Seismic Analysis of Structures: Improvements Due to Probabilistic Concepts - Dynamic Behavior of Concrete and Seismic Engineering - page 303

f ' i f with ' f 1 / T

[8.54]

i

T being the duration of the signal. According to the Shannon theorem, the studied
[0, f max ] frequency range is conditioned by the choice of the time discretization slot
1/2f max = T/2N with f max = N 'f.

The problem involves determining the A i values and T from the ORS. For this
we can show that the PSD discretized (with the 'f slot) corresponding to the F(t)
process is given by:

2

Si f ' A T/2

[8.55]

i

Thus, we are brought back to the problem in the previous section: the
relationships expressed in [8.47] and [8.49] allow us to obtain S (f) and T from the
ORS; equation [8.55] allows us to obtain the A i values.

REMARK.- Regarding the time integration methods: the seismic analysis must
result in obtaining statistical characteristics concerning the absolute maxima
(averages and possibly standard deviation) of different physical values
(displacements, accelerations, stresses). Thus, time calculation represents the
realization of the corresponding random variables. In order to carry out their
statistics with reasonable accuracy, many calculations have to be done. Therefore the
time method is cumbersome and expensive, especially in the case of non-linear
behaviors. We lay great stress on the fact that the result of one only calculation,
however complex it may be, brings only little information. The potentially chaotic
behaviors of the non-linear systems we will deal with in the next section only
reinforce that assertion.

Let us notice however that if the calculation includes several quite isolated
maxima, it will be possible to take advantage of all these maxima, not only of the
most important of them, which will allow us to decrease the number of calculations
to carry out for any given accuracy.