LTB of a Vierendeel Truss

[cleagl]

Greeting fellow eng'ers

I have a engineering problem that comes up time and time again in our industry, but as far as I know, very few people have proposed a reasonable solution. I need to be able to calculate when a vierendeel truss will buckle out-of-plane.
Context: We manufacture 2-D vierendeel trusses for the entertainment industry (these are NOT box trusses). They have no lateral bracing and are hung from the roof or gridiron by cables (tension only members). They quite often have point loads between the supports, (having typically four to six cables as supports they are suspended from). Spans between supports are anywhere from 8' to 15'. Quite often the ends are cantilevered up to 6' past the final support point.
Elastic analysis will give me vertical deflections and stresses but not buckling. The way we analyze this now is by calculating the axial compression of the bottom chord and use the outside support points as lateral bracing to calculate the Euler buckling load. Then recalculate the allowable point (or distributed) loads based on Euler.
I have this in Inventor, hoping to do a buckling analysis, but Inventor only does modal analysis, and I have no idea if the eigenvalues are the same as buckling or how to access the modal eigenvalues in this program (new to Inventor Pro).
None of the AISC LTB equations work since they are based on lateral bracing. Any ideas on how to rationally approach this?
Many thanks

Cleagl

 

[BAretired]

One approach is to analyze the compression chord using Newmark's numerical methods. This method is outlined for one particular case in the book "Theory of Elastic Stability" by Timoshenko and Gere but could be applied to other situations.

I have used it in a variety of situations and find it very helpful.

BA

 

[ishvaaag]

I really think the best way to model these things is with explicit imperfections. So you sum your fabrication tolerances plus those for damage and placement in works, and model the vierendel frame with the target deformation (s, you may need more than one model) in say autocad, pass it to say RISA 3-D or other program able to perform P-big Delta on your model, and if the result is stable (P-Delta converges) an valid in strength and deformation, it is a valid design and check for K=1 in the members. If your frame elements are relatively long and slender you may use the same process but then subdivide the members in portions with joints to capture small P-delta effects (in-member effects).

 

[dhengr]

Cleagl:
The entertainment rigging industry was pretty loose as I recall, they can do some amazingly bad (dangerous) things with your products, and then ask you to stand behind and/or below them. That’s scary. Although, maybe OSHA and city inspectors are starting to bring some control and regulation. You say: 4 to 6 cables as supports; with spacings of 8' to 15' btwn. the cables; hung from the roof structure or, or worse yet, a grid work of cables. Vertical cables stretch quite a bit, and cable nets or grid works deflect in even more indeterminate ways. Do you have any real idea what your real reactions and truss stresses are, are some assumed reactions almost zero due to cable extension or movement? This question should figure into your very first “explicit imperfections” per ishvaaag’s comments.

I’ve used the methods suggested by BA, although it’s been a long time and I didn’t dig out my text books to refresh my memory. What ishvaaag suggests sounds right to me, although he knows far more about which programs and the exact details for the use of each program.

I assume you really didn’t mean 90' long trusses, (6 spans)(15'). But, one way to make your problems easier to deal with would be to limit your truss length to 15 or 20', a few more trusses and reaction points needed, but much easier to handle in the field. And, limit the trusses to two reaction or support points, a determinate structure from this standpoint now; with restrictions on cantilever lengths and loadings and load points. Otherwise, I don’t know how you determine your reactions and their defected positions and the potential for transverse reaction components. Didn’t your industry go to triangular and box trusses in part because they were more stable and a bit more rugged?

 

[ishvaaag]

It also must be mentioned that any firm working in this field should have dynamometers -and preferably recording the given value- to see what is the real extent of the impact factor when hanging the equipment. It may turn that no allowance is being made and so loads able to pastically buckle the frames are appearing in place. A classic case of load underestimation.

 

[cleagl]

Thanks very much for all the helpful comments. I have modeled this in RISA, but don't have a good feel for the magnitude of the explicit imperfections. When I have done buckling analysis on shells, I did a separate analysis just to determine the magnitude of the imperfection. Here I can only assume the worst case which would be about 0.5% of the unbraced length. I will try what ishvaag suggests and see what it shows.
Thanks to BA for the Newmark suggestion. A little intimidating, but it might be good to have a couple of methods to compare.
It is true that our industry has a wide variety of people both with and without good judgment, but the good news is we use very conservative approaches. We use FS of 8 against rupture and a FS of 5 against yielding. Thanks again for all the help. I'll post results when I have them.

 

[KootK]

Some questions for you Cleagl:

1) What is the span/depth ratio of your typical truss?

2) How is lateral support provided to your compression chords at the supports?

If your truss is supported at the top chord and loaded at the bottom chord, is it really possible for the whole truss to glogally LTB? It seems unlikely unless your span to depth ratios are pretty large. You've got too much going for you by way of restoring forces.

If you have effective lateral support for your compression chord at the supports, you should be able to make a reasonable approximation by simply calculating regular compression buckling loads in the chord.

 

[BAretired]

If the truss is suspended with cables, the compression chord is not braced at supports.

BA

 

[KootK]

I was just noodling on that BA. I guess with no external axial reaction at either end of the compression chord, there is no P-Big Delta effect to exacerbate buckling. So... we just check P-Little Delta chord buckling with K=1.0?

I think that I may be learning something here..

 

[KootK]

And, with varying axial force along the chord, that takes me back to BA's original Newmark suggestion. Okay... I think that I'm done here.

