Environmental Engineering Reference
In-Depth Information
possible to get a clear insight of the damping distribution along the blade span, as
well as to identify the effects from the non-linearity of the aerodynamics.
7 More advanced modeling issues
The beam model presented in Section 3 is the simplest possible. Besides assuming
small displacements and rotations it also suppresses shear (Bernoulli model) which
can be important. Including shear leads us to the Timoshenko beam model while
large displacements and rotations require upgrading of the model to second order.
These two aspects of structural modelling are briefl y discussed in the sequel.
7.1 Timoshenko beam model
In geometrical terms, introducing shear consists of assuming that the cross sections
along the beam axis will no longer remain normal to the axis of the beam. This
means that
q
x
and
q
z
in eqn (3) become independent resulting in extra shear strains
g
zy
,
g
xy
. Note that in the Euler-Bernoulli model shear is solely related to torsion.
For example by suppressing torsion and bending in the
x
-
y
plane the following
strains and stresses are defi ned:
∂
=− + ⇒ =
w
(
)
g
q
t
GGw
g
=
′
−
q
zy
x
zy
zy
x
∂
∂
y
(
)
q
∂
v
(31)
′
x
e
=− ⇒= = −
z
s
E
e
E
v
′
z
q
y
0
y
y
0
x
∂
y
∂
y
which result in the following virtual work terms:
(
)
L
⎛
⎞
L
(
)
∫∫
∫
∫
′
s de
dA
+
t
dg
dA
dy
=
F
d
v
′
+
F
−
dq
+
d
w
′
−
M
dq
dy
⎜
⎟
y
y
zy
zy
y
z
x
x
x
⎝
⎠
( 32 )
0
A
A
0
∫
∫
∫
F
=
s
dA
Mz dA
=
s
F
=
t
dA
y
y
x
y
0
z
zy
A
A
A
The virtual work in eqn (32) is equal to
d
u
T
Ku
, with
u
= (
v
,
w
,
q
x
)
T
. So by assuming
that the same shape functions
N
are used for
v
,
w
and
ˆ
Nv Nw N
,
the stiffness matrix of an element takes the form given in eqn (33). Note that
K
e
is
no longer diagonal as in (7). In fact if the full (non-linear) Green strain is used,
K
e
will be fully completed. The above formulation can be similarly extended to also
include shear in the
x
-
y
plane:
ˆ
ˆ
q
:
v
=
,
w
=
,
q
= θ
x
x
⎡
⎤
⎢
⎥
∫
T
∫
T
NN
′
EA
′
dy
0
−
N
′
EAz
N
′
dy
⎢
⎥
L
L
⎢
⎥
e
e
( 33)
⎢
⎥
∫
T
∫
T
K
=
0
N
′
GA
N
′
dy
−
N
′
GA
N
dy
e
⎢
⎥
L
L
⎢
⎥
e
e
(
)
⎢
⎥
∫
T
∫
T
∫
T
T
2
−
N
′
EAz
N
′
dy
−
NN
GA
′
dy
NNN
GA
+
′
EAz
N
′
dy
⎢
⎥
⎣
⎦
L
L
L
e
e
e
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