Seismic Ground Motion - Dynamic Behavior of Concrete and Seismic Engineering - page 109

3.4.1. Spectral signature of the seismic source
The term “seismic source” refers to wave emission caused by the creep of a
finite sized fault area; this creep is neither instantaneous nor homogenous on the
failure zone, and the emission process is so complex that it forms a research field in
its own right.
When simplifying to extremes, the source of an earthquake (Figure 3.3) can be
viewed as a failure starting at a focus F (given by the ([ K ) coordinate hypocenter)
which then propagates along the fault plan ([K) at a failure speed V R , causing at
each point a sliding 'u ([Kt). At the end of each earthquake, sliding at these points
reaches a final value 'U f ([K).The source is completely determined by the space-
time sliding function 'u and the waves emitted at point ([K) have a displacement
signal proportional to w/wt ['u ([Kt)]. At present, we do not know how to
anticipate future earthquakes; nevertheless, with good instrument conditions, we can
discover its characteristics for high wavelengths (several kilometers) and low
frequencies (generally lower than 1 Hz) a posteriori. We are also beginning to
reconstruct a stochastic description of the short wavelength and high frequency part.
However, without going into all the relevant details, it is now possible to explain
certain general aspects of the spectral content of emitted waves, thanks to a few
overall features of the seismic source. These overall features together with their
effects are described below (Figure 3.3):
- the 'U f ([K) quantity is reduced to the surface F of the failure zone (LW for a
rectangular fault, Sa² for a circular fault), and to the average sliding D 0 . This enables
the definition of a fundamental quantity indicating the importance of the earthquake,
namely the “seismic moment”, M 0 = P S D 0 , where P is the shear stiffness of the
Earth's crust at the level of the fault. This quantity controls the low frequency level
of the emitted waves, expressed in N.m;
- the stress drop 'Vexpresses the shear stress relaxation between the state
immediately prior to failure, V , and the state immediately following failure end, V .
Elastic dimensional analysis allows the following rough estimate to be obtained: 'V
= C P D 0/ L c where L c is the dimension characteristic of the fault (W for a rectangular
fault, a for a circular fault), and C is a shape coefficient close to 1;
- the energy released during an earthquake corresponds to the work of shear
stresses. It can be roughly approximated by the work of the average stress E = S.
[(V + V )/2]. D 0. Thus, we can write E = M 0 . [(V + V )/2P]. Because, deep into the
ground, the shear stiffness P and the average stress (V + V )/2 do not differ
significantly, we can see that the seismic moment is a good indicator of the total
energy released;