Unbraced length of a cantilevered beam - slightly new take on an old problem 1

[cspell99]

I know that unbraced lengths have been dealt with in these forums before. I think this situation is a little different from others I have seen posted here. In addition, my question is about the cantilevered section of a beam and not the back span. Hopefully we can avoid the whole inflection point argument.

I am analyzing an existing structure. There is a W24x55 girder that supports K joists at 6'-0" o.c. The K joists connect to the top of the W beam and there is metal deck on top of the joists. The W beam has a main span of 36' and cantilevers an additional 18' for a total length of 54'. The supporting columns are cantilevered with k=2.1. At the end of the cantilevered section of the beam, there is an existing masonry building. The metal deck connects to the masonry wall, but the W beam does not. The W beam has stiffener plates at the two column connections. There are no other lateral braces along the entire length of the beam.

Is the unbraced length of the cantilevered section of the W24x55 beam:
A. 6' since appendix 6.3.1 of the 13th edition AISC says to brace the top flange of cantilevers.
B. 18' since the stiffener plates at the column and K joist at the end of the cantilever effectively brace each other.
C. 36' (or 37.8') since the situation is analogous to a fixed / free column with k=2 (or k=2.1.)
D. 45' because the unbraced length is 2.5L from figure 5.11 of Galambos.
E. 135' because the unbraced length is 7.5L from figure 5.11 of Galambos.
F. Undefined (infinity) because there is no effective brace at either end of the cantilever.
G. Something else?

Thanks in advance for your insight.

 

[CELinOttawa]

Hate to be cliché, but a sketch would be nice.

At first review, B or C, depending on the efficacy of the stiffener and K joist to act as a brace points. This can be analysed and there have been several threads dealing with this.

 

[cspell99]

Picture as requested...

 

[JAE]

It's the weekend (officially) and I have a beverage in front of me - and just off the cuff - I would design the cantilever at Lb = 18'.

 

[BAretired]

I consider the beam to have an unsupported length of 60' as shown. I would introduce bottom flange bracing as required to reduce Lb to a more acceptable value.

BA

 

[CELinOttawa]

BA: Seriously? Why? I'm lost... How would this be more than 36' long? Do you consider the points accross the support to be of no value, even with the stiffeners?

 

[BAretired]

CELinOttawa,
I may be a trifle conservative in this, but a cantilever of 18' has a support moment equivalent to a 36' simple span moment. Under unbalanced snow load, it may be possible to have negative moment across the entire central span.

The stiffeners appear to be single plate stiffeners which offer very little torsional resistance. Had they been half HSS stiffeners so that they effectively continued the column through the beam height, they could prevent beam rotation about a vertical axis and I would agree that my assessment is overly conservative.

In a buckling failure of the bottom flange, neglecting torsional resistance of the steel joists about a horizontal axis parallel to the beam and neglecting torsional resistance of the columns, I believe that the failure mode of the bottom flange would be one continuous curve from end to end of the beam (although I am aware that this is controversial in the engineering community).

Personally, I would not even consider leaving the existing structure in its present state. I believe the bottom flange should be braced twice in the span and once in the 18' cantilever, six feet from the end. In my view, the cost of adding a few bottom flange braces is a small price to pay to provide a safe structure.





BA

 

[CELinOttawa]

BA: I have been, and continue to be, a great fan of your posts. You are one of our "golden dozen", the Engineers who's opinions make me stop and re-evaluate my methods.

You have me thinking. A lot. I'm sure I'll come back to this, but it may be a while. Something tells me my thoughts are going to stew on this one.

 

[BAretired]

Thank you, CEL. I'm flattered.

It may be useful to consider a slightly different example. Consider a doubly cantilevered beam supported on two columns 6' apart with 18' cantilevers each end. Consider the columns providing vertical support, resistance to overturning but no resistance to torsion about a vertical axis. With joists at 6' centers, the "span" portion of the beam has no load at all.

Each cantilever is nearly fixed with only a small rotation at each support. But the bottom flange is not fixed against rotation about a vertical axis. If the beam fails by lateral torsional buckling, it will describe a horizontal arc over a length of 18+6+18=42'.

This is admittedly a conservative approach for a real structure because steel joists welded to the top of the beam actually provide torsional resistance. Columns, particularly HSS columns provide torsional resistance and diaphragm action of the deck prevents the top flange from bending freely in the horizontal plane. The problem is, these features are difficult to quantify so I would prefer to have them in reserve as an additional safety factor. Needless to say, not all engineers agree with me.

BA

 

[dik]

I assume that the columns are well braced...

Dik

 

[dik]

Off the top, I'd strut the joists at the columns and have a max Lu for the beam interior of 36' (I would investigate it for an Lu of 6') and would strut the free end of the cantilever with a joist or spandrel beam and have a max Lu of 12' and use and design the roof deck as a rigid diaphragm.

Dik

 

[271828]

In the SSRC Guide, there's a section on cantilever overhangs. You'd be best off taking a look at that.