Moment of Inertia 1

[slickdeals]

Folks,
Pardon my naive question.

Imagine you have a 12" deep, 1" thick plate. To this plate's bottom, 4" wide x 0.5" thick x 6" long plates are welded every 2'-0" o/c. to create a T-shape. Note that the flange is discontinuous.

Are there methods to approximate the average moment of inertia of such a section for deflection calculations? Note that strength is not an issue.

I was able to model it in a FEA program and as expected deflections are lesser than a unflanged plate. I want to be able to calculate this by hand. Any ideas?

 

[BAretired]

slickdeals,

Take the Moment Diagram and divide it by EI which in your case, is variable. Use Moment area theorems to calculate rotations and deflections at various points.

BA

 

[frv]

I second BA. Moment-area is the way to go.

 

[connectegr]

Using AutoCAD you can draw the scale cross section and AutoCAD will provide the shape properties.

http://www.FerrellEngineering.com

 

[dhengr]

BA..... I’m losing faith in you. I agree with your first sentence, but that you didn’t then blurt out, use Newmark’s Methods...., shakes me timbers. Nuff said.

 

[JStephen]

You could calculate the moment of inertia with and without the little plates, and then just take the average along the length of the beam, which would probably work as well as anything and be vastly simpler.

The beam bending equations aren't necessarily going to be accurate at a discontinuity anyway. If you treat it as a series of beams of two different cross sections, you'll still have an approximate solution, as the transition from one stress state to the other is going to be gradual, not an abrupt change like that approach assumes.

In your finite element approach, if you model it as a series of beams like that, you should get one answer, and if you model it as smaller plate elements, you should get a more correct answer.

 

[paddingtongreen]

This is a classic case for the method I was taught when I started Theory of structures.
Draw the load diagram.
Draw the shear diagram. From the left end, add the area of the shear diagram at distances "x" to the left of "x", to draw the bending moment diagram.
Draw the moment of inertia diagram
Draw the M/I diagram. Load the beam with the M/I values to draw "Secondary shear diagram" This, divided by E, yields slope. From the left end, add the area of the secondary shear diagram at distances "x". to the left of "x", to draw the secondary bending moment diagram. This, divided by E yields deflection.


Michael.
Timing has a lot to do with the outcome of a rain dance.

 

[BAretired]

In re-reading this post, I note that the bottom plates were only 6" in length and spaced at 2' centers. The moment of inertia does not change from one value to another immediately, there is a gradual transition and so I tend to agree with J. Stephen that a modified EI applied across the span is probably at least as accurate and definitely much simpler to apply.

Ordinarily I would ignore the contribution of the bottom plates and consider only the moment of inertia of the 1" x 12" plate.

BA

 

[msquared48]

Model it in RISA as a segmented beam with fixed joints. It will calculate the deflections for you at any point along the beam you need.

Mike McCann
MMC Engineering
Motto: KISS
Motivation: Don't ask

 

[BAretired]

Mike, it may calculate something, but the assumptions made by the program are not necessarily accurate. In fact, I suspect they are totally erroneous.

BA

 

[msquared48]

Just curious BA, but what have you found?

Mike McCann
MMC Engineering
Motto: KISS
Motivation: Don't ask

 

[BAretired]

What have I found? Not much. When I first responded to this thread, I had not noticed that the plates were only 6" long spaced at 24".

The change in I is not immediate and, when the plates are only 6" long, it is not clear what the moment of inertia of the Tee shape is at each end of the plate and how it varies in between.

My best guess is to ignore the welded plates and consider the shape to be 1" x 12" throughout for the purpose of calculating deflection. There will be some minor benefit as a result of the bottom plates but it is not clear how much so it is best to ignore it.

BA

 

[TXStructural]

I missed the 6" long part of the plate description also... why would you ever consider a change in I for such a thing. If anything, they only create discontinuities which focus stress (stress risers at welds). You are spot on about ignoring their effect, if any, in deflection calcs.

I guess I should read for better comprehension next time.

 

[slickdeals]

The question I posted is a simplification of a problem that I had. The real question is this:

One of my colleagues had to design a HSS12x4x5/8 jamb (size restricted by arch. requirements) for supporting glazing. Because deflections were an issue (we are in hurricane country), our detail called for a knife plate in the 12" direction to improve I.

However, the shop drawings came back with the sketch attached. They are a very reputed firm in the US and the detail said the I is now 310 in^4 (including a "we do this all the time comment").

I think that the moment of inertia number is bogus because the inner plate won't get engaged because its rotation is not prevented. The only way I see this detail working is if the rod was welded to both the HSS walls and to the plate itself to lock rotations.

Your thoughts?
PS: I should get points for attaching a sketch

 

[paddingtongreen]

Will it work? I think not, with the hole clearances the section must deflect significantly before you can be sure that the plate engages. with accurately placed holes and normal clearances, that would yield +/- 1/32" at each bar. With bowing in the middle of the span, that means 1/16" total. If it is a column with the top deflecting relative to the bottom (swaying) there is no guarantee of contact, no guarantee that the plate will contribute anything.

Michael.
Timing has a lot to do with the outcome of a rain dance.

 

[ishvaaag]

Obviously will depend on the engagement of what added to the tube, and reasonable investigation can be done with a FEM model as you have done. Mostly, floating independently is what will make the addition unsuccessful to restrain deflection and add strength. Welding at ends and lateral restraint of the inset plate is a good start for entirely counting the added inertia. If not welded at ends, the clearances in the bolt holes and maybe with the inner walls of the tube may have a say; again this can be investigated with models, not necesarily FEM.

 

[BAretired]

Even if the rods were welded to the plate, they would have to bend in order to transfer force from the HSS walls to the plate. The 10" plate is 3/4" short of the clear dimension of the HSS. If it was 5/8" x 10.75", it might contribute significantly to the stiffness, but it would be almost impossible to install.

As detailed, the plate may add some stiffness to the HSS but not much.

BA

 

[rscassar]

The "we do this all the time" argument is meaningless unless someone can actually prove to you the theory behind it.

I think the plates will stiffen the section as mentioned and help to reduce deflections. Because of the shear lag, the full stiffness of the composite section will not be engaged immediatley. Instead of modelling the composite section for 6" out of 24", I would only model the composite section for say 4" out of every 24" to allow for that shear lag effect.

This is analogous to tension stiffening in reinforced concrete design where an increase in Icr is used to allow for the stiffening contribution of the concrete cross-section between cracks.