
<Image
	src="/images/courses/multi-degree-of-freedom-dynamics-modal-analysis-and-seismic-response-simulation-in-python/overview-of-earthquake-ground-motion/img1.jpg"
	alt="Damage from the Port au Prince, Haiti earthquake in 2010. | EngineeringSkills.com"
	width="800"
	height="533"
/>
<Caption>
	Fig 1. Damage from the Port au Prince, Haiti earthquake in
	2010.[Wikipedia](https://commons.wikimedia.org/wiki/File:2010_Haiti_earthquake_damage4.jpg)
</Caption>

As we've already mentioned, our main objective in this course is to model dynamic behaviour due to ground motion. We will not focus on ground motion itself to any great extent. We will simply treat this as an input to our dynamic systems. All of the upstream mechanics, plate tectonics and seismology, are out of scope for us. However, for a more in-depth discussion of seismology, refer to **Dynamics of Structures by Clough and Penzien**. They present a practical and concise introduction.

Before diving straight into the mechanics, we ought to lay down at least some context for our analysis. In this lecture, for completeness, we will briefly review some of the key concepts relating to earthquake ground motion.

Ground motion experienced by a structure results from fractures or relative movement between tectonic plates in the earth's crust. In an earthquake event, the location in the earth's crust of the first rupture is called the **focus** of the earthquake. The **epicentre** is the point on the earth's surface immediately above the focus.

When a rupture occurs in the crust, energy is released. This results in waves that propagate through the ground. We observe these on the earth's surface as erratic ground motion. The character of the ground motion experienced at the foundation of a structure is influenced by many things, including local geology, soil conditions and distance from the earthquake epicentre.

Depending on the ground motion's magnitude, duration, and frequency content, structures may be excited and begin to oscillate. Ground motion is typically 3-dimensional in nature; however, lateral oscillations can be particularly problematic for tall structures.

From the engineer's perspective, this ground motion, typically characterised as acceleration, can be represented as an inertial force applied to the structure. This results in additional stresses developing beyond the static stresses routinely considered in design.

Our first task in designing an earthquake-resistant structure, i.e. a structure that can withstand the additional dynamic stresses developed, is understanding how it will respond to ground motion. This is what we will spend our time focusing on in this course.

### The Richter Scale

We've all heard of the Richter scale. When an earthquake occurs, the Richter scale magnitude is quoted in the news media as a measure of the severity of the earthquake. Let's now take some time to pin down what precisely the Richter scale is and what the magnitude signifies.

The Richter magnitude is based on the amplitude of the motion induced by the rupture. It is defined as,

$$
M = \log_{10} A
\tag{1}
$$

where $A$ is the maximum amplitude of the ground motion in micrometres, $100\:km$ from the epicentre, measured on a **Woods-Anderson seismograph**. Obviously, there is unlikely to be a seismograph at this specific location, so the amplitude is usually triangulated from measurements of seismographs in the region.

We can estimate the energy released as a function of Richter magnitude using the empirical formula,

$$
E = 10^{4.8+1.5M}
\tag{2}
$$

where energy $E$ is measured in Joules. Much of our knowledge of earthquake mechanics is based on empirical observation over many decades. One such observation is that earthquakes below magnitude 5 typically do not lead to structural damage. However, as the magnitude increases, we move into the realm of structural damage and widespread disruption and loss of life.

We can see the influence of increasing magnitude on energy release by noting the $\log$ relationship. The energy released, which is directly related to the resulting ground motion, **increases by a factor of 32** for every unit increase in Richter magnitude. We can visualise this with a simple plot below. Note the log scale on the y-axis.

<Image
	src="/images/courses/multi-degree-of-freedom-dynamics-modal-analysis-and-seismic-response-simulation-in-python/overview-of-earthquake-ground-motion/img2.jpg"
	alt="Plot of energy release versus Richter magnitude. | EngineeringSkills.com"
	width="925"
	height="430"
	caption="Fig 2. Plot of energy release versus Richter magnitude."
/>

The **United States Geological Survey (USGS)** maintains a list of the [largest recorded earthquakes](https://www.usgs.gov/natural-hazards/earthquake-hazards/science/20-largest-earthquakes-world?qt-science_center_objects=0#qt-science_center_objects) and lists the Valdivia Earthquake (Chile, 1960) as the largest with a magnitude of 9.5. To place this in a familiar context, the third and fourth entries on the list are the 2004 Indian Ocean earthquake and the 2011 Tohoku earthquake off the coast of Japan. Both had a magnitude of 9.1 and caused massive tsunamis that led to widespread devastation and loss of life.

