Virtual vs Real Neutral Axis
Virtual vs Real Neutral Axis
(OP)
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?
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?






RE: Virtual vs Real Neutral Axis
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/
RE: Virtual vs Real 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?
RE: Virtual vs Real 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)
RE: Virtual vs Real Neutral Axis
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.
RE: Virtual vs Real Neutral Axis
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/
RE: Virtual vs Real Neutral Axis
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?
RE: Virtual vs Real Neutral Axis
RE: Virtual vs Real Neutral Axis
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?
RE: Virtual vs Real Neutral Axis
Neither was I. I didn't mention beams.
If you mean the compression zone gets smaller, correct. (by the way, interaction diagrams can be interaction of anything with anything).
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?
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.
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/
RE: Virtual vs Real Neutral Axis
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
RE: Virtual vs Real Neutral Axis
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?
RE: Virtual vs Real Neutral Axis
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/
RE: Virtual vs Real Neutral Axis
Too bad. Windows Or Mac must have anything typed automatically saved thru a keystroke program that saved the past 30 minutes typing.
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.
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??
RE: Virtual vs Real Neutral Axis
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).
RE: Virtual vs Real Neutral Axis
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?
RE: Virtual vs Real Neutral Axis
BA
RE: Virtual vs Real Neutral Axis
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.
RE: Virtual vs Real Neutral Axis
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/
RE: Virtual vs Real Neutral Axis
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.
RE: Virtual vs Real Neutral Axis
https://newtonexcelbach.wordpress.com/2012/09/19/d...
and here:
https://newtonexcelbach.wordpress.com/2013/07/21/u...
The ULS Design spreadsheet plots the beam cross section with the neutral axis position for any given axial load. Let me know if that is what you wanted, or if you had something else in mind.
Also I happen to be working on a spreadsheet for generating moment curvature diagrams at the moment, but it will be easy to generate an interaction diagram using the same function, so I'll look at options for how the results can be displayed.
Doug Jenkins
Interactive Design Services
http://newtonexcelbach.wordpress.com/
RE: Virtual vs Real Neutral Axis
Btw.. something confused me above. You said "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"
But I know whenever axial load is increase, the concrete strain always increase.. how can it reduce initially? I'm visualizing the triangle and geometry of the strains. Thanks.
RE: Virtual vs Real Neutral Axis
The combined axial load and moment are equivalent to an eccentric vertical load, with the eccentricity on the compressive side. If you apply an additional load at the column centroid it reduces the eccentricity of the resultant load, and the width of the compression region increases, so the total compression force increases, but the maximum stress reduces. It's a similar situation to a footing with a vertical load outside the middle third. If you apply an additional load on the uplift side, the pressure on the compression side will reduce, even though the total reaction force is increased.
Doug Jenkins
Interactive Design Services
http://newtonexcelbach.wordpress.com/
RE: Virtual vs Real Neutral Axis
Thanks the great explanation. Say when you see buildings anywhere you go like in vacation. Is there a way to tell approximately where is the neutral axis location
For example. Are most columns compression region beyond the centroid at middle meaning neutral axis has mostly compressive region or is.the tension region larger. For instance. In edge columns is it possible the neutral axis has most compression region even thou axial load is not large? Is this common in building or rare? Thank you.
RE: Virtual vs Real Neutral Axis
Doug Jenkins
Interactive Design Services
http://newtonexcelbach.wordpress.com/
RE: Virtual vs Real Neutral Axis
BA. Usually how big is the percentage of moments compared to axial load did you allocate to edge columns in your gravity load only building? I know seismic need much higher allocations. Im asking for static typical framed building side columns moments where only 3 beams framed to it. What is the typical moments of these for small.houses and big houses or those you usually.build?
RE: Virtual vs Real Neutral Axis
A column which supports N levels including roof is carrying:
P = N.w.a2/2
and the typical moment is M = wa3/48
so the M/P ratio is a/24N
A 12 story building would have an M/P ratio of a/24 in the top floor and a/288 in the main floor. If a = 16', M/P = 8" in the top floor and 0.67" in the main floor. Ordinarily, a minimum eccentricity of one or two inches would apply in the lower floors.
BA
RE: Virtual vs Real Neutral Axis
Doug. If there is no concrete and only the compression steel taking the resistance at the compressive side. How do you solve for the balanced point provided the bars are of equal amount? At what moment and axial load will the strains of both the tension and compression steel be at yield at same time? Or won't it occur at all?
RE: Virtual vs Real Neutral Axis
RE: Virtual vs Real Neutral Axis
Retro. Is the formula to get the moment for the above case:
M = As.Fy.d
where As = area of steel on each face
Fy = yield stress
d = c/c distance between reinforcement
Or is it M = As.Fy.e?
where e = eccentricity from centroid to the load?
Elsewhere. I think BA seemed to believe the former. Why is it "d" and not the "e" (eccentricity)?
RE: Virtual vs Real Neutral Axis
No, if the axial load is zero then the tensile force in the tension steel is equal to the compression force in the concrete plus compression steel, so the tension steel will reach yield before the compression steel, and before the concrete reaches ultimate strain (unless the steel has a very high yield strain and high %, or the reinforcement layout is very unusual).
Doug Jenkins
Interactive Design Services
http://newtonexcelbach.wordpress.com/
RE: Virtual vs Real Neutral Axis
Doug. Retro was describing my inquiry what if there was no concrete and only bars in the compression side and tension side of a column. Just hypothetical if such structure can be invented.
Then was Retro right that in concreteless column with bars only that "If the reinforcement is symmetric the balance point will be at pure bending (ie. zero axial load)."
RE: Virtual vs Real Neutral Axis
As.Fy.d is the moment capacity with no axial load.
With axial load, the force in each set of bars would be P/2 ± M/d or P(1/2 ± e/d)
The limiting strength occurs when one or both sets of bars reaches yield.
BA
RE: Virtual vs Real Neutral Axis
First. Let's use the figure you have computed yourself in past thread. You said in another thread that:
Here's the interaction diagram for the values you have given (for the column without concrete but bars only):
The letters corresponds to the Red Letter I included in your post above.
For a week. I am unable to plot the limit curve that you see for typical interaction diagram. Where is the curve? That's why I was asking Doug how to get the balanced point of the concreteless column.
But now the following formula may be related to it and complete the interaction diagram.
Given the above values you yourself gave.
