Column Bending Stress 1

[mes7a]

To BAretired (sponton and others). This is continuation from thread

http://www.eng-tips.com/viewthread.cfm?qid=395252

because of word wrapping problem there.

BA you stated before when we were computing for pure axial load that "Note: The above calculation assumes uniform strain throughout the column. If a transformed section is used, the centroid of the combined section would shift toward the concrete portion. That would cause bending stress in addition to axial, so the condition is likely going to be worse than calculated"

I'm interested in how to exactly compute for the bending stress due to the void. In the above figure. Even without moments and pure axial load. You stated " If a column is hinged top and bottom and compressed with axial load P, the stress is uniform at every cross section, namely P/A.

If a rectangular notch is cut out of the left side of the column at mid-height, the centroid moves to the right at the notch. The axial load falls to the left of the centroid and the notched section will move right relative to the hinged ends.

If the notch had been filled with material with low E, the behavior will be similar, but the centroid will not shift as far so the filled notch will not move as far to the right as the unfilled notch because the fill is carrying some stress but not as much as the concrete.
..
If axial stress exceeds bending stress, there is no tension on any part of the cross section, simply variable compression with maximum value on the left and minimum on the right."

BA, how would the bending stress equilibrate.. there is more compression with maximum value on left and minimum on centroid. Until what moment will it steady. How would point B in the figure with concrete edge (before epoxy void) behave in the equilibrium?

Thanks.

 

[RPMG]

I recommend principle of virtual work: assume that the column is a mechanism, with a hinge.

1. What is the moment required to stabilize the column at the hinge? Pu x e = Mu
2. What is the moment capacity of the section? Phi*Mn
3. Run moment-axial interactive equation.

 

[BAretired]

It is midnight and after a few glasses of wine, it would be foolish for me to attempt any theoretical discussion at this point. Today, my engineering class celebrated its sixtieth anniversary. Thirty six of us graduated in Civil Engineering in 1955. That number has now dwindled to fifteen. Good night, sleep tight...don't let the bedbugs bite.

BA

 

[mes7a]

It is midnight and after a few glasses of wine, it would be foolish for me to attempt any theoretical discussion at this point. Today, my engineering class celebrated its sixtieth anniversary. Thirty six of us graduated in Civil Engineering in 1955. That number has now dwindled to fifteen. Good night, sleep tight...don't let the bedbugs bite.

Happy anniversary While you were celebrating and now while you were sleeping. I have used up a dozen papers trying to construct an interaction diagram of your column without concrete but with bars that is magically restrained from buckling and it seems it needs new formulas that don't have concrete stress block, etc... and still figuring out.. the other thread has word wrap problems so I'll post the interaction diagram and formulas once I've finished. Goodnyt. Many thanks.

 

[BAretired]

I have used up a dozen papers trying to construct an interaction diagram of your column without concrete but with bars that is magically restrained from buckling and it seems it needs new formulas that don't have concrete stress block, etc... and still figuring out.. the other thread has word wrap problems so I'll post the interaction diagram and formulas once I've finished.

I suspect the bars are not "magically" restrained from buckling. Would you not think that the epoxy fill around the bars would tend to restrain buckling? Even without epoxy fill, unless the gap exceeds five bar diameters, the bars are stocky columns and should manage to achieve yield stress or very close to it.

If the void is not filled, the section consists of nothing but reinforcement. If a load is located directly over the eight bars on the compression side, the failure load is simply the yield strength of eight bars. The other twelve bars do not contribute in any meaningful way. The eccentricity is d/2 where d is the c/c distance between the outer bars.

If a load is located outside those eight bars, the load in the eight compression bars is magnified and the tension bars feel stress, although not very much until the eccentricity becomes large.

If an infinitesimal load is located at infinite distance from the column, the column is subjected to pure moment with no axial component. The outer bars are equally stressed to yield, one in compression, the other in tension. The point on the interaction diagram will lie on the M axis.

