Concrete Beam Effective Inertia

The effective inertia would definitely vary along the length of the beam, so breaking it up into sections would give you a more precise estimate of the theoretical deflection. However, the difference between the theoretical cracked moment of inertia and the actual is so large that a precise estimate of the theoretical deflection is unlikely to be useful in estimating deflections for an real beam. I'm not familiar with the handbook you mentioned, but I would expect the equations approximately account for the change in moment of inertia along the beam in a typically cracked condition.

Thanks HotRod10, I agree that I'm attempting to dial in on something that make not be overly accurate to begin with. See attached for some very mediocre scans of the equations that I mentioned. It does say to take the midspan moment for a simply supported beam loaded with a uniform load to calculate effective inertia. Weighted averages for varying effective inertias are permitted for other types of support conditions. Maybe this means that the effective moment of inertia at midspan does in fact account for the varying inertia along the length of the beam.

Developing the moment-curvature graph that includes the cracked regions you want to represent, then using the area under the curve by the moment-area method may be what you are looking for. I haven't done it since college.

Thanks haynewp, that's in line with what I was thinking. Good example you posted.

"Maybe this means that the effective moment of inertia at midspan does in fact account for the varying inertia along the length of the beam."

It's hard to tell from what I could see in the scan, but I'm fairly sure it does account for the variation in I, either by a derivation similar to what haynewp posted, or by empirical testing, or a combination of the two. Most often, what I've seen done for similar situations is an iterative process, where they start with a derived equation and tweak it based on the results of some empirical testing.

Here's the commentary in ACI 318-11 on the Ie for continuous and simple beams.


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Thanks HotRod10 and JAE. Glad to know that sticking to the conventional method captures the dominant source of deflection. I'd be curious to compare the two methods to see what percentage they vary by. Reviewing the references in JAE's post would also likely be worthwhile.

Shotzie said:

Would it be possible/correct to discretion the beam into shorter sections and compute the deflection cumulatively (going back to my structural analysis notes from University)

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I've dabbled with this a bit. In the context of spreadsheet making, doing it once isn't so much different from doing it a thousand times. My vision was a MathCAD worksheet that would basically do old school double integration method, using a variable I. I bet one could get this done in a page or two of programming.

For my precast work, I use a program called ConciseBeam that basically does what you've described. I'd be happy to run some things for you if you're inclined to attempt some bench marking in the future. The advice given above regarding midspan/end moment of inertias was mostly developed for continuous beams rather than simple spans. I'd think it even more prudent for a uniformly loaded simple span as the moments tend to stay pretty flat over a good chunk of the span. And it's the chunk of the span contributing most to deflections.

I like to debate structural engineering theory -- a lot. If I challenge you on something, know that I'm doing so because I respect your opinion enough to either change it or adopt it.

@KootK It seems like a fun exercise to go through. If I end up nailing down some hand calculations and spreadsheets / SMath/MathCAD sheets it would be great to have a few bench marks to compare to. I'll keep you updated on if/when that happens.

As an aside, ConciseBeam looks pretty great. I downloaded the trial but would have to invest some time to understand what it's doing. I don't deal with prestressed or composite beams, although it looks very detailed for concrete beam design in general.

I don't know the Canadian code, but if the deflection provisions are similar to ACI 318 the problem with doing a more detailed analysis to sharpen your pencil is that the calculated deflections are already way over-sharpened, especially for lightly reinforced sections:

- The loss of stiffness immediately after cracking is much greater than given by the ACI formula.
- Significant loss of tension stiffening happens in weeks or days, rather than years.
- Shrinkage greatly reduces the cracking moment, which greatly increases deflections, even with symmetrical reinforcement.
- Differential temperature effects can also greatly reduce the cracking moment.

So if you are going to refine the beam analysis, you really need to refine the section stiffness analysis as well (unless you are confident that deflections 2-3 times greater than predicted will not have any significant adverse effects).

That said, there are plenty of free beam analysis spreadsheets out there that can easily be set up with subdivided beams, so you can vary the flexural stiffness along the length of the beam, including:

ConbeamU

Presentation on reinforced concrete deflections attached.

Doug Jenkins
Interactive Design Services