As = 20*300M = 2400
fy = 400 MPA
d= 420mm
M = 403.2 kN-m
e = 0.21 mm
Using Axial Load of 100 kN.
A = 2.As = 4800
I (moment of inertia?) = As.d^2/2 = 2400 kN * 420 mm ^2/2 = 211680000
With axial load, the force in the bars is P(1/A ± P.e/I) = 100 * (1/4800 + 100* 0.21/211680000) = 0.020843 (plus),
0.0208234 (minus)
The stress in the bars on each face, S = P/As(1/2 ± 2e/d^2) = 100/2400 * (1/2 + 2 (0.21)/420mm^2)= 0.020833 (plus),
0.0208332 (minus)
At only 100 kN. Why is the bars already near yield strain of 0.020833 (is that the formula of stress or strain (it looks like strain)? In your earlier calculations and the interaction diagram. It is supposed to have 960 kN capacity??
Anyway. In your latest post. You mentioned a formula... solving for it in excel:
"With axial load, the force in each set of bars would be P/2 ± M/d or P(1/2 ± e/d)".. solving
P/2 ± M/d = 50.9 (plus), 49 (minus)
P(1/2 ± e/d) = 55.25 (plus), 44.75 (minus)
is this the ksi or yield stress and earlier formula the strain?
But again.. at only 100kN.. why are they in yield stress whereas in the interaction diagram figures you calculated. It has capacity of 960 kN (point C). I'm very confused. Thank you.
RE: Virtual vs Real Neutral Axis
To answer the first part of your post, after writing the following post, I realized I had omitted a term:
It should have read:
With axial load, the stress in the bars is P/A ± P.e.y/I where y is d/2.
Then S = P/As(1/2 ± e/d)
Instead of correcting that post, I rewrote it expressing it in terms of bar force instead of stress.
BA
RE: Virtual vs Real Neutral Axis
In the second part of your post, the problem is complicated by the presence of four intermediate bars. For pure axial load, each of the four intermediate bars carry as much load as an outside bar but for eccentric axial load, their contribution is not too clear and was neglected in my earlier post to mes7a.
BA
RE: Virtual vs Real Neutral Axis
A column has 20 bars, 8 on each side and 4 intermediate bars. Each bar has As = 300mm2. A void has occurred in the concrete during pouring such that only the steel bars are acting for a short length near the bottom. The void was filled with epoxy which may have some beneficial effect, but this is ignored in the following analysis.
As = 20*300M = 2400
fy = 400 MPA
d= 420mm
M = 403.2 kN-m (this occurs with P = 0)
e = 0.21 mm
Using Axial Load of 100 kN.
f = P/A ± P.e.y/I
I = 300[2*8*(d/2)2 + 2*2(d/6)2] = 1233d2
y = d/2 so y/I = 1/2466d
f = 100,000/2400 ± 100,000(0.21)/(2466*420) = 41.7 ± 0.02 = 41.7 MPa << 400
M[P=100] = (400 - 41.7)2466d = 371e6 N-mm or 371kN-m (92% of M[P=0])
Or, in other words, e = 3.71m when P = 100 kN.
With just the reinforcement and no concrete, there is no tension branch to the interaction diagram. The steel on the compression side will always yield first.
BA
RE: Virtual vs Real Neutral Axis
You mean at P=100 kN. Moment capacity is 371 kN-m? how can e=3.71 meters be so big.. Why is there is no tension branch to pull it on opposite side? But in the earlier computations that makes up the following interaction diagram. When P=960kN. It has moment capacity of 202 kN-m at eccentricity of 0.21.
Are they compatible points that can be included in the interaction diagram (shown in pink dot). In the following.. is the curve more of the blue one or red one.. and how to compute for the balanced point (whatever it can be?). Million thanks Ba.
RE: Virtual vs Real Neutral Axis
After entering your formulas in Excel. I tried value Axial load of 960kN. Moment is negative. Trying 900kN. Moment capacity is 25.7kN-m at an eccentricity of 0.028 meter. But in the following where you wrote before that when the load is directly over the steel on the compression side, the capacity is 960kN. with moment capacity left of 202 kN-m at eccentricity of 210mm. You mean this isn't really the case and the real one is the 900kN and 25.7KN-m? Or maybe my excel input has typos. I'm confused. Please enlighten. Thanks.
(you wrote before:)
"When the load is directly over the steel on the compression side, those bars will carry it all by themselves. Their capacity is 8*300*400 = 960kN. That gives you another point on the diagram, i.e. 960kN at an eccentricity of 210mm which means a moment of 960(0.21) = 202 kN-m. At that point, the remainder of the bars, twelve in total, will be doing very little work."
RE: Virtual vs Real Neutral Axis
As = 20*300M = 2400 (this is wrong) should be 6000
fy = 400 MPA
d= 420mm
M = 403.2 kN-m (this occurs with P = 0)
Using Axial Load of 960 kN.
P/A =960,000/6000 = 160MPa
This leaves 400-160 = 240MPa for bending
M.y/I = 240
M = 240*I/y = 240*2466*420 = 248e6N-mm or 248 kN-m.
e = M/P = 248/960 = 0.258m or 258mm
The point on the interaction diagram, P, M is 960, 258
My earlier calculation for 100 kN is also wrong as I used the wrong area of steel. I will correct that in a subsequent post.
BA
RE: Virtual vs Real Neutral Axis
How can one tell if there is tension branch or not to the interaction diagram (why did you say there isn't?). Also if there is concrete contribution, will eccentricity be lesser?
Looking at the strain diagram in the following:
If there is no concrete contribution.. the strain of the compression bars will be more.. but the tension side will be equally more from the seasaw effect. So there seems to be tension side. But how do you get the balanced point of the interaction diagram for this concreteless column?
RE: Virtual vs Real Neutral Axis
P/A = 100,000/6000 = 16.7 MPa
This leaves 400 - 16.7 = 383 MPa for bending
M = 383*I/y = 383*2466*420 = 397e6N-mm or 397 kN-m
and e = M/P = 3.97 m
The interaction curve is actually a straight line from P = 2400, M = 0 down to P = 0, M = 414. The pure moment value is slightly higher than we had before because it takes into account intermediate bars which the earlier estimate omitted since their contribution was thought to be small.
There is no tension branch and no balanced point. Failure occurs when the bars on the compression side reach yield stress.
BA
RE: Virtual vs Real Neutral Axis
When the bars on the compression side reach yield stress.. what is the strain in the tension side.. below yield strain? I can't seem to draw it by means of the strain triangular geometric relationship. The following diagram is wrong.. isn't it?
