Braced Frame Hand Analysis 4

[Naggud]

Hi all,

I'm doing hand calculations for a one storey braced building. I need to check the deflection of the building. To do so I've done what I thought was a simple hand check (attached).

However, I did a very quick check with STAAD just to make sure my hand calc was accurate but the deflection of the frame in STAAD comes out as exactly twice the deflection that I have calculated. I'm sure there's something very simple wrong with my hand calc but I can't see it myself. I don't want to use the results of STAAD without understanding them.

I would appreciate if one of you could have a quick look and let me know if you can see the issue with my hand calc.

Also, I see that the units for area are incorrect - supposed to be cm2!

Thank you very much

 

[XR250]

You will have deflection/elongation in your right hand column as well that is unaccounted for

 

[SwinnyGG]

I am not a structural engineer so I won't attempt to diagnose your calculation problem.

I'm just here to say that I'm happy to see someone who doesn't just plug numbers into their simulator of choice and move on to the next. Checking your work and understanding why, closing the loop so to speak, seems to be an instinct that becomes ever more rare as the years pass.

 

[BS2]

It looks like you calculated the deflection along the diagonal instead of in the x-direction.

 

[jayrod12]

This was posted some time ago. I grabbed it as the best hand calc braced frame deflection reference I had come across.

 

[Naggud]

Thank you for all of your responses.

I plugged the numbers into the top formula and I got the correct answer (the STAAD answer)!

Has anyone got a derivation for the formula by any chance/the reference for the book that the formula comes from? How is the Lbrace^3/Lhoriz^2 arrived at?

 

[canwesteng]

I think deflection being exactly 2x larger in STAAD is a fluke. You need to calc axial deformation in the column as well, then determine the deflected shape of the truss, then calc deflection at the node. Axial deformation of the brace alone doesn't tell you where it's going.

 

[KootK]

I derived this myself a while back and was, frankly, suprizsed at the complexity of it owing to the factors mentioned above. It can be handled quite easily using virtual work applied to the two loaded members of the frame (brace and tension column). The rest is just algebra and trig. Give it a go and, if you get stumped, report back. I'll rederive it if need be (and I'm feeling ambitious).

I like to debate structural engineering theory -- a lot. If I challenge you on something, know that I'm doing so because I respect your opinion enough to either change it or adopt it.

 

[Istructeuk]

https://s13.postimg.org/m5cnnho3b/image.png

 

[Guest262653]

Let's see if I still get this right...

Applying the virtual work principle, the horizontal displacement would be obtained by equating work done by a unit virtual force on the final displacement of the structure to the internal work done by the internal forces on the member virtual internal displacements:

1.0*Delta_h = SUM(N*N'*L/EA)​

Applying a virtual horizontal force of 1.0kN at the top node, the horizontal displacement would be the following, assuming a constant area of 40.3cm2:

1.0*Delta_h = [(-141.42)*(-1.4142)*1.4142*5.0 + 100*1.0*5]/(210E6 x 40.3E-4) = 2.26mm​

 

[JoshPlumSE]

Some quick thoughts, when running this in FEM / STAAD:
1) Turn off P-Delta.
2) Make sure you are not using any stiffness adjustments per AISC's "Direct Analysis Method".
3) If STAAD allows it, turn off shear deformation.
4) Don't just validate your deflections. Make sure to also validate the force distribution and joint reactions that you have assumed. That might lead you to a difference between your model and your hand calc assumptions.

 

[Guest262653]

And here's the derivation for the general case using the virtual work principle:

 

[KootK]

Way to step up avscorreia!

I like to debate structural engineering theory -- a lot. If I challenge you on something, know that I'm doing so because I respect your opinion enough to either change it or adopt it.

 

[chris3eb]

Here's what I'm getting. I think avscorreia has an incorrect last term (d^2/h instead of h^3/L^2). What I have is a combination of jayrod's equation (considers brace and girder) and Istructeuk's equation (considers brace and column). Whether or not the girder influences anything is dependent on where you apply your lateral load and where you are looking at deflection. The load AND deflection have to occur away from the brace for the girder to contribute. When the load or deflection is at the brace, I get what Istructeuk had. I guess jarod's equation assumes an axially rigid column.

 

[Guest262653]

You're absolutely right. That last term seemed odd... The mistake is in the value of the axial force in the column. I used Q/sin(theta) but is, in fact, Q/cos(theta)*sin(theta) = Q*tan(theta) = Q*H/L.
Sorry. My bad.

 

[Naggud]

Thanks very much everyone, that's fantastic.
I now understand where the answer is coming from.

 

[KootK]

With the table that jayrod posted, it's important to read the fine print carefully. It's the shear deflection per story, not the total deflection per story. Unless I miss my mark, I suspect that it's intended use was for multi-story structures where a designer would follow this procedure:

1) Calculate the "cantilever" component of deflection at a given floor using the moment of inertia of the chords.

2) Calculate the shear component of deflection a a given floor using the equations cited.

3) Add 1 and 2 to get the total drift.

I think that was a more intuitive way for folks to investigate multi-story frames back in the day before computer modeling became ubiquitous. For low rise frames, the cantilever (column) contribution to drift is often negligible, as evidenced here.


I like to debate structural engineering theory -- a lot. If I challenge you on something, know that I'm doing so because I respect your opinion enough to either change it or adopt it.