STATIONARY AND ERGODIC HYPOTHESES
169
It follows that the probability of y occurring outside the ±3rru limits is (1 -
0.9974) — 0.0026, or only 0.26 percent. This is why “safe” design limits for the
structural deflection, for y = v, for instance, are often chosen as i3/7„.
To answer the second question, consider the probability that the peak value
of ÿ(t) lies within the zéro to 3
Jo
(7-9)
which is based on the Rayleigh distribution, équation (7.7). It follows that the
probability that any peak chosen at random exceeds 3rra is
P[|a| > 3
(7-10)
This shows that about one peak in 100 exceeds 3
whole range, or
F[—oo < (y or a) < oo] = 1
(7.11)
Presented in the following sections of this chapter are two methods for evaluating the standard déviation crv or erg for a structural displacement coordinate
v or rotational coordinate 0. For the first method, it will be shown that solutions
for
response functions h(t) and //( of the two and based on control theory for electrical Systems, the standard déviation is computed in closed form in terms of a three parameter spectral density
représentation of the excitation force. With the standard déviations computed
by either method, appropriate probability density functions can then be used,
as in the examples just presented, to evaluate probabilities of occurrence of
structural displacements and rotations under wave loading.
7.2
STATIONARY AND ERGODIC HYPOTHESES
With the assumption that y(t) is both stationary and ergodic, the statistical
calculations leading to the structural standard déviation in displacement are
simplified enormously. Although these assumptions are rarely checked in practice, it is nonetheless illuminating to elaborate on these two hypothèses from
an “experimental” viewpoint, as suggested by Muga and Wilson (1970). To do
this, eut a sample record y(t) such as in Figure 7.2 into J equal charts, each
of time duration tq. Again, tq is of sufficient duration that it captures the
essential character of y(t). This ensemble of J charts is denoted as
169
It follows that the probability of y occurring outside the ±3rru limits is (1 -
0.9974) — 0.0026, or only 0.26 percent. This is why “safe” design limits for the
structural deflection, for y = v, for instance, are often chosen as i3/7„.
To answer the second question, consider the probability that the peak value
of ÿ(t) lies within the zéro to 3
Jo
(7-9)
which is based on the Rayleigh distribution, équation (7.7). It follows that the
probability that any peak chosen at random exceeds 3rra is
P[|a| > 3
This shows that about one peak in 100 exceeds 3
whole range, or
F[—oo < (y or a) < oo] = 1
(7.11)
Presented in the following sections of this chapter are two methods for evaluating the standard déviation crv or erg for a structural displacement coordinate
v or rotational coordinate 0. For the first method, it will be shown that solutions
for
response functions h(t) and //( of the two and based on control theory for electrical Systems, the standard déviation is computed in closed form in terms of a three parameter spectral density
représentation of the excitation force. With the standard déviations computed
by either method, appropriate probability density functions can then be used,
as in the examples just presented, to evaluate probabilities of occurrence of
structural displacements and rotations under wave loading.
7.2
STATIONARY AND ERGODIC HYPOTHESES
With the assumption that y(t) is both stationary and ergodic, the statistical
calculations leading to the structural standard déviation in displacement are
simplified enormously. Although these assumptions are rarely checked in practice, it is nonetheless illuminating to elaborate on these two hypothèses from
an “experimental” viewpoint, as suggested by Muga and Wilson (1970). To do
this, eut a sample record y(t) such as in Figure 7.2 into J equal charts, each
of time duration tq. Again, tq is of sufficient duration that it captures the
essential character of y(t). This ensemble of J charts is denoted as