When there is no concrete.. the balance point won't shift left and right.. and will only stay at the middle?
Or don't you use any strain diagram but based it on the Moment of Inertia? Btw.. since I-beams (wide flange) are pure steel.. don't they also have balanced point? What other steel use the same computations as the concreteless column with just bars?
RE: Virtual vs Real Neutral Axis
The balanced point on an interaction diagram for a concrete column is defined as a strain of 0.003 on the compression face occurring simultaneously with a yield strain on the tensile steel. Geometry will determine where the neutral axis occurs. It is the point of zero strain. It varies according to the yield strength of the steel. A maximum strain of 0.003 is somewhat arbitrarily set and in fact there are some who believe it should be set a little higher.
Ordinarily, for a steel shape the ultimate moment is defined as M = Z.Fy where Z is the plastic modulus. In the above, I have used the relationship M = I.Fy/y which is the same as S.Fy where S is the section modulus. The reason for doing this is that, in order to use the plastic modulus, the strain in the outer bars would have to be three times yield strain or, in the case of 400 MPa (60 ksi) steel, would be in the order of 0.006 which seemed too high.
BA
RE: Virtual vs Real Neutral Axis
Depends on the eccentricity. When e = 0 (P is maximum), the strain on all bars is Fy/E. When e is infinite (P=0) the strain is equal but opposite sign. For all other values of e, the strain is between those limits.
BA
RE: Virtual vs Real Neutral Axis
"Balanced point" can have 2 meanings. One where in the interaction diagram it is point where axial load and moment fail at same time. When I mentioned "balanced points" in the column. I'm referring to the neutral axis near the middle where it is balancing between the left tension side and right compression forces of both concrete and steel. Here one can describe it as tipping in the balance. But then even if the neutral axis is not on the centroid.. one can say the column is tipping in the balance at the point of the neutral axis between tension and compression.. this is why the resultant is zero.. maybe I should avoid using the terms "Balanced point" in the column but a "point of balance" instead?
But note our concreteless column with rebars at sides are not really I-beams. It's not whole section steel I-beam. Our case is like treating the rebars as creating a force field that around it that created the column. Here is it right to analyze the column as I-beam when there is really no steel connecting the bars in our concreteless column but just bars?
This is what I've been trying to visualize for weeks. You said lately there is no tension branch in the interaction diagram. Yet when the compression side strain is increasing and yielding.. .. the tension side strain would also increase and yield. So how to reconcile this with the statement there is no tension branch in the interaction diagram. Maybe because of the presence of concrete in the normal case, the compression side has more resistance so it is not yet yielding while the tension side already yielding. But in the concreteless case.. both are yielding? But it seems the tension side would yield first and break before the compressive side. Not other way around. Have you got any drawing software.. can you please draw the strain diagram for the concreteless column where the strain is in between those limits?
Many thanks!
RE: Virtual vs Real Neutral Axis
The bars are surrounded by concrete above and below the void so that they cannot move relative to one another. If the gap was large, the bars could buckle below yield stress, but the gap is not very large, so they would tend to act like short steel columns. Also, the bars are surrounded by epoxy material which would would tend to prevent buckling. We are making the assumption that the bars are capable of carrying the full yield stress before failure. That may or may not be a valid assumption but to me it appears reasonable.
Further to your request, I include a sketch showing four strain diagrams for various combinations of P and M. In Fig. 1 the load is maximum and the moment is zero. The green line indicates an equal compressive strain for each of the four layers of steel.
In Fig. 2, the eccentricity is small and failure takes place when the right hand bar reaches yield. The remaining bars are in compression but not at yield.
In Fig. 3, the eccentricity is increased to the point where the left hand bar is in tension, but well below yield.
In Fig. 4, the eccentricity is infinite, the moment is maximum and the load is zero. The left outside bars reach yield point in tension; simultaneously, the right outside bars reach the yield point in compression. The intermediate bars are strained but only to one third of the yield strain.
I hope this helps.
BA
RE: Virtual vs Real Neutral Axis
Thanks very much for them. I finally understood what you meant. Do ALL steel columns such as I-beam (wide flange) or even HSS also have the same flat curve? So the reason concrete column has the unique curve with tension branch is simply because the concrete compressive block is giving more capacity to the compression side and also pressing on the tension side and decreasing the press as the axial load decrease below balanced point giving lesser moment capacity.. and this is what the concrete interaction diagram curve is all about.. For engineers who grow up familiar only with steel columns.. the concrete curve would not be a normal one. Generally do steel columns also need interaction diagram even if the curve is flat?
By the way. In Canada.. why is your 300M bars only 300mm.. elsewhere it's 314m (area of 20mm diameter).. also why is your yield stress 400MPA. Normal is 414MPA.. why didn't you use the value 414MPA? Just for ease of calculations? Actual yield strain of the bars are tested at 470MPA.. so inputting them in excel.. when the axial load is 960K.. moment is 328kN-m capacity instead of 248kN.. with increase of 80kN.
RE: Virtual vs Real Neutral Axis
The unusual shape of an interaction diagram is largely due to the properties of concrete; concrete does not have a well defined yield point, cannot resist tensile stress and cannot be compressed beyond a relatively small strain. In short, it is not an elastic material and that gives rise to the strange shape of the interaction diagram. Steel columns do not normally require interaction diagrams.
The bars in Canada were made slightly different than the nominal dimension, some larger, some smaller so that the area could be easily remembered, 200 mm2 for 15M, 300 for 20M, 500 for 25M etc.
Before we switched to metric sizes, the yield was 60,000 psi which is 414 MPa. Most local fabricators provide a minimum guaranteed yield strength of 60,000 psi today but they are not legally required to do so; the specified yield for 400 Grade steel is 400 MPa so that is what Canadian engineers use in design. It provides a small increase in safety factor but the cost difference is not too significant.
BA
RE: Virtual vs Real Neutral Axis
BA. What part of structural books did you get the formulas above. I was asking for edge column moment estimate. I couldn't find the formulas in the column sections in the books. But computing for your formulas.. the output doesn't look right:
unit floor load w = 100 psf pounds per square feet
a = 20 feet
axial load per floor = wa^2/2 = 100*20^2/2 = 20,000 kips (or 20,000 x 4.448 = 88,960 Kilonewton
moment shared by 2 columns = wa^3/24 = 100*20^3/24 = 33333 kips-ft (or about 33333 x 4.448 / 3.28 = 45,203 kN-m.
something is wrong.. values too much.
