Column Buckling Analysis

[stresscalcs]

I have recently been confronted with a long column buckling problem where the column ends are PARTIALLY restrained. Further, unequal bending moments are simultaneously applied to the column ends.

All the textbooks and internet documents I have been able to find do not deal with this partially restrained condition, only pinned or fixed ends being considered.

Has anybody got any suggestions where information might be found to help in the solution?
Help would be appreciated.

Note: A FE analysis is not wanted as there are many load cases to be considered which can be most easily handled in a spreadsheet.

 

[inertia4u]

stresscalcs,

A very hand reference I found for columns, beam columns etc is

William McCombs
Engineering Column Analysis
The Analysis of Compression Members

I ended up picking it up from thattechnicalbookstore for around $40US

I double checked and he does have long column with elastic retraints (Euler eqn) for various K1/K2 spring constants on page 2.10.

As I have mentioned before in the Johnson-Euler post a couple of weeks ago, the book is a collection of notes that Mr. McComb had generated while he was working in the Aerospace Business. Also note that Mr. McCombs also was involved in the development of the famous Bruhn book. (in fact, I also have a copy of McCombs "A supplement to Bruhn" which I also highly recommend.

Good Luck,

Nert

-----
Nert

 

[stresscalcs]

Hallo and thanks for the information.

I am aware of William McCombs work as I have a copy of the "Bruhn Supplement" but the column work I have not seen.

I would guess that the elastic restraints combinations is simply when you have two different restraints (considered as springs) and an equivalent length for a pin ended strut may be estimated and the pin ended column evaluated. Please correct me if I am wrong.
Such information about equivalent length factors may also be found from an ESDU data sheet.

However, the effects of the end moments have not been accounted for. They also affect the column deflection once any lateral deflection begins, which is immediately load is applied.
Is the solution a secant analysis with offset end loads with partial end restraints? Again, this condition with partial restraints is not shown anywhere. Also, the end moments I have are not equal!

I have been using non linear FE to try to understand the process of deflection of the column under end load alone with partial end restraints and it seems not to agree with the ESDU data equivalent length results.

Any comments would be appreciated.

Richard

 

[inertia4u]

Richard,

Per your first paragraph, yes, the McComb chart essentially comes up with an effective length column based off of the rotational stiffness at the ends of the column. I am not aware of how ESDU actually does this, however.

Regarding your second paragraph, yes, you are correct. The McComb chart is purely for an Euler column, however you are looking for a beam-column. I didn't notice that until I reread your question. Sorry about that.

I am also sorry that I wouldn't be able to offer additional help on this. I gather I'd have to dig into some references and try to solve using a numerical method. However, I really do like your thought of using the Secant method with the offsets. That just might work... Is it too conservative to just throw out the springs and assume pin ended?


Just some thoughts: The applied end moments force an initial displacement of the beam-column based on the rotational springs on the ends. Then, the axial load is introduced.

As far as FEM representation. I have no clue on how to set up this problem. Maybe someone here has more experience setting up a beam column analysis using FEM.


-----
Nert

 

[stresscalcs]

Hallo again

Just a few further comments and maybe somebody else will join in.

I think the equivalent length factor is also shown in Bruhn, so I am sure the method of ESDU is the same. It probably goes back to Timoschenko.

The numerical methods solution is all I can see at the moment, but I need a general solution to solve the other load cases (there are over 9000 I believe, about half have a tension axial load so can be dismissed) where moments and axial load have no relationship to each other. I have been avoiding what might be inevitable, by looking for a logical but pessimistic solution.

I need the 'end springs' as we are looking at RF's about 0.65 when we go with pin ends. We can see >2 with fully fixed ends. Peery suggests using pin ends as the end fixity is difficult to calculate. Other sources suggest 50% fixity as typical but warn of over estimation. I am not impressed with the estimation of the partial fixity but that does not affect the method of analysing the column.

The FE analysis is also not so simple. You have to put springs at the column ends.
If the column is considered straight before loading, the springs absorb part of any external end load moments, so you have to do a trial and error loop to get the required moments at the column ends.
If you assume the moments exist before the axial load is applied, I believe the then initial deflected shape is not really applicable and is an overestimate, leading to to RF <1 again.

Richard

 

[rb1957]

personally i'd probably step away from the hand calcs ... at leat this is a beam column (there are transverse loads to balance the different end moments).

i'd've thought these days that a non-linear FE analysis would solve this for you, it sounds like a simple eigne-value solution. i'd start with a simpler geometry (that you can verify) then step into the unknown !

 

[stresscalcs]

I have been doing some NL (non -linear) FE work to try to understand the situation better.

I have noticed that beam column NL models using CBeam elements do not give the same deflection results as the classic Roarkes formulas. I'm still looking for errors.
Anybody else made the same comparison?

Richard

 

[40818]

What your trying to solve is pretty hard really. The situation is identical to a wing rib vertical stiffener analysis, which has to contend with compression brazier loads with an eccentricity, fuel pressure on the side, and a variable fixity due the castellation type. There is an ESDU which relates to beam columns with standard loadings, however it also has a set of equations which (if i remember correctly) allow you to obtain fixity levels. I do have mathcad files which do exactly what your after but i cant give them to you unfortunately.

 

[rb1957]

model a problem you know the answer to, to see know the model works ... start with something like your problem ... a beam column with end moments that you can analyze by hand (rather than using Roark, which could have assumptions buried in it that you don't appreciate).

i imagine you're trying rotational springs (or something that'll give you a moment as a function of slope ... a partial rotational support). how about an iterative approach ... start with pinned ends, what slope develops (for you loads)?; try fixed, what moment ?; add 50% fixed moment on top of your loads ... find some factor of the fixed moments that you think applies. how to calculate a MS (since increasing the loads (towards critical) increases the endmoment) ... try factoring the load (and increasing the moment) untill you get a critical factor (= RF = MS+1)

 

[RPstress]

Stresscalcs/Richard: what sort of differences are you finding? Are they big?