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Cruciform LTB

Cruciform LTB

(OP)
I have an unbraced column length of 40'-0". The column is a cruciform made up of (2) W24x55 sections (one is split and then welded). How would you calculate the LTB of this section? CB = 2.24.

My co worker thinks that we can use the AISC 360 provisions for W sections, calculating Lp and Lr from rx and ry for the entire composite shape. My worry is it will overestimate the LTB capacity. Capacity comes out to be governed by the plastic section (which he gets from adding Zx from one and Zy from another W24x55) so 552 k-ft

I would think to use the LTB capacity of a single W24x55, so 135 k-ft.

Intuitively, my idea is a lower bound and my co workers is an upper bound. The gap is wide. How would you do this?

RE: Cruciform LTB

(OP)
Another note. I worry it'll overestimate the LTB capacity because the majority of the area which makes up Ry is at the flanges, so they are laterally braced. However with the cruciform, the majority of the are which makes up Ry is right at the center, not bracing the top or bottom flange.

I know it will brace it somewhat, because LTB is rotating about some point beneath the shear center (not right in the middle of the centroid) so I know my method is over conservative, but I don't know how to deal with that. Any suggestions?

RE: Cruciform LTB

You say it's a column but it sounds like you're discussing a beam. Am I confused?

RE: Cruciform LTB

(OP)
It's both :)

Column at intersection of two perpendicular moment frames.

RE: Cruciform LTB

My approach for built-up shapes (unless it conforms to a shape that is specifically covered by code) has always been to figure the limit states of the components (i.e. local buckling, LTB, etc).....and then figure stresses based on the entire [combined] section and compare.

Like everything else though....it takes a degree of common sense. For example, if you were to put a angle on the top flange of a I-beam.....you might eliminate it's LTB from the situation (because the I-Beam's is so much higher)....but at the same time, you still have to think about localized buckling or yielding in the angle (especially since the yield strengths are different).

In what you describe, I'd figure an allowable stress for a single W24 (in LTB)......and figure the stress based on the (whole) cruciform.....and compare them. Same thing about the other axis.

RE: Cruciform LTB

(OP)
WARose - I like that idea, using the allowable stress of a single W24 in LTB for the whole section. It makes sense.

RE: Cruciform LTB

(OP)
After running the calcs, it still seems overly conservative however. I increase from 135k-ft to 138.5 k-ft.

Thinking about it more, it intuitively seem overly conservative as well, that solution WARose presented assumes Lr and Lp stay at the same lengths, the only thing that is increasing is my area for which to apply the same critical stress. It makes sense that Lr and Lp would increase. It would decrease because the weak-axis W24 is providing restraint to LTB, even though the restraint is at the centroid instead of at the flange.

However, I cannot think of a logical method to increase the Lr and Lp lengths :/

RE: Cruciform LTB

Don't forget the fact you get to use the combined section modulus to figure the stress (from the applied moment). You should (at least) get that advantage from the built-up shape.

RE: Cruciform LTB

(OP)
Yes, Sx is only increased from 114 to 117 cubic inches :/

RE: Cruciform LTB

(OP)
I found this document. http://www.newsteelconstruction.com/wp/wp-content/...

It talks mostly about columns, but the last bit touches on bending. Scroll down to the bottom left corner of the 3rd (last) page to section 3.1... This paragraph states that for bi-symmetrical cruciform sections,

"The bending strength of flanged cruciform sections
is most conveniently found by calculating the slenderness,
λLT, using the method in BS 5950-1:2000. It
is recommended that when using Annex B, section
B.2.3, the value of gamma, g, is taken as 1.0 because
the value given in Annex B was derived for single
I-sections. It is also recommended that values of “u”
and “x” are calculated using the formula given for
channels with equal flanges to avoid assumptions
made for single I-sections. Tables 16 or 17 can then
be used as for lateral torsional buckling of single Isections."

Unfortunately I do not have access to the BS 5950-1:2000 :/

RE: Cruciform LTB

hmmm.....I guess another way to think of it is: the weak-axis being braced by the W24. (Maybe it would work then?) That would create something that would have to be looked at as per the bracing requirements of AISC.

The W24 in the other direction would have to be attached near the compression flange though.

RE: Cruciform LTB

(OP)
Yeah. I've thought about having (4) angles every 5'-0" or so bracing from one W24 Flange to the other, but they would still twist together so I don't know if that'd be doing anything really.

RE: Cruciform LTB

Have you tried welding some angles to the flanges? That should drive up your combined section modulus significantly.

