Origin of a Strange Column Buckling Equation

[briancpotter]

I'm reviewing a precast calculation package, and part of the analysis relies on this equation to check a column's capacity:

(phi)*Pn = 0.55*f'c*(phi)c*Ag[1 - (k*l/(32*d))^2]

It seems to be some sort of strange, normalized Euler buckling equation. It seems somehow familiar to me, but I've been unable to locate it in ACI, PCI, my PCA notes, or a concrete textbook.

Anyone happen to know where this comes from?

Brian C Potter, PE

http://simplesupports.wordpress.com

 

[CANPRO]

That equation can be found in CSA A23.3 Eq 14.1 which applies to bearing walls. I have seen a similar expression for columns (thickness is used instead of depth) but I cannot remember the source.

That equation also appears in "Design Manual - Precast and Prestressed Concrete" by CPCI (Canadian Presttressed Concrete Institute). Again, this is for walls.

I believe this is an emperical expression and is only applicable within certain slenderness limits. If I can remember where I saw the expression for columns I will post it.

 

[briancpotter]

Ah, it is in ACI afterall - eq. 14-1. It's cleverly hidden in the wall portion instead of the column portion. Thanks for the assist.

Brian C Potter, PE

http://simplesupports.wordpress.com

 

[asixth]

Nice equation, don't know that I agree with the formulations completely. Looking at the capacity curve which is generated from [1 - (k*l/(32*d))^2] I reckon it curves the wrong way (see attached). I think the capacity curve should be more like the curves in my attachement in this thread thread507-325661

That and it doesn't account for reinforcement percentage.

 

[briancpotter]

asixth, here's the text from the ACI Commentary:

"With the 1980 Code Supplement, Eq. (14-1) was revised to reflect the general range of end conditions encountered in wall designs. The wall strength equation in the 1977 Code was based on the assumption of a wall with top and bottom fixed against lateral movement, and with moment restraint at one end corresponding to an effective length factor between 0.8 and 0.9. Axial strength values determined from the original equation were unconservative when compared to test results for walls with pinned conditions at both ends, as occurs with some precast and tilt-up applications, or when the top of the wall is not effectively braced against translation, as occurs with free-standing walls or in large structures where significant roof diaphragm deflections occur due to wind and seismic loads. Equation (14-1) gives the same results as the 1977 code for walls braced against translation and with reasonable base restraint against rotation. Values of effective length factors k are given for commonly occurring wall end conditions. The end condition "restrained against rotation" required for a k of 0.8 implies attachment to a member having flexural stiffness EI/l at least as large as that of the wall.

The slenderness portion of Eq. (14-1) results in relatively comparable strengths by 14.4 for members loaded in the middle third of the thickness with different braced and restrained end conditions."

You can see their graph of capacities here:

https://www.dropbox.com/s/1t1cxjxb0h6kfgj/Photo Mar 21, 11 16 41 AM.jpg

ACI 14.4 redirects to chapter 10 (flexure and axial loads), so the dotted line should be roughly the values for normal column capacities.

Brian C Potter, PE

http://simplesupports.wordpress.com