Euler Buckling vs Moment Magnification - RC Columns 4

[fracture_point]

Can someone clarify the difference to these two methods? In school using Eurocode, I remember checking columns against Pcr (Euler buckling) and if the axial load exceeded this, the column would buckle and fail. However, in A23.3 it seems to use a moment magnification method, which calculates additional secondary stresses due to additional drifts etc. This calculation does include the Euler buckling value in the equation, but nowhere in the code can I see that P < Pcr. Can someone please explain the difference between the two methods?

 

[JoshPlumSE]

You should probably describe the two methods more completely. Some here may be familiar with the Eurocode. Ad, some may be familiar with the Canadian code. But, few will be familiar with both.

For what it's worth, I would like to clarify a couple of things about your question.

Eurocode:
You describe the Eurocode as checking against Pcr (Euler Critical buckling load) above which the column is considered to have failed or buckled. What happens to the column when it has 90% of the critical buckling load? What if it also has 10% of it's maximum moment strength. Would you consider this to be satisfactory at a code check ratio of right around 100%?

A23.3:
I no longer have a copy of the Canadian code anymore, but I don't think it's all that different from the ACI code we use here in the USA. Generally, the amplification procedure will use an axial force to amplify the moment. Something like:

M_amplified = M_original * (Cm / 1 - Pu / Pcr)

Where Pu / Pcr is the ratio of axial load to euler buckling load, and Cm is a factor related to how the moment varies over the length of the member.

Looking at this equation, (and assuming Cm = 1.0), you see that the moment amplification when Pu = Pcr is infinite (i.e. the column fails or buckles). when the column is at 90% of Pcr, the amplification is 10.0. So, your moment becomes 10 times what it was before amplification.

 

[KootK]

Yeah, that can be confusing. Here's how I think of it:

THE GOAL

- Have enough column stiffness that your moments do not grow without bound due to second order effects. That growing without bound is the very definition of instability/buckling.

ONE WAY TO ACHIEVE THE GOAL

- Classic Euler buckling check making sure that you've got enough EI to get the job done.

ANOTHER WAY TO ACHIEVE THE GOAL

- Calculate your moments using the EI that you have and then amplify those moments so that they include second order effects. If the moments remain reasonable then, by definition, they don't grow without bound. This is moment amplification. Pcr comes into it, as you noted, since the ratio P/Pcr is the metric by which we measure the loss in flexural stiffness that results from the presence of a compressive load.

It's a tricky thing to explain so we'll see how well I did. When one compares the goal with the moment amplification description, it actually makes moment amplification seem like the more direct and rational method. At least that's my opinion.

The alternate methods are two sides of the same coin. Namely: have enough column stiffness to stave off the development of out of control moments.

 

[KootK]

This came up in another thread recently and folks posted some excellent documents for understanding moment amplification. You can find those here: Link

 

[fracture_point]

JoshPlum, KootK - thanks for the responses.

I think KootK summarized my difficulties in explaining my misunderstanding, because its really difficult to communicate what my actualy confusion is. Your explanation was excellent though. Just two different methods to achieve the same goal!

 

[Settingsun]

Euler buckling doesn't actually occur in real stuctures (ie column remaining perfectly straight with zero bending moment until bifurcation failure), so you don't need to check it directly. The real failure is combined bending and axial. But the Pcr load calculated from the idealized Euler assumptions can be used to estimate the bending moments that occur in slender columns.

Although it's for steel structures, background documents about the AISC 'direct analysis method' are worthwhile reading. Richard Furlong has written some articles that show how to apply essentially the same method to concrete structures. He calls it 'rational analysis'.

 

[fracture_point]

Thanks Steveh49 - I'll be sure to check this out!