Civil Engineering Reference
In-Depth Information
and
TN
→∞ this may be written on the following continuous form
In the limit of
⎡
2
⎤
1
i
(
)
(
)
(
)
ωω
−
S
S
S
⎡
±
ω
±
ω
±
ω
⎤
x
xx
xx
⎢
⎥
⎢
⎥
(
)
(
)
2
3
(
)
S
S
i
S
±
ω
±
ω
=
⎢
ω
ω
⎥
⋅
±
ω
(2.94)
⎢
⎥
⎢
x
xx
x
⎥
⎢
(
)
⎥
sym
.
S
±
ω
4
sym
.
ω
⎣
⎦ ⎢
⎥
x
⎣
⎦
Because
(
)
S
±
is symmetric it is seen that for a stationary stochastic process
x
∞
i
Cov
⎡ ⎤
i
ω
0
0
⎡
⎤
⎡ ⎤
xx
(
)
∫
S
d
=
⋅
±
ωω
=
(2.95)
⎢
⎥
⎢ ⎥
⎢
⎣ ⎦
x
3
Cov
ω
⎣
⎦
⎣ ⎦
xx
−∞
Thus, the spectral density of time derivatives of processes may be obtained directly from
the spectral density of the process itself. Since the single sided spectrum is simply twice
the double sided, Eq. 2.94 will also hold if
(
)
(
)
(
)
S
,
S
and
S
are replaced
±
±
±
x
x
x
()
()
()
by
S
,
S
and
S
.
ω
ω
ω
x
x
x
()
()
()
From
and
the average zero crossing frequency
of the process
S
ω
S
ω
f
0
x
x
()
()
2
()
x t
may be found. Referring to Eq. 2.34, 2.56 and introducing
S
ωω
=
S
ω
, the
x
x
following applies:
1/2
⎡
∞
⎤
2
()
∫
Sd
ωωω
⎢
⎥
x
1
σ
1
1
μ
⎢
⎥
x
x
()
0
2
f
0
=⋅
=⋅
=⋅
(2.96)
⎢
⎥
x
2
2
∞
2
πσ
π
π
μ
⎢
⎥
0
()
∫
Sd
ωω
x
⎢
⎥
⎣
⎦
0
∞
th
n
()
∫
where for convenience the so-called
n
spectral moment
μ
=
ω
⋅
S
ω
d
ω
has
n
x
0
been introduced.
2.10 Spatial averaging in structural response calculations
A typical situation in structural engineering is illustrated in Fig. 2.14. A cantilevered
tower-like beam is subject to a fluctuating short term (stationary) and distributed wind
load. The problem at hand is to predict a load effect, e.g. the bending moment at the
base. It is for simplicity assumed that the beam is so stiff that it is not necessary
