the required strength development at the 'critical section' which can be fracture, yield, or below yield
This doesn't necessarily equate to the highest moment point as you've highlighted

So, what saves us from nobody calculating this correctly?
All the code conservatisms I assume

 

[phamENG]

Very true. Looking at it on my phone, your ε[sub]t[/sub]>>ε[sub]s[/sub] looked like ε[sub]t[/sub]=ε[sub]s[/sub].

An interesting question that I hadn't really considered. Thanks for bringing it up.

 

[Smoulder]

Run some actual numbers to understand it. Firstly the lever arm doesn't change too much between elastic stress strain profile and rectangular ULS profile. Second the bending moment falls away very quick in this example. Third is the safety factor that Phameng mentioned. Put it all together and a cogged end does the job. No cog and I check partial development against actual bending moment diagram.

 

[DaveAtkins]

First yield occurs where the moment is greatest, at the face of the wall. Am I missing something here?

Are you suggesting that the bar yields across some length, and that the bar is not fully developed where the bar stops yielding? That is not something design methodologies consider, so I assume that is not a known failure mode.

DaveAtkins

 

[ChickenBake]

Yes, first yield occurs at the face of the column. But the yield moment is lower than the ultimate moment, because after the steel yields the neutral axis can be pushed up and the curvature increased until the concrete crushes.

That means that when the face of the support reaches the ultimate moment, the yield moment will occur at some distance away from the support. And development doesn’t seem to be checked there.

 

[bones206]

Definitely a worthwhile thought experiment. My first take is that even if a length of the bar has yielded, the nubs on the bar are still engaged in the concrete along that length, therefore still providing some degree of anchorage capacity. I think the development length equations must empirically take this into account. In other words, tests have shown that as long as Ld is provided past the section of interest, it can be relied upon to provide enough anchorage to develop Fy at that section (plus additional strain up to code limits). Whatever is going on in that yield zone is what it is, but it's baked into the equations. That's my hypothesis at least.

 

[hardbutmild]

1) Bending moment of a section at yield is almost the same as the bending moment at failure so it would be a vary close section (as Smoulder mentioned).
2) If you intend to use longitudinal bar for shear capacity increase, it should be anchored from the section that is d away from your section (where diagonal crack can form), I've seen it checked like this sometimes.

Just to show this here's a quick example of a footing. The difference between My and Mu is 2 % in this case. Also note at the bottom how bond stresses change. Most of the bond is achieved closest to the crack so if you have 5 cm less bond length at the far end because of yield penetration you'll lose much less than 5 cm / bond length because you're removing length from the least stressed part.

In the case of a hook, it's pretty obvious that most of the development is clustered at one end

If the critical crack is so far away from the hook it actually has no influence on the behaviour.

EDIT: For this example above, at failure load which is very high, yield has penetrated 1 cm away from critical sections (I calculated it with the assumption that stresses in the soil are constant), practically nothing.

 

[BridgeSmith]

That means that when the face of the support reaches the ultimate moment, the yield moment will occur at some distance away from the support. And development doesn’t seem to be checked there.

Yielding of the reinforcement is considered failure. Development length does not account for behavior beyond failure.

Also, I'm not sure if your assumption that "the yield moment will occur at some distance away from the support" is necessarily true, in the sense that steel within the concrete at some point farther out will yield before tensile rupture of the steel at the cracked concrete section.

 

[ChickenBake]

Thanks for showing that example hardbutmild. That confirms my understanding that ultimate moment is higher than yield moment but not much higher. I wonder if there was a generic way to prove that is always true. That would put this to bed in my mind.

BridgeSmith, how can yield be considered failure if we design for a moment that can only occur after substantial yielding?

 

[hardbutmild]

Yielding of the reinforcement is considered failure.

If this was true we would have a very hard time making RC work in seismic.

 

[hardbutmild]

I wonder if there was a generic way to prove that is always true. That would put this to bed in my mind.

Do it for minimum reinforcement (like my example) and for maximum reinforcement. Anything in between should work if you can prove that it's true for those extremes.

 

[BridgeSmith]

Ok, I should have been more specific. We consider the yield stress to be the limit of the bar capacity for design (even for seismic, right?) We assume the strain can exceed the elastic limit (the reinforcing can deform plastically), but the stress is assumed constant at the yield stress. At least that's how we do it.

With that assumption, we also assume that when the concrete cracks at the highest moment location (in the case under consideration, the face of the column), and that crack just gets wider.

AASHTO does include provisions for a 25% increase in development length for longitudinal reinforcement in columns, but that appears to be for overstrength considerations.

 

[Celt83]

ACI 408R-03: Bond and Development of Straight Reinforcing Bars in Tension may be a good document to review for this: