Environmental Engineering Reference
In-Depth Information
In the case of a 3D beam structure of general shape, the same approach is
followed but now the derivation involves the introduction of curvilinear co-ordinates
which makes things a little more complicated. Such a need is a direct consequence
of the fact that cross sections in the deformed state are no-longer normal to the
beam axis and therefore the variables
x , h , z
we use in defi ning the Green strain are
non-orthogonal (for additional reading see [8, 14]).
7.2 Second order beam models
Current design trends suggest that wind turbines will get bigger in future and this
will eventually lead to more fl exible blades. Therefore it could turn out that the
assumption of small displacements and rotations will be no longer suffi cient. One
option is to upgrade the model into second order as already developed for helicop-
ter rotors in the mid 70s. The derivation is too lengthy so we will only outline the
main ideas (for further reading, see [28, 29]).
The formulation is carried out in the same three steps as in the fi rst order model:
fi rst the displacement fi eld is defi ned which is next used in order to determine the
strains. Assuming linear stress-strain relations, the stress distributions are readily
obtained. Finally the stresses are integrated over the cross sections of the beam
material, and so the sectional internal loading is obtained. The main complication
originates from the form the displacements take.
With respect to the [O;
xyz
] beam system, the elastic axis of the beam will lie
along the y axis only in its un-deformed state. In order to describe the geometry of
the beam in its deformed state, a local [O
x h z
] system is defi ned that follows the
pre-twist and elastic defl ections of the beam (Fig. 8). The
h
axis of this co-ordinate
system follows the deformed beam axis at any position whereas
x
and
z
defi ne the
local principle axes of each cross section. At the un-deformed state
y
and
h
will
coincide while
x
-
z
will differ from
x
-
z
by the angle defi ning the principal axes of
each section. The position vector of any point of the deformed beam with respect
to the beam system [O;
xyz
] is given by
′
(
)
T
(
)
T
ru vw
=+ + −′
2
,
h
,
S
x
,
f
q
,
z
( 34 )
w
Figure 8 : Beam co-ordinate system defi nition.
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