Civil Engineering Reference
In-Depth Information
0
0
0
0
0
0
⎡
⎤
⎢
⎥
0
0.002164
0
0
0
0
⎢
⎥
⎢
0
0
0.035
0.21
0
0
⎥
−
ˆ
Cov
=
⎢
⎥
RR
24
0
0
−
0.21
1.26
0
0
⎢
⎥
⎢
⎥
0
0
0
0
0
0
⎢
⎥
0
0
0
0
0
0
⎢
⎥
⎣
⎦
The covariance matrix of a chosen set of cross sectional forces is given in Eq. 7.53.
7.4 The resonant part
Fluctuating cross sectional forces at an arbitrary spanwise position
x
that are
exclusively ascribed to the resonant part of the response may in general be extracted
from the derivatives of predetermined modal displacements
(
)
(
)
( )
x
,
t
x
t
r
=
Φη
⋅
(7.62)
r
r
r
as defined in Eqs 4.7 and 4.8 (see also Eq. 4.79). The direct transition from the variance
of the fluctuating displacement response quantities to the variances of corresponding
dynamic cross sectional forces is therefore presented below. The procedures for the
calculation of response displacements are in a general format shown in Chapter 4. For
the special cases of buffeting or vortex shedding induced dynamic response the
procedure is shown in Chapter 6. For simplicity it is in the following as usual assumed
that we are dealing with a line-like horizontal (bridge) type of structure where axial
forces may be disregarded, in which case the force component
F
in Eq. 7.7 may be
omitted. (Axial forces may in general be determined by the product of the axial stiffness
of a beam type of element and the difference between the axial displacements at its end
nodes.)
It follows from the definition of cross sectional forces in Fig. 1.3 that the connection
between the fluctuating force vector at an arbitrary spanwise position
x
and the
corresponding cross sectional stress resultant is
T
T
[
]
(
)
xt
,
FFFFF
⎡
VVMMM
⎤
F
=
=
⎣
(7.63)
⎦
r
2
3
4
5
6
y
z
x
y
z
where
VV
are the shear forces in the direction of the
y
and
z
axes,
,
M
,
M
are
y
z
y
z
the bending moments about the same axes, and where
M
is the torsion moment.
x