Anyway. You mentioned the M/P ratio or eccentricity is lower in the ground floor.. This is because the moments in the column above can make it straight and less tendency to bend versus when the column above can translate in any direction as in top floor column.
The bottom line of what it seems to be saying is.. moments from unbalanced loading in edge columns in ground floor are not as high as top floor. Therefore it is the seismic moments induced in the ground floor edge column that would dominate than edge column gravity moments, isn't it.
RE: Virtual vs Real Neutral Axis
The M/P ratio is lower in the ground floor because P is larger; it is the sum of all the upper story loads. M from unbalanced loading is approximately equal in all floors. In most tall buildings, lateral forces are resisted by shear walls, elevator shafts or moment resisting frames, not by simple bending of columns.
We are fortunate in Alberta that we don't get significant seismic events. We don't get typhoons either but we do design for some wind forces. In the Philippines, I would expect lateral forces resulting from seismic events and wind load to be much more of a concern than gravity load.
BA
RE: Virtual vs Real Neutral Axis
Ok. Thanks for all assistance. To summarize and for my conclusion of it all after 2 years of trying to understand the behavior of epoxy repair in column voids.
Instead of the designed 4 storey with concrete roof and all concrete walls. We would just make it 3 storey with very light roof and very light wall. That's like reducing it by 2 solid storeys (building only half the intended floor). This should produce lower seismic base shear. This should hopefully put the seismic load to within 890kN axial load capacity of the bars only column section and 328kN-m moment capacity when load is directly over the compression bars (when seismic moment is more.. then axial capacity would be less.. that's where the epoxy may help (see below)). Edge column already used 33kN from the unbalanced load, there would be 297 kN-m capacity left of the moments in the pure reinforcement column alone. If you'll add the contribution of epoxy. Then one has to use the interaction diagram formula that includes the bars and epoxy acting together, right? It's then capacity of 863 kN axial load and 488kN-m moment capacity at balanced point with eccentricity of 565mm. Improving from the sole bars only column capacity of 960kN and 328 kN-M when load is directly at compression bars with eccentricity of 0.258 meters.
reviewing the computations of the interaction diagram of the epoxy.
fs (tension steel) = strain Es (d-c)/c
fs' (compression steel)= strain Es (c-d')/c
C (compressive resultant) = C = 0.85fc'ab
cb (neutral axis balance failure) = d (strain concrete ultimate/(strain concrete ultimate + strain bar ultimate)
a= 0.85 cb
Pn= 0.85 fc' ab + As'fs' - As fs
M = Pn e = 0.85 fc' ab (h/2 - a/2) + As' fs' (h/2 - d') + As fs (d-h/2)
To relate it to epoxy. I'll use stress 1350 Psi (0.003 strain x 450ksi (epoxy)) instead of concrete
4000 Psi. because let's treat the entire compression block to be composed
of epoxy
given:
epoxy strain 0.003 (although it can be pushed higher but let's use it as standard meantime)
steel unit strain is 60/29,000=0.0021,
column dimension 19.685" x 19.685 " (0.5 x 0.5 mtr)
area steel = 8 x 0.46 = 3.68
from excel input of formulas and values
cb (neutral axis) = 10.10885"
a (stress block depth)= 8.59"
fs' = 65.48 ksi but <= 60 ksi
C = 0.85 x 1.35 (psi) x 8.59" x 19.685" = 194 kips = 863 kN
Pn = 863 + 3.68 x 60 - 3.68 x 60 = 194 kips = 863 kN
Mn = 4318.946 in-kip = 359.9 ft-kip = 488 kN.m
This means with the epoxy as compressive block in addition to the bars.. axial load capacity is 863 kN and Moment capacity is 488 kN.m. eccentricity is 22.25" or 565mm.
Since the compression block is 0.2 mtr or about 8 inches.. then the entire compression block is really epoxy.
Notice that at the balanced point of the epoxy or even normal concrete column. The tension bars and compression bars cancel out in the formula of axial load which is Pn= 0.85 fc' ab + As'fs' - As fs.
This means at eccentricity of 565mm (see 2 paragraph above). The compression bars axial capacity really cancel out to the tension bars? This is the part that still confusing me a bit when I tried to reconcile both the contribution of bars only and epoxy. Does it mean if there is no epoxy or concrete, the behavior of the bars only and contribution of the bars at compressive side to axial load is different than with either concrete or epoxy? Or if same.. it means at eccentricity of 565mm.. there is almost no axial capacity of the bars? Or is it because the bars only column can't reach balanced point that it won't happen? Meaning before the tension side yield.. the compression side is already yielded and even rupturing later on. Just to confirm this one.
Thanks a lot again BA!
RE: Virtual vs Real Neutral Axis
BA
RE: Virtual vs Real Neutral Axis
Ok. For years I always just wanted to focus on concrete structures and not steel structures to avoid more understanding and spending time studying it. But this epoxy repair in void is finally making me think of steel especially when you said the bars in compression can act like steel. After spending some time today on it (til my eyes strain reading many materials). Just some quick question about Section Modulus and Moment of Inertia (I know their basics but just verifying my understanding). I know it is one properties that concrete structures don't have or focus. My questions is (don't worry I won't drag it long just the following part). For example. The steel
from http://www.engineeringtoolbox.com/american-wide-fl...
W12x30 has Moment of Inertia of 38.6 and Section Modulus of 6.2
W16x26 has Moment inertia of 38.4 and section modulus of 3.5.
(notice the shorter W12X30 has slightly bigger moment of inertia and bigger section modulus.
Using L/25 for W16x26 and say 10 meters span
If L/25 = 10 meters /25 = 400mm (16")
So the depth of W16x26 is optimal for 10 meters span for less deflection. But the shorter W12x30 has less depth and yet it's moment of inertia and section modulus is larger than the W16x26. Does it mean when put in a 10 meter span. The W12x30 would be as good or even better than the W16x26 because of bigger section modulus and slightly bigger moment of inertia too? Would they deflect about the same at maximum load when put in identical 10 meter span? Since the Moment of Inertia is almost the same, or would the smaller W12x30 deflect bit less since it has higher moment of inertia and section modulus too than the W16x26 under the same 10 meters span? Just this important question. I know I'll read the rest of concept in text books in weeks ahead but need to know just this one now. Thanks.