RE: Cruciform LTB

(OP)
I would just upsize the column to a W24x76, and then it would work assuming it's just a single W24x76. I was just hoping there was a logical way to go about designing a cruciform column for LTB :/. I can't think of a better method than what you presented, but it still seems overly conservative. Oh well! Thanks for your help :)

RE: Cruciform LTB

>>>Yes, Sx is only increased from 114 to 117 cubic inches :/ <<<

Should be a tad more; shouldn't it be 114 + 8.30 = 122.3?

RE: Cruciform LTB

(OP)

Quote (Archie264)

Should be a tad more; shouldn't it be 114 + 8.30 = 122.3?

I'd be happy with that if it was! I'm afraid you can't simply add section modulus together like you can plastic modulus or moment of inertia.

S = I / c

For the combined section, S = (Ix + Iy) / (d/2) = (1350+29.1) / (23.6/2) = 116.9

RE: Cruciform LTB

(OP)
I'm just surprised by the lack of information I can find on the subject. You would think someone has thought of this before!

RE: Cruciform LTB

Hmm, I guess I'm so used to working with doubling and tripling up wood 2x members of the same depth (where it does work out) that I forgot my fundamental strength of materials. Glad I'm not designing cruciform beams...

RE: Cruciform LTB

Cal91, I feel like I have seen an article on the topic. I couldn't find it, though.

RE: Cruciform LTB

Quote:

I'm just surprised by the lack of information I can find on the subject. You would think someone has thought of this before!

I think there are so many different permutations of section combinations that it probably makes it impossible to have a one-size-fits-all. IIRC, the formula for figuring the LTB value will vary from shape to shape.

A few common situations have caused some investigation in the past. Probably one of the most common: crane runways [I-Shapes] with a channel cap. A (2nd quarter) 1998 AISC Journal article investigated the LTB value for these sections.

RE: Cruciform LTB

I've looked at this in the past -- can't seem to find those old calcs or references though... sorry.

I recall trying the method mentioned above -- using the "weak axis" W24 to brace the "strong axis" W24 about the axis of rotation (per AISC brace provisions). As I recall, it was a lot of work without much payoff.

I also recently had a project where the flange tips of a built-up cruciform column were battened together as relative bracing. I wasn't able to justify those as providing restraint for LTB, only traditional weak axis buckling.

Sorry to not be more help.

----
The name is a long story -- just call me Lo.

RE: Cruciform LTB

I don't believe that this shape can laterally torsionally buckle (LTB). Ix = Iy, which means that rotation of the beam does not produce added deflection and, therefore, does not result in a lowering of the system potential energy (mathy definition of buckling). In fact, under uniaxial bending, I believe that your effective Ix would actually increase when this particular section rotates. Super stable.

I think that even your coworker's proposal is conservative.

As described in your article, it's important to note that LTB and torsional column buckling are different animals. That said, torsional buckling's off the table here too. It's a non-issue for normally braced wide flange columns and, by inspection, therefore even less of an issue for a flanged cruciform section which is really quite great at resisting torsion. Not HSS great. But nowhere near angle/tee/plate cruciform bad.

I also think that the article's proposal to interconnect the flanges with stiffeners intermittently is junk. It's the same misguided logic that lead folks to erroneously stiffen the crap out of wide flange beams in the hope that will meaningfully increase torsional strength or torsional stiffness. Boo I say. Stiffen 'til you're blue in gills, all the action will just take place between stiffeners.

I like to debate structural engineering theory -- a lot. If I challenge you on something, know that I'm doing so because I respect your opinion enough to either change it or adopt it.

RE: Cruciform LTB

Ix=Iy, but are those the minimum I values for any rotation?

I don't have access to a shape property calculator at the moment. But my gut feel is that there is probably a less stiff orientation somewhere in between there.

Could be wrong!

----
The name is a long story -- just call me Lo.

RE: Cruciform LTB

Even though this is made up of open sections (wide-flanged beams), the composite section is doubly symmetrical. Similar to a closed tube, wouldn't that mostly remove lateral-torsional buckling from consideration? And the more lacing bars you provide, the closer that would approximate a closed tube? Instead of bars, maybe use plates with extended length - that would really approximate a tube.
Dave

Thaidavid

RE: Cruciform LTB

Quote (Lomarandil)

I've looked at this in the past -- can't seem to find those old calcs or references though... sorry.

I recall trying the method mentioned above -- using the "weak axis" W24 to brace the "strong axis" W24 about the axis of rotation (per AISC brace provisions). As I recall, it was a lot of work without much payoff.

I also recently had a project where the flange tips of a built-up cruciform column were battened together as relative bracing. I wasn't able to justify those as providing restraint for LTB, only traditional weak axis buckling.

Sorry to not be more help.

