Concrete Beam Discrepancy

[StrEng007]

Is there a term/phrase or discussion by ACI to account for the fact that these two things are not equivalent (at least not equivalent when I calculate them).

•Calculating the depth to the neutral axis by taking moments about the N.A. of a cracked beam with transformed sections
•Using the stress equations where neutral axis is c = a/β1

Values are close but they differ. Anybody else ever get tripped up by this? I expect it's to do with the first method being a true calculation by transformed areas. While the second method is the conversion of a non uniform curve into an equivalent uniform stress via a single factor (β1) and 0.85f'c.

I recall seeing the transformed sections in text books, but I don't think it was specified in ACI. I'm not hung up on it... I just figured this is the perfect place to talk about the couple % difference.

 

[lexpatrie]

I think beta1 is kind of a curve fit to the test data.

Back when the stress block was parabolic as I recall and they switched it to rectangular somewhere along the line, maybe 1980-1990, as it was computationally easier and didn't lose meaningful accuracy. The two methods probably don't have the same underlying assumptions so they produce slightly different values. Plus the beta 1 is for a cracked section. I'd poke around the oldest commentary to ACI you can find and try to track it down that way. It's maybe the difference between the rectangular stress block and the parabolic/linear in the transformed section?

 

[Celt83]

The cracked transformed section is based on a service level elastic triangular stress block and conversion of the steel to “concrete” via the modular ratio.
Sigma = M y/Icr = M (yi - yna)/Icr = M/Scr
Sigma = 0 at the Neutral Axis
S = first moment of area, which is 0 at the neutral axis.
So you can solve for the cracked elastic neutral axis depth by finding the location where the first moment of the concrete area = the first moment of the transformed steel area.

The beta,1 formula is for the ultimate limit state with peak compression strain of 0.003 and calibrated so that the resultant compression on a rectangular cross-section, the Whitney block, aligns with the true compression resultant of the ultimate parabolic-ish stress field and no transformation of the steel area. This approach actually breaks down if the cross-section itself is not rectangular.

 

[IDS]

It is possible to calculate the two parameters for a rectangular stress block so that it will give exactly the same results as any specified parabolic-rectangular stress block, for a rectangular section. If you do that the results are significantly different from any of the code values that I know of. Even the Eurocode parabolic-rectangular stress block will give parameters different to the Eurocode rectangular stress block values.

But it's really not that significant, since variation in the actual properties of the concrete will be much more than the variation in code values.


Doug Jenkins
Interactive Design Services

http://newtonexcelbach.wordpress.com/

 

[haynewp]

You may be comparing linear-elastic inertia equations versus an assumed ultimate stress distribution.

 

[Celt83]

StrEng007:

putting it in different terms in your first computation you are finding the elastic neutral axis of the cracked section and in the second, c=a/β1, you are finding the plastic neutral axis.

 

[StrEng007]

Celt83, thanks for the information.
There is a typo in the text reference you show. After factoring the binomial, and solving for x, it should be " -2.25 + √(76.5 + 2.25²)"

But it's really not that significant, since variation in the actual properties of the concrete will be much more than the variation in code values.

I figured as much.

putting it in different terms in your first computation you are finding the elastic neutral axis of the cracked section and in the second, c=a/β1, you are finding the plastic neutral axis.

I'm not typically messing around with the elastic N.A. anyway, but thank you for pointing this out.

 

[Celt83]

"typo in the text reference"
yep good catch, there are a bunch of other errors like that throughout that textbook as well.

 

[StrEng007]

good old McCormac

 

[lexpatrie]

Celt, what book and edition is that calculation from, just for completeness?

 

[Celt83]

Chapter 2, McCormac has some of the better examples for computing the cracked moment of inertia.