RE: Virtual vs Real Neutral Axis
Size------------- d - - - b - - - t - - - - Area - - W- - - Ix- - - Iy- - Wx- - Wy
W 16 x 26 -- 15.69 -- 5.5 -- 0.250 -- 7.68 -- 26 -- 301 -- 9.6 -- 38.4 -- 3.5
W 12 x 30 -- 12.34 -- 6.52 -- 0.26 -- 8.8 - - 30 -- 238 -- 20.3 -- 38.6 -- 6.2
I copied two lines of the table you provided. The justification leaves something to be desired, but you can see that Ix is about 26% greater for the W16 than the W12. The strength of the two shapes is almost identical shown by Wx (section modulus). In Canada, we normally use the symbol Sx to denote section modulus about the X axis.
The beams will carry nearly the same moment if they are continuously braced against lateral torsional buckling (LTB). If they are unbraced against LTB, the W12x30 will do much better, particularly for longer spans.
Deflection is inversely proportional to the moment of inertia Ix, so the W16x26 will deflect 79% of the W12x30 deflection.
BA
RE: Virtual vs Real Neutral Axis
Size----------------- d - - - b - - - t - - - - Area - - W- - - Ix- - - Iy- - Wx- - Wy
W 14 x 22 ----13.74 --- 5 ------ 0.230 -- 6.49 -- 22 -- 199 -- 7 -- 29.0 -- 2.8
W 10 x 45 ---10.10 -- 8.020 -- 0.350 -- 13.3 -- 45 -- 248 -- 53.4 -- 49.1 -- 13.3
In the above example. The shorter w10x45 has both bigger section modulus and moment of inertia compared to the deeper w 14 x 22.
1) first question, why manufacturer a deeper wide flange when the moment of inertia and section modulus are smaller than a less deep one? loading requirement?
2) from say L/25... if L=350" or 29.166 feet.. then w14x22 fulfill it because L/25 = 350/25 = 14". Now my question is. What if your use the w10x45 on the same 29.166 feet span. What would deflect more.. the w14x22 or the w10x45? I'm asking because I want to know if its possible the w10x45 can deflect less even if its only for 250" or 20.83 feet if its use on the 29.166 feet span used by the w14x22?
2)If the answer is yes to question 2.. and i'm expecting it (please confirm).. can you consider the span to depth ratio deflection some kind of tension stress? this deflection is supposed or calculated to occur at maximum load isn't it. So if you load is just one half.. you can use a smaller size wide flange even if it's only half the required span/depth ratio?
RE: Virtual vs Real Neutral Axis
The W10x45 would deflect less, actually 80% of the deflection of the W14x22 but would cost twice as much in material.
The span to depth ratio is not a tension stress or any other kind of stress. Economical design requires that a beam having the least weight be selected provided it satisfies all of the requirements, namely strength and deflection. The span to depth ratio is a measure of deflection when the beam is stressed to a particular stress under service loads.
BA
RE: Virtual vs Real Neutral Axis
What is the resultant then of this epoxy compressive block? In the concrete one, the resultant is the average of the concrete compressive block (the C).. or you mean one must not even use resultant in the epoxy block? Or how to compute for it.. how do you handle this say the stress variation in the epoxy is really triangular in shape.. For concrete, it's like more of slight sloped square? I'm trying to compute for this epoxy axial and moment contribution. Thanks.
RE: Virtual vs Real Neutral Axis
We now have an unsymmetrical section consisting of 20 bars located as previously discussed plus an eccentrically located 10" x 0.31" steel plate. The center of gravity of the transformed section, the transformed area, the transformed moment of inertia and the transformed section modulus can all be calculated. With those properties, P and M can be calculated for various eccentricities and an interaction diagram can be drawn.
The maximum stress in the epoxy is the maximum stress in the steel plate divided by the modular ratio, 64.4. The compression attributed to the epoxy is the average pressure multiplied by the area of epoxy. It will be located at the center of gravity of the stress distribution which is easily calculated because the stress distribution is linear.
BA
RE: Virtual vs Real Neutral Axis
If there is load in the 2nd floor top. The moments in the ground floor got lower (only 2.75kN in the lower left.. notice this is just test column and beams and not my building etab file which is with my designer). See:
Notice the moment in the lower left (in ground floor) decrease from 6.26kN to 2.75kN (when load of 2nd floor top is added). This is really true.. ah? what is this principle called... Equilibrium? Where in the book is this mentioned. Because if true it means if I removed the waterproof topping above the second floor. It would do the opposite of what I intended.. it would increase the moments in the ground floor.. so maybe I shouldn't touch it and leave it as it is. Hmm....
Any profound thing about this with regards to Neutral Axis?
RE: Virtual vs Real Neutral Axis
Removal of topping on the second floor is not warranted given the current state of knowledge about the design and would likely do more harm than good. Leave the structure alone until you know what you are doing.
BA
RE: Virtual vs Real Neutral Axis
That's just a more extreme example (omitting entire column and beam in the analysis). Initially I have reduced the load in the second floor top and there is an increase in the column moment at ground floor. For example.. this is the dead load loading of my structure (dead load of slabs connected to the beams, the left lower column is the one with epoxy void):
If there is additional 10kN uniform load on top of the 2nd floor.. the moment in the ground floor column is lower at 7.48kN-m. Whereas if there is no additional 10kN uniform load.. the ground floor column moment is higher at 9.39kN-m.
With more load in the 2nd floor top. The base moment of the 2nd floor is higher at 16.87kN-m instead of 13.05kN-m. However this makes the ground floor top column moment smaller. This is what I'm pointing out. What is the technical explanation for this. It's as if the 2nd floor column base moment takes some moment away (shares more) from ground floor top column.
RE: Virtual vs Real Neutral Axis
I think the building should be left as it is until there is good reason to take remedial measures. At the present time, I am unaware of any such reason. Design and review during construction has been carried out by a local structural engineer who appears to be unwilling or unable to expend any effort in resolving the issues which are of concern to the owner (mes7a). If there are serious concerns about the safety of the building, another engineer must be retained to review the design and to inspect the facility. It cannot be done in a forum such as this.
BA
RE: Virtual vs Real Neutral Axis
That output is not meant to accurately depict the exact structural framing and loading because it is the designer who is good in that especially seismic. I'm asking just how opposite moments seem to interact. And I showed it to you pinned bases because I thought most are more familiar with pinned bases because to make really fixed bases in reinforced concrete, there must be massive top reinforcement which is not usually done in footings. We spent over $10,000 extra for the top reinforcement...
