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

Finding the Neutral Axis ANGLE for Biaxial bending of concrete members

Finding the Neutral Axis ANGLE for Biaxial bending of concrete members

(OP)
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.

RE: Finding the Neutral Axis ANGLE for Biaxial bending of concrete members

Is the member symmetrical about both axis?

RE: Finding the Neutral Axis ANGLE for Biaxial bending of concrete members

(OP)
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?

RE: Finding the Neutral Axis ANGLE for Biaxial bending of concrete members

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.

RE: Finding the Neutral Axis ANGLE for Biaxial bending of concrete members

(OP)
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.

RE: Finding the Neutral Axis ANGLE for Biaxial bending of concrete members

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.

RE: Finding the Neutral Axis ANGLE for Biaxial bending of concrete members

(OP)
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.

RE: Finding the Neutral Axis ANGLE for Biaxial bending of concrete members

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.

RE: Finding the Neutral Axis ANGLE for Biaxial bending of concrete members

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

RE: Finding the Neutral Axis ANGLE for Biaxial bending of concrete members

(OP)
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?

RE: Finding the Neutral Axis ANGLE for Biaxial bending of concrete members

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

RE: Finding the Neutral Axis ANGLE for Biaxial bending of concrete members

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.

RE: Finding the Neutral Axis ANGLE for Biaxial bending of concrete members

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

RE: Finding the Neutral Axis ANGLE for Biaxial bending of concrete members

(OP)
@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...

RE: Finding the Neutral Axis ANGLE for Biaxial bending of concrete members

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.

RE: Finding the Neutral Axis ANGLE for Biaxial bending of concrete members

Mar2805.  KootK is correct.  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.  The spreadsheet also accommodates user-defined stress-strain curves for both the concrete and the steel.  It can be downloaded from my web site (rmniall.com).

RE: Finding the Neutral Axis ANGLE for Biaxial bending of concrete members

2
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/

RE: Finding the Neutral Axis ANGLE for Biaxial bending of concrete members

(OP)
@Kootk
Understood! smile

@Denial

Quote:

[ 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 sad

RE: Finding the Neutral Axis ANGLE for Biaxial bending of concrete members

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.

RE: Finding the Neutral Axis ANGLE for Biaxial bending of concrete members

Mar2805,

I have just checked my website, and it seems to be accessible and working fine.  Browse to it, then take downloads followed by software downloads.
Alternatively, browse directly to
http://rmniall.com/downloads/software-downloads/#C...

If you still cannot access the spreadsheet, please give me a description of the difficulty you are having.  Preferably send it to my e-mail address which you will find on my website.  (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.  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).

RE: Finding the Neutral Axis ANGLE for Biaxial bending of concrete members

@denial / @IDS - what is the programing like for something like this? Do you iterate the NA location?
I'd be interested in learning how to create such an algorithm for this problem.

EIT
www.HowToEngineer.com

RE: Finding the Neutral Axis ANGLE for Biaxial bending of concrete members

Ryan,

There's probably no "standard" approach, and I had to machete my way though my own personal numerical jungle.  The underlying assumption is that plane sections remain plane, from which it follows that the strain at any point (X,Y) on the cross-section is given by the "strain equation"
A + BX + CY
and the stress at any point follows directly from the strain equation and the material's stress-strain curve.
Once the three unknowns (A,B,C) are known the problem is solved, and the equation of the neutral axis is
A + BX + CY = 0

Start with an initial guess for (A,B,C).  Use numerical integration to calculate the applied forces (P,Mx,My) that would be in equilibrium with that strain distribution.  Compare these equilibrating forces with the actual applied forces:  the difference (δP,δMx,δMy) is a measure of your current error.

If this "error" is not sufficiently small, you need to revise your estimates for (A,B,C).  But how?  In effect, the method I used was to apply (δP,δMx,δMy) to the cross-section, thereby calculating a "correction" to the strain equation's (A,B,C).  In practice this was not as easy as it sounds, particularly for cases where the cross-sectional behaviour was highly non-linear, and I had to calculate the cross-section's "marginal stiffness" and apply (δP,δMx,δMy) to that.

I have some very rough notes from when I was developing the method, which I will e-mail to you on a "no responsibility" and "no discussion" basis.

RE: Finding the Neutral Axis ANGLE for Biaxial bending of concrete members

@Denial: I'd love to see your rough notes as well.

Attached is a printout from a MathCAD sheet that I wrote for this almost a decade ago (algorithms showing). It creates a graph of the plastic section over the cross section at the end. I'd be happy to share a live version if anyone's interested. Originally, I'd planned to do something as ambitious as Denial's spreadsheet. Unfortunately, the rabbit hole got too deep and the solver stopped running. I found the most difficult part to be simply handling the geometry of an arbitrary section and figuring out where the proposed NA crossed it mathematically. I used formulas from surveying class to calculate areas and such based on a matrix of corner points which was very helpful.

