Seismic Ground Motion - Dynamic Behavior of Concrete and Seismic Engineering - page 100

characterizations are used by two distinct communities: the “producers” and “users”.
Seismologists prefer using Fourier spectra, due to mathematical practicability and
the possibility of establishing useful relationships with wave emission and
propagation physics. On the other hand, “users” like civil engineers (structure,
geotechnology) have become accustomed to reasoning in terms of response spectra,
as they are easily adapted to the simplified modeling of civil engineering structures.

3.3.2.1. Response spectra

The origin and interest in response spectra lie in the fact that they reduce (as a
rough approximation) the seismic behavior of a building to a simple oscillator with 1
degree of freedom. Representation as a response spectrum directly accesses the
motions of a structure's center of gravity.

Let us consider a 1-dof linear viscoelastic oscillator, characterized by its
frequency f and damping ]. Due to an earthquake characterized by an acceleration
a(t), the oscillator will undergo a relative displacement x(t), and an absolute
acceleration x"(t) + a(t). Response spectra are defined as the time maximum
responses of the oscillator for a relative displacement, relative speed and absolute
acceleration:

Sd (f, [ ) = Max {x(t)}

t

S (f, [ ) = Max {x'(t)}

v t

S (f,

[ ) = Max t.{x"(t) DD + a(t)}

On varying the frequency (f) of the oscillator with a constant damping (]), we
obtain three curves, Sd, S v , S a defining respectively the response spectra for
displacement, speed and acceleration. These are usually calculated for damping
discrete values: 0%, 2%, 5%, 10%, and 20%, with 5% as the most frequent value.

With the following expression for the basic equation of a mass (m), stiffness (k),
and damping (c) of a simple oscillator with 1 degree of freedom:

m [ a(t) DD + x"(t) ] = [ c x '

Ȧ k/m ;ȟ c/ km

1

0 2