## Double vs Single Curvature Bridge Columns

## Double vs Single Curvature Bridge Columns

(OP)

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

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

## RE: Double vs Single Curvature Bridge Columns

## RE: Double vs Single Curvature Bridge Columns

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.

## RE: Double vs Single Curvature Bridge Columns

## RE: Double vs Single Curvature Bridge Columns

## RE: Double vs Single Curvature Bridge Columns

## RE: Double vs Single Curvature Bridge Columns

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

## RE: Double vs Single Curvature Bridge Columns

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?

## RE: Double vs Single Curvature Bridge Columns

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

## RE: Double vs Single Curvature Bridge Columns

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).

## RE: Double vs Single Curvature Bridge Columns

Ben

## RE: Double vs Single Curvature Bridge Columns

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,

## RE: Double vs Single Curvature Bridge Columns

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

## RE: Double vs Single Curvature Bridge Columns

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).

## RE: Double vs Single Curvature Bridge Columns

Euler and Bernoulli may disagree.

## RE: Double vs Single Curvature Bridge Columns

## RE: Double vs Single Curvature Bridge Columns

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

## RE: Double vs Single Curvature Bridge Columns

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!

## RE: Double vs Single Curvature Bridge Columns