### Modified Mercalli scale

The Richter magnitude gives us a helpful shorthand for grasping the relative power of earthquakes. This is why it is so widely reported in the media. However, Richter magnitude alone is not a sufficient indicator of intensity. Remember, the Richter magnitude is related to the energy released at source. As we've already mentioned, this energy is dissipated through the ground and propagates radially outwards in wave form, generally weakening as it travels.

To quantify the severity of an earthquake at any specific point, we introduce another lesser-known scale - the **Modified Mercalli scale**. This is an intensity scale ranging from one (ground motion not felt) to twelve (total damage and destruction) based on the level of disruption/destruction observed.

<Image
	src="/images/courses/multi-degree-of-freedom-dynamics-modal-analysis-and-seismic-response-simulation-in-python/overview-of-earthquake-ground-motion/img3.jpg"
	alt="Modified Mercalli scale, reproduced from USGS. | EngineeringSkills.com"
	width="1000"
	height="542"
/>
<Caption>
	Fig 3. Modified Mercalli scale, reproduced from
	[USGS](https://www.usgs.gov/media/images/modified-mercalli-intensity-scale).
</Caption>

The scale above has been abbreviated. For completeness, intensity points 11 and 12 are defined as:

- 11 Few, if any (masonry) structures remain standing. Bridges destroyed. Rails bent greatly.
- 12 Damage total. Lines of sight and level are distorted. Objects thrown into the air.

Although damage assessment is somewhat subjective, this scale is a helpful tool in understanding the impact of an earthquake. This is particularly the case when the intensity is represented graphically on an **Isoseismal map** indicating the regions over which different levels of intensity have been observed.

<Image
	src="/images/courses/multi-degree-of-freedom-dynamics-modal-analysis-and-seismic-response-simulation-in-python/overview-of-earthquake-ground-motion/img4.jpg"
	alt="Isoseismal map of Tohoku Earthquake of the coast of Japan (03-11-2011). | EngineeringSkills.com"
	width="800"
	height="763"
/>
<Caption>
	Fig 4. Isoseismal map of Tohoku Earthquake of the coast of Japan (03-11-2011).
	([USGS](https://earthquake.usgs.gov/earthquakes/eventpage/official20110311054624120_30/map))
</Caption>

### Ground motion measurements

As we mentioned above, earthquake ground motions are recorded using seismographs. These are essentially just accelerometers that record acceleration in three orthogonal axes. These axes are typically a north-south axis, east-west axis and vertical axis. The records produced are referred to as accelerograms and are simply time-histories of the acceleration in each axis.

In our analyses, we will confine ourselves to studying lateral motion of planar structures. This means we will ignore the influence of out of plane and vertical motion. However, if required, superposition of dynamic response in multiple axes can be used, provided linear structural behaviour is maintained.

Throughout this course, we will use both synthetic ground motions in the form of simple sinusoids and publicly available recorded ground motions from historical earthquakes.

One famous ground motion used extensively in the study of earthquake engineering is the North-South component of the **El Centro earthquake** that occurred in California in 1940. This record is as synonymous with the study of earthquake engineering as the single degree of freedom system is to fundamental dynamics. It is readily available online and is reproduced below from [this source](http://www.vibrationdata.com/elcentro.htm). You can download the .csv file directly from the lecture resources above. We'll write some code to import this record into our Jupyter Notebook a little later.

<Image
	src="/images/courses/multi-degree-of-freedom-dynamics-modal-analysis-and-seismic-response-simulation-in-python/overview-of-earthquake-ground-motion/img5.jpg"
	alt="Plot of the North-South component of the El-Centro ground acceleration record. | EngineeringSkills.com"
	width="800"
	height="389"
/>
<Caption>
	Fig 5. Plot of the North-South component of the El-Centro ground acceleration
	record.
	([USGS](https://earthquake.usgs.gov/earthquakes/eventpage/official20110311054624120_30/map))
</Caption>

We could perform numerical integration on this time history to obtain the corresponding velocity and displacement. However, for dynamic analysis, it will usually be convenient to work directly with ground acceleration records.


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