Anyway. My concern is the physics principle. Because if moments all lined up together, no building can stand up because all the edge columns would bend and bend from increased moments until they just collapse. But in reality, the moments are S shaped.. the lower part of the S can seem to cancel the higher part of the moment of the lower floor. This is unexpected physics principle I learnt. I just want to know if you have another way of explaining it.. why the moments seem to cancel. In the following case. Even with fixed bases, the column moments in the ground floor is lower with additional load in the eccentric beam of the second floor top. Normally you'd expect the moments in the ground floor to increase when eccentric loading are done in the upper floor. But it seems to have equilibrium effect.
With no additional load applied on second floor top:
WITH additional load applied on second floor top (moment in ground floor got lower.. becomes 7.93kN-m from 10kN-m. see below.. Before I thought moment should increase.. but some physics principle seems at work.. what is your other way to state this physics principle.. something to do with opposite moments cancel? ):
RE: Virtual vs Real Neutral Axis
BA
RE: Virtual vs Real Neutral Axis
RE: Virtual vs Real Neutral Axis
I don't know what you mean by your question. Your language is not precise. Moments do not subtract or cancel, whatever that means.
It is normal for an edge column under gravity load to have stress reversal from top to bottom in each story. It bends in the shape of an "S"; the top is stretched on the outside while the bottom is stretched on the inside. Somewhere near mid-height, the moment is zero. The moment diagram is a straight line which peaks at top and bottom and passes through zero at the point of inflection. The moment at the top does not cancel the moment at the bottom or vice versa. They each must be reinforced as required by a frame analysis.
BA
RE: Virtual vs Real Neutral Axis
(Edit here are clearer illutration
S
S
Imagine the 2 S above are moments of columns in each storey. You will notice that the moment at bottom of second storey is opposite to the moment of column in top of ground floor. Or in your earlier statement. The above column can share the moments used by first storey beam. In the case of normal edge column with the 2 S. Load above a storey can lessen the moments below it bec it seems to restrain the beam in opposite direction. Gets?)
RE: Virtual vs Real Neutral Axis
BA
RE: Virtual vs Real Neutral Axis
Here are clearer illustration
S
S
Imagine the 2 S above are moments of columns in each storey. You will notice that the moment at bottom of second storey is opposite to the moment of column in top of ground floor. Or in your earlier statement. The above column can share the moments used by first storey beam. In the case of normal edge column with the 2 S. Load above a storey can lessen the moments below it bec it seems to restrain the beam in opposite direction. Gets?)
Post Edited
RE: Virtual vs Real Neutral Axis
The red is the third floor column position with the epoxy void at ground floor. Originally the sides of the front rafter (see blue arrows) has the sides rested on the perimeter w8x21 wide flange (rafter is also w8x21). But I told designer it may deflect and the column may take majority of the load. I told him what if I rested the rafter on the columns at side. He said it's up to me.. because he didn't want to consider the epoxy soft modulus (he didn't want to review the meaning of modulus.. he is a civil engineer.. in our country.. our structural engineers are mostly civil engineers not familiar with the physics of it). So I and the contractor kept discussing these past 2 days about this slant implementation right on the column tops. Do you think there is problem if rafters were put on the slant at front resting on all columns?
Now the most important part I'm concerned these past 3 days is.. is it possible to balance (or diminish) the ground floor unbalanced column moments by the position of the rafter on the front column. This is the reason I'm asking what is the effect of column moments at third floor on ground floor and whether it is possible to diminish the moments. In your 60 years of practice.. how often do you need to diminish the edge ground column moments with loading in the third floor. Please share your experience on this.
This is related to the heart of this thread when Doug mentioned earlier "The combined axial load and moment are equivalent to an eccentric vertical load, with the eccentricity on the compressive side. If you apply an additional load at the column centroid it reduces the eccentricity of the resultant load, and the width of the compression region increases, so the total compression force increases, but the maximum stress reduces. It's a similar situation to a footing with a vertical load outside the middle third. If you apply an additional load on the uplift side, the pressure on the compression side will reduce, even though the total reaction force is increased."
I'm wondering if you can diminish the ground floor column moments by loading in the third floor? Thanks.
RE: Virtual vs Real Neutral Axis
Where the two "S"s meet, the moment acting on both upper and lower columns are clockwise. They are not opposite. They are the same. The only thing opposite about them is that they put tension on different sides of the column, but that is a consequence of geometry.
Load above a story does not lessen moments below. If anything, it increases them as a result of the P-delta effect.
Moments applied to a joint above a story can lessen or increase moments elsewhere depending on the direction of the applied moment. If you want to see how an applied moment affects your structure, remove all load and apply a single moment to any joint of your choosing and see how that affects moments elsewhere.
BA
RE: Virtual vs Real Neutral Axis
In response to your last post, the roof will be a light structure which will have no significant effect on the moment on the first story column at X2/Y4 irrespective of how you frame it.
I would be much more concerned about moments resulting from earthquake or typhoon. Without any shear walls, your structure relies totally on bending of nine columns to provide stability. Are the exterior walls capable of acting as shear walls?
BA
RE: Virtual vs Real Neutral Axis
Tomorrow we will spend about 5 hours to bring it up because they are so heavy. W8x21 is 21 lbs per foot.. so per piece (21 x 20 feet)is about 420 lbs.. can't be carried by even 3 people. They need to pulley chain it up. 420 lbs is about 1.868253072 Kilonewton. So 1 complete rafters composing of 2 pcs are 3.736kN. Now there are many C-purlin sized 2x6" 1.8mm crisscrossing it spaced at 0.6 mtr apart.. so weight can reach 4.5 kN.. added the roof can reach 5 kN (do you have formulas to get weight of rafters and purlins over a rafter or column?). I'm worried this can further strain the column with epoxy void. We even plan to just put rafters at the sides and just welded it at middle without any support (meaning not connected to middle front column) but they don't seem competent for such connections and we are nervous about if such will even be stable. What you think? Is this connection stable?
