Civil Engineering Reference
In-Depth Information
()
rx
may be obtained from simple static equilibrium conditions.
The standard deviation of the fluctuating part of the response
The mean value
r
()
σ
may either be
obtained from a time domain integration of the dynamic load effects from the fluctuating
flow field and possible vortex shedding, or from a modal approach in frequency domain.
The former alternative is computationally a demanding task, as it requires the time
domain simulation of a wind field that is usually broad banded and spatially un-
correlated (such a simulation procedure is shown in appendix A). In the following it is
the alternative of a modal frequency domain approach that is presented. As illustrated in
Fig. 6.2.b, the main steps involve the transfer from a wind field cross-spectral density via
a corresponding modal load spectrum to the final sought response spectrum. The area
under the response spectrum is then the variance
rr
2
r
σ
of the response.
As shown in Eq. 1.6 (and illustrated in Fig. 1.3.b) the cross sectional displacement at
a position
x
that has been chosen for the relevant response calculation is in general a
vector containing three components:
r
in the along wind horizontal direction,
r
in the
across wind (for a bridge) vertical direction and the cross sectional rotation
r
θ
. Since
these describe a combined cross sectional displacement in a plane perpendicular to the
span, the peak factor in Eq. 6.1 is equally applicable to each of the components, and thus
⎡ ⎤
σ
r
r
⎡⎤ ⎡⎤
r
y
y
y
⎢ ⎥
⎢⎥ ⎢⎥
()
x
r
r
k
r
=
=
+
⋅
⎢ ⎥
σ
(6.2)
⎢⎥ ⎢⎥
max
r
z
z
p
r
z
⎢ ⎥
⎢⎥ ⎢⎥
r
r
σ
⎢ ⎥
⎣⎦ ⎣⎦
⎣ ⎦
θ
θ
r
max
θ
As mentioned above (and further discussed in chapter 1), the wind induced response
of a slender structure is assumed stationary, and then the total response may be split into
a mean (static) and a fluctuating (dynamic) part. What can in general be expected in the
case of a slender structure is illustrated in Fig. 6.3. The static part is proportional to the
mean velocity pressure, i.e. to the mean wind velocity squared, until motion induced
forces may reduce the total stiffness of the combined structure and flow system, after
which the static response may approach an instability limit (torsion divergence). The
dynamic part of the response may conveniently be separated into three mean wind
velocity regions. Vortex shedding effects will usually occur at fairly low mean wind
velocities, buffeting will usually be the dominant effect in an intermediate velocity
region, while at high wind velocities motion induced load effects may entirely govern
the response. Such a partition should not be taken literally, as there are no tight borders.
E.g., important motion induced load effects may also occur in what seems like a typical
vortex shedding or buffeting behaviour. In the vicinity of a certain limiting (critical)
mean wind velocity the response curve may increase rapidly, i.e. the structure shows
signs of unstable behaviour in the sense that a small increase of
V
implies a large
increase of static or dynamic response, indicating an upper stability limit (
V
).
cr
