Seismic motion model

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Seismic motion model





Seismic motion model

Seismic action arising in any specific site is fully characterized once the history of progression of accelerations, velocity and soil displacements are known.

Clearly it is not possible to arrive at such detail merely by knowledge of macroseismic parameters such as magnitude M  and focus R.

These two values apart from being of empirical nature (as opposed to the values desired that are wholly physical) do not distinguish the specifics of the various mechanisms that can generate the seismic event.

Moreover, the local event is profoundly influenced by the geologic and morphologic conditions of the crust that is penetrated by the waves, and by the stratigraphic and geotechnic properties of the site.

However at present, the two macroseismic parameters are the only ones of which it is possible to obtain a degree of information concretely useful for the analysis of seismic risc.

In order to arrive at a definition of the local seismic movement one is coerced into adopting simplified schemes, in which the macroseismic parameters are integrated with empiric information (Statistics on recordings of earlier quakes) or in their absence, by elements based on on appropriate to the data specific to the problem: Distance from the potential sources, local terrain characteristics , etc.

A simplified model of local seimic motion (e.g. of time/acceleration history) adapted however to numerous concrete situations is represented by the expression:


(1) a(t) = a×Cn cos (ωn× t-ϕ n)


in which :

-         a stands for the intensity parameter, more precisely the peak terrain acceleration that is a rather uncertain value, whose distribution can be derived from the also uncertain values of magnitude M and of the focus distance R;

-         The terms Cn are coefficients of the development of normalised Fourier summations  They describe the frequency content of the motion in as much as they furnish the relative importance of the various elementary components of frequency ωn. The diagram of the Cn coefficients as function of frequences ωn represents the FOURIER spectrum of the sesmic event under consideration.


The Cn coefficients are considered normalised such that the the sum of the second member of (1) give a maximum unit value so that the following:




in accordance with the definition of a;

-         terms ωn=n2π⎜D are pulses (in rad/s) of the various harmonic components, multiples of the minimum frequency: ω1=2π/D, where D is the duration of the vibration;

-         terms ϕn are the phase angles, one for each harmonic component, ranging from 0 and 2π.



Rise Time

Time to reach maximum acceleration


rise time