Post Buckling Deflected Shape
Post Buckling Deflected Shape
(OP)
Is there an approximate way to find the deflection or deflected shape of the a member which has buckled elastically? Or is this something that requires full on Non-linear FEA. Seams like if it has buckled elastically there should be some sort approximation, may be if the load is within a few percent of buckling perhaps?
Thanks!
Thanks!






RE: Post Buckling Deflected Shape
BA
RE: Post Buckling Deflected Shape
1) You know the 1/2 wavelength for the sine curve = column length - imposed axial displacement. So you can get lambda in the sine wave equation.
2) You set up an equation for the arc length of a half sine wave and set it to the original length of the column.
3) You use the relation established in #2 to back calculate "A", the amplitude of the sine wave.
4) Now that you have lambda and "A", the shape of your sine curve is defined and your done.
Trouble is, there's no anti-derivative for the arc length of a sine wave. As such, you're forced to result to numerical methods like the one that BA mentioned.
Luckily, there's a much easier way that's sure to be sufficiently accurate for any practical situation. Use the imposed displacement and one of the models below to work out "A" in the sine wave equation. This replaces steps 2 & 3 above and get's you straight to a complete equation for estimating lateral displacement along the column.
I like to debate structural engineering theory -- a lot. If I challenge you on something, know that I'm doing so because I respect your opinion enough to either change it or adopt it.
RE: Post Buckling Deflected Shape
EIT
www.HowToEngineer.com
RE: Post Buckling Deflected Shape
Unless it is the following document: Numerical Procedure for Computing Deflection , Moment and buckling loads by N. M. Newmark which I did find via google.
Thanks again!
EIT
www.HowToEngineer.com
RE: Post Buckling Deflected Shape
Yes, I have noticed that the Newmark procedures which I posted were subsequently omitted. There are quite a few pages, so I will have to scan them again. Can't do it right now but will try to do it in the next couple of days.
BA
RE: Post Buckling Deflected Shape
https://engineering.purdue.edu/~ce474/Docs/Newmark...
Also, Article 2.15 of "Theory of Elastic Stability" by Timoshenko and Gere covers the case of elastic buckling of a stepped column using the numerical procedure devised by N. M. Newmark.
BA
RE: Post Buckling Deflected Shape
P/PE = 1+(pi2/8)*(delta2/L2)
RE: Post Buckling Deflected Shape
https://newtonexcelbach.wordpress.com/2010/06/19/e...
Doug Jenkins
Interactive Design Services
http://newtonexcelbach.wordpress.com/
RE: Post Buckling Deflected Shape
Thanks BA!
EIT
www.HowToEngineer.com
RE: Post Buckling Deflected Shape
Using your method -> is it possible to iterate the amplitude A until the arc length of the half sine wave matches the original length of the column. Or is the problem that you can't determine the arc length of a half sine wave (I should probably know this).
EIT
www.HowToEngineer.com
RE: Post Buckling Deflected Shape
Given the information supplied so far, I'd run with this:
1) Use Hokie's equation to estimate A.
2) Plug A into the sine curve equation.
3) Use one of my suggested approximations to fact check Hoki's equation.
I like to debate structural engineering theory -- a lot. If I challenge you on something, know that I'm doing so because I respect your opinion enough to either change it or adopt it.
RE: Post Buckling Deflected Shape
I guess my recommendation is still the mopethod that I proposed initially, using the sketches to approximate "A".
I like to debate structural engineering theory -- a lot. If I challenge you on something, know that I'm doing so because I respect your opinion enough to either change it or adopt it.
RE: Post Buckling Deflected Shape
Buckled shape: LINK
Interestingly you get a lot of lateral deflection for not much vertical deflection.
Thanks again for the input!
P.S. Math software is free and is found at SMath Studio, great stuff if you ask me.
EIT
www.HowToEngineer.com
RE: Post Buckling Deflected Shape
Looks tight to me. That's exactly the procedure that I had in mind but with software doing the numerical stuff.
I'm not surprised at all. That's just what we discussed here: Link
That software does look pretty sweet. Does it have any advantages over MathCAD express? Can you program?
I like to debate structural engineering theory -- a lot. If I challenge you on something, know that I'm doing so because I respect your opinion enough to either change it or adopt it.
RE: Post Buckling Deflected Shape
EIT
www.HowToEngineer.com
RE: Post Buckling Deflected Shape
I like to debate structural engineering theory -- a lot. If I challenge you on something, know that I'm doing so because I respect your opinion enough to either change it or adopt it.
RE: Post Buckling Deflected Shape
RE: Post Buckling Deflected Shape
I like to debate structural engineering theory -- a lot. If I challenge you on something, know that I'm doing so because I respect your opinion enough to either change it or adopt it.
RE: Post Buckling Deflected Shape
Another interesting find - I wanted to find a transverse load (point load applied at midspan in the weak axis) that would cause a deflection equal to that due to a load just above the buckling load. It turned out to be about 0.05*Pe
EIT
www.HowToEngineer.com
RE: Post Buckling Deflected Shape
I like to debate structural engineering theory -- a lot. If I challenge you on something, know that I'm doing so because I respect your opinion enough to either change it or adopt it.
RE: Post Buckling Deflected Shape
BA
RE: Post Buckling Deflected Shape
"The value of P at which a straight column becomes unstable is called the critical load. When the column bends at the critical load, it is said to have buckled. Therefore, the critical load is also called the buckling load. At the critical load the column is extremely sensitive to increase in load, in the sense that a very slight increase is accompanied by a large lateral deflection. Thus, if the critical load P = 4K/L for the column of Fig. 4-1a by only 2/3 percent, delta is 10 percent of the column length L (Fig. 4-1c)". Figure 4-1a is two rigid bars of length L/2 connected together by a rotational spring with a spring constant K.
The equation provided previously is an approximate solution that provides very similar results for lateral deflection to the two-bar with rotational spring model.
RE: Post Buckling Deflected Shape
But, respectfully submitted, that "ain't gonna happen" UNLESS the buckling reaction is distributed evenly through the column mid-section AND the column cross-section through the mid-section of the curve starts off uniform and even, AND stays uniform and evenly misshapen during the entire period of movement.
Instead, the column will be uniformly deflected until the buckling collapse begins. At that point in time, and at that point along the column the where a weakness or flaw has happened, the column will "kink" and yield suddenly. The rest of the column - once the kink has started will restraighten slightly back from its ultimate yield (plastic - >2% yield) position.
You will end up two slightly curved ends and a sharp bend in the middle where the coulmn cross-section is flattened completely.