Biaxial bending on columns

[Mixtli]

Hi everyone,
Please refer to the attached.
So the stress make the cross section bend about one line, that could possibly be the NA.
But a) I drafted the moment vectors and the resultant and b)drafted a line between the centroids of the compression and tension stresses an both are co linear, which is consistent with the forces applied to create those stresses.
Judgement says that the NA should be perpendicular to the resultant moment. The thing is that this NA is totally different from other NA (From stresses diagram)

What am I missing here?

Thanks a lot guys

M.

 

[rb1957]

please refer to the notes "no student posting"

Quando Omni Flunkus Moritati

 

[BAretired]

Actually rb, we don't know that he is a student. In any case, I believe we touched on this issue on a recent thread. If I remember correctly, the neutral axis is not always parallel to the resultant moment vector (not perpendicular either, but I think mixtli(OP) meant to say parallel).

BA

 

[BAretired]

Yes, I think I have found it. It is thread507-365763.

BA

 

[IDS]

The thread linked by BA has a link to a spreadsheet that will calculate the resultant moment angle for any specified NA angle, for any shape defined by corner coordinates.

If the section is not symmetrical about a perpendicular line through the mid-point of the NA then there will be a moment about the perpendicular as well as the moment about the NA, so in general the resultant moment axis will not be parallel to the NA, except for circular sections and certain angles with rectangular sections or other regular shapes.

Doug Jenkins
Interactive Design Services

http://newtonexcelbach.wordpress.com/

 

[rb1957]

I picked on "assignment" in the attachment title, and assumed ...

Quando Omni Flunkus Moritati

 

[Mixtli]

Thanks BA.
RB, I just took that assignment from and old course and update it to what I'm doing at work. I should have changed the filename, lol.
Anyways, I had gone through that thread before I posted here with no luck. To me the NA is the one the moment spins around, no? And therefore is perpendicular, on the other hand the resultant moment could be the right hand rule where the thumb points to the direction the spinning force goes?

Anyways I will read more carefully that thread again and see if I can find more sense to it. Like a movie, the more I watch it the more things I discover about it.

I'm sure I'll come and ask more questions.

Cheers

 

[rb1957]

i think the problem is your moment vector (which should be normal to the NA) ... what happens when you plot Mx along the x-axis and My along the y (it looks to me that you have Mx along the y-axis)?

Quando Omni Flunkus Moritati

 

[BAretired]

Using the right hand rule, Mx is a moment about the Y axis and My is a moment about the X axis. Mx and My are vector quantities shown by a double headed arrow pointing in the direction of the right thumb when the right fingers are curled in the direction of the applied moment. When Mx acts alone, there is uni-axial bending and the NA is parallel to the vector Mx, not normal to it.

BA

 

[rb1957]

but not by his calc, Mx*y/Ix ... maybe that's the problem ?

and yes, I know, My is typically sagging in the x-z plane which usually puts it in the +ve z-axis (using typical FEA convention, x- along the element) and Mz is then along -ve y-axis.

in any case the moments as plotted don't match the calc.

Quando Omni Flunkus Moritati

 

[Mixtli]

Hi guys,
I revised my sketch.... I was a little lost with the right hand rule, I think I got it with BA's explanation.
I think I got it right now, but I still have a some doubts about the magenta line (pink) which don't align with the resultant vector.
Now, in uni-axial bending, the NA is parallel to vector Mx, I have no doubt about it, but the same applies in biaxial bending?, i.e. the NA is parallel to the resultant vector?

The calcs I did with the general stress formula were to determine the stresses at the corners and find the line where about the cross section pivots, which I assumed is the NA.

I am missing something here and dunno what it is...

Cheers!

 

[IDS]

The confusion over the convention for the bending moment is a side issue.

Judgement says that the NA should be perpendicular to the resultant moment.

If you rotate and translate the section so that the NA (the line of zero stress) is on the X axis, with the Y axis through the mid-point of the NA, then in general the centroids of the compressive and tensile forces will not be on the Y axis, so there is a moment about the Y axis, so the resultant moment axes are not coincident with the NA and NA-perpendicular axes, however you want to define them.

Doug Jenkins
Interactive Design Services

http://newtonexcelbach.wordpress.com/

 

[BAretired]

Using the right hand rule, Mx is a moment about the Y axis and My is a moment about the X axis. Mx and My are vector quantities shown by a double headed arrow pointing in the direction of the right thumb when the right fingers are curled in the direction of the applied moment. When Mx acts alone, there is uni-axial bending and the NA is parallel to the vector Mx, not normal to it.

I'm sorry, I seem to be getting everyone confused including myself. Mx is the moment about the X axis and My is the moment about the Y axis. On your diagram, X is the major axis and Y the minor axis. So Mx = 480 and My = 240 kn-m due to a 600 kN point load at corner D. Your vector diagram is correct except that the notations Mx and My should be swapped.

If your column is an elastic material, your stresses appear to be valid but if you are talking about concrete, you cannot sustain a tensile stress of 5556 kPa at Corner A. The concrete would crack and the reinforcement would have to be taken into account.

If you are investigating the ultimate biaxial bending behavior of a concrete column, the strains will be linear but the stresses will not, so the location of the NA is different than it is for the elastic range.

For an elastic material in the elastic range, your magenta line represents the NA and the resultant moment vector is not parallel to it. For a material such as reinforced concrete, the result would be different but you still would find that the NA was not parallel to the resultant.

Why don't you check out the IDS program and see what you get?

BA

 

[BAretired]

800 kN load, not 600 kN. Sorry...bedtime for me.

BA

 

[rb1957]

why would you plot a +ve My along the -ve y-axis ?

from your equation Mx is the moment along the x-axis, so stress = Mx*y/Ix.
both momenta are positive, no??

Quando Omni Flunkus Moritati

 

[rb1957]

btw, if Mx is the moment along the x-axis, then the bending stress should be Mx*y/Ix (causing tension on the +ve Y-axis), but it should be -My*x/Iy ('cause +ve My causes compression on the +ve X-axis).

Quando Omni Flunkus Moritati

 

[BAretired]

It's easier to mark up the drawing. See link.

BA

 

[Mixtli]

Thanks for the markup BA, and the explanation before that; I'm gonna revise it and send it over again, if you don't mind, but I will revise the moment diagram and keep the X axis where it was before, it's where it should be as standard practice.

I have tried Doug@IDS's spredsheet, impressive by the way, but excel crashes if I have no rebar, or if I try putting the rebar on the CG, I'm trying to omit the rebar cause there is no rebar in my diagram, but now that Im thinking, I guess the tension stresses of -5556 kpa=-5.56 MPa are too high for the strength of the concrete, any comments Doug?

Cheers

 

[IDS]

The spreadsheet is set up to find the ultimate capacity of a reinforced concrete section, so there are a few significant differences from bending of a homogenous material:
[ul]
[li]Concrete is treated as having zero tensile strength, so all the concrete below the NA is ignored. That's why it won't work if there is no reinforcement in the tension zone.[/li]
[li]It's an ultimate analysis with the concrete stress treated as uniform over part of the compression zone, and zero over the remainder.[/li]
[/ul]

So it won't analyse a section under elastic bending with tension included, but it does illustrate the principle that the direction of the resultant moment is not parallel/perpendicular to the Neutral Axis.



Doug Jenkins
Interactive Design Services

http://newtonexcelbach.wordpress.com/

 

[BAretired]

Mixtli,
The usual practice when designing concrete beams using WSD (Working Stress Design) is to use a transformed section. Tensile stresses in concrete are assumed to be zero which means the reinforcing steel takes all of the tension. If you are trying to emulate WSD, I think you must use similar assumptions.

BA