Calculating Natural Frequency of cantilever column

[structuresguy]

So I am working on a cantilever steel tower where I need to accurately calculate the natural frequency of the tower. Current design is round pipe, 12 feet diameter, 80 feet tall. Base of pipe is assumed rigidly attached to infinitely stiff foundation. So it will behave as a cantilever column, free to sway at the top.

I can't find any equations for cantilever columns, but have an equation for cantilever beam with uniform load I got from Roark's. I have modeled the tower in Staad using plate elements, and asked for mode shapes and frequencies.

In staad, I get frequency of about 45 Hz. Using the roark's equation, it is about 6 Hz.

I need help figuring out why the difference is so large.

Thanks

 

[rb1957]

i guess one thing would be you've got two loads to consider wind (lateral) and weight (axial)

 

[structuresguy]

oh, i should mention, it is inside another structure, so no wind loads on it at all.

 

[charliealphabravo]

Is there a load at the top of the column or just the evenly distributed self weight?

Also make sure that the self weight is not being lumped at the nodes.

 

[JoshPlumSE]

The issue has to be either with the mass or the stiffness, obviously.

The first thing I would do is sub-divide the STAAD model into smaller pieces. A single column with mass lumped at the top and bottom may not be very accurate. Therefore, if you can get the mass to distribute more evenly through the structure, then you should have more luck.

 

[charliealphabravo]

Also make sure that Staad is reporting natural frequency (fn) and not radian frequency (wn). To obtain natural frequency divide the radian frequency by 2*pi.

 

[trainguy]

Can you describe the mode shape that STAAD produces for the first mode?

tg

 

[structuresguy]

I'll try to answer everyone:

There will be a load at the top (5kip) but right now I am just trying to justify my analysis method versus known equations. So I am using tube of uniform dimension (12ft diameter) and uniform wall thickness (1 inch). So the self weight is uniformly distributed.

I have it modeled in staad using square plate elements with approximate dimensions of 1ft x 1ft. I am using 36 segments around the perimeter, 1 ft high each, so each is 1.047 ft wide. At the shaft is 80 feet high, so have 80 segments in the vertical direction. At the base, every node of the plates are assigned a fixed support.

How do I know what type of frequency (fn or wn) is being reported? I added the command "MODAL CALCULATION REQUESTED" into the self-weight load case. If the output is radian frequency, then once I divide by 2*pie, it would be very close to the equation i am using. Here is excerpt from the Staad output file giving the mode shape data:

NUMBER OF JOINTS/MEMBER+ELEMENTS/SUPPORTS = 2916/ 2880/ 36

SOLVER USED IS THE OUT-OF-CORE BASIC SOLVER

ORIGINAL/FINAL BAND-WIDTH= 71/ 71/ 432 DOF
TOTAL PRIMARY LOAD CASES = 1, TOTAL DEGREES OF FREEDOM = 17280
SIZE OF STIFFNESS MATRIX = 7465 DOUBLE KILO-WORDS
REQRD/AVAIL. DISK SPACE = 145.8/ 373330.0 MB


NUMBER OF MODES REQUESTED = 6
NUMBER OF EXISTING MASSES IN THE MODEL = 2880
NUMBER OF MODES THAT WILL BE USED = 6

CALCULATED FREQUENCIES FOR LOAD CASE 1

MODE FREQUENCY(CYCLES/SEC) PERIOD(SEC) ACCURACY


1 51.699 0.01934 2.193E-14
2 51.699 0.01934 5.489E-14
3 51.832 0.01929 2.744E-16
4 107.494 0.00930 1.147E-08

The first two mode shapes are lateral deflection, in X and Z directions. The 3rd mode shape is axial compression. The 4th mode shape is tube x-section warping.

BTW, the equation I am using is from Raorks Formualas eq 3b from TAble 36 (6th ED), and is

Fn = Kn/(2*pi) * sqrt( EIg / wL^4)

Using this equation, I get Fn = 6.11 Hz.

Thanks for your help guys. I'm just not sure that a cantilever beam equation should be the same as a cantilever column, since gravity is acting in different directions.

 

[structuresguy]

I guess I found my problem. While reading some posts on the Bentley tech support forums, I was reminded that I need to define the mass in the direction of possible motion. I had only applied selfweight in the Y direction. Once I defined it in the X direction, the frequency went down to about 5.9 Hz, just a little under the 6.1 Hz I calculated using the formula. Yeah! Now I just have to figure out how to make it over 20 Hz, and I'll be good.

