Column Bending angle and Failure Mode 1

[mes7a]

Please see figure below of a column pinned on top and bottom. If there is a hole at the side at midspan, the column would deflect more than if there hole is at the base. Is this right? What is the principle called where hole at midspan would make it deflect more than if it occurs at bottom near base? Is it related to K factor or what?

 

[BAretired]

If the base is fixed, there should be no rotation at the bottom, so your curvature diagram is inaccurate.

The axial load is constant from top to bottom, namely P. The moment varies linearly from top to bottom according to your diagram. A point of zero moment occurs somewhere between the top and the bottom.

Since both end moments are clockwise, equal and opposite horizontal reactions are required top and bottom to maintain equilibrium.

Ht = Hb = (Mt+Mb)/h
where Ht, Hb are the horizontal reactions at top and bottom respectively and h is the height of column.

BA

 

[mes7a]

If the base is fixed, there should be no rotation at the bottom, so your curvature diagram is inaccurate.

The axial load is constant from top to bottom, namely P. The moment varies linearly from top to bottom according to your diagram. A point of zero moment occurs somewhere between the top and the bottom.

Since both end moments are clockwise, equal and opposite horizontal reactions are required top and bottom to maintain equilibrium.

Ht = Hb = (Mt+Mb)/h
where Ht, Hb are the horizontal reactions at top and bottom respectively and h is the height of column.

You mean at the same distance say 1 meter above column base, the compression block 0.85 fc' a b of both pinned and fixed column are same value but on opposite side? This is different concept from buckleness of slender column vs short column isn't it. What topic in structure books does the pinned vs fixed moments concept highlighted?

Etabs show this:

Why is the moments not equal and the zero is not right at midspan?

Again. Very important to know. You mean at the same distance say 1 meter above column base, the compression block 0.85 fc' a b of both pinned and fixed column of same loadings and eccentric at right side are same value but on opposite side? Many thanks.

 

[BAretired]

You mean at the same distance say 1 meter above column base, the compression block 0.85 fc' a b of both pinned and fixed column are same value but on opposite side?

No, that is not what I mean and is incorrect! A column pinned at the base will have a moment varying linearly from a maximum value at the top to zero at the bottom.
Edit: At some point above the base, the moment in the pinned base column will be equal and opposite to the moment in the fixed base column.

This is different concept from buckleness of slender column vs short column isn't it.

Completely different.

What topic in structure books does the pinned vs fixed moments concept highlighted?

Any elementary structural analysis book will refer to boundary conditions.

Why is the moments not equal and the zero is not right at midspan?

Mt and Mb can be equal under certain load conditions, but a column fixed against rotation at the base and free to rotate but fixed against translation at the top will have Mb = Mt/2. If the top moves horizontally, that will not be true.

BA

 

[mes7a]

No, that is not what I mean and is incorrect! A column pinned at the base will have a moment varying linearly from a maximum value at the top to zero at the bottom.

See:

How do you compute where in height of the two boundary conditions the compression block 0.85 fc' a b (at ultimate strain 0.003) will be similar for same loading and eccentricity to right side.. at what height for both conditions. This is for the column:

 

[mes7a]

I make a graphics of the above to illustrate the idea. see:

and to zoom graphics see:

http://imagizer.imageshack.us/v2/1280x1024q90/909/QCZwjX.jpg

At Ultimate Strain 0.003 at the balanced point. Moment in the section is PROPORTIONAL to the stress block size 0.85 fc' a b where a is the size of the compressive block, right? This means where there is greater moment, the stress block is smaller (from the strain distribution triangle).

Now since the moment varies in both the pinned and fixed boundary conditions. Then in the fixed, the moment at bottom which you said varies by one half to the above would have twice larger compression stress block than at top, right?

And in the pinned condition. Since there is larger moment at the top, then the compression block at ultimate strain 0.003 is smaller and becoming bigger as it goes down the bottom?

Is the idea generally correct?

Thanks so much.

 

[sponton]

Also, why is there so much bending in the column, is it only connected in one side? if it's connected in 4 sides supporting similar spans then by default the transferred moment to the column is zero [moment balanced] at least in the vertical load case. Second is eccentricity, you cannot get an eccentricity of 30 cm, especially in reinforced concrete, the whole thing was probably monolithically casted in place, moment unbalance would give you a moment which could be turned into a virtual eccentricity but you cannot just go and assume there's an eccentricity like that, unless it's steel or wood and it'd still be arguable at some point. These are things to take into account, no point in losing time with numbers if you don't understand whether they're applicable to the situation.