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

RE: Finding the Neutral Axis ANGLE for Biaxial bending of concrete members

KookT - you might like a look at:
http://newtonexcelbach.wordpress.com/2014/05/10/di...

which has a spreadsheet that will rotate a shape defined by coordinates, and then divide it into layers parallel to the X axis.

It's actually the start of a bi-axial bending program, but the rest is still in production.

Doug Jenkins
Interactive Design Services
http://newtonexcelbach.wordpress.com/

RE: Finding the Neutral Axis ANGLE for Biaxial bending of concrete members

@ Denial / IDS,

R-E-S-P-E-C-T. That's what your algorithms mean to me! Seriously... cudos & thanks.

@ Denial: Your notes seem to suggest a finite element-ish approach. Is that how you tackled the implementation of the algorithm? Or did you do it as layers like IDS? Now that I think of it, I suspect that the layering approach is how this is handled in commercial packages. Also, did you program your own solver routine rather than attempting to use the Excel solver?

@ IDS: Awesome. I want to replicate what you've done in my MathCAD worksheet. Can you give me a brief, in English, summary of the algorithm? Something like:

1) Identify all vertices and draw layer lines there.
2) Split remaining section into more layers if desired.
3) Figure out where layer lines intersect section polygon.
4) Use ray tracing to identify the insides of the layers.
.......

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

RE: Finding the Neutral Axis ANGLE for Biaxial bending of concrete members

Wow, Yeah, very impressive thanks guys.

I also would like to try something but in SMath Studio, the poor man's version of MathCAD (but I must say SMath is starting to impress)

EIT
www.HowToEngineer.com

RE: Finding the Neutral Axis ANGLE for Biaxial bending of concrete members

KootK,

The approach is not FE-ish.  It might, initially, appear that way because it breaks the concrete shape into a series of triangles, with each triangle being defined by the origin and one of the polygonal edges of the concrete boundary.  The integration over each triangle is exact (within the limitations of Gaussian Integration).  Nor does the approach use "layers", except insofar as it needs to establish where on the cross-section the strain values correspond with the defining points of the stress-strain curve(s).

While I am a great fan of the Solver I cannot see how it might be used on this problem.  The solution process is entirely coded in VBA within the spreadsheet.  This VBA is protected behind a password, but the password is given on the "Documentation" worksheet.  (I use password protection on my VBA only in order to avoid confusing the user if something goes wrong:  a "normal" engineer running the spreadsheet does not want to suddenly find himself deep within the VBA Editor, and an "abnormal" engineer can always remove the protection.)

RE: Finding the Neutral Axis ANGLE for Biaxial bending of concrete members

KootK - Here is a brief algorithm for generating the trapezoidal layers for any shape defined by node coordinates:

Add node index numbers to the node coordinate list
Sort node coordinates and index numbers by increasing Y, then increasing X
List Y values for the bottom/top of each layer
For the base point(s) of the first layer:
_Count nodes with minimum Y value
_Check for vertex points; add a dummy node for each vertex
_List X values for each node
For each subsequent Y value
_Find IP X values for each line segment at or crossing this Y
_Where more than one end node has the same Y check for:
__Horizontal line segments defining the base or top of a layer
__Vertex points at the base or top of a layer
__Count nodes for the layer below and above Y (if any)
_List X values for the top of the layer below level Y
_List X values for the base of the layer above level Y, if any
Sort X values for each row in ascending order

I have also added that to the spreadsheet.

The basic algorithm is quite simple, but the code gets fairly complicated in checking situations where there are horizontal line segments or a top or bottom vertex. The code requires a check of the direction of the preceding and following line segments to check if the line forms part of the trapezium above or below the line. It also gets complicated at the end getting the node numbers right.

Doug Jenkins
Interactive Design Services
http://newtonexcelbach.wordpress.com/

RE: Finding the Neutral Axis ANGLE for Biaxial bending of concrete members

Thanks for taking the time to describe your algorithms guys. This should be more than enough to get me started down the right path.

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

RE: Finding the Neutral Axis ANGLE for Biaxial bending of concrete members

Are we talking about ultimate strength calculations? Are we talking about biaxial bending alone or biaxial bending combined with axial forces?

BA

RE: Finding the Neutral Axis ANGLE for Biaxial bending of concrete members

The approach I took in my spreadsheet allows for an axial force in addition to biaxial bending.  And since it uses user-defined stress-strain curves the question of whether or not it is "ultimate strength" does not arise.

RE: Finding the Neutral Axis ANGLE for Biaxial bending of concrete members

That sounds like a good approach. Spreadsheets are wonderful things, although I never developed much expertise in them before retiring.

BA

RE: Finding the Neutral Axis ANGLE for Biaxial bending of concrete members

BA - the end purpose intended for my spreadsheet splitting a rotated shape into "horizontal" layers is to allow the section to be analysed with a closed form solution for any given rotation angle of the NA, so iteration will only be required to find the angle. I should be able to plug this routine into existing spreadsheets for both ULS and elastic analysis, including axial load. The only additional calculation required will be to find the moment about the Y axis, which should be straightforward.

