Seismic Analysis of Structures: Improvements Due to Probabilistic Concepts - Dynamic Behavior of Concrete and Seismic Engineering - page 278

where H n and q n are respectively the modal damping and the seismic participation
factor (the projection of the MU force of inertia on mode q n = () n , M U)).

The ORS data allows the maxima reached by the absolute values of modal
contributions to be obtained without calculation from:

ª º

2

a ¬ q / m Ȧ Sf,İ [8.3]

n max n n n pa n n

¼

where f n = Z n /2S = resonance frequency.

Obtaining the variables useful for strength diagnostics (relative displacements
and stresses in certain points of the structure) requires an assumption.

Actually, if we are interested in the relative displacement at a specific point r, for
instance, the latter will be a linear combination of the modal contributions:

¦

xr, t a t ) r [8.4]

n n

n

To obtain the absolute value of x, the so-called simple quadratic combination
rule is applied:

¦

2 2

X(r) a ) r [8.5]

max n

2

nmax

n

This rules implies, as will be shown later, the statistical independence of the
different terms of the sum. In particular, the previous formula cannot be applied with
modes whose frequencies are beyond the seismic range.

The summation is then restricted to the first N modes (the higher limit of the
seismic range) and the contribution of the higher modes is represented by a
“pseudo-mode”, determined by virtue of an associated static solution:

2

N

-

N

½

° °

¦ ¦

ª º

x r a ) r Ȗ X r q / m Ȧ ) r [8.6]

® ¾ smax s n n n

2 2 2

¬

2

¼

max n max n n

° °

¯

n 1

¿ n 1 n 1

with the X s (r) static solution given by K X s = M U.