Non-symmetric beam analysis

[Dinosaur]

Due to a fabrication error, we are looking at a non-symmetric beam. This made me reflect on a philosophical question about design of bridge girders.

The exterior girder is non-symmetric due to the presence of the concrete slab with an overhang not equal to the effective flange width toward the interior. We never analyze this unsymmetric condition.

Is this because the cross frames prevent the distortion from causing additional stress due to the non-symmetry? Are we just comfortable the exterior girder has enough "beef" to carry all the loads? I say this because usually the interior girders take more of the live load than the exterior girder. Or are we making a big mistake?

 

[Umassengr]

My State DOT generally likes to limit the overhang so that the effective flange width on the exterior beam matches the interior beams though so I can't say I've thought too much about it. In general most of the bridges that I have designed have very little live load going to the exterior beam because of the lane configuration. This load is never as high never as for an interior beam.

AASHTO 3.23.2.3.1.4 requires that the exterior beam to have the same or greater capacity than interior beams. I have always thought this was due to the possibility of future widening. Is it possible that 3.23.2.3.1.4 was also concieved to deal with the probability of the asymmetrical exterior beam?

 

[bridgebuster]

Another reason for limiting the overhang is if the deck was designed by the empirical method.

As I understand it, the intention of AASHTO 3.23 etc is to allow for future widening, if widening was possible. I had a field job where we were adding stringers because the fascia stringers had less capacity than the interior ones, but it was a waste of money - it was a truss bridge.

 

[VoyageofDiscovery]

More and more as I deal with rehabilitation of bridges, I am analyzing and designing exterior girders to take full live load as excessive cantilevers are required with no sidewalks. Mind you I do a full FEA for each case.

 

[Dinosaur]

Thanks for your input. I agree that in a typical situation, analysis of the exterior beam is unnecessary. However, I am questioning if there is an analytical reason that includes consideration for the non-symmetric conditions that justifies this approach? Or, are we just comfortable the interior giders govern and use the design for the exterior girder as Umassengr explains?

In my particular situation, the bottom flange was not placed properly and is eccentric to the centerline of the web. I believe in this case the error is small enough to let go, but I am wondering how much analysis is justified in this circumstance when the exterior girder is non-symmetric anyway. Are the diaphragms able to restrain the girder enough to neglect this non-symmetry, for example?

VoyageofDiscovery,

I imagine it takes some time to create a coherent FEA model. What is your secret?

 

[VoyageofDiscovery]

There is no secret, being conversant and quick in FEA and spreadsheets goes a long way.

 

[miecz]

Dinosaur,

I tend to agree with you, that for a fascia girder, a small ecccentricity is of little concern, due to the reduced distribution of live load. Not to mention that the parapet adds section that is often ignored in design.

However, if there is a possibility that the bridge may be widened in the future, this member becomes an internal stringer, and carries the same live load as the others. I wouldn't leave it to the rehab engineers to pick up on this.

For this case, it would seem that one effect would be that the shear center of the section has shifted slightly. This would induce a small torsion to the stringer. The stringer could be analyzed with torsion like a curved stringer. I think it could be proven that the diaphragms provide enough resistance to handle the torsion. The only other concern would be the b/t ratio of the outstanding flange.

 

[Dinosaur]

meicz,

I didn't think to mention that this may become an interior stringer someday, but this is the basis for making exterior girders the same as interior girders.

I did an analysis of the non-symmetric section with the steel alone. Based on this I determined the maximum bottom flange stress would increase 11% due to loads applied to the steel section alone (e.g. selfweight and deck slab). Then when I started to study the second stage of loading, I began to question whether it was necessary since I hadn't analyzed the original section for its non-symmetric effects. I could still do the analysis, but it made me think about the basic assumptions.

The b/t is going to be fine. The flange sticks out only 9.5 inches, and the thickness varies from 1 inch to 1.5 inches. The flange is only in tension along this section.

 

[gwynn]

The one thing not mentioned above, though it may be obvious, is to make sure that if the construction engineering is being done by others they are aware of the fabrication errors. Depending on construcion methods it could have a significant impact on temporary bracing requirements and edge of soffit elevations.

 

[miecz]

I must have missed something. I thought your bottom flange wasn't centered on the web, so that the vertical centerlines were slightly offset. I don't see how that results in a calculated stress increase. Does it rotate you principle axes?

 

[Dinosaur]

A non symmetric section does not follow all the rules used in normal beam analysis. The eccentricity of the bottom flange causes the resultant forces to be unaligned with regard to the plane of the load. To predict the change in stress, you need to compute Ixy and Iyy also. I am using the formula from the book "Advanced Mechanics of Materials" by Boresi and Sidebottom to compute the change.

As I mentioned above, after getting through step one I realized we never consider this for step two (the loads applied to the composite section) on a regular project. I am wondering if we are just comfortable it doesn't control or is there an analytical basis for the decision not to analyze this.

Thanks for your help.

 

[miecz]

The short answer is: I don't know. But let me make a few observations:

1. The code is mum on this issue.
2. Looking at my two references on unsymmetrical beams, the subject beam is free to deflect as it will. For the case of a composite stringer, the attachment to the deck and other stringers won't allow this. That is, the composite stringer is going to deflect vertically. I believe this justifies ignoring the assymetry for composite fascia stringers in bridges.
3. For the non-composite state, it's a gray area, so I would include the effect for this load condition.
4. Eleven percent seems like a high number for a small eccentricity.
5. If you ignore the assymetry for composite condition, the total stress increase (for the widened bridge) will probably be less than 3%. Hopefully, it would still meet the code. Even if you are slighty over, you'll have stringers on either side that share the load. If there ever was a case for allowing a slight overstress, this is it. Not saying that I would, but others might.

 

[Dinosaur]

The official designer has already said it is OK. He says the original design had something approaching 25% reserve capacity. I don't want to go any deeper into that since my official role is to supervise the fabrication inspection. It took the designer some time to respond to the initial question, so I decided to get out a pencil and paper and do a quick check myself, comparing the change in the Sx to get a feel for how it might play out.

I am wondering if the presence of the cross frames also reduces this effect as you note the slab does also.

I agree that because of the restraint of the slab, the effect is reduced, I would just like to put a number on it. Based on my experience, I believe the change in the bottom flange stress will be between 3-5%.