Steel Portal Frame Frequency 2

[JKW05]

I have attached a sketch and calculations for a simple frame which will have a very small impact load applied downward at the mid-span of the beam. The required minimum natural frequency of the structure measured along the vertical axis at the mid-span of the beam is 200 Hz. From the attached calculations, it appears that column deformation alone makes this unattainable. (For simplification, I have not included the applied load in the calculations (200 lbs.) as that would not have an impact on my conclusion). Am I missing something here? Any assistance would be appreciated. Thanks.

 

[WARose]

The first question I have is: where is this "required minimum natural frequency" of 200 Hz coming from? That's off the scale in the reference you cite (i.e. see Fig. 2.1 for starters).

 

[271828]

I agree with WARose. What is the reason behind the 200 Hz limit? It seems extremely high.

 

[JKW05]

ANSI Z359.7 "Qualification and Verification Testing of Fall Protection Products", Section 4.1.1:

"The required minimum natural frequency of the drop test structure shall be 200 Hz when measured along the vertical axis through the point of the test anchorage or anchorage connector to which the test specimen is attached"

I have been asked to modify an existing structure to meet this criteria.

JKW

 

[patswfc]

For the axial shortening of the column do you need to account for shortening due to frame self weight? This would occur prior to the impact load.
You may be able to meet the 200Hz limit (very high as noted by others) if this can be discounted and only considering the impact load, by using a
beam approx. twice as stiff as currently shown and columns with 50% more area.

 

[WARose]

You might try some braces going to the point where the testing apparatus is hung. (Down to the base plates.)

Speaking realistically, I would doubt (unless you have piles under the columns as the foundation) you could hit that natural frequency with any reasonable estimate of the vertical (dynamic) spring constant. Not sure if they have that included/in mind when they refer to the "drop test structure". But just something I would like to point out.

 

[JKW05]

patswfc:

We have discussed the axial shortening condition in our office. I agree the deformation is already in place, especially since it is an existing frame. But the question to us then becomes do (can) we also ignore the beam self-weight deflection which has also already occurred. The AISC Design Guide specifically says to include actual live and dead loads, which to me suggests the beam/column weight is part of the dead load, and the Design Guide provides the provision for the inclusion of column deformation.

WARose:

Your bracing concept is exactly where I started from. But the column self-weight deformation seems to be the controlling factor. If these deformations which have already occurred are neglected, I think the criteria can be met with some improvements.

Thanks for your input!

JKW

 

[271828]

The Dunkerley relationship that you used is pretty conservative. I recommend building a model of this in a frame analysis program and computing the frequency. That should be very easy. It might be quite a bit higher than you computed. 200 Hz is still very high, though.

 

[271828]

The masses of the members and anything attached must be included in the calculations. This isn't like strength analysis. The frequency would be the same if this frame was turned on its side or if it was on the moon.

 

[JKW05]

I had modeled the frame in an analysis program with the following results:

Beam Deflection: Model = 0.00048"
Hand Calc = 0.000394" (simple beam = 5wL^4/384EI)

The frame analysis program includes shear deformation which I have calculated to be 0.000094"

Beam Deflection: Model = 0.00048"
Hand Calc = 0.000488" (including shear deformation)

Column Deformation: Model = 0.000617"
Hand Calc = 0.000616"

So the deformations appear to be quite consistent.

Attached are derivations I have developed for frequency calculations for 3 cases:

1) Uniform Load; Simple Beam
2) Concentrated Load; Simple Beam
3) Axial Load; Column

In all three cases my derivation concludes f = 0.16 x (g/delta)^0.5.

AISC Design Guide 11 uses f = 0.18 x (g/delta)^0.5

Curiously, the coefficient (Kn) in Roark's Formulas for Stress and Strain is consistent with 0.18 (Kn = 9.87) for uniform loads but 0.16 (Kn = 6.93) for concentrated load.

[RE: Table 16.7, Page 769:

http://materiales.azc.uam.mx/gjl/Clases/MA10_I/Roark's formulas for stress and strain.pdf

]

So I have two new questions I am seeking assistance with:

1) What accounts for the difference between my derivation(s) and the AISC Design Guide/Roark (assuming my derivation is accurate).
2) What other methodology would a frame analysis program use that would yield different results in the calculation of the frequency? (I am a bit reluctant blindly accepting results from a "black box").

