Civil Engineering Reference
In-Depth Information
In order to focus on the most important aspects of wind induced dynamic response the
calculations presented in Chapter 6 (and the ensuing calculations of cross sectional
forces presented in Chapter 7) have been performed in a modal format where each
displacement component has been given a separate representation. These calculations
may also be performed in a finite element format. Such a procedure is presented below.
It has the advantage that it will comply with the computational methods usually applied
elsewhere in structural mechanics. Thus, from a computer programming point of view,
all the well known stiffness and mass properties from other types of structural dynamics
problem will be applicable. The only difference is that the wind and motion induced
loads need special attention. On the other hand, it should be noted that due to the fairly
short correlation lengths and sharply dropping coherence properties of the turbulence
components there will be demanding requirements for the choice of largest element
length, i.e., the number of degrees of freedom in the finite element system may become
cumbersome from a computational point of view. The same applies to the choice of time
stepping increment in a time domain solution. In general, convergence should be
checked. The overall problem of a structural system of line like members in a turbulent
wind field, defined by the wind velocity components
() (
)
(
)
Vz
uy z t
,,
vy z t
,,
+
,
f
f
f
f
f
(
)
wy z t
, is illustrated in Fig. 9.1. At an arbitrary position on an element
n
,,
and
f
f
(between nodes
p
and
k
) the wind field and the interaction between flow and structural
motion will generate three load components, one in the direction of drag, one in the
across wind direction (vertical or horizontal depending on the orientation of the element
in relation to the flow) and one torsion (pitching) moment. Adopting a system of six
degrees of freedom in each node, there is a load vector
T
RRRRRR
[
]
1
2
3
4
5
6
T
rrrrrr
in each node, as
shown in Fig. 9.2. It is taken for granted that the global axis
X
,
Y
and
Z
coincide
with the flow axis
[
]
and a corresponding displacement vector
123456
− ,
x
and
z
, i.e. that the structural system is two-dimensional
and perpendicular to the main wind flow direction. Unfortunately, a two-dimensional
system is at the moment a necessary restriction as the experimental support of the wind
load on an element at an arbitrary attitude in the flow is insufficient. Strictly speaking,
the theory below is only sufficiently supported by experimental data if the elements of
the system are either horizontal or vertical. Nonetheless, the possibility of a yaw angle
has been included below. It should be noted that experimental data from structural
aerodynamics usually complies with the force and displacement definitions given in Fig.
1.3, where pitching moment and cross sectional rotation are defined by windward edge
up. It is in the following assumed that this definition applies to all the aerodynamic data
(e.g. load coefficients, aerodynamic derivatives, etc.) that are adopted for numerical
calculations. However, in the finite element theory it has been chosen to strictly comply
with the usual convention that all external and internal forces and displacement degrees
of freedom are vectors in global as well as local coordinates (see Figs. 9.2 and 9.3).
y
f