 

[UcfSE]

At least you had the sense to check the computer output instead of taking it for granted.

 

[JStephen]

The direction of gravity doesn't affect the natural frequency of a beam.

You could also calculate a natural frequency in the vertical direction which is presumably of no interest here, but if you're getting the answer from software, make sure which one you're dealing with.

You'd want to reduce the thickness at the top to increase the natural frequency. It may be very difficult to change the frequency that much without getting into some unreasonable dimensions, so take a second look at the frequency requirement as well.

 

[Qshake]

Roark's book also gives natural frequencies of various structural arrangements.

Your problem is Case No. 3 in the 6th addition on page 714. Please note that wherein it says continuous system that means it's accounting for the weight of the beam over the length rather than a lumped mass. As a result, this should be as accurate as you'll need.

Regards,
Qshake

Eng-Tips Forums:Real Solutions for Real Problems Really Quick.

 

[structuresguy]

JStephen: You are right, getting to the "required" frequency is going to be near impossible. I suspect that the requirement is somewhat arbitrary. They have also provided some rotational and torsional deflection criteria, which I will also be checking. But they are all VERY small, like 1 arc-second rotation of the top of the tower under a 50 lb lateral force. BTW, in case you are wondering, it is to support a telescope.

Also, we are actually toying with a few different structural concepts, like a trussed tower. But I wanted to verify the software was giving me the correct values before i started in with the complex geometry. Easier to validate it with a simple round tube, than some trussed tower.

 

[271828]

You can approximate the natural frequency by pretending that your column is turned on its side like a beam, and apply the self weight and force due to the lumped mass. Compute the deflection at the tip in in. The natural frequency in Hz is approximately 0.18 times sqrt(g/Delta) where g=386 in./sec.^2

That formula (with 0.179) is nearly exact for a simply supported beam, but still works "OK" for some other cases. I've used it to check the period coming out of a program for a LFRS model. Apply gravity sideways, compute the deflection at the roof, and use the same formula. It's usualy pretty close.

BTW, I wouldn't have jumped all the way to a shell (I hope you're using "shells" and not "plates" or that's a big error right there.) for the first STAAD model. Create a stick model with frame elements first. Work your way up.

Good job digging into the documentation and trying to verify, BTW. Seems to be somewhat of a lost art.

 

[saplanti]

Just be careful about the purpose of the calculation of the natural frequency.
That diameter of shell might have lower natural frequency on the shell plate than the beam element natural frequency.

 

[slickdeals]

ASCE 7-10 commentary lists the formula for the natural frequency as well. This is on Page 521.

An approximate formula for cantilevered,
tapered, circular poles (ECCS 1978) is
n1 ≈ [λ/(2πh^2)]√(EI/m) (C26.9-12)

 

[Hickory]

To Structuresguy:

If you have the natural frequency at 6 Hz, why would you want to increase it to 20 Hz? You say the column is to support a telescope, right? I would think that the ideal frequency in that case would be zero, meaning the telescope was rigidly locked to the planet. If I am right in that assumption, then you should happy that you can get the frequency down to 6 Hz, because that is a lot better than 20 Hz.

What I'm saying is that you need to check back with your client and make very sure that he really wants the frequency to be greater than 20 Hz. It sounds like a typo: maybe he really wants it to be less than 20 Hz.

If I am wrong about that, I would be very interested in hearing the reason why.

 

[IDS]

I just ran this in Strand7 using 80 beam elements. For the first mode I get:

6.10 Hz exluding shear stiffness
5.91 Hz including shear stiffness

I didn't check the Roark formula, but I assume it wouldn't include shear stiffness, so those numbers tie in well with both Roark and the revised STAAD analysis.

The third mode is vibration along the vertical axis and has a frequency of 51.7 Hz. Deactivating the mass in the horizontal direction this becomes the first mode.

It makes very little difference whether the mass is lumped or distributed with this number of beams.

Does STAAD allow you to look at the mode shapes? I find that is useful to understand what is happening.

Doug Jenkins
Interactive Design Services

http://newtonexcelbach.wordpress.com/

 

[IDS]

I happened to have a cylinder model handy, so I scaled that to the given dimensions and get:

Mode 1 and 2 (squashing of the cylinder at the top):5.26 Hz
Mode 3 (first cantilever mode) 5.92 Hz

See attached file for mode shapes

Doug Jenkins
Interactive Design Services

http://newtonexcelbach.wordpress.com/