I think you misunderstand as someone pointed the use of interaction diagrams or their development,

 

[mes7a]

Also, why is there so much bending in the column, is it only connected in one side? if it's connected in 4 sides supporting similar spans then by default the transferred moment to the column is zero [moment balanced] at least in the vertical load case. Second is eccentricity, you cannot get an eccentricity of 30 cm, especially in reinforced concrete, the whole thing was probably monolithically casted in place, moment unbalance would give you a moment which could be turned into a virtual eccentricity but you cannot just go and assume there's an eccentricity like that, unless it's steel or wood and it'd still be arguable at some point. These are things to take into account, no point in losing time with numbers if you don't understand whether they're applicable to the situation.

I think you misunderstand as someone pointed the use of interaction diagrams or their development,

It's eccentric loading. Etabs show:

The building was already built 2 years ago. Just trying to understand it by computing everything manually. The designers used software to model it and they forgot how to manually compute.

 

[sponton]

ok first of all, what do you mean by eccentric loading in etabs, how are you applying this, are you shifting the center of mass of the structure and applying loads there? cause moment imbalance due to vertical loads alone can produce moments in the columns...
what load cases are you running, what are the loads? which one is the column you're analyzing?

Etabs may show many things but if you're not sure what you're inputting and what the program is doing, it might as well be a black box.

 

[mes7a]

ok first of all, what do you mean by eccentric loading in etabs, how are you applying this, are you shifting the center of mass of the structure and applying loads there? cause moment imbalance due to vertical loads alone can produce moments in the columns...
what load cases are you running, what are the loads? which one is the column you're analyzing?

Etabs may show many things but if you're not sure what you're inputting and what the program is doing, it might as well be a black box.

The etabs capture is just to show it is eccentric loading. I'm not trying to model anything in etabs. I'm trying to just understand where is the stress block in the column for different boundary conditions. The column concerned is the eccentric column:

Do you think the stress block magnitude is dependent (inversely proportional) on the moment magnitude like the following. I'm still waiting BaRetired reply about this.

 

[sponton]

depends what you mean by stress block

the concrete stress blocks needs to balance the stresses in the steel stress block [if you will] to accomodate the moment [by shifting the position of the neutral axis from top to bottom of the section]. A balanced section would give you a balanced stress block between the two, as you shift the neutral axis you get different conditions [meaning a moment and an axial load due to the difference between the resultants of the concrete and steel stress blocks], if you have a point (P,W) in the diagram, you can get a virtual eccentricity e=w/p
so if you're given a problem like you have a load P at e cm from the center of the support you can say W=P*e so you can find it in the diagram.

I don't think you know what you're looking at in etabs, you're showing a column that has moment at the top and moment at the bottom because the conditions where set like that, you basically have a moment frame with fixed supports. It has nothing to do with eccentric loading, the unbalanced moment that the column takes can be turned into a virtual eccentricity [i don't know for which purpose you'd do that] but it makes no sense whatsoever except for the example i stated above.

 

[BAretired]

In the following, BLACK by mes7a...RED by BA:

At Ultimate Strain 0.003 at the balanced point. Moment in the section is PROPORTIONAL to the stress block size 0.85 fc' a b where a is the size of the compressive block, right? The size of stress block is related to both moment and load. It is not directly or inversely proportional to moment. This means where there is greater moment, the stress block is smaller (from the strain distribution triangle). Not necessarily. It depends on the M/P ratio.

Now since the moment varies in both the pinned and fixed boundary conditions. Then in the fixed, the moment at bottom which you said varies by one half to the above would have twice larger compression stress block than at top, right? No, it is not that simple.

And in the pinned condition. Since there is larger moment at the top, then the compression block at ultimate strain 0.003 is smaller and becoming bigger as it goes down the bottom? If moment causes a strain of 0.003 at the top, it will cause a lesser strain further down the column, so the comparison is not valid.

Is the idea generally correct? No!


BA

 

[mes7a]

The size of stress block is related to both moment and load.

Oh. I was familiar with beam where the neutral axis can migrate towards the compression edge as the load and steel stress
increase. But reviewing the formulas for column. It seems the compression block is more fixed depending on ultimate strain
of both concrete and bars. For example. If concrete unit strain is 0.003 and steel unit strain is 60/29,000=0.0021, and
neutral axis for the example in the book with 20" column across is cb = 17.5 x 0.003/0.0051 = 10.3". stress-block depth
a=0.85 x 10.3 = 8.76" and concrete compressive resultant is C= 0.85 x 4 x 8.76 x 12 = 357 kips.

