Beam Stability Running Away ...

I just did the coolest problem.  Actually, it was frustrating in the process.  I gave my students an assignment of determining the required beam size for a problem we did in class.  In class the beam was fully braced laterally.  Their homework problem was to determine the size if unbraced along its entire length. In class we solved for the fully braced case in glulam (24F-V4 DF), and also steel W shape (A992, 50 ksi).  Here are the conditions:

SL 60 psf
DL 15 psf
Span 30 ft
Spacing (trib width) 12 ft
Deflection limits: L/240 for SL and L/180 for TL (excl creep)
An additional 5 psf area load was assumed to accommodate the beam self weight.

Here is what we got for a fully braced beam:

5-1/8 x 25.5 24F-V4 DF
6-3/4 x 22.5 24F-V4 DF

and

W 16 x 26 A992 Gr. 50 ksi

We were able to use the AITC Span Tables (after getting line load from area load, of course) for the glulam.

For the W shape we did LRFD plastic hinge design, compact section, etc., blah, blah ...

Then in class we released the top of the STEEL beam from lateral support ... the example being, perhaps, of a wall coming down on a beam without some kind of ceiling or something at beam level to prevent lateral movement.

And we got a beam of some DOUBLE the weight ...

W 10 x 54 ... W 12 x 53 ...

The W 10 x 54 barely cleared the deflection criteria.

The W 10 x 54 is a stinking stout beam ... as wide as it is tall ...

With this, then, I sent them home to find the unbraced glulam sizes ... with the suggestion they start with a `guess' on C_L of ... 0.50 ... (after all, we were needing twice the steel beam, unbraced).

Instead of getting some other work done - I decided to solve what I had asked my students to do.

Amazing ... the guess of C_L of 0.50 for the 5-1/8 in. wide beam undershoots ...

In other words, if we let f_b = F_b' with a guess of C_L of 0.50 and solve for the required section modulus, we get a section that is 33 in. deep and drives C_L to 0.414 (using the 2005 NDS provisions and the Emin's of AITC 117 2010).

Resolving using an even deeper beam ... C_L continues to drop Fb'.

The deeper section reduces f_b ... but not fast enough ...

We go off the charts ... looking for a 5-1/8 in. wide beam (depth) that will work.  And we can't find one!  At least in any reasonable depth.

FINE ... a tiny bit surprising ... certainly interesting



BUT WHAT I WOULD DISCOVER NEXT WAS EVEN MORE SO!


Turning to the 6-3/4 in. wide beam ...

The guess of C_L ... OVERSHOOTS!

It results in a depth of about 30 in.

But the 30 in. depth results in a C_L of 0.73.

So I tried a lesser depth ...

27 in. depth gives C_L of 0.788

Fb' = 2175 psi

And ... fb = 1580 psi ... STILL OVERSHOOTS.

Trying 25.5 in. ... STILL OVERSHOOTS.

Hmmmmm ... 24 in. ... STILL OVERSHOOTS.

W.T.F.!!!

Trying the 22.5 in. depth - the depth we got with full bracing ... it works for the unbraced condition.

I DID NOT EXPECT THAT!

Especially with what the unbraced length did to the other beam selections (5-1/8 wide glulam and W shapes) ...


The Volume Factor ...

For the 6-3/4 x 22.5 beam the C_L calcs out 0.866 for the unbraced length of 30 ft.

For the fully supported beam C_L is, of course, 1.00.

But in both cases C_V is 0.882.

So for the braced beam the 22.5 in. depth had a penalty due to volume nearly equivalent to what the stability effect would be when unbraced.

The Beam Stability factor took over with the release of the bracing, BUT NOT BY MUCH.


Answer:


                                      Braced                             Unbraced

24F-V4, 5-1/8 x ...        25-1/2                                 None

24F-V4, 6-3/4 x ...        22-1/2                                 22-1/2

W A992 50 ksi            W 16 x 26                          W 10 x 54??? or W 12 x 53      

WILD!!!

jrf

(when I get the scanner working I may include the calcs ... maybe)

Update: here they are ...

...