Virtual vs Real Neutral Axis 3

[hocho]

The neutral axis in the interaction diagram doesn't really correspond to the physical location of the column. Or do they?

For example. Below the balanced point. At zero moment. The neutral axis is very small (or infinitely small). At zero moment (edit: I meant at zero axial load). It acts like a beam. Yet in a beam. The real neutral axis is the middle of the beam. So the interaction diagram neutral axis doesn't correspond to the physical location of the neutral axis. What is the formula that relates the virtual and real neutral axis (what official terms distinguish these two)? Is there a software that can show or distinguish them and plot them both?

 

[IDS]

In a member with axial load and no moment the neutral axis is at infinity.

For a member with no axial load and no moment the neutral axis is not defined, or everywhere if you prefer, since there is no strain anywhere.

For any non-zero moment, with zero axial load, the neutral axis passes through the centroid of the section (assuming linear elastic behaviour).

There is no virtual neutral axis.

Doug Jenkins
Interactive Design Services

http://newtonexcelbach.wordpress.com/

 

[hocho]

In a member with axial load and no moment the neutral axis is at infinity.

For a member with no axial load and no moment the neutral axis is not defined, or everywhere if you prefer, since there is no strain anywhere.

For any non-zero moment, with zero axial load, the neutral axis passes through the centroid of the section (assuming linear elastic behaviour).

There is no virtual neutral axis.

Im studying the derivations of the interaction diagram. At neutral axis magnitude below the balanced point.. Axial load decrease because the neutral axis distance is actually multipled by the fc' and others and it's decreasing. So neutral axis is always on the right of the balanced point. It is not about axial load on the tension side. Yet the explanation is the moment increase or decrease due to the the concrete compression of the tension side. Why didn't the formula directly model the compression of the tension side?

 

[hocho]

In a member with axial load and no moment the neutral axis is at infinity.

For a member with no axial load and no moment the neutral axis is not defined, or everywhere if you prefer, since there is no strain anywhere.

For any non-zero moment, with zero axial load, the neutral axis passes through the centroid of the section (assuming linear elastic behaviour).

There is no virtual neutral axis.

Second comment on "For any non-zero moment, with zero axial load.". Remember between the balanced point and zero axial load in the tension control portion of the interaction diagram. The neutral axis gets smaller and smaller.. so how can you say that it passes through the centroid of the section? I'm asking where exactly is the location of the neutral axis in the physical column as the neutral axis in the diagram gets smaller in the tension side (making the C smaller affecting and making Pn smaller)

 

[Retrograde]

so how can you say that it passes through the centroid of the section?

In your original post you did not state that you were considering a reinforced concrete column. What Doug said is correct given his assumption of linear elastic behaviour.

 

[IDS]

OK, so you are talking about reinforced concrete and the ultimate moment- axial load interaction diagram. I probably should have guessed because you talked about the balance point, but you also said that the NA in a beam was at the section centroid, which it isn't for cracked reinforced concrete.

If you start at zero axial load and move the NA towards the compression face the compression force on the concrete will reduce (constant stress, but reducing area), but the tensile steel will be yielded, so have a constant force and the nett force on the section will be increasing tension.

At some point the steel on the compression side will go into tension, and then yield, so the moment about the centroid due to the steel will be zero. As the depth of the NA approaches zero the compression force on the concrete approaches zero, so the moment about the centroid also approaches zero, and the nett axial force approaches the yield force on the bars.

Of course in reality the bars would rupture long before the depth of the NA got close to zero.

Doug Jenkins
Interactive Design Services

http://newtonexcelbach.wordpress.com/

 

[hocho]

[OK, so you are talking about reinforced concrete and the ultimate moment- axial load interaction diagram. I probably should have guessed because you talked about the balance point, but you also said that the NA in a beam was at the section centroid, which it isn't for cracked reinforced concrete.

If you start at zero axial load and move the NA towards the compression face the compression force on the concrete will reduce (constant stress, but reducing area), but the tensile steel will be yielded, so have a constant force and the nett force on the section will be increasing tension.

At some point the steel on the compression side will go into tension, and then yield, so the moment about the centroid due to the steel will be zero. As the depth of the NA approaches zero the compression force on the concrete approaches zero, so the moment about the centroid also approaches zero, and the nett axial force approaches the yield force on the bars.

