Double vs Single Curvature Bridge Columns 3

[bpiermat]

While analyzing an existing bridge in CSi Bridge, it came to my attention that we don't (or at least I don't) have a good working definition of single vs double curvature.
Here are the overall analysis steps:
1. Determine the Response Spectra (RSA) demand values
2. Perform Pushover analysis with an assumption of single vs double column curvature (in my case I chose double)
3. Record moment diagram at RSA displacement and for any hinging that occurs

4. Determine if column curvature assumption is correct.

I am having some difficulty with step 4. Is the definition of positive and negative moment in a single column the absolute definition of double curvature? Or is there a ratio, where it is primarily one or the other? OR is this completely wrong and all that matters is the hinging, if so, what if no hinging occurs during the push to the RSA?

Thanks,

Ben

 

[BridgeSmith]

Single or double curvature is the deformed shape, which will be dictated by the moment values along the column, which in turn is typically a function of the restraint at the ends of the column. If both ends of the column have moment restraint, and the top of the column is displaced, there will be moment reversal and double curvature. If the moment capacity of the column is exceeded, a hinge will form, the moment at the hinge is relieved, and the column is now in single curvature (and the displacement increases due to the increase in moment at the other end of the column).

 

[Agent666]

Hotrod, once the hinge forms it doesn't go into single curvature. The moment stops increasing at the location of the hinge, and is sustained at the limiting moment capacity of the member (in practice it might increase marginally dependant on strain hardening, etc), but its still in double curvature, even after the second hinge forms at the other end its still in double curvature.

bpiermat, usually as hodrod10 noted, you'd make an assumption on the fixity and verify that your detailing achieves this (fixed/pinned/spring, etc). I feel like you are sort of coming at it from the wrong direction with the train of thought you've laid out your question.

 

[BridgeSmith]

Agent666, you're correct for a steel column, but for a concrete column (which is what I assumed from the question), hinging is usually accompanied by significant concrete spalling and breakage, resulting in an almost complete loss of strength. This is especially true in the case of seismic loading, where the forces are cyclic.

 

[Agent666]

Well I'm going to disagree with that in its entirety (I also assumed a concrete column). Ductile concrete moments frames are used all over the world, including seismic regions. Provided theres sufficient confinement, maintaining the nominal capacity at high ductility isn't an issue. Almost every code I'm familiar with allows them and has design and detailing rules for them. I'm not aware of any requirement in a code or otherwise that discounts all flexural strength contribution once you form a plastic hinge (shear contribution from the concrete is another matter).

 

[BridgeSmith]

Well, the design codes may allow you to consider post-hinging strength of concrete columns (I'd have to take a closer look at what the AASHTO code provisions are in that regard), but for the bridges I've designed in higher seismic regions, I haven't accounted for it in my displacement calcs. Theoretically, with adequate confinement the core may continue to function through multiple reversals, but I haven't seen enough empirical evidence to convince me that the theory will match the reality. OTOH, I've seen plenty of photos of bridges after past earthquakes that didn't perform as expected.

 

[bpiermat]

Thanks for all the responses. I am coming from the other direction, because I am rating an existing bridge, therefore its capacity and hinging is not something I cannot control, rather something I need to consider.
My bridge is a single pier/column bridge. Which I know many of you will say...oh its single curvature. But its not that simple, the bridge is curved, basically has two columns on the same line. Therefore CSi is indicating double curvature, Or at least I think it does, with a positive moment of lets say 100 kip-ft at the top and a negative moment of 1000 kip-ft at the bottom.

The detailing of the column and footing/superstructure moment capacity, is part of the CSi model/or we have checked it or accounted for reduced capacity. (For example, we softened the foundation springs to account for an undersized footing). Therefore, this maybe one reason why the results are lopsided/not clear if it is double or single.

Oh and yes this is a Concrete Column/Bridge

 

[hardbutmild]

Or at least I think it does, with a positive moment of lets say 100 kip-ft at the top and a negative moment of 1000 kip-ft at the bottom.

But what are the curvatures? If plastic hinge forms on top and not on the bottom, curvature might be similar on top and bottom while the difference in moments is large.

Also, you said it's a curved bridge so it has two columns in a line. I can see it in one direction, but how does that happen in two directions? Or are you only considering a certain direction of an earthquake?

