Wolfram Computation Meets Knowledge

Built to Last: Understanding Earthquake Engineering

Last week, the world was shocked by the news of massive earthquakes and devastating tsunamis in Japan. The event is still unfolding and could become one of the most tragic natural disasters in recent history.

Scientific understanding and modeling of complicated physical phenomena and engineering based on such analysis is imperative to prevent unnecessary loss of life from natural disasters. In this post, we’ll explore the science behind earthquakes to better understand why they happen and how we prepare for them.

Note: The dynamic examples in this post were built using Mathematica. Download the Computable Document Format (CDF) file provided to interact with the simulations and further explore the topics.

First, let’s start with locations. The following visualization is created from the U.S. Geological Survey (USGS) database of earthquakes that occurred between 1973 and early 2011 whose magnitudes were over 5. As you can clearly see, the epicenters are concentrated in narrow areas, usually on the boundaries of tectonic plates. In particular, there are severe seismic activities around the Pacific, namely the “Ring of Fire”. Unfortunately, Japan is sitting right in the middle of this highly active area.

Earthquake map

Earthquakes are oftentimes caused at the boundaries of tectonic plates, which form huge faults on the surface. When large enough forces are applied in two different directions, they overcome the friction between the boundary and cause a sudden movement. The phenomenon, also known as strike-slip, is one of many mechanisms that causes earthquakes. The following animation simulates the strike-slip at a fault and seismic waves caused by it.

The scale of an earthquake is measured by the amount of released energy during the seismic activity. The magnitude scale is logarithmic. The following chart shows the relation between the scale and the energy released by the event, with a few recent major earthquakes indicated.

Mouse over the image to see the logarithmic graph.

PJ, or petajoule, may not be the most familiar measurement unit of energy for many people. We can use Wolfram|Alpha to convert it to more popular measures, such as megatons of TNT explosive. The Sendai earthquake was measured at magnitude 9, and the energy released was approximately 2000 petajoules:

2000 PJ to megatons

478 megatons of TNT

To put it into some context, the largest nuclear explosive ever tested (Tsar Bomba) was roughly 58 megatons. We are talking about an order of magnitude more energy than that of the largest nuclear bomb.

The magnitude scale is a base-10 logarithmic scale. For instance, the difference in energy released between the current Sendai earthquake (magnitude 9) and the Haiti earthquake in 2010 (magnitude 7) is:

Energy difference

Energy difference

Energy difference

Thus, the energy released during the Sendai earthquake is 1000 times more than that of the Haiti.

So, how is all the energy from earthquakes transferred? Most of the energy is converted into heat generated by friction, but some of it is transformed and radiated as seismic waves. There are two types of seismic waves: body waves and surface waves.

Body waves travel through the earth and can be differentiated by the direction of oscillation during propagation. In P-waves, the particles vibrate in the same direction as wave propagation, similar to sound waves. On the contrary, particles move perpendicular to the direction of the propagation in S-waves.

Surface waves travel only through the crust or the surface of the Earth. However, they are responsible for much destruction due to their properties. Again, the surface waves can be divided into two: Love and Rayleigh waves.

Love waves create motions similar to S-waves, but horizontally. This horizontal movement is particularly damaging to the foundations of buildings.

Rayleigh waves traverse the way that waves roll across lakes or oceans. During propagation, particles move in elliptical paths. Sometimes called ground rolls, Rayleigh waves are low frequency (less than 20 Hz) and oftentimes are detected by animals, but not humans.

Understanding seismic waves is essential for minimizing or preventing structural damages to buildings during earthquakes. So the question is, what you can do?

If external forces are applied, buildings will sway. The structural dynamics are quite complex and depend on a lot of parameters. But as an example, I used a simple damped harmonic oscillator equation to simulate the sway of a building in the following animation (the gray cylinders represent the foundations of the building):

The sway helps to dissipate energy from the seismic waves. Through structural design, material selection, and construction techniques, engineers have tried to reduce the effect of earthquakes on buildings. One such example is a tuned mass damper, a device installed in buildings to reduce harmonic resonance.

Tuned mass damper

Tuned mass damper on display in Taipei 101. Photo courtesy of Armand du Plessis.

Another option that is widely considered today is a technique called base isolation, or seismic isolation. In nutshell, you put some devices, such as a lead rubber bearing, at the base of a building to isolate the building structure from the vibration from earthquakes.

This technique can help reduce structural stress significantly. It is also considered to be a good option for short buildings with high stiffness, or for retrofitting existing structures. In fact, many historical monuments in the U.S. have already been retrofitted with base isolation systems to reduce the damage from earthquakes.

We are providing the downloadable CDF for the post. Most examples are dynamic, and you can interact with simulations. Feel free to use Mathematica or Wolfram CDF Player to explore the content.

Download the CDF file


Join the discussion

!Please enter your comment (at least 5 characters).

!Please enter your name.

!Please enter a valid email address.


  1. What a fantastic blog post! I wonder given W|A’s wind direction inputs, can one model the radioactive cloud movement in real time?

  2. I don’t know Samuel. Just simulation can be done I guess (we have done some small scale pollutant simulation before), but I am not sure of the quality of data in particularly the source of radioactive materials…. It would be just falling under rough modeling or speculation, not really prediction. I will take a look. Thanks!

  3. Cool post. The ability of systems to absorb energy may be the least understand principles of structure. There is a YouTube video showing the effect of the 2008 Sichuan, China earthquake on the Taipei 101 tower (approximately 1150 miles away): http://www.youtube.com/watch?v=NYSgd1XSZXc . The 250-ton structure is so massive but moved fluidly during the earthquake.