I'm sorry, I don't follow. How would battening together the flange tips not provide LTB restraint in this case? It may be old school, but I've always been told that only 2% of the compression flange force is required to classify as lateral bracing. You must've had some incredible loads to not be able to get the battening to check out.

Just so we are on the same page, when you say battening I am imagining a section similar to the cruciform shown as Figure 7 from page 3 of the document that cal91 linked to above.

edit: I guess if the battening was made of thin plate only, I could see it not working out. I was imagining angles as the shape at first.

RE: Cruciform LTB

(OP)

Quote (Lomorandil)

Ix=Iy, but are those the minimum I values for any rotation?

I don't have access to a shape property calculator at the moment. But my gut feel is that there is probably a less stiff orientation somewhere in between there.

Could be wrong!

No matter how you rotate it, the I value is the same. Ix1 + Iy1 = Ix + Iy. For any shape where Ix = Iy, no matter how you look at it I is always the same. Because of this I agree with KootK on the first half. LTB won't happen here. However, column torsional buckling will. I wasn't asking about it because I feel I already had a good handling on it. But if KootK disagrees maybe we'll open that discussion up :)

Quote (KootK)

As described in your article, it's important to note that LTB and torsional column buckling are different animals. That said, torsional buckling's off the table here too. It's a non-issue for normally braced wide flange columns and, by inspection, therefore even less of an issue for a flanged cruciform section which is really quite great at resisting torsion. Not HSS great. But nowhere near angle/tee/plate cruciform bad.

Here I disagree with you. Yes, they are different animals but, It's NORMALLY a non-issue for braced wide flange columns because usually their weak-axis bracing points will also brace torsionally. However we can think of this cruciform section as two W-sections, each strong axis is bracing the other's weak axis along the entire length. However, they do not brace each other from torsion because they will rotate about the same axis. So the unbraced length for weak axis bending is basically 0, while the unbraced length for torsion and strong axis bending is 40'-0".

I've already ran the calculations for this shape's capacity as a column. KL/rx = 52.7, flexural buckling capacity is 539 kips. Torsional Buckling is 167 kips.

Having said that, I do agree with you about providing intermittent stiffeners does not help with torsion. But they do help with Flexural buckling. If the cruciform were bending about a 45 degree axis, the flanges would want to flatten out to the neutral axis. Providing these intermittent stiffeners prevents that.

RE: Cruciform LTB

(OP)
KootK, you have me thinking now about the columns torsional buckling in terms of minimizing energy in the system. With flexural, and even flexural torsional buckling, this is made apparent because as the column buckles the load on top of the system is lowered and thus lowering the potential energy.

However with purely torsional buckling such as in angles and as I would expect in this cruciform shape, I do not see how torsion lowers the potential energy of the load... I'm starting to wonder if there is even such thing as purely torsional buckling. I always thought angles did purely torsional buckling, but now I'm thinking it's actually torsional flexural buckling. But then AISC 360 Section E4 distinguishes between torsional and flexural-torsional buckling... Any thoughts?

RE: Cruciform LTB

Quote:

(cal91)

I'm starting to wonder if there is even such thing as purely torsional buckling.

There is....but it is precluded for most of the shapes in the manual (at lengths that would be used in construction) where the limiting width to thickness ratios of flanges and so forth (in the code) are followed. It can become critical in singly symmetric shapes or plate girders (forgetting those limiting ratios for a moment).

That's the danger of getting away from the code in this: you are inviting in buckling modes that are not easy to quantify. (Which goes back to my reasoning in my original post.)

You aren't getting a great deal more by figuring this (and risking a great deal) in my experience. For example, I was looking at the AISC article I mentioned in a previous post on this thread......and in one case: the buckling stress went from about 5 to 9 ksi (going from the I shape to the built up one). Doesn't justify that time and risk to me.

RE: Cruciform LTB

Quote (cal91)

Any thoughts?

I hear 'ya on this. I find torsional buckling harder to "feel" somehow. Try telling yourself this story for your particular column: it's effectively four tee shaped columns buckling in unison and each deflecting much as a normal column does when it buckles. I think the logical hangup there is still the center of the column where it's hard to visualize anything moving vertically downwards. In the wild, I would assume that buckling of the four tees would be followed shortly by ordinary lateral buckling possibly combined with yielding of the central bits.

Any chance you'd want to share your calcs on this? I'm surprised by your results obviously. I suspect this could be a learning opportunity for me.

Without running any numbers I have the following, anecdotal doubts about torsional buckling governing:

1) This shape is a popular shape for mega high-rises. Hard to see that working out if there's a glaring torsional buckling issue.

