Fundamental Mode - Vertical Direction Monumental Stair

[sticksandtriangles]

I am designing a monumental stair very similar to that as shown in this post:

https://www.eng-tips.com/viewthread.cfm?qid=455434

I am trying to hit a vertical natural frequency higher than 5hz and a lateral natural frequency greater than 2.5hz.

Reviewing my mode shapes and participation factors, I am little lost as to whether I can ignore the first mode of my stair for my vertical natural frequency check.
First mode image does show both x and z deformed shapes:

Output of mass participation for the modes:

Based on the results both the x and the z participation are very close for the 1st and 2nd modes.
Note that the 1st mode has a Z mass participation of 0.1873 and the 2nd mode has a Z mass participation of 0.2.

Am I allowed to call my fundamental Z frequency 4.658hz (1/0.214 secs) as it has the highest Z mass participation?
If so, I am close on my stair design, but if I utilize the first mode, I am pretty far off on my stair design.

Thoughts appreciated.






S&T

 

[r13]

Try add two supports at the landing on the building side, same as those shown on the photo. It should improve your design.

 

[JoshPlumSE]

What is the mass participation for each of those modes? Is it 0.2% or 20%?

If it's 0.2% then I don't think either of them can be considered fundamental modes.

Could this be a case where a lot of the mass is directly associated with boundary condition locations? If so, then you might want to discretize your mass.... Or, you might replace your rigid support with something that more realistically models the stiffness at that support location.

 

[271828]

Frequency response functions would be immensely helpful here. The frequency response function for vertical load and vertical acceleration would probably have a big peak at one of the lower modes -- that would be the fundamental mode for vertical vibration. Similar with the fundamental lateral mode.

From your screen shots, it looks like you're using SAP2000. It computes the FRF using the Steady State Analysis feature.

 

[ldeem]

I am not familiar with the program you are using so I will not comment on the results but what caught my attention are your design criteria. I deal with vibrating equipment more often than floors but the floor reference I have used in the past is ATC Design Guide 1. ATC-1 section 2.2.1 advises limits close to 10 Hz.

There are more advanced methods as 271828 points out that you should consider if vibration is a concern and you are not sure of the accuracy of your results. Speaking of accuracy it has been my experience with vibration to maintain a very large safety factor since the problems are difficult to model accurately. Try making some changes to your support rigidity and you may see a significant change in the results.

 

[sticksandtriangles]

Retired, I am trying to see if I can get away without the lateral supports as shown in the photo. As of right now, I seem to be pretty good in regards to my lateral natural frequencies, so I was hoping I could survive without.

Josh, the mass participation is 20%. I have the mass being discretized at each node shown in the image, so I do not believe it to be a mass discretization issue.

271828, I will look into the frequency response functions, never heard of this before, I am using SAP2000.

ldeem, the vibration criteria comes from AISC Design Guide 11 which recommends the 5hz and 2.5hz check on vertical and lateral natural frequencies. There is also one additional check that it recommends which I have calc'd out and the stair appears to be passing.

S&T

 

[Agent666]

Be aware on a stair that lateral deflection/excitation is almost more disturbing to a user traversing the stair flights than vertical excitation.

Provide the lateral support at midheight landings.

 

[MIStructE_IRE]

Interesting that you have pinned connections shown in the model. Do these things not rely on torsion in the members and torsional fixity at the supports?

 

[Guest262653]

Unless you have a true pinned connection at the supports, for the vibration analysis I would consider moment restraints as well. The deflection amplitude isn't enough to consider a fully pinned restraint.

 

[glass99]

Your landing beam looks too flexible. It seems to be hinging at the inner stringers.

 

[r13]

Even this model can pass vibration and strength criteria, it will fail under service criteria.

 

[glass99]

Even this model can pass vibration and strength criteria, it will fail under service criteria.

We have built stairs like this before. It comes down to the "twisting" movement of the landing, which becomes a function of either the torsion stiffness of the stringers or generating vierendeel action through the treads.

 

[r13]

An pin supported A frame turns side way. I'm not doubting the feasibility, but good luck.