Wherever the axial load is located, the force in the eight compression bars can be calculated by a simple relationship. If it is at distance x from the tension steel, the force in the compression steel is P.x/d and the force in the tension steel is P(1-x/d) which is negative when x>d, signifying tension.

If you wish to consider the contribution of the epoxy filler, it gets a little more complicated.

BA

 

[mes7a]

I suspect the bars are not "magically" restrained from buckling. Would you not think that the epoxy fill around the bars would tend to restrain buckling? Even without epoxy fill, unless the gap exceeds five bar diameters, the bars are stocky columns and should manage to achieve yield stress or very close to it.

If the void is not filled, the section consists of nothing but reinforcement. If a load is located directly over the eight bars on the compression side, the failure load is simply the yield strength of eight bars. The other twelve bars do not contribute in any meaningful way. The eccentricity is d/2 where d is the c/c distance between the outer bars.

If a load is located outside those eight bars, the load in the eight compression bars is magnified and the tension bars feel stress, although not very much until the eccentricity becomes large.

If an infinitesimal load is located at infinite distance from the column, the column is subjected to pure moment with no axial component. The outer bars are equally stressed to yield, one in compression, the other in tension. The point on the interaction diagram will lie on the M axis.

Wherever the axial load is located, the force in the eight compression bars can be calculated by a simple relationship. If it is at distance x from the tension steel, the force in the compression steel is P.x/d and the force in the tension steel is P(1-x/d) which is negative when x>d, signifying tension.

If you wish to consider the contribution of the epoxy filler, it gets a little more complicated.
BA

Thanks for the relationship or formula. After reviewing trigonometry and geometry. I kinda derive it after a day and your paragraph solidified the cconcept.

About the column subjected to pure moment with no axial component. Even on normal column with concrete. It's there too. . At balanced point and using statics and resultant.. Pb = 0.85fc'ab + As'fs' - Asfs

And As'fs' is same value to AsFs but opposite reactions. In other words the opposite reaction literally cancel out leaving just the concrete or epoxy filling to resist the load. So at balanced point. How can you still rely on the bars to resist the axial load at the compression side when the tension side reactions cancel it out??


Anyway going to the main topic of this thread. And this is just repost of message #1.

BA you stated before when we were computing for pure axial load that "Note: The above calculation assumes uniform strain throughout the column. If a transformed section is used, the centroid of the combined section would shift toward the concrete portion. That would cause bending stress in addition to axial, so the condition is likely going to be worse than calculated"

I'm interested in how to exactly compute for the bending stress due to the void. In the above figure. Even without moments and pure axial load. You stated " If a column is hinged top and bottom and compressed with axial load P, the stress is uniform at every cross section, namely P/A.

If a rectangular notch is cut out of the left side of the column at mid-height, the centroid moves to the right at the notch. The axial load falls to the left of the centroid and the notched section will move right relative to the hinged ends.

If the notch had been filled with material with low E, the behavior will be similar, but the centroid will not shift as far so the filled notch will not move as far to the right as the unfilled notch because the fill is carrying some stress but not as much as the concrete.
..
If axial stress exceeds bending stress, there is no tension on any part of the cross section, simply variable compression with maximum value on the left and minimum on the right."

BA, how would the bending stress equilibrate.. there is more compression with maximum value on left and minimum on centroid. Until what moment will it steady. How would point B in the figure with concrete edge (before epoxy void) behave in the equilibrium?

Many Thanks.

 

[BAretired]

To determine stress at point B under elastic conditions, get the transformed section properties; then determine stress by the usual expression:

f = P/A +- M.y/I
where A, I relate to the transformed section and y is the distance from the c.g. of transformed section to point B.

To determine how stress at point B affects the interaction diagram, it would be safe to ignore it as it will have virtually no effect on ultimate strength.

BA

 

[mes7a]

To determine stress at point B under elastic conditions, get the transformed section properties; then determine stress by the usual expression:

f = P/A +- M.y/I
where A, I relate to the transformed section and y is the distance from the c.g. of transformed section to point B.

