Beams braced with a welded flat steel plate 1

[BWally]

I have a pair of parallel beams that I'd like to brace with a flat steel plate (of at least 3/16" thickness) spanning between them. This plate would be continuously welded to the top flanges of the beams, and also to any cross-beams that might exist. This plate would also have angles welded beneath it to stiffen it for carrying live psf gravity loading. (For the purposes of this analysis, I'd like to ignore the contribution to beam bracing offered by the cross-beams and the angle stiffeners since there may be a situation in the future where these don't exist.)

So, I would like the plate to act as a beam brace and also be able to directly carry transverse loads itself.

I have researched the topic of "diaphragm-braced beams" by ordering and reading several papers published in the late 1960s to mid 1970s in the ASCE Journal of the Structural Division. The paper "Columns and Beams Braced by Diaphragms" by Errera, Pincus, and Fisher gives formulas for calculating the critical lateral-torsional buckling moment "Mcr" when diaphragm bracing is present. However, their equations are dependent on a quantity called "Q", the shear rigidity of the diaphragm, which is the product of diaphragm cross-sectional area "Ad" and effective shear modulus "Geff". The authors state that "Shear rigidity ["Q"] of diaphragms was determined experimentally because no general theory of diaphragm behavior was available. Such theory is necessary for the general application of the results of the investigation to practical design situations".

None of the other papers I reviewed presented a general theory of diaphragm behavior applicable to my welded flat steel plate.

Have any theories or data been published since then that I could use for my situation? The most helpful information would be values of "Q" to use in the equations presented by Errera, Pincus, and Fisher in "Columns and Beams Braced by Diaphragms". Also, is there any information on how the bracing capability of the diaphragm is diminished when it is loaded transversely?

 

[JAE]

AISC has information on bracing of beams against LTB that is much more up-to-date and relevent than papers from the 60's and 70's. Have you read through the AISC steel specification?

 

[BWally]

JAE,

Thanks for responding. I've read a lot of different posts, and I always find the quality of your responses quite high. I’m glad that someone of your caliber has responded to my post. (I hope this doesn’t scare off other responders—I want to hear all thoughts on this topic.)

We design using the AISC Steel Construction Manual. I use Allowable Stress Design specifically. It's not the general bracing of beams that I'm concerned about. I have a pretty good understanding of how to design beam bracing, thanks to Dr. Joseph Yura's paper "Fundamentals of Beam Bracing". What I'm concerned with here is the specific situation of bracing beams with a welded flat steel plate. The paper that I referred to above would be extremely helpful if only I knew how to evaluate the shear rigidity of the diaphragm "Q". In this paper, there exists an equation for calculating the lateral-torsional buckling moment of a diaphragm-braced beam. If I knew “Q”, I could plug it into that equation, solve for “Mcr”, divide by 1.67 if it’s in the elastic range, and that gives me my allowable moment. As I’m sure you know, the AISC ASD Manual makes no mention of diaphragm-braced beams, and I presume the new 13th edition black Manual makes no mention of beam bracing with a welded flat steel plate. That's why I took off on my own and tried to find published papers on the topic of diaphragm-braced beams.

I know it would be easier to just use a bracing system that's easier to evaluate, but let’s pretend I’m constrained into bracing my beams with a welded flat steel plate and can’t use any other bracing configuration.

 

[WillisV]

Bwally - Actually the AISC 13th edition does have information that would prove useful to you. You can download the specification for free at

http://www.aisc.org.

Take a look at Appendix 6, section 6.3, part 2A nodal bracing. Equation A-6-10 provides the "required cross frame or diaphragm bracing stiffness" needed at each point of restraint. In your case you would simply calculate this required stiffness and compare it to the 6EI/L provided stiffness of the flat plate (assuming double curvature in the plate - reffer to the commentary for Appendix 6).

 

[JAE]

Very nice post there, WillisV. BWally - does that answer your questions?

 

[BWally]

JAE and WillisV,

According to the research I've done, a flat plate diaphragm can brace a beam in 2 ways: one is with its weak-axis bending stiffness as a continuous torsional brace; the other is with its "in-plane shear rigidity". The later 3 of the 4 papers that I read neglect the weak-axis bending stiffness of the diaphragm because the authors claim that its contribution to bracing is much less important than the contribution from in-plane shear rigidity.

So I could use the equations from Appendix 6, but I'd be greatly underestimating the bracing capability of the plate by not counting its in-plane shear rigidity.

