Concrete Column Interaction Diagram, c>d'

[sklev]

I am designing a concrete column and am trying to calculate the interaction diagrams so I can play with different sizes and amounts of steel and be able to see it visually. I have set up an excel spreasheet to give me values for Mo, Po, Mb, Pb to start my diagram off with.

The problem is not concerning the Blanced Condition, but the condition that solves for where M and P = zero, which follows in the spreadsheet.

The problem is that I end up getting a negative value for f's, which results in a negative value for Cs (boxed in red). I can see that it is caused because my c value is less than my d'.

I was thinking that to solve the problem I would play with my values to get a positive number for f's and Cs, but then I got to wondering whether or not it was necessarily a bad thing that these values came up negative.

I am struggling conceptually. I think that this is saying that the compressive steel is not doing any work and the concrete is taking all of the compressive force.

Can someone please help shed some light on whether or not this is a bad thing, and what excatly is going on conceptually here?

 

[ash060]

If the depth to your netural axis is less than the depth to the top steel than you top steel is in tension not compression.

Treat as a beam with two layers of steel.

 

[Spanky7]

Try using one of the files I have attached. They are as follows:

For Concrete:
RECTBEAM (318-05)_Ver1.2.xls -- Based on ACI 318-05
RECTBEAM_Ver3.2.xls -- Based on ACI 318-99
For either of these used the Uniaxial or Biaxial Tabs to get the Interaction Diagrams.

For Masonry:
Axial_and_Flexure.xls
This is a spreadsheet my Masonry professor gave us to use when I was getting my Master. Use the Allow Stress Interaction or Strength Interaction Tabs. Note: using the Strength Interaction Tab with em=0.003 and fm=f’c will give you a semi-close approximation of the interaction diagram for concrete.

Hope these help.

 

[Spanky7]

One other thing I forgot. The tab if the files may be protected but there is no password used. You'll just have to unprotect the sheet to view equations.

 

[ishvaaag]

As mathematical constructs when establishing sectional equilibrium you may find that you obtain some alternative equilibriums or apparently paradoxical results. I found some of these cases when doing my Mathcad worksheets for sectional analysis. I remember in one case two actual possibilities of equilibrium were given as answers, and in fact could both be true depending of the way the loading was bringing the section of the beam to the specific equilibrium. Other more curious case, in this case in a problem of stage loading in huge composite beams, showed analytically according to the evaluation of moment strength for such particular cases that in no way the addition of the top slab was adding anything to the limit strength already shown by the steel girder before... think about it as if trying to restrain the deflection of some steel beams with a compression head of butter, well, that concrete was butter to the steel beams.

Quite curiously, as in the 2 previous cases, these unusual outcomes of analysis seem to have actual counterpart in the physical behaviour of what analyzed; other well known case is that, even when we lay our rebar at the bottom for simply supported beams "to gain mechanical arm" beams having the same amount of steel distributed in the side faces are shown in some cases to attain the same moment strength and superior ductilty (the service level behaviour maybe likely not being as good).

 

[sklev]

ash060 - So if both layers of steel are acting in tension, would my d value be the distance to the centroid of both layers acting as one, i.e., the center of the column?

Spanky7 - thank you. I will take a look at these.

 

[ash060]

d could be measured to the centroid, but it should not be in your case because the steel stress in the top may not be past yield.

I would use two d's for the case you mentioned one to the top steel and one to the bottom steel.

 

[sklev]

ash060 - Okay. After thinking a bit about what you said, it's making sense. Thank you.

 

[Spanky7]

What ash060 says is correct. You should be using a d for each layer of steel you have. I have some exaples at home I will have to look for them tonight and see if I can post them for you.

 

[sklev]

That would be great Spanky. All the more resources, the better.

 

[Spanky7]

Spoke too soon. Found the example sitting here in my book shelf at the office. this will be a multi part upload as there seem to be a file size limit.

Part 1

 

[Spanky7]

Part 2

 

[Spanky7]

Part 3 - Final part

 

[Spanky7]

If you need the interaction diagram for a circular column check out a program called DT Column at

http://www.dtware.com/

It will give you an interaction diagram for a circular column. However, I have never tried to verify the result by hand so use at your own risk.

 

[sklev]

Thanks Spanky.

 

[IDS]

I would be interested to see a numerical example of a non-composite beam that has two different ultimate load solutions for a given load combination, depending on how the load was applied.

Doug Jenkins
Interactive Design Services

http://newtonexcelbach.wordpress.com/

 

[ishvaaag]

IDS,

The first question is that I was making a quite wild variety of sectional analysis worksheets, and between then not only those providing the ultimate strength of the section in compatibility of deformations, but also those that show the status of the section under some solicitation, different concrete stress-strains laws etc. It surely was for these that the referred case was found.

I found the issue quite casually whilst doing work in sectional analysis in the Mathcad worksheets (I spent around 2 years making almost exclusively the Mathcad worksheets, sectional analysis and other) it may be difficult for me to duplicate the then observed issue, and give a proper comment.

How it was produced is not, for in establishing equilibrium I was establishing guess values, say, of the depth of the neutral axis. You can start with a guess higher or lower in the section and then such input for the referred cases provided two different solutions, both of which satisfied equilibrium, and showed nothing that made them physically improper.

Maybe it was a case where tension in the concrete was taken into account. Then you have that till the cracking moment the neutral axis is at some height, then the crack develops and snaps to other position. Depending of your guess you find one or another; or you may find such thing as a result of considering or not tension in the concrete (pears with apples). If this was the case it would be only that I met the cracking moment in casual way, for intently was not, a situation where two neutral axis' depths were involved and different steel and concrete stresses develop, but I can't attest if such was the case, for I do not remember the thing as cracking moment related, only that I found the thing not be a problem from the viewpoint of reliability for use of the worksheet, since I found acceptable (to me) explanation to the alternative equilibriums presented.

 

[ishvaaag]

It may also be that the specific shape of the concrete stress-strain law (I was testing maybe over 20 if simplified ones are counted) was amenable to two different positions of equilibrium for the neutral axis. But I simply do not remember the actual case, except that it did happen.

 

[IDS]

OK - if pre and post cracking are treated differently, or if the concrete stress/strain diagram has a softening region after the peak stress then there could be different NA positions for the same load.

For an ultimate load analysis ignoring concrete tension and with a rectangular or parabolic/rectangular concrete stress block there should be only one NA position for any given axial load.

Doug Jenkins
Interactive Design Services

http://newtonexcelbach.wordpress.com/

 

[ishvaaag]

True IDS, even the mathematical procedure would produce that even between peer solutions, since a process of maximization of the sustainable moment.