Concrete Column Effective Moment of Inertia

[ron9876]

I am reviewing an existing condition where a building has concrete columns supporting steel trusses. The trusses have both top and bottom chords tied into the concrete columns. My analysis shows large moments/shears in the columns however the magnitude depends on the relative stiffness of the columns. My question is does anyone know of a method to determine the effective stiffness of columns with both axial tension and compression and moments.

 

[JoshPlumSE]

There are many different ways to approximate the stiffness of a concrete column. The methods that I use are usually one of the following:

1) Simple assumptions per ACI 10.11.1... I_cracked = 0.7 * I gross. This assumes strength level loads. It probably also assumes the columns will be always in compression. At service level loads, you would be able to assume that I = Igross.

2) ACI section 9.5.2.3 which calculates an "effective" moment of inertia based on I gross, Icracked, and the ratio actual moment to cracking moment. It looks like it's more intended for beams. But, if you calculate Icracked and the cracking moment on the actual axial load in the colummn then it can still be used for columns. Though you would have different values for the different axial loads.

 

[IDS]

I'm guessing that axial loads are a small proportion of the axial capacity, and that the section will be cracked under service loads. In that case using I_cracked = 0.7 * Igross will give a conservative estimate of the bending moments at the connection, but will underestimate forces in the truss, and greatly underestimate rotations at the connection. The spreadsheets in the link below will calculate cracked stiffness values, including the effect of axial loads. You can get a wide range of answers, depending on the assumptions and method used, so check results for lower and upper bound stiffness values.

http://newtonexcelbach.wordpress.com/2009/11/22/reinforced-concrete-moment-curvature-1/

Doug Jenkins
Interactive Design Services

http://newtonexcelbach.wordpress.com/

 

[BAretired]

The effective stiffness of the columns depends to a large extent on the nature of the load. For sustained load, concrete columns will creep whereas the steel trusses will not (at least not significantly). So there is no simple answer to the question and engineering judgment is required.

BA

 

[ron9876]

BA that is the rub. I know how to determine the boundary conditions and if I was designing the project I would use that approach. However I am reviewing an existing structure and the assumption of column stiffness will determine the moments/forces in the column and the forces in the steel to concrete connections. It will likely be the determining factor in whether I have to tell the owners they need to repair their building or not.

The axial tension/compression on the columns are between 0.05f'c and 0.1f'c. When I calculate Ieff based on service moments only I come up with around 0.35Ig. That reduces the moments/forces significantly. I was hoping someone knew of an analytical approach to determine Ieff since I will likely be asked to provide calculations to the engineer-of-record to review.

 

[BAretired]

ron9876,

When confronted with a situation such as that, you must be open about your assumptions. It is not possible to state the stiffness of the columns with absolute certainty. In fact, I[sub]eff[/sub] of a column varies throughout its length because creep and cracking occur where moment is high. After creep and cracking, moment decreases.

You can only say what you believe to be true. It should not matter whether you are designing the project or reviewing an existing design. If your best assumptions suggest remedial action is required, then that is what you must tell the owner. If the EOR does not agree with your assumptions, that can be the subject of further discussion, but should not be the basis for diluting your recommendations.

BA

 

[ron9876]

Valid input BA. Thanks. I have come to that conclusion myself. I will give the owner my results and suggest they go back to the EOR for their response.

 

[JAE]

ron,

I agree with BA's comments above - I would add that I have done "analytical" calculations in the past for tilt up panel walls that get rather slender.
In those cases, I would break the panel (in your case your column) up into bite sized lengths (perhaps 1 ft. lengths) and get the axial and bending results from a first order analysis.
Then, with these results, you can determine the Ie values.

Note that the Ie value must be adjusted a bit if you have axial compression or tension. The ACI formula for fr = 7.5 sqrt(f'c). It then sets Mcr = fr(Ig)/yt.
But with axial, you would adjust fr. If compression exists, then the cracking moment is higher (more moment required to overcome the bending and create tension on the face).
So the adjusted formula - Mcr = (fr + P/a)Ig/yt. Keep the sign convention such that a compressive P value is positive. The higher Mcr will increase the Ie value.

With these first run (first order) Ie values you insert them into your model and then re-run.
This creates new axial and bending forces along the length which alters the Ie values again. This requires multiple iterations until the member Ie values converge.

The code requires application of multiple load combinations so you need to go through this process for each individual load combination (at service load level - deriving the Ma values).
Then, with each combination you will end up with DIFFERENT Ie values for the member. This is the challenge here because technically the ACTUAL condition of the member depends on its past loading
and the degree to which it has cracked.

However, in your case, you need to find a safe means of analyzing the column and the connection to the truss. To be "safe" the higher your column Ie the higher are
the connection and column forces. If you underestimate the column stiffness you have a potential unsafe system where the connections might fail before the column cracks.

One thought would be to try to back-calculate the condition of the column (its Ie values along its length) right at the safe capacity of the connections and derive a column stiffness from that - re-run the model with that column stiffness to check other parts of the system.

 

[ron9876]

JAE I see what you are saying. However this neglects the contribution of the reinforcing steel. Both tension and compression reinforcement. I am surprised that there isn't a more exact method to calculate this.

 

[JAE]

No it doesn't neglect the reinforcement. The Ie value is based on Icr which includes the rebar in its derivation.

 

[IDS]

I imagine that ron was reffering to the calculation of the cracking moment, which is normally calcualted based on the gross concrete dimensions neglecting the reinforcement content. It's quite simple to include the reinforcement in that calculation for any known cracking stress, but since other variables will have much greater effect on the stiffness, it's debatable whether that is a good idea. In particular shrinkage has a significant effect on both the cracking moment and the moment/curvature relationship (even with symmetrical reinforcement). Also the ACI method of calculating the cracked flexural stiffness gives much too high a stiffness just above the cracking moment, particularly for lightly reinforced sections.

Doug Jenkins
Interactive Design Services

http://newtonexcelbach.wordpress.com/

 

[BAretired]

It should be clear by now that the use of concrete columns with structural steel trusses cannot be analysed with any degree of precision. Thus, conservative assumptions are recommended.

BA

 

[JAE]

Reinforcement doesn't have much to do with cracking moment.
The strains at concrete cracking are much too small to significantly engage the rebar prior to cracking.

For ron9876's situation, having a too high effective moment of inertia is on the conservative side of the problem.

With a stiffer column comes higher stresses in the connection between truss and column, which as an engineer, you want to be on the safe, conservative side of the true, but unknowable, solution.

I agree with BA that you can't perfectly find the Ie of the column, but you maybe can find an Ie that gives you assurance that you are on the high side.

 

[ChrisMagadiaCom]

it depends on the behaviour and distribution of loads...as mentioned above the aci recommendation is based from tests and previous experience and not every projects or structures are the same

the most important thing to remember in this subject is knowing or predicting if the column is gonna crack...columns in tension mean you will end up with cracked sections unless you can provide more reinforcement which is not practical...columns in compression i would go with full uncracked moment of inertia

chris magadia

http://www.chrismagadia.com

ChrisMagadia.Com - The Structural Engineers' Forum and Resources Website. Civilizations owe its existence to Structural Engineering. Do you Agree?

 

[JAE]

you will end up with cracked sections unless you can provide more reinforcement

Reinforcement does not prevent cracking - only larger crack widths.

 

[ron9876]

Based on code D+L the columns will definitely crack. Also when wind load combinations are considered they crack easily. I am going to have to make an educated guess and look at a higher Ieff and a mid-range Ieff.