Seismic Analysis of Structures: Improvements Due to Probabilistic Concepts - Dynamic Behavior of Concrete and Seismic Engineering - page 283

The imposed acceleration will be considered as a zero-average stationary time
random process *(t), characterized by its PSD S(f). The structure is characterized by
its by-pulse response G (r, W) (response at point r and instant W to an anchoring
acceleration corresponding to a unit pulse exerted at instant W = 0).

Let us consider the resting structure at W = 0 and let us exert the *(t) process from
that moment. The relative displacement of the structure at point r and instant W can
be considered as a stationary random process of time t: X (r, W, t), the PSD of which
is expressed by:

2

x S , IJ, f

H r,IJ,f S f [8.13]

H (r, W, f) has the meaning of a transfer function and is given by:

³

H r,IJ,f G r, u exp 2iʌfu du

[8.14]

0

If Wof, the transfer function tends to the conventional expression of the
Fourier transform of the by-pulse response, written h(r, f).

Formula [8.13] then gives:

2

S r, f H r, f S (f) [8.15]

x

which represents the PSD of the structure response set in r.

In [8.15] we can express the H function owing to the natural mode basis of the
structure:

¦

[8.16]

H (r, f) = H f ) r

n n

n

2 2

with

Hf q/m/Ȧ 2iİȦȦ + ȦȦ = 2ʌf

.

n n n n n n

Hence: