Finding the Neutral Axis ANGLE for Biaxial bending of concrete members 6

[mar2805]

Hi!
Im interested how to find the angle of the neutral axis of a concrete members subjected to moments aboth 2 axis, x and y?
Could the simple formula of tan(alpha)=Mx/My be used?
Thank you.

 

[EngineeringEric]

Is the member symmetrical about both axis?

 

[mar2805]

What if not, and what if yes?
I suspect your going to say that if the section is symetrical aboth both axis then formula tan(alpha)=Mx/My can be applied, correct?
But what if not?

 

[dhengr]

Mar2805:
We don’t actually teach Engineering Mechanics and Strength of Materials here on E-Tips, they do that in college. You would do well to get some text books on these subjects and study them. You should have learned this material during your uni. engineering education. The variations and manipulations are complicated enough that you really have to study this a bit on your own to get the hang of it.

 

[mar2805]

Would you be kind enough to direct me to a book you know of and highly recommend that deals with this subject on RC design?
Thank you.

 

[KootK]

My version of the Salmon & Wang textbook addresses this to some degree.

It's important to note that the answer will be different depending on whether you're looking at an elastic, serviceability case or a plastic, ultimate strength case. In general the NA angle won't coincide with the angle of the resulting moment, particularly for plastic design.

For the plastic state, you want to use the same methods that are used for biaxially loaded reinforced columns. For your sake, I hope that you've got access to some software for that.



The greatest trick that bond stress ever pulled was convincing the world it didn't exist.

 

[mar2805]

KootK egsacty my point!
For servicability if we assume linear behaviour of stress in the section (non cracked state) finding angle of N.A. is not a problem.
But for ULS (cracked state) of an non symetrical section in bending the problem is complicated.
As you mentioned Biaxialy loaded columens are the way to solve this.
But one thing that is confusing me. Please look at the file attached.
Laste few pages there is an explenation of how to solve Biaxial loaded columns.
There is a formula thats used for defining a resultatnt Moment and the angle that this moment makes with axis.
The whole section is rotated and the problem is solved as Uniaxial bending of non symetrical section.

How is it possible that the formula for finding N.A. angle is tan(alpha)=Mx/My?!?!

This forumla only applies to square sections since moment of inertia aboth two principle axis are the same.

 

[KootK]

If you're refering to pages 36 & 37, I believe that what is shown there is either a) incorrect b) misleading or c) should be acknowledged as an approximation.

The greatest trick that bond stress ever pulled was convincing the world it didn't exist.

 

[msquared48]

Also covered in my 1980 vintage Fourth Edition of the CRSI Handbook on pages 2-27 to 2-33 for a rectangular column in Biaxial Bending..

Point is that this is not anything new...

Mike McCann
MMC Engineering

 

[mar2805]

Yes, pages 36 and 37.
The problem is that this way is mentioned in many other places also!
So this is NOT the way to do it?
So besides varying the neutral axis depth for biaxialy loaded columns one should also vary the N.A. angle?

 

[mar2805]

@mssquared
DO you have any link to your book?
Thank you

 

[KootK]

Correct. All my sources sources are clear on this. NA angle <> resultant moment angle in general.

MS^2 was refering to an older version of this publication: Link

The greatest trick that bond stress ever pulled was convincing the world it didn't exist.

 

[BAretired]

The link suggests three different approaches to the problem of biaxial bending. None are exact and you probably can't do better than a reasonable approximation. Try all three and compare without spending an excessive amount of time on any of them.

BA

 

[mar2805]

@KootK
Im sorry but I didnt understand you.
What do you mean by "Correct. All my sources sources are clear on this. NA angle <> resultant moment angle in general."

The link to the book you gave me, does it have an explenation of dealing with biaxialy loaded columns and the proper procedure of how to solve this problem?
I wish to order it.
I already have book from MacGregor and White "RC mechanics and design" it has an solved egsample of Biaxialy loaded column wich is veyr well explained but N.A. angle is given and its not said that it should vary...

 

[KootK]

You asked me these two questions:

1) So this is NOT the way to do it?
2) So besides varying the neutral axis depth for biaxialy loaded columns one should also vary the N.A. angle?

The statements that you've made in both cases are correct. That is not the way to do it and one should expect variation in the NA angle.

Don't order the CRSI book. It's a great book but it's very expensive and won't tell you anything more than your MacGregor book.

I have three versions of MacGregor's book and he's one of my favourite concrete authors. I'm quite sceptical that his book is claiming that the NA will always align with the resultant moment.

You can sort this out relatively easily on your own:

1) Assume that the NA and resultant moment axis align.
2) Run your calculations and determine the tension and compression forces required for equilibrium.
3) Sum the moments that your tension and compression forces induce about an axis perpendicular to the NA.
4) Baring some serious luck, note that the moments calculated in step three do not balance. SUM(My) <> 0.




The greatest trick that bond stress ever pulled was convincing the world it didn't exist.

 

[Denial]

Mar2805.[&nbsp;] KootK is correct.[&nbsp;] The assumption on pages 35 & 36 of that reference, that the neutral axis is parallel to the arrow-vector of the resultant applied moment, is not correct and it becomes increasingly incorrect as the column's "Ixx/Iyy" ratio moves away from unity.

If you have a particular column that warrants detailed investigation, I have developed a spreadsheet that analyses arbitrarily-shaped reinforced concrete columns under the actions of biaxial moments and axial force.[&nbsp;] The spreadsheet also accommodates user-defined stress-strain curves for both the concrete and the steel.[&nbsp;] It can be downloaded from my web site (rmniall.com).

 

[IDS]

First - those people who suggested that this is a simple question that any engineering graduate should be able to answer should perhaps do some revision of the basics themselves. It's not a simple question at all.

Second - because it isn't a simple question, in practice people mostly avoid doing it, using code provisions for combining the moment capacity about the principal axes.

Third - if you decide to do it anyway (and there are situations where it is worth doing) the procedure is basically to iterate through trial NA angles to find the NA position and direction where the reaction forces and moments are in equilibrium with the applied forces and moments.

Doug Jenkins
Interactive Design Services

http://newtonexcelbach.wordpress.com/

 

[mar2805]

@Kootk
Understood!

@Denial

[ is not correct and it becomes increasingly incorrect as the column's "Ixx/Iyy" ratio moves away from unity.]

Yes, this is correct. From simple Mechanics, its only valid if you moment of inertia aboth both axis are THE SAME. quadrant shape in this case.
But this is something that I would like to ask more.
Please look ata the picture attached.
YOu have two shapes, quadrant Ix=Iy and a rectangles Ix>>Iy
From Mechanics you can find the angle of N.A. using formula writen in the picture.
The moment is acting on an angle.
Main principle axis are alinged with coordinate axis.
For a quadrant shape Ix=Iy the formula will tell you that you N.A. is purpendicular to you resultant Moment.
If the shape is rectangle or any other, where Ix>>Iy, N.A. forms an angle thats NOT purpendicular to Moment.

This is the formula thats used for linear state of behaviour.
For ULS, I guess I couldnt not be used...due to plastic state of stresses in the cross section and tension of cross section excluded...

@Denial
Could you try using you spredsheet and just see how the N.A. angle deviates from this linear elastic aplied formula for finding N.A. angle? Please.

I went to your site you gave me but could not open it :-(

 

[Guest262653]

I totally agree with IDS regarding the complexity of an "exact" section analysis.

And I add that if people get bullied every time they make a question, I don't expect a long life for this forum.

 

[Denial]

Mar2805,

I have just checked my website, and it seems to be accessible and working fine.[&nbsp;] Browse to it, then take downloads followed by software downloads.
Alternatively, browse directly to

http://rmniall.com/downloads/software-downloads/#CrossSection

If you still cannot access the spreadsheet, please give me a description of the difficulty you are having.[&nbsp;] Preferably send it to my e-mail address which you will find on my website.[&nbsp;] (Yes, I am aware of the potential circularity of this request.)

I do not have the time to run any cases for you, and even if I did you have not given me anywhere near enough information to specify a precise specific case.[&nbsp;] But why not simply apply the formula you give in your 3Jun14@2:58 attachment, which is fully correct for linear elastic material behaviour (after allowing for modular ratios).