In beams. The compressive resultant is not analyze that way but can get smaller as it gets to the compression edge. Any
illuminating idea why? I'd reread the book after getting this critical bird eye view why. Many thanks!

 

[BAretired]

I normally use Mozilla FireFox as browser, but something has gone haywire with this thread and word wrap is not working properly so I have switched to Google.

We seem to be drifting a long way from the topic introduced in the original post. You have a problem with a column in an existing building. In particular, you have a void near the bottom of a column where concrete did not fully fill the full area during the pour. The void, if I remember correctly, has been filled with epoxy.

To check the strength in the vicinity of the void, you could neglect the concrete entirely and assume that the steel bars carry all the load. If you do that, you cannot have a tension failure so for the purposes of this discussion, behavior of a beam is not relevant. You are of course, entitled to consider the reduced moment in the vicinity of the void.

Have you considered this approach and if so, have you come to any conclusions regarding the adequacy of the column in question?

BA

 

[mes7a]

I normally use Mozilla FireFox as browser, but something has gone haywire with this thread and word wrap is not working properly so I have switched to Google.

We seem to be drifting a long way from the topic introduced in the original post. You have a problem with a column in an existing building. In particular, you have a void near the bottom of a column where concrete did not fully fill the full area during the pour. The void, if I remember correctly, has been filled with epoxy.

To check the strength in the vicinity of the void, you could neglect the concrete entirely and assume that the steel bars carry all the load. If you do that, you cannot have a tension failure so for the purposes of this discussion, behavior of a beam is not relevant. You are of course, entitled to consider the reduced moment in the vicinity of the void.

Have you considered this approach and if so, have you come to any conclusions regarding the adequacy of the column in question?
BA

I'm not drifting. The reason I'm interested in the stress block size whether it's always one half of the diameter of the column is at least
if it's one half.. the compression block won't become so tiny that it falls right in the void. In the following column, the red is
the part without concrete but filled with epoxy.

But I'm worried that in beams, the compression block can become so tiny near the compressive edge. Can column also do it? I want
to avoid that kind of loading because I want to avoid compressive failure of the bars where it can fracture.

I'm interested in the moments because i'm trying to gauge whether the following really work. By enclosing it with 1 meter concrete
pedestal.. can the column moments be suppressed at that portion and transfer upward (above void):

bars before concrete poured around void with epoxy

concrete poured

 

[mes7a]

depends what you mean by stress block

the concrete stress blocks needs to balance the stresses in the steel stress block [if you will] to accomodate
the moment [by shifting the position of the neutral axis from top to bottom of the section].

position of neutral axis from top to bottom?? Is it not from side to side.. or is your column section upside down
when you state it.
See:

A balanced section would give you a balanced stress block between the two, as you shift the neutral axis you get
different conditions [meaning a moment and an axial load due to the difference between the resultants of the concrete
and steel stress blocks], if you have a point (P,W) in the diagram, you can get a virtual eccentricity e=w/p
so if you're given a problem like you have a load P at e cm from the center of the support you can say
W=P*e so you can find it in the diagram.

Sponton, what codes are you using because your terms seem foreign. I'm using ACI. Here we don't use steel stress
block. In fact. At balanced point, the steel Asfy at tension and compression side cancel. In your code it doesn't
cancel?

I don't think you know what you're looking at in etabs, you're showing a column that has moment at the top and moment
at the bottom because the conditions where set like that, you basically have a moment frame with fixed supports.
It has nothing to do with eccentric loading, the unbalanced moment that the column takes can be turned into a
virtual eccentricity [i don't know for which purpose you'd do that] but it makes no sense whatsoever except for
the example i stated above.

But the column with moment at top and bottom only occur if it's eccentric loading. If it's not, then you earlier
said the 4 sides balance it. See:

from

http://www.iitk.ac.in/nicee/EQTips/EQTip17.pdf

Anyway see my previous message to BA above to see picture of the concrete pedestral retrofit. Maybe you can comment on it. Thanks.

 

[mes7a]

I have reviewed the book on the interaction diagram for half day. And the following is what I comprehend. Please
correct me if i'm wrong.

1. Concrete stress-strain diagram is for elastic range only at 0.0005.. for near strain limit of 0.0021.. you can no
longer use the stress-strain curve

2. The compressive stress block in the interaction diagram is not the real section of compression when the column is
bending at great moments. The compressive stress block or resultant C=0.85 fc` a b is related to moments only for the
capacity of the P above and below the balanced point of the interaction diagram. Right? This is what is very tricky to
understand. I'm realizing this because when the neutral axis is below the balanced point value.. the C resultant must
be larger.. instead the amount is smaller.. so it is only related to the moment interaction diagram.

