Seismic Ground Motion - Dynamic Behavior of Concrete and Seismic Engineering - page 104

3.3.2.3. Relationships between response and Fourier spectra

A qualitative correspondence exists between these two spectral representations,
which both express the frequency repartition of energy of the seismic signal.
However, no strict mathematical relation between the elastic response spectra and
the Fourier spectra exists. The only relationship that can be demonstrated is an
inequality, given by:

S(f,0) t F (f)

v

a

3.3.2.4. Generic spectral shapes

These spectra are characterized by simple shapes (Figure 3.1): once smoothed, a
Fourier spectrum A(f) is characterized by a plateau shape between two frequencies f c
and f max . The first (the corner frequency) is inversely proportional to the size of the
failure area, but it is also linked to the stress drop caused by the earthquake and to
the directivity process, that is, to the position of the receptor at the front or at the
back of the failure propagation direction. For destructive earthquakes, it is generally
less than 1 Hz, and can drop well below 0.1 Hz for very high magnitude
earthquakes. The second frequency is linked to inelastic attenuation phenomena
which affect the waves during their in-depth and surface propagation, and to
processes that govern failure dynamics on the crack. This frequency is generally
higher than 4-5 Hz, and can sometimes exceed 20 Hz. Below f c , the acceleration
Fourier spectrum modulus varies as f ² , which corresponds to a plateau for the
displacement Fourier spectrum, usually written : 0 , and is linked to the properties of
the source by the following relationship:

3

: M.R /(4ʌȡ R c )

0 0 șij

where M 0 is the seismic moment P.D 0 . S, Pisthe shear stiffness at focal depth, D 0 is
the average reactivation on the fault, S is the total failure area one the fault plane, R
is the focal depth, and c is the wave propagation speed (and thus generally that of
the S waves if we are considering horizontal component forces).