Of course in reality the bars would rupture long before the depth of the NA got close to zero.

No. I'm not referring to beams. I'm talking about column above zero axial load (I know zero load is become beam case). I thought when one mentions interaction diagram. It automatically is connected to column. Let's start with the NA at balanced point. As the NA moves to the compression side. The neutral axis gets smaller and smaller.. this makes the axial load capacity smaller and smaller and also moment capacity. All along, it is the right side of NA that is in compressive stress. The left side of NA is in tension. Yet is is mentioned in books that in the range of tension failure, a reduction in axial load may produce failure because it is pressing on the tension bars less and less. But in tension failure, the entire tension side never touch (because it is left of the neutral axis). Is this correct? If you say it is touching, then the NA in the interaction diagram doesn't correspond to real NA in the column. What is the case?

 

[Retrograde]

I think that you may be confusing the meaning of the neutral axis with that of the centroid. For a reinforced concrete column the centroid is normally taken as the center of mass of the concrete section assuming it is uncracked and (normally) ignoring the reinforcement. For a rectangular column the centroid is D/2 (D = depth of the member).

 

[hocho]

I think that you may be confusing the meaning of the neutral axis with that of the centroid. For a reinforced concrete column the centroid is normally taken as the center of mass of the concrete section assuming it is uncracked and (normally) ignoring the reinforcement. For a rectangular column the centroid is D/2 (D = depth of the member).

But the neutral axis can move back and forth in the physical column section.. is it not?

Or just answer this. When the moment and axial load is below the balanced point of the interaction diagram and it is in so called tension control region.. is there any compression in the tension side? Or none at all?

 

[IDS]

No. I'm not referring to beams.

Neither was I. I didn't mention beams.

I'm talking about column above zero axial load (I know zero load is become beam case). I thought when one mentions interaction diagram. It automatically is connected to column. Let's start with the NA at balanced point. As the NA moves to the compression side. The neutral axis gets smaller and smaller.. this makes the axial load capacity smaller and smaller and also moment capacity.

If you mean the compression zone gets smaller, correct. (by the way, interaction diagrams can be interaction of anything with anything).

All along, it is the right side of NA that is in compressive stress. The left side of NA is in tension. Yet is is mentioned in books that in the range of tension failure, a reduction in axial load may produce failure because it is pressing on the tension bars less and less. But in tension failure, the entire tension side never touch (because it is left of the neutral axis). Is this correct?

I don't know what you mean by "the entire tension side never touch". The tension side is all in tension, yes. What conclusions are you drawing from this?

If you say it is touching, then the NA in the interaction diagram doesn't correspond to real NA in the column. What is the case?

There is only one NA. It is the line where axial strain is zero. The NA used in constructing an interaction diagram is the same as the NA in the real column. Again, it isn't clear to me why you think they might be different.

Or just answer this. When the moment and axial load is below the balanced point of the interaction diagram and the tension bars is in tension.. is there any compression in the tension side? Or none at all?

The tension side is all in tension, and the compression side is all in compression; from the definition of the Neutral Axis.


Doug Jenkins
Interactive Design Services

http://newtonexcelbach.wordpress.com/

 

[BAretired]

When the moment and axial load is below the balanced point of the interaction diagram and the tension bars is in tension.. is there any compression in the tension side? Or none at all?

By definition, the NA is the point where stress and strain are zero. Everything to one side is in tension while everything on the other side is in compression.

BA

 

[hocho]

I don't know what you mean by "the entire tension side never touch". The tension side is all in tension, yes. What conclusions are you drawing from this?

