Moment of Inertia
Moment of Inertia
(OP)
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?
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?






RE: Moment of Inertia
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
RE: Moment of Inertia
RE: Moment of Inertia
http://www.FerrellEngineering.com
RE: Moment of Inertia
RE: Moment of Inertia
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.
RE: Moment of Inertia
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.
RE: Moment of Inertia
Ordinarily I would ignore the contribution of the bottom plates and consider only the moment of inertia of the 1" x 12" plate.
BA
RE: Moment of Inertia
Mike McCann
MMC Engineering
Motto: KISS
Motivation: Don't ask
RE: Moment of Inertia
BA
RE: Moment of Inertia
Mike McCann
MMC Engineering
Motto: KISS
Motivation: Don't ask
RE: Moment of Inertia
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
RE: Moment of Inertia
I guess I should read for better comprehension next time.
RE: Moment of Inertia
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 :)
RE: Moment of Inertia
Michael.
Timing has a lot to do with the outcome of a rain dance.
RE: Moment of Inertia
RE: Moment of Inertia
As detailed, the plate may add some stiffness to the HSS but not much.
BA
RE: Moment of Inertia
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.