The contractor wants to connect the sides at the perimeter w8x21 but I'm not allowing him. If he does that and the perimeter beam is flexible. How much load would the column sees? (remember rigidity can take more load than flexibility)
No the exterior walls are just composed of hollow blocks with mortar thrown into them. These mortars don't even mostly fill up the gap between column and hollow block for example. So it's effect on seismic may be insignificant.. or is it? We can't put braced frames either. What is your opinion. Also I plan not to use hollow blocks walls at the third floor to avoid more seismic base shear (our headache now is now to put up light wall walls). We still don't know what kind of wall to put up. (remember we were discussing how to put hollow blocks parapet above.. but decided not to do it anymore because of twisting problem). It must surely be lightweight for less base shear. Last if we remove the waterproof topping which becomes unnecessary because the presence of the roof. It's about 2 inches thickness so we will be removing unnessary SD load. But if I remove them all. I notice the moments in the ground column (one with epoxy) can increase. I'm still trying to understand your explanation of the 2 S put in top of each other signifying floors. You said a beam is shared by lower and top column. What scenario when the top column will have decrease in moment by certain pattern of column above the joint in next floor. This is what I've been trying to show you and understand.. You still haven' see the slight moments decrease between them?
RE: Virtual vs Real Neutral Axis
You are modifying the design on the fly? You can't do that! If you want to change the roof framing, get the approval of the design engineer.
I don't understand what you mean and I don't wish to comment. Check with the design engineer.
BA
RE: Virtual vs Real Neutral Axis
Why don't you get all these ideas on the drawing before proceeding with the work? What is the point of having architects and engineers if you are going to change everything as you go? Perhaps you could suggest a steel stud wall system to the architect; much lighter than masonry.
That does not sound right to me. I suspect your input is wrong.
Consider any joint in your frame. Assume that a clockwise moment is applied to it. All members meeting at the joint rotate in a clockwise direction. The upper column feels tension on the right face. The lower column feels tension on the left face. The left beam feels tension on top while the right beam feels tension on bottom. That is all I was saying, nothing very mysterious. Since the opposite ends of those members are not free to rotate, a moment is introduced at the far end of each member which distributes among the members meeting at that joint according to their relative stiffness. That moment is counterclockwise.
I have seen your moment output but I haven't seen your input so I cannot draw any firm conclusions.
BA
RE: Virtual vs Real Neutral Axis
The contractor wants to follow the original plan where the sides of the rafter rested on the perimeter beam. I asked the designer if I can rest the rafter sides at columns instead. Designer said I can do either because it's just light roofting anyway and depends on the skill of the contractor. designer is just 22 years old.. and just operator of Etabs.. not really a pure blooded structural engineer. He has only 3% of your knowledge. Anyway. I'm talking of the following concept shared by Kootk in other threads where he stated:
"1) if an end deflects downwards, it will decrease the reaction at that end, increase the reaction in the middle, and decrease the reaction at the opposite end.
2) if an end deflects upwards, it will increase the reaction at that end, decrease the reaction in the middle, and increase the reaction at the opposite end.
3) if both ends and the middle support deflect the same amount, you're back to 5P/16 & 11P/8. "
So if the rafters ends at the perimeter beams deflect downwards, it will decrease the reaction at that ends, increase the reaction in the middle column.
Anyway. We'll just put the ends on columns to equally distribute the load.
I'd like to go back to this double S curvature... because with roofing put, the waterproof 2" topping below is useless and it's unnecessary SD load. The 22 year old designer said I could remove it or not.. depends if I want increased expenses of removal. We won't use jackhammer but just manually lifting the topping piece by piece..
This decision depends on my understanding of the moments redistribution of columns between floors. Back to the column curvature in the 2nd and second floor.
S
S
Note if the value of the 2nd floor S moment curvature is much larger.. there would be decrease in the moments at the top of the 1st floor column. You said earlier "Where the two "S"s meet, the moment acting on both upper and lower columns are clockwise. They are not opposite. They are the same. The only thing opposite about them is that they put tension on different sides of the column, but that is a consequence of geometry."
But there is clearly a decrease in the moments right below the joint. I was asking yesterday the explanation for the decrease. Again note you said a beam moment is shared by the column below and above it. So if the moment above is big it's because the bigger moment curvature above the joint makes it share more load from the beam and why the moments at the column below that joint decrease?? Of course we won't mean removing the reinforcement (because they are already casted in concrete.. just want this understanding as the last concept in this thread).
But this may not affect the moment much in the column base with epoxy void at ground.. isn't it. So I'm thinking it's better to remove the topping to lessen seismic load. Btw.. the columns are all rested on very big combined foundation designed for 4 storey (see below). So the other moment resisting system may hopefully compensate for the pinned like condition at the epoxy void (esp when almost 1.5 storey were not actually built and designer said the forces in the columns are much lower).
RE: Virtual vs Real Neutral Axis
Let's just create a very simple beam span supported by 2 columns. The 3D of it is the following:
The following shows the beam and column moments.
In the following. Uniform load of 5 kN were added to the top beam. You can see that the moments of the lower columns decrease.. I simply want to understand why. Are you saying the counterclockwise thing you mentioned makes it looks like it decreases?
RE: Virtual vs Real Neutral Axis
The upper sketch on the attached file shows column deformation when the Second Floor beam is loaded. The lower sketch shows column deformation when the First Floor beam is loaded. The blue shading indicates load.
The curved dashed lines show the column deformation under each load. It is apparent that a uniform load on the Second floor beam produces the opposite effect to a load on the First floor beam, that is the curvatures are reversed. That would explain why your base moment decreased with the addition of 5kN to the upper beam.
BA
RE: Virtual vs Real Neutral Axis
Gee. Thanks. I couldn't have figured this out myself for many months. It's like the concept of superposition, isn't it. In superposition, there is really cancellation effect. Can't you say the decrease is due to some cancellation. So more load in the 2nd floor top can lessen the moments in the ground floor column. Now i'm in dilemma. If I remove the useless waterproof topping below the soon to be built roofing system above second floor slabs (not adhesively connected to topping), the moments in the ground would increase (making the epoxy void even more compressed). But if I retain it. It would be less bent even fulfilling what Doug described in the following because by lessening moments in the epoxy void.. more concrete can be mobilized and axial capacity increased much more. Note in gravity mode, if your column are more straight.. even in seismic base mode.. it won't be as bent.
Doug stated:
"The combined axial load and moment are equivalent to an eccentric vertical load, with the eccentricity on the compressive side. If you apply an additional load at the column centroid it reduces the eccentricity of the resultant load, and the width of the compression region increases, so the total compression force increases, but the maximum stress reduces. It's a similar situation to a footing with a vertical load outside the middle third. If you apply an additional load on the uplift side, the pressure on the compression side will reduce, even though the total reaction force is increased."
