Slenderness ratio for column having different cross sections

Newmark's Numerical Methods is a very straightforward method to use.

BA

I have an idea for a Mathcad worksheet that may produce numerical single-step direct solution of the critical limit state load for a variable section of varying inertia buckling restrained as doubly hinged in single curvature, that it may function or not (to investigate), and may correspond to either Newmark's or eigenvalue methods -I would love to retain all these things in my mind but it seems impossible with everything else-, that if I develop will post.

Basically the procedure will divide in segments the column where moments and curvatures will be reconciliated under the loading between themselves and to satisfy end conditions, and through investigation of progressing convexity of the single curvature find when the first limit state under axial load and standing moment (at any investigated point) is found, that for general application in ductile materials could be a plastic condition of failure. Most likely Mathcad should be able to do that quite efficiently and maybe I will try to produce such worksheet,eventually.

Other than as above for the fun of learning and get a nice worksheet, ordinary analysis programs may be directly well suited to find a practical limit load for the case by the mere artifice of dividing the column in segments and see when limit state is met, that you may do even for columns with initial imperfections.

There is an AISC / MBMA design guide for tapered member design that is being released soon. Perhaps there is a draft available from their website. Or, perhaps they have finally started selling it.

From what I have seen, this guide is geared mostly towards built-up I shaped members. However, my impression is that the design procedures (which are far more detailed than I have found anywhere else) would be applicable towards other shapes as well. In fact, they would probably be a bit simpler.

I'd recommend taking a look at that manual / design guide if possible.

I have adapted my frame analysis spreadsheet to generate the necessary column data and carry out an iterative analysis of buckling load. The column may be divided into any number of segments with differing properties, allowing a good approximation of either stepped or tapered sections.

More details and a download link here:


Comments or suggestions for improvement are welcome.

Doug Jenkins
Interactive Design Services

Doug,

Just looking at the deflected shape, I wonder why the column in your link has a reverse curvature. I would have thought it would be single curvature.

BA

thanks, Doug, have not looked at yet.

I was making mine's in Mathcad but finding some difficulties in the implementation; then a bad Save file and as old mathcad 2000 still does all lost. Anyway was not going very well, so I stopped for now; it also made me to remember how much I miss mathematics enough for my things. Yet I may take again the subject, some day.

Doug,

I see now...the top is fixed against rotation, hence the double curvature. I thought the top was free.

BA

Ishvaaag & Doug:
I agree with BA; Newmark’s Numerical Methods handles this problem very nicely. I suspect he and I did these problems long hand, on paper; but I am sure this could be set up very nicely either in a spread sheet or in MathCad. Correct me if I’m wrong BA, but it was numerical integration, and the shorter you made your beam/col. elements, thus better defining your changing section properties, the closer you approached the exact solution.

dhengr,

Yup, that's right. I included a 26 page commentary on the method in an earlier thread:


Page 23 of that commentary illustrates the calculation of the buckling load for a tapered column varying from EI at one end to 7EI at the other.

BA

dhengr & BAretired - I actually started out using a method similar to Newmark's method (i.e. a numerical integration of the slope-deflection equations), but had troubles with the different end conditions, so I ended up taking the easy way out and just plugging into the frame analysis spreadsheet I already had set up. No doubt it is overkill for a straight column, and it does get a bit slow with anything over 80 segments in the column, but it is still much quicker than doing it by hand! (and 20 segments is plenty for practical purposes anyway). Another advantage of using the frame analysis is that it can be easily adapted to analyse buckling of 2D frames, using the same method.

Regarding the shape - I'm pretty sure it comes up with the right buckling shapes, corresponding to the specified end conditions, but having the plot allows that to be checked easily. This was another reason for using the frame spreadsheet, which already had the plot function built in.

Doug Jenkins
Interactive Design Services

BAretired - I had a play with the tapered column example in the paper you linked. Initially I had trouble getting the spreadsheet to converge, but the problem seemed to be that linear shortening of the column was interfering with the analysis. Since the area is only in there because the frame analysis program needs it, I just made the area a very high number, so the column was very stiff axially, and only bending deflections had an effect on the result.

Doing that I found a buckling load about 3% higher than that given in the paper, using 24 segments, so that's pretty good agreement. I then calculated the I value based on a linear variation in diameter, rather than a linear variation in I. Doing that the buckling load was reduced to about 25% lower than the figure in the paper, which also makes sense, since the average I would be considerably lower than the value of 4 used in the paper.

Doug Jenkins
Interactive Design Services

In the end I have ordered one copy of Numerical Analysis of Beam and Column Structures," by William G. Godden, N.M. Newmark (startingly valued between $6 and 250, of course I took the cheap one, even if worth the dearest) hoping to get it by september or even later... hope find time enough to learn it; I think is a problem of current professionals, at least where I live, to become distraught by the many things everyone, professionals or not, is expected to do, what is not conducive to deep knowledge in any field.

In my case part surely is my fault because I always try to be jack of a number of trades, also interesting, but with akin effect.

It all started with this... (see attachment)

Fancy how we can recall with precision something of 46 years ago and not at all many more intently learnt in-between.

ishvaaag,

That is because the Meccano set was a more entertaining way to learn than reading books.

BA

Doug,

Sounds like excellent agreement between the two methods. I agree that a linear variation in radius is likely a more realistic problem.

Referring back to your link. On Line 5 you say "Top Y must be free". Line 6 says "At least one of the other freedoms must be fixed".

For a flagpole type of column, neither of the other freedoms is fixed. Can your program handle this situation?

BA

Referring back to your link. On Line 5 you say "Top Y must be free". Line 6 says "At least one of the other freedoms must be fixed". For a flagpole type of column, neither of the other freedoms is fixed. Can your program handle this situation?

Click to expand...

Yes, I meant at least one of:

- Rotation at the base
- X direction at top
- Rotation at top

must be fixed.

Fixing any combination of one or more of those will work.

I'll change the note to make it clearer.


Doug Jenkins
Interactive Design Services

Doug,

I think I understand what you are getting at now. My preference would be to allow the user to select any degree of freedom he chooses, top or bottom.

If he chooses one which is unstable, the program could alert him with a message (perhaps a raspberry).

If he chooses one which results in zero moment or zero axial force, he should be afforded the opportunity of reviewing his input to discover his error, thereby enhancing his understanding of the issue at hand.

BA

The spreadsheet does allow you to enter whatever end conditions you want.

The words tell you what you should enter if you want to get sensible answers!

It assumes that the column is fixed in position at the base. I could have allowed either end to be fixed, but since it is easy to input the column dimensions "upside down" I didn't think it was worth the effort.

Doug Jenkins
Interactive Design Services

Doug,

Okay, I think I understand.

BA