Doug Jenkins
Interactive Design Services

In page 272 of the book by Nilson. It is written:

"It is important to observe, in Fig. 8.10, that in the region of compression failure the larger the axial load Pn, the smaller the moment Mn that the section is able to sustain before failing. However, in the region of tension failure, the reverse is true; the larger the axial load, the larger the simultaneous moment capacity. This is easily understood. In the compression failure region, failure occurs through overstraining of the concrete. The larger the concrete compressive strain caused by the axial load alone, the smaller the margin of additional strain available for the added compression caused by bending. On the other hand, in the tension failure region, yielding of the steel initiates failure. If the member is loaded in simple bending to the point at which yielding begins in the tension steel, and if an axial compression load is then added, the steel compressive stresses caused by this load will superimpose on the the previous tensile stresses. This reduces the total steel stress to a value below its yield strength. Consequency, an additional moment can now be sustained of such magnitude that the combination of the steel stress from the axial load and the increased moment again reaches the yield strength.
The typical shape of a column interaction diagram shown in Fig. 8.10 has important design implications. In the range of tension failure, a reduction in axial load may produce failure for a given moment"

<book snip ends>


Is the above description not wrong? Because in the tension region below the balanced point.. the tension bars are not being pressed by any axial load.. the axial load is being taken solely by the compressive stress at the right side of the neutral axis. The book assumes the left side has axial load imposing on it. This is what confuses me to no end. What is your say?

 

[IDS]

I just lost a reply through the net connection dropping out just as I hit submit, so this will be a bit abbreviated.

I think the text quoted is correct. One thing to remember is that it is not describing the situation at ultimate moment/axial load. If the steel is just at yield, and the axial load is below the balance point, then the concrete will not be at yield.

Some things to remember when considering this:

Plane sections remain plane. If you apply an axial compression, with no change in curvature, it will both increase the compressive strains and reduce the tensile strains.
Compressive stress cannot be transmitted through the concrete across a crack, but reductions in tensile stress can be transmitted through the steel.


Doug Jenkins
Interactive Design Services

http://newtonexcelbach.wordpress.com/

 

[hocho]

I just lost a reply through the net connection dropping out just as I hit submit, so this will be a bit abbreviated.

Too bad. Windows Or Mac must have anything typed automatically saved thru a keystroke program that saved the past 30 minutes typing.

I think the text quoted is correct. One thing to remember is that it is not describing the situation at ultimate moment/axial load. If the steel is just at yield, and the axial load is below the balance point, then the concrete will not be at yield.

At below the balance point. The axial capacity is directly on the compressive side of the neutral axis. For example. For a column that is 20" in size (it's actually the example in the book), the neutral axis for the balanced failure conditions is:

cb = 17.5 x 0.003/0.0051 = 10.3 "

stress-block depth a = 0.85 x 10.3 = 8.76 "

concrete compressive resultant = C = 0.85 x 4 ("ksi") * 8.76 x 12 = 357 kips

Note it is the concrete compressive resultant at the right side of the neutral axis that gives the axial load capacity. Now the other side of the neutral axis is in tension. Therefore the 357 kips axial load capacity can't be pressing on the tension side.

Some things to remember when considering this:

Plane sections remain plane. If you apply an axial compression, with no change in curvature, it will both increase the compressive strains and reduce the tensile strains.
Compressive stress cannot be transmitted through the concrete across a crack, but reductions in tensile stress can be transmitted through the steel.

But since the neutral axis is supposed to be the dividing line between tensile and compressive strains and stress. The axial compression should be on the compressive side only.. and not press on the tension side. But then I just realized now your statement plane sections remain plane. So you mean as the resultant resultant got bigger, the strain in the yielding side gets smaller. I think this description makes more sense than saying the axial load is directly pressing and making the concrete of the tension side touching. It doesn't touch.. but the yielding strain is decreasing.. maybe this is all it means??

 

[Retrograde]

What Nilson is saying is this:

If you have a moment and no axial load you will have a particular location of the neutral axis with tension on one side and compression on the other. If you then add an axial load to the section, the neutral axis will move and you will have less tension on the tension side and more compression on the compression side (which is what Doug was saying).

 

[hocho]

What Nilson is saying is this:

If you have a moment and no axial load you will have a particular location of the neutral axis with tension on one side and compression on the other. If you then add an axial load to the section, the neutral axis will move and you will have less tension on the tension side and more compression on the compression side (which is what Doug was saying).

Ok. I finally understood it now after trying to comprehend it for many months. Thanks to the Eng-tips folks.

Now question. When you guys design columns. Do you make the axial load close to the balanced point? Usually where in the tensile region below the balanced point do you usually locate the axial load in your practice?

