Slender Concrete Walls -

[RWW0002]

I am in the process of re-writing some spreadsheets for changes to ACI318-08. While re-looking at slender walls, I have been using the PCA notes on ACI318 for worked examples to check by. When calculating Icr (ACI 318 Eq 14-7), PCA calculates "c" using the traditional nominal strength relationship between the neutral axis and the compression block (a/beta1). I have been using c corresponding to cracked transformed section properties. I feel like c corresponding to cracking makes more sense, but since the Ch 14 method is somewhat empirical I thought I would see what others thoughts were.

 

[ishvaaag]

Without entering on the whole applicability of one as a substitute for the other, it is clear that finding c as a/beta1 must mean that was the way the implied correlation was established and it might mean that you wouldn't be getting a better agreement with the actual behaviour when using c from a compatibility of deformation analysis.

 

[RWW0002]

Ishvaaag

Just to be clear, c = a/Beta1 is found in PCA notes on ACI318, not in ACI318 itslef. The "implied correlation that was established" by ACI 318 is what I am looking for, (and what I have always assumed to correspond to cracked section properties) but it is not implicitly clear in the code or commentary provided by aci.

It is worth noting that the Icr equation (ACI 318 Eq 14-7) form matches that derived from transformed section analysis of a fully cracked section.

 

[spats]

You should use c = a/beta1. This is borne out in the design examples of ACI 551.2R-10, "Design Guide for Tilt-up Concrete Panels", page 7.

 

[RWW0002]

I agree that many design examples use a/beta1 (TCA manual, ACI 551, and PCA). This would be more than enough to convince me if it was not so counterintuitive. I know the method is somewhat empirical, but the formula for Icr is not (see attached sketch and calculations). It comes directly from a transformed section analysis, which is only valid assuming linear elastic behavior. By using a/beta1 the linear-elastic assumption is out the window. It makes me wonder if the typical formula for c was plugged in somewhere along the way and it was not thought out. Any thoughts?

 

[ishvaaag]

Well, it is late here, but tomorrow will try to elucidate (if able) the matter. You seem to have proven through reverse engineering that the formulation is consistent with a c for a cracked section; maybe using c=a/beta1 what is saying is to use a specific c consistent with some particular status of the (cracked) section, one that you also could determine through compatibility of deformations in akin way than for the rectangular stress block with whatever other law ruling concrete strees-strain behaviour.

 

[ishvaaag]

Going to the code

"c = distance from extreme compression fiber to
neutral axis, in., Chapters 9, 10, 14, 21"

and seeing the comment at R14.8.3

"The neutral axis depth, c, in Eq. (14-7) corresponds to this
effective area of longitudinal reinforcement."

my interpretation of this is that you set a compression block of equal compression capacity than the effective area of longitudinal reinforcement and find then the depth of neutral axis for such condition of equilibrium of the section.

Now, whichever the nature of the stress-strain law we use to determine the depth of the neutral axis for such balanced condition we are complying with the code. So you may elect a rectangular stress-block as the c=a/beta1 implies, or one "elastic" triangular stress block. Quite likely observing c=a/beta1 in the examples means just that it is seen as a simple and generally consistent with the recommended practices in the code to establish the depth of the neutral axis.

So in my view you are correct (from the derivation you have come to state) and may use if so you want a triangular stress block to find the depth of the neutral axis.

 

[RWW0002]

Thanks for the thoughtful response.

I have a hard time justifying a neutral axis depth any "shallower" than that corresponding to yielding of the steel. (which would be Ccr per the example I posted). After that the section is no longer in the elastic range and the assumptions used to develop the Icr equation are bust.

Additionally is does not make sense that the section would gain stiffness as loading continues past steel yield (as is usually the case if you decrease c for a wall section since the contribution of the steel to stiffness is usually much more than the concrete). As the section progresses past yield, the stiffness would decrease if anything. I don't think any stiffness greater than the actual cracking MOI should be used for nominal strength level bahavior.

Although the difference in the two c values may not be significant for most wall sections, I think I stand by the fact that Ccr should be used for C. It makes more sense with the assumptions of the Icr equations and with the theory for a P-Delta type analysis.

I apologize for badgering my point. I really did come here seeking the advice of others, not to offer my own opinions, but the more I have thought about it the more I am convinced. Thanks again for the help.

 

[ishvaaag]

I am not sure on if you are doubting to count on any bigger stiffness than that provided by the cracked section with the tensile steel at Fy, i.e., you are deciding not to count the bigger stiffness allowed by the code due to the concomitant presence of Pu.

Yet we can remind how for columns many times 0.7 of gross is allowed where only 0.35 of same for beams.

So, in my view, your concern would be granted IF that Pu is one prior to the consideration of geometrical nonlinearities (and, if so pertain, material ones, through stress-strain laws); but if your Pu and Mu are both coming from an analyisis that readily holds the effects of material nonlinearities plus P-Delta and P-delta, the status at the section will be such that (path of loading permitting) no further degradation of stiffness occur than that the concomitant existing Pu and Mu allow to resolve from cracked status, if any.