 

[bpiermat]

I.C. I will check out the moment-curvature values, I think your implying that one should be negative while the other one positive? If my assumption of double curvature is correct?

In the longitudinal direction, there is frame action, so it is in double curvature.
In the transverse direction, it would be traditionally single curvature, except for the fact that the superstructure has a significant curve that aligns several columns in the transverse direction, which could lead to double curvature in theory.

Ben

 

[hardbutmild]

Oh, okay. Now it makes sense when you described the structure.

Yes, I was suggesting that one of them should be negative and the other positive to say it's double curvature. But also it depends on the values. Personally, I'd probably check where the zero curvature is and if that location is "inside a plastic hinge" I'd say it's single curvature. It's a tricky thing to see that since your pushover has hinges in a point. I'd say it's inside a plastic hinge if it's inside a length where you'd usually put a large amount of stirrups. I hope you understand what I wanted to say. Also if one curvature is 100 and the other is -1 then it's pretty obvious that it's not double curvature (I personally would probably say one has to at least be 10% of the other, but you might see it differently).

 

[bpiermat]

Thank you Hardbutmild, this is exactly what I was looking for.

Ben

 

[bpiermat]

One question for HardbutMild, So in summary, you would calculate Curvature as M/EI for the column at points where no hinge has formed and where hinges have formed you would record the plastic curvature and add it to the elastic curvature for comparison purposes.

So, since the hinge represents a sudden jump in curvature, the point of zero moment would be the same point as the point of zero curvature?

Thanks,

 

[hardbutmild]

point of zero moment should be the point of zero curvature.

I think you could easily construct the moment-curvature diagram (make an excel, it's not hard at all) and for a given moment simply read the curvature from the diagram. It's not really as simple as M/EI since column will probably be cracked at the bottom also.

I'd create the moment-curvature diagram for a section at top and section at bottom (maybe you get it from your software) and from known moments determine curvature at bottom and curvature at top. I would compare them and if bottom/top > 10 I'd say it's single curvature. I'd also check how high the zero moment point is. If it's close to the top edge I'd say it's single curvature.

In addition to this I'd check is the critical length (length at which confinement is provided) is long enough and few other things I guess.

But I'm just telling you my idea, it might be wrong or overly simplified, keep that in mind

 

[Agent666]

Curvature for evaluating plastic hinge rotation is usually a function of plastic rotation at a hinge over hinge length. M/EI is the rotation, not the curvature.

The point of zero moment isn't a point of zero curvature. Unless its zero moment over some finite length of the member (i.e. a straight member with no rotation).

 

[Ingenuity]

M/EI is the rotation, not the curvature.

Euler and Bernoulli may disagree.

 

[Agent666]

Sorry I mean slope not rotation!

 

[bpiermat]

Last time I checked M / E I has units of 1/Length. Not degrees or unit-less (slope)

Yes, I left out the part where you have to multiply the hinge length by the rotation to get the hinge curvature. This would be added to M/EI

 

[Agent666]

Maybe I'm dreaming but isn't slope = θ = ∫(M(x)/EI(x))dx was what I was getting at and probably getting my wires crossed!

multiply the hinge length by the rotation to get the hinge curvature

Pretty sure this is incorrect, curvature has units radians/length (or /length), so it should be the rotation divided by hinge length as I noted a couple of posts above.

Typically (round here anyway) you are not working it out from M/EI as you cannot tell how much is plastic vs elastic portions are. The way we work it out here is simply by saying the center of the hinges are connected by a straight member, then work out the rotation based on the geometry at the ultimate interstorey drift (the analysis drift is factored up by another factor to take into account that our analysis drift from our linear elastic model is lower than the peak drift values as shown by non-linear time history studies, this factor varies 1.2-1.5 and is a function of building height).

From the total rotation divide by the effective plastic hinge lengths. This gives you the total curvature (you can do it for beams/walls/columns in a similar manner if you propose where the hinges will form to give you your ductile mechanism).

We have another formula for the initial yield curvature, this is the elastic bit basically. So you can work out the plastic curvature and compare to the code limits from this. Though I think our limits now can be compared directly to the total curvature as they rewrote it a while ago because people found it far too confusing. But they wrote it in a way that is still confusing, bless those code writers!

 

[hardbutmild]

Couldn't you determine the curvature of a section from internal pair of deformations? If you need an average curvature on some length you could just find an average value. Or am I missing something here?