Thank you in advance for any input.

JKW

 

[WARose]

I don't mean to get away from the questions you are asking.....but I think this whole thing is going to be just about impossible to achieve (with this frame). Your model(s) so far are taking into account only the flexural/axial behavior of the members involved. What about the contribution of the connectors that tie the testing apparatus to the frame? Or the connections within the frame? Or the spring constants for the foundation itself (that I mentioned before)?

If you really have to have 200 Hz.....you may have to face the possibility that it is just about impossible to achieve with you trying to modify this existing frame.

 

[SocklessJ]

To account for connection and foundation stiffness, you could always verify it with testing. I bet they've apps that use the accelerometer in your phone and can convert from time to frequency domain. But probably nothing that can deal with 200 Hz. That's really, really high.

Your analysis is a bit confusing because frequency of vibration is based on mass and stiffness, not load. However some approximate methods involve applying loads, and I think that's what you're doing. It's been a while since I've done the Rayleigh method. If you have staad, you could have it run an eigenvalue analysis and get a more accurate frequency. Breaking the columns into a series of lumped masses may help you out.

For an SDOF system, frequency is proportional to √k/m. You've figured out that as you add stiffness, you also add mass. So if a more accurate analysis doesn't work, the only fix is to shorten the columns.

 

[JKW05]

Thanks for your responses guys.

SocklessJ, you indicate that my analysis is based on load, not mass. But all 3 of the derivations I posted use the basic equation of f=[1/(2 x pi)] x (k/m)^0.5 where m=W/g. Except for the discrepancy between the 0.18 and 0.16, this is also consistent with what the AISC Design Guide presents. So I thought I was using mass. The example I posted in the OP does not have any loads applied, just the deformations (stiffness) resulting from the selfweight (mass) of the structure.

Is f=[1/(2 x pi)] x (k/m)^0.5 just considered an "approximate" analysis?

JKW

 

[ThomasH]

What is the actual impact load that the structure shall handle?

I think you have two options:

1. If the requirement 200 Hz is non-negotiable, I would give up. It seems impossible to meet with a reasonable design.

2. Work with the actual impact loading. Mass and velocity when it impacts the structure. Can the structure withstand the impact with acceptable deformations?

Just a thought

Thomas

 

[JKW05]

The impact load is 200 to 300 pounds. The 200 Hz requirement is from the ANSI Z359.7 standard referenced above, although I do not know what that requirement is based on.

 

[WARose]

The 200 Hz requirement is from the ANSI Z359.7 standard referenced above, although I do not know what that requirement is based on.

Most likely they are trying to have a system where the effects of a fall are entirely attributable to the "protection" product being tested. 200 Hz works out to a very fast transfer time (of impact).

 

[JKW05]

I agree that is likely the intent, but why not 100 or 150 .....? (rhetorical question).

 

[WARose]

I agree that is likely the intent, but why not 100 or 150 .....? (rhetorical question).

And it's a good one. I don't know. Maybe they aren't comfortable with the transfer time (of the support frame) going from a few milliseconds to double digits. That may be eating into the range of these devices.

 

[Agent666]

Can you post the clause that notes the 200Hz requirement, so we have the context in which it is stated (sorry not familiar with the standard referenced), as others have stated it seems overly onerous so it might be something has been interpreted incorrectly?


I've always taken the simplification of 18/sqrt(9.81) as shown here for a uniformly distributed mass & stiffness case:-

 

[JKW05]

The referenced requirement from the standard is quoted in the 4th post of this thread.

"I've always taken the simplification of 18/sqrt(9.81)as shown here for a uniformly distributed mass & stiffness case:-"

Where did the 9.81 come from? Did you mean 18/sqrt(delta)?

The first two forms appear to be consistent with each other and with my derivations posted on 11/5. Not clear how it gets from [1/(2 x pi)] x (g/delta)^0.5 to 18/(delta)^0.5.