 

[BAretired]

Most concrete columns in practice fall into the compression branch of the interaction diagram. No special effort is made to be close to the balanced point. As a matter of fact, it is desirable to be as high as possible on the interaction diagram; that means the column is doing what it is intended to do, namely to carry axial load.

BA

 

[hocho]

Most concrete columns in practice fall into the compression branch of the interaction diagram. No special effort is made to be close to the balanced point. As a matter of fact, it is desirable to be as high as possible on the interaction diagram; that means the column is doing what it is intended to do, namely to carry axial load.

In the columns located at the sides of the houses. It bends. In your experience.. what is usually the amount of column moments you encounter. For axial loads. We can easily estimate it because of the simple formulas of 23.56 kN/cube meter of concrete. So when you have beams of say 0.2x0.4. You can estimate the weight by 23.56x0.2*0.4 = 1.88 kN. And let's say the slab is 0.15 thick.. we can easily compute for dead weight of it by 23.56x0.15 = 3.534 kN square meter and getting area and tributary load, etc. and we can easily get axial weight by area of concrete and bars multiply each by the fc, fs to get factored and ultimate axial.

But how about moments. What is the easiest way to determine moments of an edge column of a house. Do you compute for the center of gravity of the beam and getting its axial load and turning the distance into eccentricity? If not.. what is the easiest way to estimate or compute for moments manually? Many structural engineers nowadays use software in design that we got confused when asked to to it manually. And manual good way to check what's on the computer output.

 

[IDS]

The dimensions of a column really depend on the application and the applied loads. If the moments are small, or if bending moments are about equal for two perpendicular axes, then it makes sense to make a column square or circular (or any regular shape in between), then the dimensions will be controlled either by the axial force or the moment, depending on their relative magnitude.

If you have a large moment in one direction, and no restrictions on the column shape, it may be efficient to choose a shape so that the design actions are near the balance point, but even in this case this may not be the optimum solution. This really needs a trial and error solution, taking into account costs of steel, concrete and formwork, and architectural requirements (if applicable).

One point from the earlier discussion that I think needs clarification is you said:
"Note it is the concrete compressive resultant at the right side of the neutral axis that gives the axial load capacity"

The axial load in a column is the difference between the compression forces in the concrete (plus compression steel), and the tension in the tension steel. When the applied loads are on the interaction diagram limit both the concrete and the steel will be at their yield stress. If this point is below the balance point, remember that the axial load is the minimum required to avoid failure, not the maximum. If the axial load is increased, with no increase in moment, the depth of the compressive stress block will increase, and the tensile strain in the steel will reduce. As the axial load is increased the concrete strain at first reduces, then when the depth of the stress block is deep enough the concrete strain starts to increase again, until it returns to the design yield strain, when the axial load crosses the upper interaction diagram line.


Doug Jenkins
Interactive Design Services

http://newtonexcelbach.wordpress.com/

 

[hocho]

The axial load in a column is the difference between the compression forces in the concrete (plus compression steel), and the tension in the tension steel. When the applied loads are on the interaction diagram limit both the concrete and the steel will be at their yield stress. If this point is below the balance point, remember that the axial load is the minimum required to avoid failure, not the maximum. If the axial load is increased, with no increase in moment, the depth of the compressive stress block will increase, and the tensile strain in the steel will reduce. As the axial load is increased the concrete strain at first reduces, then when the depth of the stress block is deep enough the concrete strain starts to increase again, until it returns to the design yield strain, when the axial load crosses the upper interaction diagram line.

Yes, I have already understood this part yesterday. Also note below the balance pint, if the axial load is increase, the moment capacity has to decrease to maintain a steady capacity. Also because of the formula Mn = 0.85 fc' ab (h/2 - a/2)... when the compressive stress block increases in value (the axial capacity increases).. a/2 becomes larger so h/2 - a/2 becomes smaller.. hence moment capacity gets smaller. I have tried to master all the derivations.

Anyway. What interaction diagram software do you know that plots it nicely and superimpose it on the column so you can visualize the neutral axis moving in the actual column for certain axial loads and moments.

I tried playing with the excel formulas. At balanced point.. if you lower the strain, moments and axial loads decrease in value yet they are still in balance. As the column is moved to higher moment in one direction, there will be changes in the neutral axis. I want the software to show the varying neutral axis for even more mastery of it.