Calculating Period and WEIRD Buildings.

Outside of California, few places in the country are known for having earthquakes. But because the possibility for one exists nearly everywhere, earthquake loads nearly always need to be considered in the design of a structure. During an earthquake, a building will vibrate back and forth, much like a weight attached to a spring. How fast it vibrates dictates how much force the building experiences – this quantity is known as a building’s period, and determining is an important part of calculating the seismic load on a building.

Harmonic motion. My job would be a lot more fun if we were hanging buildings from springs.

ASCE 7 allows the use of several possible formulas for calculating period, depending on the exact sort of building being designed. These equations are very simple, and don’t seem to resemble the equations of simple harmonic motion, the concept that seismic load calculations are based on. So where do they come from?

Since 1993, ASCE’s earthquake provisions (commentary and all) have been based on provisions from the National Earthquake Hazard Reduction Program (NEHRP). The NEHRP Provisions, in turn, are based on published research and available data on seismic loads in buildings.

And this is where it starts to get tricky.

To collect useful data on building period, the building must have a permanent accelerograph installed. And the best data comes from buildings shaken strongly enough to undergo harmonic motion, but not so strongly that structural components begin to fail and deform inelastically. Despite how much work does into designing for them, earthquakes that size rarely occur. In America, there have only been a handful over the past 40 years, the largest being the 1971 San Fernando earthquake and the 1994 Northridge earthquake. The data we have to work with is thus limited to a very small population of buildings.

Specifically, it’s limited to 106 buildings. That’s the size of the database used for the development of the period prediction equations*. Not only is this an uncomfortably small number of buildings, but it’s a small number of a certain type of building – specifically, the type of building that was built in urban California in the mid 20th century to survive large earthquakes. And as anyone who’s glanced at at the requirements for Seismic Design Category ‘D’ knows, a building that can survive large earthquakes is a much different animal from one that can’t.

There’s a famous paper in psychology titled “The Weirdest People in the World”. It points out that most psychology research gets performed in American universities, and the subjects are generally students. Because of this, there’s the risk that conclusions from these papers which meant to apply to all types of people, in fact only apply to the sorts of people that are college undergraduates. Specifically, people that are Western, Educated, Industrialized, Rich, and Democratic – WEIRD.

There’s a similar situation going on with building period calculations. Equations that are meant to generalize to all sorts of buildings, may in fact only apply to the types of buildings that populate the database.

To it’s credit, the NEHRP recognizes this:

It is generally accepted that the empirical equations for Ta are tailored to fit the type of construction common in areas with high lateral force requirements. – NEHRP 2003 commentary, pg. 69

Fortunately, building period doesn’t need to be predicted exactly. As building period goes down, lateral force goes up – thus, as long as the equations reliably underestimate the period of a building , the buildings will still be safe, merely slightly overdesigned. The period equations take this into account, and are calibrated to deliberately underestimate period by 10-20%. And all else being equal, a building designed for high lateral forces such as earthquakes will be much stiffer (and thus have a lower period) than one with lower lateral force requirements. Thus “normal” buildings should have higher periods than earthquake-resistant ones.

Close enough for government work.
Close enough for government work.

However, to properly follow the spirit of the code, the period equations should be used only for preliminary design – as soon as possible, period should accurately calculated be calculated using computational methods. Unless you live in California or design skyscrapers for a living, you probably don’t do this, and instead just crank through the equations the code gives you. Still, it’s worth remembering that the dataset that produced these equations is somewhat sparse, and that they’re not designed to be correct – they’re designed to NOT be dangerously incorrect.

*Data from the 1971 San Fernando Earthquake was used in the development of the equation Ta = Ch^x as part of ATC3-06, which is still part of the code today as the most “general” method. This data was combined with data from later earthquakes to develop the equations for specific framing systems and included in NEHRP-2003. The Ta = 0.1N equation has drifted in and out of the code over years, originating (I believe) in the SEAOC Blue Book. It disappears for a while, then shows up again in NEHRP-94. It’s currently authorized for use on concrete or steel moment frames.

Sources:

Heinrich, J. et al (2010) The weirdest people in the world?

ASCE Standard 7-10: Minimum Design Loads for Buildings and Other Structures

NEHRP Recommended Provisions and Commentary for Seismic Regulations for New Buildings and Other Structures. 2003 Edition

NEHRP Recommended Seismic Provisions for New Buildings and Other Structures. 2009 Edition

Goel, R. and Chopra, A. (1997) Vibration Properties of Buildings Determined from Recorded Earthquake Motions

Ghosh, S.K. (2004) Update on the NEHRP Provisions: The Resource Document for Seismic Design

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