My problem is. If I didn't remove the waterproof topping. It would attract greater seismic load. Given similar situation. Would you remove the topping or not?
RE: Virtual vs Real Neutral Axis
If the column appeared inadequate, I would study the effects of a redistribution of moment to other members of the frame. If that appeared reasonable, I would take no further action.
If the nine columns acting together provide insufficient lateral resistance, I would try to find a way of improving lateral resistance.
BA
RE: Virtual vs Real Neutral Axis
Ok. We'll first put the roofing before deciding whether to remove the waterproof topping (the reason for the topping was to let rain flow to drain). Btw.. you were wondering yesterday in the message
"Why does C1 have a large moment at Story 3 (-22.17 or -20.88kN-m), C2 has less than half as much (about 8) and C6 has zero moment?"
The following is the layout in the third floor for the graphics you commented the above on. I got an old file 2 years ago from the designer when the building was designed for 4 storey. I removed the 4th storey and partially removed some beams (middle longitudinally) in the third storey to test out the concept of column superposition (so the values were not meant to depict the real structure.. but the superposition curvature is still there). Right now in actual it's only columns that actually rise up above second storey. Roof will be deliver tomorrow and put on Monday.
Anyway. In your drawing of the curvature from third floor to ground. Why didn't etabs show the 2 storey combined curvature? etabs only show the column moment in each storey. Your drawing shows curvature for 2 storey. Maybe it's called second order moments? What is the exact term for it that I could find in books? I have 5 days to read the book to decide whether to remove the topping. The designer said it's up to me to decide because he believes the epoxy void functions like concrete.
RE: Virtual vs Real Neutral Axis
Is the curved lines supposed to show the moments? Why are they showing it pinned? What would happen if you draw it double curvature in one beam that is originaly depicted in etabs (I'm referring to the simple beam column span sample. not the complex one.. just ignore the complex one to avoid confusion).
Again
with 5 10kn load on top of second beam:
Etabs is showing the S curvature, you are showing one curvature only. Or is your curved lines not moments but deformation? But isn't moments curvature already how they actually deformed?
RE: Virtual vs Real Neutral Axis
I am not familiar with etabs but some frame programs show deformation if you request it. The moments we have been discussing are not second order. Second order moments result from deformation of the structure which changes the value of some moments because the geometry of the structure has changed. Second order effects are usually small and are often neglected in practice.
There is no special term for moments resulting from a frame analysis. Each part of the structure simply responds in accordance with its stiffness. Any book on the analysis of statically indeterminate structures would be suitable. You can likely find a lot of information online.
The designer is wrong on two counts. First, it is not "up to you to decide" because you are not the designer, you do not have the training to make such a decision and you do not have full access to his design calculations or computer printouts. It is up to him to make a recommendation. Responsibility for design belongs to the designer and if there is a problem with the structure, he is the one who must accept liability for his design, not you.
Second, epoxy does not perform like concrete. It has adequate strength but it has a much lower modulus of elasticity; a much larger strain is required to develop that strength. The presence of the epoxy filled void changes the stiffness of the column, bringing into question the results of the frame analysis.
If the slab was intentionally roughened before placing the topping, they will be well bonded together. Removal will not be easy without damaging the slab below.
BA
RE: Virtual vs Real Neutral Axis
So those moments already take into account the deformation as one of the superpositions. Makes good sense. How many structural engineers intentionally makes the second floor having more load to kinda decrease the moments in the columns in ground floor? Have you done it?
Between the topping and slab is the waterproof membrane. So they are not really connected. In our country. We mostly use Bitumen membrane. See:
http://www.ikogroup.co.uk/Products/Flat-Roofing/Bi...
It's described as "Torch-on membranes have an underside which has been pre-treated with a covering of thermofusible bitumen. This covering is then heated with a propane gas torch and the membrane is applied to the surface of the roof while the bitumen is still hot."
On top of the slab is the waterproof membrane then we put wire mess so concrete topping (to let rain sloped to drain) can bind together. To remove the topping. I don't want to use jackhammer but lifting the topping (again it's not directly connected to slab because of the waterproof member between them) but don't know how easy when the wire mesh bind all topping together. Any ideas or experiences?
After removing the topping. I wonder if the waterproof would still be smooth. Because the tenant want to put wooden tile and they want smooth surface.. I wonder if the membrane could still be smooth surface and if rough.. I wonder if thin concrete can bind to the membrane (have you tried it in 60 years?).
Since the total roof area is about 175. And each 2" of topping weights 23.56x0.05 = 1.178 MPA * 175 = 206 kN.
206 kN of unnecessary SD load is large, huh? (unnecessary SD load because roof will be put into the floor next week) But then.. since the building has 8 edge columns. The weight above the second floor can decrease the moments of the 8 columns in the ground floor. Again. Are their structural engineers anyway who took advantage of this moment decreasing effect of increasing loading in the second floor?
Many many thanks.
RE: Virtual vs Real Neutral Axis
No! I don't know anyone who has.
No experience with that.
Not that I can remember.
Not so far as I am aware.
BA
RE: Virtual vs Real Neutral Axis
Ok. But with the simple beam and column sample hit with lateral force USB97. The one with additional 5kN uniform load in the top beam has lower ground floor column seismic moment.
See:
Without the additional 5kN uniform load in top beam, there is greater ground seismic column moment. See:
In the example. The top beam need to have twice the loading of the lower beam for there to be significant lowering seismic moments in the ground columns. So I guess this is not practical in real application. The topping may not significantly affect it. So I guess I have to remove the topping. But tenant want smooth surface for wooden tile. Anyway. In your experience. What is the minimum thickness of smooth cement polish surface for it to be stable on top of raw rough slab surface? Do you just use adhesive or wire mesh? I think half inch of topping may break easily. I don't want to end up duplicating the topping by putting a new wiremesh and putting another topping. If one inch is the minimum thickness. I may not have to remove it or just grind some part to make it even. Thanks.
RE: Virtual vs Real Neutral Axis
It seems foolish to remove existing topping, then add a new one. I would think it best to grind the surface flat, then apply the tile on an adhesive.
BA
RE: Virtual vs Real Neutral Axis
BA.. thank you so much for all the help and enlightening tips on the theoretical side. I couldn't have understood some concept for many months without you. Appreciate it so much. I guess I now have to ask all local contractors about grinding and polishing.
Beers and cheers to you :)
RE: Virtual vs Real Neutral Axis
BA