 

[Dinosaur]

I must be missing something. I was under the impression that you were welding a flat plate to the top flanges and the plate was spanning the beams so it could carry transverse loads. Diaphragms are attached to webs and don't carry gravity loads. Could you describe your geometry once again? Thanks.

 

[BWally]

Dinosaur,

You are correct in your re-statement of the geometry. The plate will carry transverse loads AND diaphragm-brace the pair of beams.

Diaphragms are not necessarily attached to webs. In the paper "Columns and Beams Braced by Diaphragms" by Errera, Pincus, and Fisher published in ASCE Journal of the Structural Division in 1967, their derivations are based on diaphragms attached to one or both flanges of the columns or beams they are bracing. The tests they performed were with corrugated diaphragms attached by fasteners to both flanges of a pair of columns and to the top compression flange of a pair of beams. They do not attach the diaphragms to the webs.

 

[Dinosaur]

BWally,

Thanks for the clarification. These diaphragms described in your paper, are they attached to both flanges? I think a plate attached to the top flanges only would provide a modest amount of support but does not qualify as a diaphragm. Likewise, a flat steel plate of 3/16 inch thickness will not carry much load as compared to highway loadings I am familiar. Good Luck.

 

[BWally]

Dinosaur,

The authors' formulas address diaphragms attached to one or both flanges. Their beam tests were performed with the diaphragm attached to the top compression flange ONLY.

The authors state the following at the end of their paper: "Present theory and test results demonstrate conclusively that shear-resistant diaphragms, properly attached, can be highly effective as lateral bracing for slender columns and beams." And: "Diaphragm bracing is also effective in supporting slender beams against lateral buckling. The yield moment of beams appears to be readily attainable using shear-rigid diaphragm bracing."

Based on everything I've read so far on this topic, I think a plate attached to only the top (compression) flanges could possibly provide a great deal of bracing strength and stiffness. Could you elaborate a little as to why you don't think it would?

I agree that the plate I end up using will not carry much transverse load unstiffened. It will have to be stiffened with angles welded beneath it to carry the psf loading.

Thanks for your input.

 

[BWally]

Come on, fellow engineers. Somebody out there has to have dealt with this situation before.

 

[Dinosaur]

My experience is that it is recommended that diaphragms be as deep as possible and if possible connected to both flanges, sometimes indirectly through a connector plate. I would ask the author(s) of the paper for guidance on their recommended methods.

 

[BWally]

Good idea, Dinosaur. I hope I can locate them. The papers are old, so the authors may be retired or deceased.

Do you know anything about the Diaphragm Design Manual published by the Steel Deck Institute (SDI)?

 

[Dinosaur]

I work in the bridge industry. I know of a few studies including one mentioned by Dr. Dennis Mertz at an HPS steel conference that reasserts the need for attachment to both flanges. Good Luck

 

[miecz]

The word "Diaphragm" in the bridge industry has a completely different meaning than the same word in the building industry. I believe we have two structural engineers discussing two different structural elements with the same name.

Dinosaur-

In the building industry, a "diaphragm" is a plate attached to the top flange of a beam. Examples are: roof sheathing, metal deck on steel joists.

BWally-

In the bridge industry, a "diaphragm is a vertical member between stringers, often attached to the web, usually spaced 5 to 15 feet along the stringer. Sometimes these are channel sections, other times, cross bracing.

It doesn't appear to me that Appendix 6 answers your question. I don't see a solution in the deck design manual, either. I would look in the AISI Cold Formed Steel Design Manual, and the Aluminum Association Manual.

The "Q" term wouldn't seem to apply in this case. As you say, "Q" is the shear rigidity. For typical roof decks, flexibility is mostly related to shear. In this case, I believe your bending deflection will be equally significant.

 

[BWally]

Thanks, jmiec. I will look into the 2 manuals you suggest.

I agree--Appendix 6 does not completely answer my question. I've about decided that if I use the torsional brace method in Appendix 6, adapted for a continuous brace like my plate, it will provide me with a "lower bound" for the bracing capability of my plate. If I choose my plate thickness based on the method in Appendix 6, it will without doubt provide my beams with the bracing strengh & stiffness they need because I wouldn't even be considering the in-plane shear rigidity of the plate, which is far more significant than its weak-axis bending resistance. I plan on reducing the stiffness term "6EI/L" to account for the plate's deflection at the beam caused by the transverse loading on the plate. (The transverse loading will reduce the stiffness available at one end of the plate to resist the bracing moment "Mbr", but will increase the available stiffness at the other end. I will use the smaller value for my brace stiffness.)

Could you elaborate a little on your last paragraph?

 

[BWally]

Dinosaur,

I found out that one of the authors of the paper I mentioned earlier still teaches, here in the same city I live in! I found his email address, emailed him, and he said he no longer practices engineering and that since the paper was almost 40 years old, he wouldn't be able to help me. Oh well, at least he wrote me back.

 

[miecz]

BWally

Steel roof beams often carry a metal roof deck. While I don't have the referenced paper, I'm guessing that the authors (Errera et al) were studying the ability of roof deck to provide lateral bracing to the beams.

Roof deck is stitched together and is shaped like an accordian. As such, it's stiffness is typically expressed in terms of effective shear modulus and cross sectional area. It sounds as though the authors were relating lateral brace effectiveness in these terms.

In your case, you have a solid plate. It's not stitched together as roof deck is and it's not shaped like an accordian. Not only that, but it's welded continuously to the beams that it braces. The result is a composite horizontal beam. I think this is a whole different animal. My gut tells me this is stiff enough to consider the beam continuously braced.

I ran into a similar condition a few years back with parallel beams crossing a circular concrete tank. I remember thinking that a plate should work, but couldn't find the research to back it up.

By the way, make sure you prevent rotation of the beams at the supports. I would put a bridge style cross brace (bridge diaphragm) at the support points.

Have you checked the Structural Stability Research Council Guide to Stability Design Criteria for Metal Structures?

 

[BWally]

jmiec,

Yes, the authors (Errera et al) were studying the ability of decking to provide lateral-torsional bracing to beams. But the mathematical derivations they began their paper with did not assume anything about how the decking was attached to the beams (other than defining it as “continuous” connection) or to itself, and did not assume anything about its profile. In other words, they kept their derivation very general. The equations that resulted from it could then be applied to any kind of diaphragm bracing, be it corrugated decking stitched together as you described, or solid plate welded continuously to the beams.

You really got me to thinking when you mentioned “composite horizontal beam”. I started to wonder if I could treat my 2 parallel beams + plate as a composite beam with the beams as the flanges and the plate as the web. The distributed brace force would then be like a distributed load on the composite beam, creating compression or tension in my 2 parallel beams and shear in my plate. (I suppose it would also create torsion in my 2 beams because the plate is connected at the top flanges and not at mid-depth.) But the more I think about it, the more I think that this would not be a correct treatment of the situation. I think the treatment presented by Errera et al is the proper way to analyze it.

I bought the Guide to Stability Design Criteria for Metal Structures a couple of months ago. It was what actually referred me to all the research papers on diaphragm bracing. The Guide doesn't really go into much detail, just a brief discussion and presentation of a formula that involves that old "Q" term that keeps rearing its head.

Your 2nd-to-last paragraph brings up another thing I've been puzzled about until recently. AISC always states that rotation about the beam's longitudinal axis must be prevented at beam supports. I finally found something in the 13th edition, where they mention that, in regards to rotation prevention at beam support points, “the strength and stiffness requirements of Appendix 6 can be applied to ensure the stability of the assembly”. The only other mention I’ve seen of this is in a 1972 paper by Nethercot and Rockey called “A Unified Approach to the Elastic Lateral Buckling of Beams”, where they recommend a torsional stiffness at the support = 20 x GJ/L. Not to put you on the spot, but in your practice, when do you consider that rotation at beam supports has been prevented?

Also, what’s your thinking about how the transverse loading of the plate affects its bracing ability?

 

[miecz]

BWally,

If the authors were studying the ability of decking to provide lateral restraint to beams, then the solution only applies to metal deck. Under shear load, metal deck doesn't behave like a solid steel plate. It deflects uniformly in shear, as though the moment of inertia were infinite.

However, if this was a general solution that requires a shear rigidity, you may be able to derive one. Say your plate were cantileved half the length of your beam with a unit load on the end. How much would it deflect, due to bending and shear? The load divided by the cantilever length and deflection is an effective shear modulus. Multiply by the area of you plate and you have "Q". Note that the derived "Q" is a function of the cantilever length. I think half the length of the beam is appropriate.

In the above derivation, I think you'll agree that you should use the moment of inertia of the composite horizontal beam.

Rotation at the beam supports is prevented by a cross brace, bridge style, with horizontal elements at the top and bottom of the X. Alternately, if both the top and bottom flange are prevented from lateral movement.

I'm not sure I understand what you mean by transverse loading of the plate. If you mean load out of the plane of the plate, and the resulting tendancy to draw the beam flanges together, I think it's effect is small and can be ignored, assuming, of course, that the plate has been designed to carry the out of plane forces. I would probably overdesign the plate for the transverse forces.