2) I feel that each column in the set should be capable of dealing with its own torsional stiffness needs independently. If one column would be good for P, then two columns ought to be good for at least 2P I'd think.

I'm curious, if you looked at a single w245x55 here, would torsional capacity govern over strong axis? If so, maybe things are shaking out as they have because this simply isn't a good column section, regardless of whether it's used alone or built up. Perhaps this is part of the reason that we consider certain sections to be column appropriate in the steel manual and others not.

I like to debate structural engineering theory -- a lot. If I challenge you on something, know that I'm doing so because I respect your opinion enough to either change it or adopt it.

RE: Cruciform LTB

(OP)

Quote (KootK)

hear 'ya on this. I find torsional buckling harder to "feel" somehow. Try telling yourself this story for your particular column: it's effectively four tee shaped columns buckling in unison and each deflecting much as a normal column does when it buckles. I think the logical hangup there is still the center of the column where it's hard to visualize anything moving vertically downwards. In the wild, I would assume that buckling of the four tees would be followed shortly by ordinary lateral buckling possibly combined with yielding of the central bits.

I see it now. At first it's purely torsional. The four WT's all buckle clockwise about their intersection point. Their flanges move vertically downwards but the center does not. As this happens the effective column section is reduced to just the intersection point, which will then yield/flexurally buckle and thus move vertically down (thereby lowering potential energy).

Quote:

Any chance you'd want to share your calcs on this? I'm surprised by your results obviously. I suspect this could be a learning opportunity for me.

No problem. The numbers I gave you before was for a single W24x55 braced in weak axis but not braced flexurally or torsionally for 480 ft, and then multiplied the capacity by 2. I realize that this isn't entirely accurate so I reran it based on the cruciform shape. The torsional buckling capacity exactly doubled, but what I didn't expect is for the flexural buckling capacity to increase by 70%.




Quote:


Without running any numbers I have the following, anecdotal doubts about torsional buckling governing:

1) This shape is a popular shape for mega high-rises. Hard to see that working out if there's a glaring torsional buckling issue.

Yes, I think that is most likely because we have a beam shape instead of a column shape. W24x55 has the largest Ix to Iy ratio in the steel manual haha. But we have much larger moment demands than axial demands so that is why.

Quote:


2) I feel that each column in the set should be capable of dealing with its own torsional stiffness needs independently. If one column would be good for P, then two columns ought to be good for at least 2P I'd think.

I'm curious, if you looked at a single w245x55 here, would torsional capacity govern over strong axis? If so, maybe things are shaking out as they have because this simply isn't a good column section, regardless of whether it's used alone or built up. Perhaps this is part of the reason that we consider certain sections to be column appropriate in the steel manual and others not.


Agreed! It's just that in both scenarios it's torsional buckling that governs.

Thanks all! It's been fun diving into the nitty gritty for cruciform columns, column torsional buckling, and lateral torsional buckling.

RE: Cruciform LTB

I've only used box sections at columns shared by orthogonal frames, and I suspect that most would approach this problem by ignoring the weak-axis components of the cruciform. That said, I think I mostly agree with your coworker. Drawing from some notes I have from the recent AISC Night School course on stability, the user note equation in AISC 360 F2.2 can be rewritten in a way that helps elucidate the behavior and makes me feel more comfortable using the composite section properties:

Mcr2 = (π2 E Iy / Lb2)(G J + E Cw π2 / Lb2)

The first term represents the compression flange trying to produce minor axis buckling of the entire cross-section. The second term represents the effect of the tension flange resisting this buckling by creating a resisting torque, which includes St. Venant (G J) and warping components (E Cw). For your flanged cruciform section, Iy is the minor axis moment of inertia of the composite section, and J and Cw are twice the values of those used for a single beam.

Lr can be found by using equation F2-6, which sets Mcr equal to 0.7 Fy Sx and solves for Lb. I don't have any insight into how Lp is determined so I would be hesitant to apply it to your section, but it would be conservative to use Lp for a single beam.

RE: Cruciform LTB

Thanks for the calc. It suggests a possible reinforcement strategy:

- Weld some batten plates between the flanges for 2' at the bottom of the column and 2' at the top.

- Now call your column thing torsionally fixed at the bottom and the top. K = 0.5 theoretically but we'll roll with 0.65 for good measure.

- This should increase your torsional capacity by about 2.36 and get you within about 15% of your flexural capacity.

Not bad for a buckling mechanism we barely even believe in.

I like to debate structural engineering theory -- a lot. If I challenge you on something, know that I'm doing so because I respect your opinion enough to either change it or adopt it.

RE: Cruciform LTB

(OP)
Excellent. Thanks Deker and KootK.

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