Arch buckling

[ido3778]

Hello,

I need to design a steel pedestrian bridge that the middle span is hanged on an arch that pass over it.

How do I take into account the arch geometry for buckling calculation?

I think I need to reduce the effective length, but I an not sure, and if it is the effective length I do not know the amount of the reduction.

Please advice.

Any reference will be great.

Thanks.

 

[rowingengineer]

intresting, a sketch would help.

An expert is a man who has made all the mistakes which can be made in a very narrow field

 

[ido3778]

Hope This is help.

http://files.engineering.com/getfile.aspx?folder=1264fdc4-f4e4-4b51-ab0c-310d09add9c8&file=1.JPG

http://files.engineering.com/getfile.aspx?folder=89494dbd-2a81-4241-9b8b-856ca7fb011a&file=2.JPG

Thanks

 

[BAretired]

For in-plane buckling, take the straight line distance from the springing point to the midpoint of the arch as the effective length.

For out of plane buckling, you can reduce the buckling length by adding bracing between the arches.

BA

 

[JStephen]

Are you concerned about buckling in the arch or in the suspended span?

 

[ido3778]

Thank you all for the replays.

BAretired: Can you please point me to a reference that I can see what you suggests? I need a solid reference in case I want to include these solutions in my static calculation notes.

JStephen: My concern is about the arch buckling.

Thanks again.

 

[prex]

Timoshenko & Gere's Theory of Elastic Stability treats the buckling of arches.
For a uniformly compressed hinged circular arch it is found that the critical compressive stress is the same as for a hinged prismatic bar with a length equal to the developed length of half arch: this only in part supports and explains the suggestion by BAretired (but is only true when the half opening angle is small in comparison to [π]).
Timoshenko also treats what is likely your case, the parabolic arch with a load uniformly distributed along the span, but this requires tabulated values that are in the book (page 303 2nd ed.).
These treatments do not account for the interaction of the arch with the bridge, that's not necessarily negligible, though neglecting it is on safe side. IMO this interaction comes out because the arch buckles with half going downwards and the other half going upwards: so some bending occurs in the bridge influencing the buckling load. However I don't know if such a treatment can be found anywhere in the literature.

prex

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[BAretired]

I was speaking from memory of "Theory of Elastic Stability" by Timoshenko and Gere. For a circular arch the critical length is the developed length of half the arc as stated by prex. See attached for the relevant text.

BA

 

[JStephen]

You might check to see if either load case is included in Roark's Formulas for Stress and Strain.

Another load case that might be of interest is circular rings subject to in-plane compression. It sounds similar to the first arch case above.

 

[ido3778]

Thank you all.

Just to make sure, I have fixed arch at both ends,
If I use le=0.5*l I think it is conservative.

l represents the WHOLE arch length and not just projection line.

Please advice.

 

[BAretired]

Have you considered unbalanced live load on your arch? I would expect that to govern your design rather than in-plane buckling.

BA

 

[ido3778]

I considered unbalanced live load, yes.
I considered live load only on half of the middle span.

 

[IDS]

At the risk of annoying BAretired again, I'd suggest analysing the structure in a frame analysis program including geometric non-linearity, using conservative lower bound values for the member stiffnesses, including the foundations.

In this case I would suggest that this is the only sensible way to do it, since any results using standard formulas or tables will only give an order of magnitude result at best. Also deflection and/or dynamic behaviour are quite likely to control the design, and you are going to need an analysis including geometric non-linear effects for that as well.

Doug Jenkins
Interactive Design Services

http://newtonexcelbach.wordpress.com/

 

[BAretired]

Doug, I do not find your suggestion at all annoying. I had not thought about dynamic effects, but they should certainly be considered for a pedestrian bridge. An army platoon would break step when crossing a bridge, but civilians might be tempted to find its resonant frequency by applying rhythmic motion to the deck.

I'm not sure what you mean by geometric non-linearity or how you would go about analyzing it...with or without a frame analysis program.

BA

 

[IDS]

BAretired - a basic frame analysis ignores the change in shape of the structure due to the applied loads, and will thus never show buckling, but it is possible to include this effect (sometimes know as the p-delta effect) in a frame analysis, which will then model buckling behaviour. If the program provides this option it is just a matter of ticking a box, but it is important to review the model to make sure it will capture potential buckling modes. Long members need to be subdivided to model the geometry properly.

Doug Jenkins
Interactive Design Services

http://newtonexcelbach.wordpress.com/

 

[BAretired]

Okay Doug, I think I get the picture. You subdivide the arch into a number of straight line segments, say twenty or thirty of them. You calculate the X and Y coordinates of each node and apply point loads to every node.

You run one analysis and come up with [Δ]X and [Δ]Y for each node as well as axial forces, shear forces and bending moments on each of the twenty or thirty members.

You modify the X and Y coordinates by adding [Δ]X and [Δ]Y to each X and Y coordinate and carry out a second analysis. You get new [Δ]X and [Δ]Y which you add to the revised coordinates from the previous step.

You continue the process for as many iterations as you wish and if the coordinates converge, you have stability. If they diverge, you have buckling.

Is that the way it works?

BA

 

[IDS]

Is that the way it works?

That's one way to do it, and in my opinion the best way, especially in this case where you may want to incorporate non-linear (no compression) behaviour in the cables from the arch to the walkway. It is also possible to incorporate the effect of deflections in the frame stiffness matrix, but I would have to remind myself on the applicability of that approach.

Doug Jenkins
Interactive Design Services

http://newtonexcelbach.wordpress.com/

 

[BAretired]

Thanks, Doug.

BA

 

[JoshPlumSE]

Regarding IDS's / Doug's suggestion of using frame analysis. I saw a brief presentation of something very similar. Essentially, the engineer used the P-Delta (or geometrically non-linear analysis) to analyze the structure for buckling.

However, there was one major caveat. He modeled in slight geometric irregularities into his model. I believe he must have run two or three models. Because the irregularities (on the order of L/500 at mid span) were intended to approximate the buckled shape of the structure.

The drawback to this is that you have to have a good idea of what the buckled shape is BEFORE you do your analysis. But, the major benefit is that the frame analysis should provide an excellent estimation of the elastic buckling of the structure.

For what it's worth, the Direct Analysis Method described in the AISC code tries to do something very similar with it's modeling of initial displacements or notional loads. The biggest difference is that for buidling type structures the buckled shape is basically just a cantilever and is extremely easy to estimate. An arch, on the other hand, is not quite so simple.

 

[JoshPlumSE]

Something I did not mention in my previous post, but which is important.

A frame analysis needs an "initial deflection" in order to iterate the P-Delta and predict buckling. Sometimes this can be provided by applied lateral loads, or applied gravity loads. But, other times, it must be introduced into the model by the user. Especially, for cases where the controlling buckled shape may be complex.