To determine how stress at point B affects the interaction diagram, it would be safe to ignore it as it will have virtually no effect on ultimate strength.

But at ultimate strength.. the concrete compressive block capacity is 0.85 fc' a b.. and if the fc' is changed to epoxy.. it compressive strength can become lesser.. so why did you say there is virtually no effect on ultimate strength.

Also what do you mean by bending stress? I was thinking of it initially P-delta like effect where the column would deflect more as a result of the void..

Anyway. To summarize the situation and what I learnt so far. Please see pictures below:

In the above. I'm worried about P-delta like effect. When you mentioned bending stress. I was thinking there would be more moment induced that can bend it more.. or what is the other meaning
of "bending stress"?

above is close up image of voids.. at middle it goes deeper. The following is estimate of the void size..

the following is the whole building and the red below front column showing the void.

the distance between columns are all 5.5-6 meters and beam depth used in all is 0.3 x 0.5 mtr and slabs used is 4". Structure is originally designed for 4 storey with roof slab but now it's only built up to 3 storey with light metal roof. I'm afraid to add the 4th storey because of the void.

The following is what I learnt so far:

1. The moment at bottom is only one half that of top because it's fixed base where all bars are
directly connected to the foundation bottom in L bent. .. Mb=Bt/2

2. I understand what is compressive resultant in the interaction diagram. I was hoping the compression block would at least really be one half diameter of column so it can use the existing concrete.

3. I think the compressive resultant in the section on top (with 2 times more moment) is twice lesser than the one at bottom near ultimate strength right? Because if so, the top would fail first before bottom because the top would reach ultimate strain first.. does this make any sense?

4. The reason the column is bent is because of unbalance moments from beams framing on one side (it is at edge in the structure plan (see above drawing). There is really no eccentric loading as what sponton emphasized (sponton please comment on all this too). Now in software, there columns were shown bent because there are no bars yet.. when you put bars.. would the bending actually be suppressed.. because I was hoping the compressive stress block at bottom in void would cover the
entire section to make use of the concrete parts.

5. I learnt the above lately but now what I need to learn is how the bending would become more as a result of void and possible P-delta effect. But you said it has no effect on ultimate strain.. this is what confused me. But I think you meant something else. Also what is the formula or method to compute for the more bending induced bec of the void.

So many thanks. Understanding this more bending induced by void is last thing I want to know.. I know we are near wearied already of all this.. lol..

 

[BAretired]

We seem to be having a communication problem.

How would point B in the figure with concrete edge (before epoxy void) behave in the equilibrium?

I thought you were asking how point B would behave before putting epoxy in the void. My response was that the stress at point B would have little effect on ultimate strength of a column with an unfilled void.




BA

 

[mes7a]

I thought you were asking how point B would behave before putting epoxy in the void. My response was that the stress at point B would have little effect on ultimate strength with just a void.

Ah. When I said (before epoxy void). I was referring to before as in position.. not before as time Now I understood why you said there was no effect. I was reading about transformed section and kept thinking what you were thinking. Anyway. If you have time later, pls. go to the message above with 4 pictures for my final inquiries. Many thanks!

 

[BAretired]

1. The moment at bottom is only one half that of top because it's fixed base where all bars are directly connected to the foundation bottom in L bent. .. Mb=Bt/2

Seismic moments behave differently. Moment at the bottom due to a seismic event is likely to be at least as large and possibly larger than the moment at the top.

BA

 

[BAretired]

2. I understand what is compressive resultant in the interaction diagram. I was hoping the compression block would at least really be one half diameter of column so it can use the existing concrete.

Depends on M/P ratio.

BA

 

[BAretired]

3. I think the compressive resultant in the section on top (with 2 times more moment) is twice lesser than the one at bottom near ultimate strength right? Because if so, the top would fail first before bottom because the top would reach ultimate strain first.. does this make any sense?

