Seismic Ground Motion - Dynamic Behavior of Concrete and Seismic Engineering - page 110

- the failure speed typically ranges from 2 to 3.5 km/s, but it can sometimes
exceed the speed of the waves S. In such cases, the failure is termed “supersonic” or
“super shear”. Together with the dimensions of the failure area, this speed controls
the wave emission time and the length of near-field strong motions. Moreover, it
plays a part in directivity phenomena which are similar to the well known Doppler
effect: a receiver located at the front of the failure propagation direction will receive
waves emitted for a shorter time the nearer the failure speed is to wave propagation
speed (S), which involves more intense motions. The opposite is true for a site
located at the rear of the failure front;
- the final “global” parameter used is the “rising time” (W). This is the time
necessary for sliding at a specific point to pass from 0 (pre-failure) to its final value
'U f . . In reality, W varies from one point to another on the failure area. Nevertheless,
for simple models, we assume the sliding function is identical on all points, i.e. that
'u([Kt) = U f ([K S(t - d/V R ), S(t) being a function the derivative which has a
bounded support [0, W], and d being the distance between the current point ([K and
the hypocenter ([ K The simplest function is a linear ramp function (see Figure
3.3).
Figure 3.3. Schematic representation of the failure process. The diagram on the left
represents the real history of the sliding field on the fault D (x,y;t) (isochrones of the failure
front at successive instants), whereas the diagram on the right illustrates its simplified
representation as a (L,W) dimensioned rectangular fault, in which the dislocation function
D(t) follows the same time history on each point (ramp function characterized by its rising
time W and its uniform final dislocation D 0 )
The analysis of these parameters for numerous earthquakes demonstrates they do
not vary from one another, and therefore that the following three “scaling laws”
exist:
- a “geometric” law; W/L | c 1 , with c 1 | 0.5 (except for important strike-slip
earthquakes, where width W is limited by the thickness of the brittle part of the
Earth's crust, i.e. 15-20 km);