Inelastic Buckling 1

[MegaStructures]

If inelastic buckling is defined as buckling of a section after parts of the section have yielded, how does inelastic buckling occur at a load lower than the compressive "crushing" strength of the column? The only way I can think that this is possible is that residual stresses from welds/fabrication procedures "pre-yield" the section and then a compressive load placed in the field can cause buckling of this yielded section. If this is the case how do these yielded portions of the column cross section reduce the buckling capacity of the member?

“Any idiot can build a bridge that stands, but it takes an engineer to build a bridge that barely stands.”

 

[JoshPlumSE]

When a column is very slender and buckling will occur prior to yield then this is elastic buckling.... because the buckling involves no plastic yielding of the cross section.

If a column is significantly less slender then the act of buckling may cause the compression fibers to yield. This would be the "Double Modulus" theory.... Where the modulus of elasticity on one side of the column is different from that on the other.

In addition to this you have the concept that you suggested.... Where residual stresses play a part in the extreme fiber stresses and can greatly change the inelastic buckling behavior.

Both effects (Double Modulus and Residual Stresses) play important parts.

 

[MegaStructures]

So if I understand correctly in the Double Modulus Theory buckling is not initiated by yielding of the cross section, it is initiated by elastic buckling, and is exacerbated by additional yielding of the extreme compression fibers?

“Any idiot can build a bridge that stands, but it takes an engineer to build a bridge that barely stands.”

 

[JoshPlumSE]

I think that's correct. I'm not sure if "initiated" is the correct term. Think of it this way. There will always be a SMALL amount of bending in the column due to axial load. Just because of the eccentricity or out-of-straightness. That results in a larger compression stress on the fiber that has flexural compression and a lower compression stress on the other side. This is the cause of the Double Modulus type of behavior.

Then residual stresses exacerbate this so much further because you've got portions of the cross section (flange tips) that yield so early. You no longer have a constant E over the whole cross section. And, if buckling is based on EI then everything gets a little more complex.

The best texts I have seen on the subject are:
1) Salmon and Johnson's "Steel Structures"
2) the SSRC's "Guide to Stability Design of Metal Structures" (note I may not have the title exactly correct as I don't have a current copy of this book).

 

[r13]

inelastic buckling. [‚in·ə′las·tik ′bək·liŋ] (mechanics) Sudden increase of deflection or twist in a column when compressive stress reaches the elastic limit but before elastic buckling develops.

 

[MegaStructures]

I've noticed that most of the mechanical textbooks do not consider inelastic buckling, Roark's being the main reference I am using to analyze buckling capacity of steel bar. I thought maybe the reason that inelastic buckling isn't checked there is because there are no residual stresses from the fabrication process (if that were the main cause for inelastic buckling), but from the explanation above it seems that there is a possibility for inelastic buckling (given that the bar is not excessively slender).

See also this calculator for power screw buckling . Again here they only consider elastic Euler buckling.

“Any idiot can build a bridge that stands, but it takes an engineer to build a bridge that barely stands.”

 

[r13]

I think you can find many articles about inelastic buckling. Here explains by a stress-strain curve.

 

[Agent666]

Inelastic buckling is due in part to a loss of stiffness associated with the the progressive yielding of the cross section as you increase the load. Residual stresses impact on this progressive yielding causing some regions of the cross section yielding before others. Simplistically flange tips of I section start to yield first for example, resulting in stiffness reducing which impacts on the stability and buckling occurs prior to the theoretical elastic buckling load being reached.

 

[r13]

The intermediate column tends to buckle inelastically. Please have a look on the linked PPT (p.11). Link

 

[patswfc]

Link to youtube playlist which has a few videos on elastic and inelastic buckling

https://www.youtube.com/watch?v=jNwvub87l8o&list=PLXdvc710ImyKoCSALYq0XLAmb9cmPoZy0