For seismic load, the top and bottom moments are probably close to equal. Moreover, the top does not have a huge void which reduces the effective section.

BA

 

[mes7a]

Seismic moments behave differently. Moment at the bottom due to a seismic event is likely to be at least as large and possibly larger than the moment at the top.

Oh no. This is bad news. My designers forgot the meaning of stress-strain curve so don't know the behavior of epoxy. I'll have to convince them by sharing the principles and computations I learnt so they can try to look into this.

Depends on M/P ratio.

at what ratio for example is the moment right at zero so the axial load can use all the concrete compression stress resultant? any example?

2 years ago.. the contractor put pedestral over the epoxy to make it stronger after getting approval from designer when I complained to them i'm worried. The designer kept saying epoxy is even better than concrete and just laughed of the unnecessary expense.

void epoxy injected:

epoxied portion pedestal retrofit bars

concrete poured over retrofit bars.

My hope is the maximum moments can transfer above the retrofit by making the column one meter above foundation moment fixed.. maybe the steel is not sufficient isn't it? Should have put epoxy fiber around it. If you were to fix the column so there would be no moment in that section, what could you have done 2 yrs ago? There is tenant on top of it now so I can't touch it anymore.. but just want an idea in case in future they leave and it is necessary to add 3rd retrofit over the 2 retrofit.

 

[BAretired]

4. The reason the column is bent is because of unbalance moments from beams framing on one side (it is at edge in the structure plan (see above drawing). There is really no eccentric loading as what sponton emphasized (sponton please comment on all this too).

Now in software, there columns were shown bent because there are no bars yet.. when you put bars.. would the bending actually be suppressed.. because I was hoping the compressive stress block at bottom in void would cover the entire section to make use of the concrete parts.

The column bends as a result of moment applied by the beam resulting from gravity load. It also bends from shaking in a seismic event.

Bending is not suppressed by reinforcing the column. The size of stress block in the vicinity of the void is not known.



BA

 

[BAretired]

5. I learnt the above lately but now what I need to learn is how the bending would become more as a result of void and possible P-delta effect. But you said it has no effect on ultimate strain.. this is what confused me. But I think you meant something else. Also what is the formula or method to compute for the more bending induced bec of the void.

Who said that the P-delta effect has no effect on ultimate strain? Not me!

BA

 

[BAretired]

Another problem seen in the above photo is that the epoxy is stepped up from right to left so that its height and thickness are variable. This further complicates the problem of calculating the contribution of the epoxy to the strength of the column.

BA

 

[mes7a]

Who said that the P-delta effect has no effect on ultimate strain? Not me!

Oh it was that misunderstanding about "before epoxy void" where you thought I was referring to prior to epoxy put.. but I was saying the position before epoxy void. But I understood that part now here you were mentioning about stress at point b where as you put it.. "stress at point B would have little effect on ultimate strength of a column with an unfilled void.".. no confusion with it now.. thanks..

 

[mes7a]

Another problem seen in the above photo is that the epoxy is stepped up from right to left so that its thickness is variable. This further complicates the problem of calculating the contribution of the epoxy to the strength of the column.

That's right. When looking at side view. Consider the concrete cover is 40mm.. stirrups is 10mm.. bars is 20mm.. void is really between 100 to 200mm.

 

[mes7a]

I edited above. When looking at side view. Consider the concrete cover is 40mm.. stirrups is 10mm.. bars is 20mm.. void is really between 100 to 200mm so 0.2x0.5 would be a conservative figure for approximate computations.

Earlier I was asking about moment magnification or p-delta. When there is a void and the column is already bending towards the void from
load. What formulas do you use to estimate how the moment would be increased from the void. This is my original question about column bending stress. I thought bending stress means even more moments from voids. Could any formula in moment magnification effect be used or others? Let's consider the void is uniform (for sake of knowing what formulas are used in situation like voids). Thanks.