Don’t try this until a Professional Engineer has checked things out

I’ve been out dancing several times in the past month or so – New Year’s Eve was spent with a Latin band which played salsa, bachata, and merengue.  A few weeks later my wife and I went to a ballroom gala, getting an opportunity to trot out our fox trot, rumba, cha-cha, and swing.  And every few Fridays we manage to squeeze in an hour or so of our favorite, Argentine tango.

Social dancing is a great hobby: it gets me out of the house, provides a bit of exercise, is a very portable vacation activity (no golf clubs to carry), and is easy for engineers – after all, counting 1-2-3-4 is child’s play compared to the calculations we do every day on the job.

The nice thing about dancing is seeing – and feeling – everybody act as one.  Sure there may be the occasional couple dancing to a different drummer, but in general the entire crowd moves together to the tempo of the music.  In fact it is really hard not to be in synch – a good, strong rhythm penetrates everything, forcing your body, the floor, and everything else to move in time with it.

I often think that a disco classic like “Brick House” is like the dynamic forcing function that I learned about in my structural dynamics classes during my college days.  And our bodies represent the single-degree-of-freedom oscillators responding to those forcing functions.  And how do they respond?  Often, if the relationship between the forcing function (music) and the dynamic characteristics of the responding oscillator (us) is just right, resonance is achieved and the response of the oscillator greatly exceeds what it normally would be if that same force was applied statically.  And that, my readers, is the scientific explanation as to why the booty shakes.

Two Single-Degree-of-Freedom Oscillators in Action

It is not just the human body that can experience an amplified response, but the structural body (the building) as well.  On occasion we see this effect taken to a horrific extreme, as in 1981, when the cat walks of the Kansas City Hyatt Hotel fell as the crowd enthusiastically danced to “Satin Doll” (killing 114); or the collapse of the Versailles Wedding Hall dance floor in Jerusalem in 2001 (killing 23).

To avoid tragedies such as these, society counts on the structural engineer to adequately design our buildings against all loads.  In order to do this, the engineer needs a strong knowledge of structural dynamics.

Without going too deeply into the theory of dynamic analysis, a summary of the salient points are:

1. Any dynamic load (i.e., where the load varies with time), when applied to a given structure, will cause a response that may be less than, equal to, or greater than the response of that same structure under that same magnitude of load applied in a static, unchanging manner. The ratio of the dynamic response to the static response is known as the Dynamic Load Factor, or the DLF.
2. Any combination of dynamic load and structure will have an associated DLF, but the highest DLFs tend to occur when the imposed load follows a harmonic pattern (i.e., a pulse applied at a steady interval or beat), and the pulse frequency is close to the natural frequency of one of the dominant modes of vibration (i.e., tendencies under which the structure vibrates). The natural frequency can be simplistically calculated (in radians/sec) as √K/M, where K and M are the stiffness and mass of the structure/mode respectively.  (Any good structural analysis software – like CloudCalc, of course – can be used to calculate the natural frequencies of the modes of vibration of the structure.)

More traditional Single-Degree-of-Freedom Oscillator

So what does this mean?  Although any dynamic load may present an engineering design problem, structures in which people are moving rhythmically in unison (harmonic actions such as dancing or exercise) present the biggest potential danger, as the dynamically amplified response to a multitude of footsteps, if not designed for, can on rare occasions grow large enough through its elevated DLF to bring down the building.  Offering guidelines to avoid this, AISC has published Design Guide 11, “Floor Vibrations Due to Human Activity” – Chapter 5 specifically covers designing for dancing and other rhythmic activities.

Given a dynamic response problem, the engineer’s solution is simple – change the DLF by changing the relationship between the imposed load and the responding structure, which means changing either 1) the timing of the imposed load or 2) the natural frequency of the structure.

A good example of the former is an army’s practice of “breaking stride” http://www.livescience.com/34608-break-stride-frequency-of-vibration.html when crossing a bridge.  Although the soldiers may march in unison, and with perfect cadence (an ideal harmonic loading), as they approach a bridge, they are instructed to switch to a random step pattern once they set foot on that bridge, assuring there is no harmonic load to potentially resonate with a natural frequency of the bridge.

An example of changing the structural characteristics would be to change the stiffness K (add or remove a support column) or mass M (increase or reduce the number of people attending) of the structure in order to move the natural frequency away from an area of high DLF to an area of lower DLF.  If the time profile of the forcing load is known, it is relatively easy for the engineer to construct a graph, or curve, of DLF vs structural natural frequency, and so be aware of which natural frequencies are better to avoid.

AISC Design Guide 11 provides advice on natural frequencies to avoid when designing dance floors in general.  But I began thinking of all of the dances that I have tried (over 20 to date) and realized that the time profiles, and thus the DLF curves, of these dances differ from each other significantly.

For example, what if you are designing a salsa club?  Salsa has a steady 1-2-3-hold 5-6-7-hold step pattern, with a tempo in the range of 180-200 beats per minute (BPM).  When the salsa steps are run through DLF curve generating software, it shows that salsa would amplify footfalls by a factor of up to 8 in a structure with natural frequencies in the vicinity of 3 Hz (cycles per second) and also, to a lesser extent (DLF approaching 4), around 12 Hz.  Other natural frequencies would see no significant response.

The Two Step (the national dance of my home state), with a tempo (180 BPM) very similar to that of salsa, but a step pattern of Quick-Quick-Slow-Slow, would generate a DLF curve with high amplification (also around 8) in structures with natural frequencies in the range of 3 to 9 Hz.

How about traditional ball room dances?  The waltz has a much slower tempo (90-100 BPM) but a more regular step pattern: 1-2-3, 1-2-3.  The corresponding DLF curve shows a high response tightly focused around a natural frequency of 1.5 Hz.

And finally, how about Argentine Tango?  That dance is my favorite, as it gives the most opportunity for creative variation of the steps – quick steps, slow steps, long pauses, plus some time in which the lady crawls up the gentleman’s leg.  From the various step patterns that Argentine Tango offers, I have selected one here, the first part of a basic cross-system salida:

Here we see that the peak DLF is lower than it is for the other, more harmonic, dances, with DLF near 4 in the natural frequency range of 2-4 Hz.  And that is only if all of the milongueros dance the same patterns in unison.  In reality the creative freedom of the tango has a similar effect to soldiers breaking stride – the multitude of patterns occurring simultaneously may broaden the responsive range of the DLF curve (see the red line above), but more importantly brings down the peak DLF to a level that barely affects any structure.

So if designing a traditional dance hall (ballroom, salsa, C&W), one would probably be well advised to avoid structural natural frequencies which induce high DLFs (i.e., the vicinity of 1.5-9 Hz).  In fact, a good rule of thumb is to design the lowest structural natural frequency to be greater than 10 Hz.  But for tango?  Dynamic amplification appears not to be much of a concern in any structure – so put down your calculations and join the fun!

Gomez and Tish, keeping DLF to a minimum
(Yes I know they danced a different type of tango!)

Dancing with the Stars?  Or just sharpening up your Macarena for your cousin’s wedding?
Don’t miss a step, be fully prepared by dynamically analyzing that dance hall using www.cloudcalc.com – Structural Analysis on the Cloud.

By Tom Van Laan

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