# Explanation of Parameters

What is PGA? | What is %g? | What is PE? | What is SA?

## What is "Peak Acceleration" or "Peak Ground Acceleration" (PGA)?

### 1. What is "acceleration"?

When you push on the gas pedal in your car, you experience the increase in velocity as a force pushing you back into your seat. Technically, then, acceleration is the rate of increase in velocity, that is, how much the velocity changes in a unit time. Personally, we are most aware of acceleration by the experience of an applied force.

So, consider a car increasing in speed from a stop to 60 miles an hour. 60 miles per hour is 88 feet per second. If the acceleration is uniform (constant) while the car increases speed, we could say that if the car reaches a velocity of 88 feet per second in 8 seconds, the velocity changes by 11 feet per second every second, and the acceleration is 11 feet per second per second. If the acceleration were not uniform, but started off small, achieved a maximum, and then decreased as we approached 60 miles an hour, the largest value of the acceleration would be the "peak" acceleration.

### 2. What do we mean by "peak" acceleration as a measure of earthquake ground motion?

A small particle attached to the earth during an earthquake will be moved back and forth rather irregularly. This movement can be described by its changing position as a function of time, or by its changing velocity as a function of time, or by its changing acceleration as a function of time.

Since any one of these descriptions can be obtained from any other, we may choose whichever is most convenient. Acceleration is chosen, because the building codes prescribe how much horizontal force building should be able to withstand during an earthquake. This force is related to the ground acceleration. The peak acceleration is the maximum acceleration experienced by the particle during the course of the earthquake motion.

## What is "% g"?

When acceleration acts on a physical body, the body experiences the acceleration as a force. The force we are most experienced with is the force of gravity, which caused us to have weight. The units of acceleration of the map are measured in terms of g, the acceleration due to gravity. An acceleration of 11 feet per second per second is 11 * 12 * 2.54 = 335 cm/sec/sec. The acceleration due to gravity is 980 cm/sec/sec, so an acceleration of 11 feet/sec/sec is about 335 / 980 = 0.34 g. Expressed as a percent, 0.34 g is 34 % g.

### What is the relation to building damage?

Pre-1940 dwellings are likely to perform poorly in earthquake shaking. Pre-1975 dwellings are likely to have some vulnerabilities to earthquake shaking. Some post-1985 dwellings, built to California earthquake standards, have experienced severe shaking (60% g) with only chimney damage and damage to contents. Simple retrofit measures can greatly reduce the vulnerability of dwellings:

1. Bolt house to foundation
2. Provide resistant paneling or bracing to cripple walls
3. Reinforce brick chimneys or replace with patent metal chimneys

NOTE: The 10 % g value was chosen because on the average it corresponds to Modified Mercalli Intensities VI to VII, levels of threshold damage, in California, for ground motions within 25 km of the earthquake epicenter. This value should not be used in the case of particular buildings, because:

1. the relation between intensity and peak acceleration is quite variable,
2. for more distant sites, longer duration ground motions may cause damage at lower acceleration values, and
3. buildings differ greatly in their vulnerability.

Furthermore, the hazard maps combine near and distant ground motions indiscriminately.

## What is "probability of exceedance" or PE?

For any given site on the map, the computer calculates the ground motion effect (peak acceleration) at the site for all the earthquake locations and magnitudes believed possible in the vicinity of the site. Each of these magnitude-location pairs is believed to happen at some average probability per year. Small ground motions are relatively likely, large ground motions are very unlikely.

Beginning with the largest ground motions and proceeding to smaller, we add up probabilities until we arrive at a total probability corresponding to a given probability, P, in a particular period of time, T. The probability P comes from ground motions larger than the ground motion at which we stopped adding. The corresponding ground motion (peak acceleration) is said to have a P probability of exceedance (PE) in T years.

The map contours the ground motions corresponding to this probability at all the sites in a grid covering the U.S. Thus the maps are not actually probability maps, but rather ground motion hazard maps at a given level of probability.

In the future we are likely to post maps which are probability maps. They will show the probability of exceedance for some constant ground motion. For instance, one such map may show the probability of a ground motion exceeding 0.20 g in 50 years.

## What is "spectral acceleration" or SA?

PGA (peak acceleration) is what is experienced by a particle on the ground. SA is approximately what is experienced by a building, as modeled by a particle mass on a massless vertical rod having the same natural period of vibration as the building.

The mass on the rod behaves about like a simple harmonic oscillator (SHO). If one "drives" the mass-rod system at its base, using the seismic record, and assuming a certain damping to the mass-rod system, one will get a record of the particle motion which basically "feels" only the components of ground motion with periods near the natural period of this SHO. If we look at this particle seismic record we can identify the maximum displacement. If we take the derivative (rate of change) of the displacement record with respect to time we can get the velocity record. The maximum velocity can likewise be determined. Similarly for response acceleration (rate of change of velocity) also called response spectral acceleration, or simply spectral acceleration, SA (or Sa).

PGA is a good index to hazard for short buildings, up to about 7 stories. To be a good index, means that if you plot some measure of demand placed on a building, like interstory displacement or base shear, against PGA, for a number of different buildings for a number of different earthquakes, you will get a strong correlation.

PGA is a natural simple design parameter since it can be related to a force and for simple design one can design a building to resist a certain horizontal force.

PGV, peak ground velocity, is a good index to hazard to taller buildings. However, it is not clear how to relate velocity to force in order to design a taller building.

SA would also be a good index to hazard to buildings, but ought to be more closely related to the building behavior than peak ground motion parameters. I presume design might also be easier, but the relation to design force is likely to be more complicated than with PGA, because the value of the period comes into the picture.

PGA, PGV, or SA are only approximately related to building demand/design because the building is not a simple oscillator, but has overtones of vibration, each of which imparts maximum demand to different parts of the structure, each part of which may have its own weaknesses. Duration also plays a role in damage, and some argue that duration-related damage is not well-represented by response parameters.

On the other hand, some authors have shown that non-linear response of a certain structure is only weakly dependent on the magnitude and distance of the causative earthquake, so that non-linear response is related to linear response (SA) by a simple scalar (multiplying factor). This is not so for peak ground parameters, and this fact argues that SA ought to be significantly better as an index to demand/design than peak ground motion parameters.

There is no particular significance to the relative size of PGA, SA (0.2), and SA (1.0). On the average, these roughly correlate, with a factor that depends on period.

While PGA may reflect what a person might feel standing on the ground in an earthquake, I don't believe it is correct to state that SA reflects what one might "feel" if one is in a building. In taller buildings, short period ground motions are felt only weakly, and long-period motions tend not to be felt as forces, but rather disorientation and dizziness.