 

[BAretired]

A sketch of a typical truss would help.

BA

 

[vandede427]

Doesn't it take at least a little torsional restraint at the ends to get LTB in the middle? If they're only hung by cables it seems like global stability is more of an issue than local stability.

 

[BAretired]

I don't believe global stability is an issue. It is similar to a lifting beam.

BA

 

[cleagl]

So here is a drawing of the truss I am working on right now. 54'-0" long x 1'-0" high.
Suspended by (4), 1/4" dia 7x19 cable. The ends cantilever 7.5' past the last support points. There really is no lateral bracing at all. Top and bottom chords 1.5" sched 40 (or sched 80) pipe. verticals 1/4" x 3" flat. Typical loads would be 15-50 lbs plf

For this particular case we have 2 load cases, a) a UDL of about 17 lbs /lf for the entire length, and b) 60 lbs /lf on just the two cantilevers. In real life what happens to these is in load case b) is that the ends sag, the two center lines go slack and the center of the beam flexes out-of-plane. But at what value is difficult to predict.
As above what we do now is Euler buckling on the lower chord using the outside two support points as the unbraced length (in this case Lb=40'-0").

 

[KootK]

That IS a long & slender truss! I can see why you're concerned about it.

With your truss, as well as with lifting beams, I believe that it is the restoring forces generated by the load / support conditions that prevents LTB. However, I think that only works with fairly low L/d ratios.

With your situation, I suspect that LTB is a legitimate concern, particularly at the cantilevers. Loading the compression flange will increase stability but not eliminate the possibility of LTB altogether.

Could you generate faux solid beam properties for your truss and use those to calculate LTB as you would for an I beam? Ix, Iy, Cw etc would be based on the chords only.

I do suspect that you can assume rotational support -- at least some -- at the locations that you mentioned. I'm having a hard time rationalizing why however...

 

[BAretired]

There is no rotational restraint at the supports, so the shape buckles over the its full length.

In this case, the unbraced length of the bottom chord is 54', not 40'. But it is not loaded at the ends. The load increases gradually from each end to the end suspension points. It can be solved using Newmark's procedures.



BA

 

[KootK]

I tentatively disagree BA.

If our concern is simply the buckling of the compression chord alone then yes, the unbraced length is 54'.

However, simply preventing column style buckling of the compression chord does not preclude LTB of the composite section (the same holds true for the lower WT portion of I-beams). This is because the section can still become unstable by the tension chord rotating around the un-buckled bottom chord.

Since structures like lifting beams and Cleagl's truss are able to sustain some load without LTB, some form of rotational restraint must be present by virture of the loading / support conditions.

As a thought experiment, consider the reverse of this problem. Same truss but with the support cables attached to the bottom chord and the loads applied to the top chord? What's the capacity of that system? Zero, right? Zero, even though you could still do that buckling check on the bottom chord and determine a capcity.

It may be that I'm missing something here. I've got an older version of Timoshenko's book at my desk (pre gere). Which section is Newmark's method found?

 

[BAretired]

Having the two end support cables fastened to the top chord and the loads hanging from the bottom chord is, I agree a necessary and sufficient condition preventing rotation about the axis of the truss.

If our concern is simply the buckling of the compression chord alone then yes, the unbraced length is 54'.

The unbraced length is 54' because there are no lateral braces anywhere and 54' is the extent of compressive stress in the chord. The location of the support cables affects the bending moment in the truss and hence the magnitude of compression in the bottom chord, but has no bearing on unbraced length. The buckled shape of the chord must be a continuous curve over the full length.

Theory of Elastic Stability by Timoshenko and Gere - Article 2.15. This book can be downloaded for free.

BA

 

[KootK]

Really? Where? I paid good money for mine and I'm afraid to read it because it's so crusty and frail.

Are you of the opinion then that your compression chord check IS the LTB check? That's the crux of my objection.

 

[dhengr]

Cleagl:
Why aren’t your end supports at a panel points 7'± from the ends, instead of just near them? Seems you have enough problems without secondary stresses in the top chord. For this load configuration adding some of your own dead load in the middle of the truss would allow you to adjust the stresses in the t&b chords, maybe to your advantage or disadvantage.

And, better yet, at the two panel points 7'± from the ends, weld a 3' piece of pipe to the top chord, pointing up, in the plane of the truss, and attach support cable to the top of this 3' piece of pipe. Also, maybe the vert. chord member should be pipe too, not 1/4" x 3" bar. You want to develop some moment cap’y. Do this same thing in the middle of the truss, but hanging below the bottom chord, and affix your own dead loads, you pick the weights. Voila! You have some rotational resistance, which increases as the truss starts to rotate. Ask ishvaaag how to introduce these as variable rate torsional spring restraints to your computer model. These 3' pipes, or some such, with a 1'+ back span, to engage the opposite parallel chord, could just be clamped over (around) a standard truss.

Add these to your idea pile: If you had a triangular pick-up frame above the top chord, with the cable attached at its high point, 3' above the top chord and centered over my 7' panel point, and this frame slopes down to grasp the top chord 4' from the truss end and 10' from the truss end. This frame may have some moment cap’y. across the t&b chords too. This would certainly change the stresses in you standard truss. I think, for the better, although I haven’t analyzed that as I type this. I think you would learn a great deal testing one of these trusses and determining what loads, at different locations, start to cause some amount of lateral rotation of each chord.