Cruciform LTB 2

[cal91]

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?

 

[cal91]

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?

 

[Archie264]

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

 

[cal91]

It's both

Column at intersection of two perpendicular moment frames.

 

[WARose]

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.

 

[cal91]

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

 

[cal91]

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 :/

 

[WARose]

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.

 

[cal91]

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

 

[cal91]

I found this document.

http://www.newsteelconstruction.com/wp/wp-content/uploads/TechPaper/NSCApril06_Tech.pdf

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 :/

 

[WARose]

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.

 

[cal91]

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.

 

[WARose]

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

 

[cal91]

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

 

[Archie264]

>>>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?

 

[cal91]

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

 

[cal91]

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

 

[Archie264]

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...

 

[Archie264]

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

 

[WARose]

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.