I'm hoping that this approach will be faster than finding both the angle and position by iteration, but I'm really only doing it for my own interest. The RMNial solution looks like it should do just about anything anyone might require.

Doug Jenkins
Interactive Design Services
http://newtonexcelbach.wordpress.com/

RE: Finding the Neutral Axis ANGLE for Biaxial bending of concrete members

(OP)
There seems to be difefrent procedures involved when caculating Moment Capacity of a column.
Some seem to use the "centroid of a concrete section" others use "plastic centroid".
Seem that plastic centroids are ONLY used if section is not symetrical (reinforcement included).
Is this correct?

Having a symetical section reinforced with different ratios in top and bottom portion should use P.C. wich would only have y coordinate. Correct?
Having a non-symetrical section with diferent rations in top and bottom portion should use P.C. wich would have x and y coordinates. Correct?

RE: Finding the Neutral Axis ANGLE for Biaxial bending of concrete members

@Doug of IDS,
I can see how the layer system would be helpful for analyzing unsymmetrical sections under elastic conditions. It seems to me that, for ultimate conditions that after a trial Neutral Axis (N.A.) has been selected, the compression block is defined and the concrete stress will be uniform throughout that block, i.e. 0.85f'c. The centroid of the compression block can be calculated without using layers.

Steel stresses may vary according to the distance from the N.A. but cannot exceed fy in tension or compression.

As rectangular columns are commonly used in buildings, a useful application for a spreadsheet might be to focus on rectangular columns with symmetrical reinforcement and perhaps to prepare a chart of ultimate moments and axial capacity for commonly used column sizes.

BA

RE: Finding the Neutral Axis ANGLE for Biaxial bending of concrete members

BA - yes, for any trial NA position the ultimate capacity can be calculated from the section properties of the stress block, without dividing the section into layers, you can do an elastic analysis as well. The purpose of dividing into layers is that I have a routine that will find the position of the NA without iteration, so I'm hoping the process of finding the NA angle and position will be faster, as iteration will only be required to find the angle. It will also handle triangular or parabolic-rectangular stress distributions, as well as rectangular.

I work mainly on bridges, which often have non-rectangular columns and deck beams are usually not rectangular, so I'd like the procedure to be as flexible as possible.

mar2805 - I'm not sure that I totally understood your question, but I hope the following points will answer it. For any member with non-zero axial load it is important that moments are taken about a well defined axis. It doesn't matter where that axis is, so long as the same axis is used for the load eccentricity, and the resulting reaction eccentricity. Load eccentricity is usually measured from the centroid of the concrete section (ignoring reinforcement), and in a frame analysis the columns are normally placed on the centroidal axis of the concrete section, so it usually makes sense to use this as the reference axis for analysis. Note that any reinforced concrete section will be asymmetric about the bending axis if the concrete is cracked in tension.

For a section that is symmetrical about the Y axis (taking the NA as the X axis), the centroid of the concrete stress block will always be on the Y axis, so we don't have to worry about bending about the Y axis. If the section (either concrete or steel) is not symmetric about the Y axis (either because of the section shape or biaxial bending) then we need to check for moment equilibrium about both axes.

Doug Jenkins
Interactive Design Services
http://newtonexcelbach.wordpress.com/

RE: Finding the Neutral Axis ANGLE for Biaxial bending of concrete members

@IDS:

I do a lot of sport programming too. It's my favorite way to learn something complex if I can find the time. If I can program something, I own it. Those surveying algorithms that I mentioned may be an elegant and robust way for you to complete your next step. I doubt that you need any help in the programming department but, if you're interested, I'd be happy to hook you up.

When I retire, I want to make a spreadsheet that will find the shear center of an arbitrary section, even a multiple, closed cell section. I consider that to be the holy grail of sectional analysis spreadsheets.

KootK

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

RE: Finding the Neutral Axis ANGLE for Biaxial bending of concrete members

@Mar2805,

There are almost no practical design scenarios where you would want to use anything but the cracked section properties of a concrete section. Certainly, it the dominant way to go for strength design. And, even for deflection calculations, you're usually dealing with a cracked section.

Other than academic exercises, the only applications that that I can think of for elastic section properties are:

1) situations where concrete members are pre-stressed to the extent that they will remain uncracked.
2) in some codes, the column slenderness procedures require the calculation of an elastic moment of inertia for use in estimating an Euler buckling load.

KootK

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

RE: Finding the Neutral Axis ANGLE for Biaxial bending of concrete members

As mentioned above, I have incorporated the function to split a shape into layers with a routine to find the ULS moment capacity of a reinforced or prestressed concrete section under combined axial load and bending, so that it will now work for biaxial bending and/or non-symmetrical sections.

See:
http://newtonexcelbach.wordpress.com/2014/06/19/re...

for more details and a link to the download file, which is free and open-source.

The spreadsheet will work to Australian codes, ACI, or Eurocode 2, but at the moment only for rectangular stress blocks.

Doug Jenkins
Interactive Design Services
http://newtonexcelbach.wordpress.com/

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