    An episode of “Build It Bigger” showed construction techniques for the bridge that’s replacing the Oakland Bay Bridge in the SF area. The single tower on the new segment contains linkages which are designed to absorb the energy during an earthquake. The wikipedia article ( http://en.wikipedia.org/wiki/Eastern_span_replacement_of_the_San_Francisco_%E2%80%93_Oakland_Bay_Bridge ) calls them sacrificial box structures (or mechanical fuses). If these “fuses” are expended during an earthquake, they are designed to be rapidly replaced in the days after the incident.

  4. The Technocrats of Earth quakes have totally failed for the prediction of Earth Quakes. The reason behind is we are using the techniques of mathematics of 19th century i.s. y=f(x) which does not provide correct definition &structure of function pointed out by Logicians since 1924 to onward. Also, Applied Physicists have pointed the same deficiencies as pointed out above(see,http://www.opmisra.com). On the otherhand,not a single country has given attention on such disasters and some of countries like Burma might has financial problem due its own problems. Only persons take interest when loss of life occured otherwise they never want to solve such required problem. If you want to know more about prediction of Earth Quakes I can be able to discuss and one can send mail to me. Thanks, O.P.Misra

  5. Part of the problem is the requirement to solve some highly nonlinear stress-tensor-based partial differential equations that no one really knows how to solve, nor can the errors be constrained as we do not have a good grasp of the degrees of freedom. Without the ability to constrain the volume of the phase space, we are left with a ver chaotic dynamical system that almost forbids predictability.

  6. Tuned mass dampers are not a good idea for resisting seismic forces – they rely on the mass & damper being tuned closely to the frequency of the building. However the design of most buildings relies on them taking damage in a large event, and this will shift the frequency of oscillation. This would be bad news with a tuned mass damper as it would now be out of tune and still very heavy. I believe that the Taipei 101 mass is locked down in the event of an earthquake.

  7. Correction… there are two types of TMDs in Taipei 101. The pinnacle tuned mass dampers are indeed locked down, but, the main TMD is restrained by a snubber ring if its motion exceeds 1m.

    To quote the designers: “Approaching events with mean recurrence
    intervals of 1000 to 2500 years, the design challenge was to keep the TMDs from damaging the structure, and to
    remain in place and intact after severe event had passed and the vibration of the structure ended.”

    “In strong seismic events, e.g. those with return periods up to 100 years, the designed response characteristics of the
    building TMD are quite tame. The steel ring and lower set of 8 dampers, shown in Figure 1 on the floor at elevation
    374m, is a secondary system (named a “snubber ring”) designed to engage the TMD only at relative amplitudes
    which exceed 1m.”

    “It was at last decided to passively “lock out” the Pinnacle TMDs with
    robust secondary mechanisms. In this manner, the TMDs will travel as inert mass locked to the Pinnacle structural
    system, and avoid any damage that would occur from strong internal collisions – in effect, they will “ride out” the

  8. The 8.8 magnitude earthquake from Chile occurred on 2010, not 2008.

  9. That was an oversight. I will fix the post and the notebook. Thank you.

  10. Sorry professor, very interesting article, but I have two notes that I would do:
    – As Ignacio said, the earthquake in Chile was in 2010
    – Why not consider in their measurements of the earthquake in Valdivia (Chile) in 1969?

  11. There’s a mistake on the propagation of the Rayleigh waves, the propagation is similar to the sea waves, as stated, but the particle motion is the opposite

  12. In my view Scientists want to make use of their investigation which are based on 17th century Mathematics and that is y=f(x) which have totally failed in the forecasting of Natural Disasters. The 20th century mathematics is known as Functional Analysis which is not Abstract mathematics due to Theory of Distributions by French Professor Laurent Schwartz(1950-51) together with more theory of Value & Limit about A Point as An Operation on Schwartz Theory(1971-2002) by O.P.Misra having discussions and suggestions with Professors Laurent Schwartz,H.G.Garnir,J.L.Lavoine(all French) and one Logician Professor Henk Barendregt(Dutch & Buddhist Philospher).. However, only Logician Professor Henk is present. In fact this work replaces the present system of science & technology having old base and thereby Scientists do not want to make use of new technology. Even, seismology is a young science and its scientists can not make use of such modern technology as stated above. Thereby, from one point of View Italian Govt has started a case of their Scientists not to able predict Earth Quake in 2004 recently. Accordingly, such cases should be followed in Human Right Courts sothat the old technology can be replaced by new one. I do not see another way of such Predictions. The whole ststed work is based on Logic(since1924) and experimental Physics(since1962). I can be able to reply any of the questions provided any one asks. Thanks, Misra

  13. Regarding Prediction of Earth Quake,new 20th century theory of Laurent Schwartz together Value About A point(theory) pioneered by Professor Misra(2002) can be able to provide the results as desired. However, An American Logician Alonoz Church(1930,1941) about deficiencies in y=f(x) have been kepping in mind to remove by Schwartz Distribution & Its Value About A Point. Accordingly, such Natural Disasters like Earth Quakes etc are predicable.

  14. Regarding the results of An American Logician A. Church(1941) he has pointed out Correct Definition Of Function has been proved fully by me in (2002) in my book as Distributional Lambda Calculus in Chapter 7. Thus before that period either in Chilli or elsewhere there was no valid reasons and proof for Prediction of such said Disasters. Even I am not getting support for Prediction of such Disasters. Thanking you to asking important question.

  15. Whoah this blog is fantastic i love reading your articles.
    Keep up the great paintings! You know, a lot of persons are hunting around for this info, you could aid them

  16. One of the methods of contrast with earthquakes is seismic retrofitting of buildings.
    Therefore, old and unsafe buildings should be strengthened by methods such as strengthening with FRP in accordance with ACI 4402R or retrofitted by classical methods such as FEMA 360.