Seismic Analysis of Structures: Improvements Due to Probabilistic Concepts - Dynamic Behavior of Concrete and Seismic Engineering - page 284

¦¦

S r, f H (f) H f ) r ) r S (f)

[8.17]

x n m n m

nm

For a wide frequency band source signal (quasi-random noise), it is possible to
integrate [8.17] into the frequency domain. For example, if we take S (f) = 1, we
obtain the response variance:

¦¦

2

[8.18]

ı r ı ) r ) (r)

X

2

n m

nm

nm

with V nm 2 = 1/[2Z n Z m (H n Z n + H m Z m ) (1+O nm 2 )] and O nm = «Z n Z m «/ (H n Z n + H m Z m ).

In the case of quite distinct modes (O nm !! 1 for n different from m), the non-
diagonal terms in the double summation can be neglected:

¦

ı r ı ) r

[8.19]

2 2 2

X n n

n

with V n 2 = 1 / (4 H n Z n 3 ).

Equations [8.18] and [8.19] can be interpreted in the following way.

Inside the natural mode basis of the structure, modal contributions are the motion
variables. Their evolution over time results from the random stimulation (here the
unit random noise). Thus, they are A n (t) random processes and we obtain:

¦

Xr, t A t ) r

[8.20]

n n

n

The V nm 2 values represent the correlation coefficients of the A n (t) values with one
another (for the same value of t), whilst V n 2 values represent their variances.
Equation [8.19] shows that if the modes are very distinct, the A n (t) will be
statistically “decorrelated”.