Partial Steel Beam Reinforcement Anchor Force 7

[Baffled Engineer]

Hello,

I'm working on a steel beam reinforcement consisting of a new W-shaped beam welded below an existing W-shaped girder, which looks like this:

I'm trying to determine the anchorage force and extension required for partial reinforcement. According to my reference below from the Canadian Steel Handbook, the formula provided consist of the area of the reinforcement times the distance from the centroid of the reinforcement to the centroid of the entire combined section, which is the same variable (Q) used in shear flow calculations. My question is, would this formula still apply to my W-shaped reinforcement? Or is it limited to cover plates?

I'm concerned that there's an implicit assumption that the plate has uniform stress if assumed to be thin, and with the W-shaped reinforcement, there is a considerable stress distribution across the depth of the section. Any thoughts on this? Thanks.

 

[Settingsun]

KootK, it's very tidy how simple the Vratio equation turns out. I was happy enough to integrate the shear stress since the question doesn't come up often.

You'll probably kick yourself at how close you were to the simplified form. In your 8Apr 20:04 post, replace Q/Ar with y_crc (distance between composite section centroid and reinforcement centroid) then rearrange the Srb terms to Srb(y_cbc - y_crc) = Srb.y_cbr = Ir.

What I call the St Venant length is from the St Venant principle, approximately the beam depth. But no fixed number which is why I left it at St V length. There will probably be different opinions. Celt83's stress plots give an indication for the rectangular cross section.

 

[BAretired]

Well, I really learned something in this thread.

Firstly, KootK alerted me to the fact that there was a reaction of the reinforcing beam which had to be taken by a hanger from the original beam. I didn't see it for a long time, but he was right.

Secondly, even after I did see it, I calculated only part of the reaction, namely the reaction from the axial force component, neglecting the moment component. Eventually, I could see that my answer was wrong because the reaction of the two beams did not add up to the reaction of the composite beam.

KootK came up with the correct answer, but steveh49 provided a simplification which I am embarrassed to admit, I missed entirely, namely:

Vreinf / Vtotal = (Ir + Qr * Cr) / Ic

Ir = I of reinforcement about own centroid
Qr = Q of reinforcement when considered as part of composite section (the usual Q value)
Cr = distance from centroid of reinforcement to interface with the main section
Ic = I of the composite section

It is beautiful in its simplicity.

So, thank you KootK and Steveh49.

BA

 

[dik]

Concur... great thread... thanks, Baffled.

Rather than think climate change and the corona virus as science, think of it as the wrath of God. Feel any better?

-Dik

 

[dik]

For Plate reinforcing... have to modify my programs for T,W and L... same methodology...

Rather than think climate change and the corona virus as science, think of it as the wrath of God. Feel any better?

-Dik

 

[dik]

*.pdf file

Rather than think climate change and the corona virus as science, think of it as the wrath of God. Feel any better?

-Dik

 

[dik]

SMath file

Rather than think climate change and the corona virus as science, think of it as the wrath of God. Feel any better?

-Dik

 

[KootK]

@BA: thanks for reporting back to let us know that we've reached a substantial consensus. A degree of closure is a boon for both this thread's active participants and for posterity I think. Being pressed to defend my ideas has also been a boon for me personally. I've been chipping away at the various aspects of elastic, composite design theory on and off for the better part of five years now (mostly with wood). The insights that I've gained here have me dangerously close to complete-ish understanding I suspect.

 

[Baffled Engineer]

Thanks for the help guys.

I really appreciate the effort that went into this discussion and into deriving a simplified formula for everyone.

I suppose my next question would be the effect of preloads prior to reinforcing...but I would leave that for another time as I have decided to shore and unload the girder before reinforcing. Thanks again guys much appreciated.

 

[dik]

Your question was great... I learned a bunch.

Rather than think climate change and the corona virus as science, think of it as the wrath of God. Feel any better?

-Dik