Civil Engineering Reference
In-Depth Information
However, the mean values in this situation are time invariants, and the response
calculations have inevitably been based on predetermined values taken from standards or
other design specifications. They have been established from authoritative sources to
represent the characteristic values within a certain short term weather condition chosen
for the special purpose of design safety considerations. Therefore, in a particular design
situation time invariant quantities may be considered as deterministic quantities, and
thus, the mean values of displacements or stress resultants may be obtained directly from
simple linear static calculations. I.e., it is only the fluctuating part of the response
quantities that requires treatment as stochastic processes. It may be shown (see chapter
2.4) that if a zero mean stochastic process is stationary and Gaussian, then its extreme
value is proportional to its standard deviation
σ
, i.e.
r
k
(
)
rxt
,
k
σ
⎡
⎤
=⋅
(1.10)
⎣
⎦
k
pr
k
max
where
k
is a time invariant peak factor between about 1.5 and 4.5, and thus
()
( )
()
r
r
x
r
x t
,
r
x
k
=
+
⎡
⎤
=
+
⋅
σ
(1.11)
⎣
⎦
k
k
k
k
p
r
k
max
max
Similarly, the extreme values of bending moment and shear force stress resultants may
be expressed by
MMk
=+⋅
σ
where
k
=
xyz
,,
(1.12)
k
k
p
M
k
max
Therefore, the main focus is in the following on the calculation of the standard deviation
to fluctuating components,
, whether they contain dynamic amplification
or not. However, in many design situations it is necessary to consider the combined
effects of stresses or stress resultants, and therefore, it is not only the standard deviation
of processes that are of interest in structural design considerations but also the
covariance between fluctuating components. For instance, let a fluctuating (dynamic)
displacement response at arbitrary position
x
σ
and
σ
r
k
M
k
()
()
()
(
)
r
⎡ ⎤⎡ ⎤
r
x
r
x t
,
,
,
⎡⎤
⎢
⎥⎢ ⎥
y
y
y
⎢⎥
=
(
)
r
r
x
r
x t
+
(1.13)
⎢⎥
⎢ ⎥⎢ ⎥
⎢ ⎥⎢ ⎥
z
z
z
⎢
⎣⎦
⎣ ⎦⎣ ⎦
(
)
r
r
x
r
x t
θ
θ
θ
be associated with corresponding cross sectional moment and shear force components
