diaphragm and beam theory for lateral analysis 2

[conradlovejoy]

I was taught to treat a structure's horizontal diaphragm(s) as if they were bending members (beams) being loaded by wind or seismic forces. I have always been more inclined to obtain forces by using "beam theory" equations as opposed to just "using the trib".

Imagine a beam with three supports, a 60 foot span and a 30 foot span, with a uniform load of 500plf. A beam analysis shows that the reactions are 12.2 kips, 30.9 kips, and 1.88 kips. If one were to use the "tributary" method of resolving diaphragm forces, they would simply take the 500 plf load and multiply it by the tributary widths between the supports, yielding reactions of 15 kips, 22.5 kips, and 7.5 kips. With certain building conditions, the solutions for analyses yielding 1,880 lb shear force versus 7,500 lb shear force could be vastly different. In easy cases where the building is a box with two exterior shear walls, the reactions yielded from both methods will be the same since it is simply wl/2. But when the shear wall layout is irregular, the results from the two methods begin to diverge. Then there are cases of cantilevered diaphragms and how that force is considered to move back into the shear wall at the cantilever. Should the shear wall force be obtained by simply multiplying the cantilever width with the diaphragm force, or should it be analyzed like a true cantilever beam, the shear wall force being equal to the reaction at the beam support? The selection of analysis begins to unravel for me when the results can vary so significantly.

It is my opinion that neither of these methods can be considered to give perfect answers and the real results lie somewhere in the middle. This got me thinking about rigidity/stiffness and the support conditions. In simple beam theory (I don't think) the beam stiffness is considered when computing the reactions. It isn't necessary to determine whether the beam (or diaphragm in this case) is rigid or flexible when the supports are considered pinned and the beam is determinate. I am assuming whatever amounts of actual support deflection and rotation at the supports are negligible, so in theory they are zero and the reactions are determined without and regardless of that consideration. This is why I am concerned that treating the diaphragm like a simple beam may not always be the best analysis method for obtaining lateral forces since there is so much debate as to the diaphragm's rigidity.

I know there is debate whether to treat a wood building's diaphragm as flexible or rigid, but I would like to TRY to not veer completely into that debate for this example. I just want to know if anyone has had similar thoughts when trying to compute the forces that their shear walls will be resisting form the diaphragm.

 

[struct_eeyore]

Bernoulli beam theory ignores shear deformation when considering deflections and reactions - in a deep beam shear deformations come into play, and the result is that force distributions at support will not necessarily follow that of a simple beam. You want to read up on Timoshenko beam theory if you are interested in this - it's represented in partial diff. equations, so rather dense.

 

[phamENG]

I have never used beam theory to determine LFRS shear. It boils down to diaphragm stiffness, as you alluded to. If the diaphragm is flexible (unfilled steel deck in most cases, wood in most cases, etc.), the shear deflection (which is typically ignored in "typical" beams but is the dominant factor in diaphragms) is such that the loading really does distribute roughly along tributary lines. If you have a rigid diaphragm (concrete slab, concrete filled steel deck, etc.), then the shear deflection is minimal and the distribution of forces is dependent upon the relative stiffness of your LFRS elements. I disagree with your assertion that beams are not designed based on stiffness at support conditions - this is a very important aspect of structural analysis and you ignore it at your peril if you're doing much more than a simple span beam. Same thing applies here. Stiffness is king in diaphragm force distribution.

There's also the gray area in between - the Steel Deck Institute refers to them as "Semi-Rigid Diaphragms" (and I think that shows up in ASCE 7 and other places as well). In this, the diaphragm deflects enough that the forces don't distribute only based on support stiffness, but doesn't deflect enough to distribute based on trib. In this case, you have to consider both your diaphragm stiffness and your support stiffness.

The SDI Diaphragm Design Manual (I think it's version 4 now) is a great resource for these calcs.

 

[conradlovejoy]

Thanks. The explanation that the loading actually more closely follows tributary lines for flexible diaphragms helps with my confusion.

 

[KootK]

This was a really neat thread on shear deflection and some of the Timoshenko stuff was included there: Link

While diaphragm stiffness issues make for interesting theoretical discussion fodder, my personal approach is to pick whatever path I find to be a little bit reasonable reasonable and a lot expeditious calculation wise. You go back a couple of decades and diaphragms basically weren't getting designed at all so we've come a long ways. So just having a load path is a pretty big improvement.

And there's just sooo much complexity involved in the accurate modelling of a diaphragm that making any serious attempt at it strikes me as a fool's errand. We are, after all, constructing buildings, not pianos. There's some irony in diaphragm design, I think, in that designers will often engineer the crap out of a big box building diaphragm because it's simple enough that you can and then mostly punt on the design of a complex museum floor diaphragm because it's so complex that it pretty much defies rational analysis.

At this very moment I'm working an a large precast podium diaphragm with multiple buildings on it and, therefore, areas that are topped and areas that are untopped. Good luck "knowing" much of anything in that scenario. It's also a two span diaphragm where an enveloped design including rigid behavior would make the middle shear line completely untenable.

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