3. But then you still have to deal with the real compression section when the column is bending at greater moments.
Here there is a point when the left side of the column is no longer in touched (like in tension). Here one can use some
idea in the derivation of the interaction diagram. Let's pretend the left side of column is no longer in compression but
only the 0.5x0.2 mtr right portion in compression (so all the axial load is taken up by this). Is this possible? see again:

For my column concern above. Let' say 0.5x0.2 mtr section is replaced with epoxy and composing of 8 20mm bars (314 mm^2 area)
And the column is bending so much only it is taking the compression portion (the other side in tension). Then.

strain = 0.003
Epoxy modulus = 450ksi
stress = 1350 PSI or 9.3 MPA
epoxy area = 0.5x0.2 = 0.1 sq. mtr
P = 9.3 MPA x 0.1 sq.m = 930 kN

so axial load capacity of 0.5x0.2 mtr epoxy section is 930 kN.

the 8 bars contribution is
strain = 0.0021
bars Modulus = 29,000ksi
stress = 60900
bars area = 0.002512
P (of bars only) = 1054 kN.

BA.. 2 yrs ago I was computing like the above but my understanding wont be complete without computing for
the bending stress of the composite concrete epoxy. At that time my understanding of moment is less and at
least more now.

You mentioned earlier about the column with only bars present and no concrete and the bars prevent from buckling
by magic. There you mentioned about eccentricity of 210mm and moment of "d=420mm, then M = As.Fy.d =
8(300)*400*420 = 403,000,000 n-mm OR 403Kn-M (bars here is 300mm instead of 300 in Canada but that will do
for rounding off). And you said "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.".

Now the capacity of the epoxy is 930 kN. Can you use similar concept where the eccentricity is 210mm
so the moment is 930 (0.21) = 195.3 kN-m?

Or more accurate. Since the center of the 0.5x0.2 mtr portion at right side is 0.15 meter away from
center of column Then moment is 930 (0.15) = 139.5 kN-m?

If can't apply the same concept as your example of the concreteless bars. Then how do you compute for
the moment capacity of the epoxy section with 930 kN P capacity? How do you convert it to moment
capacity?


After learning this. Last thing is to consider the centroid has moved to the left side of the harder
concrete (against epoxy) and it has natural inkling for bending stress. Here just give me a clue how
to compute for it and I'll read the book for another day. Thanks so much BA!

 

[mes7a]

After trying many numbers in excel for the interaction diagram. It seems clearer now how to relate
the compression block, the compression bars and the tension bars. They are interrelated and the
following formulas and computations seem to relate them all and it may actually be able
to solve for the moment capacity of the epoxy section.

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. This is not incorrect right.. 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 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.. 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

I tried changing strain since epoxy has greater strain-stress block and change fs (tension) up to tensile strength
and the capacity seems to increase (ductile) but first is the above method correct BAretired or spanton?

Thanks a whole lot!

 

[BAretired]

I don't have time today to go over the numbers, mes7a but the general idea seems okay to me. In other words, you should be able to consider a contribution from the epoxy in addition to the steel.

BA

 

[mes7a]

Because of the word wrap problem. Let's continue in this thread BAretired, sponton and others

http://www.eng-tips.com/viewthread.cfm?qid=395487

Btw.. in the computations above. I entered all formulas in excel from the book and I crosscheck it via the book examples to see
that I entered the formulas correctly and it produces final Pn and Mn, e for different neutral axis values like in the book.

 

[mes7a]

Also, why is there so much bending in the column, is it only connected in one side? if it's connected in 4 sides supporting similar spans then by default the transferred moment to the column is zero [moment balanced] at least in the vertical load case. Second is eccentricity, you cannot get an eccentricity of 30 cm, especially in reinforced concrete, the whole thing was probably monolithically casted in place, moment unbalance would give you a moment which could be turned into a virtual eccentricity but you cannot just go and assume there's an eccentricity like that, unless it's steel or wood and it'd still be arguable at some point. These are things to take into account, no point in losing time with numbers if you don't understand whether they're applicable to the situation.

sponton, I've been pondering on what you said above. Yes. I'm talking about moment unbalance from column at sides of building because of
beams framing into it at one side only.. it has nothing to do with actual eccentricity.. only virtual.. virtual in the sense that
you can only relate moments to axial load via eccentricity in formulas. If no virtual eccentricity.. how else would you able
to compute moments and axial loads and interrelate them?

In actual columns. What is the maximum virtual eccentricity at most by having unbalance beams framing on one side of it only? what you think?